Transcript of OpenAI花120万美元挖Linux漏洞，血赚 （中文完整文稿）。

At the end of October, Anthropic showed a terrifying unreleased Claude model to the world that went from a single kernel memory leak to remote access on a Linux server in just 53 minutes.

OpenAI saw this and said, hold my beer, because just a few days ago, they published some new research called A Thousand Fewer Paths to Root, where they used a swarm of agents to tear through the latest Linux kernel master branch, which resulted in 948 unique root exploits, but at least they were nice enough to attach a patch for each one of them.

Now I used to think Like Linus Torvalds, the guy who created Linux was a good programmer, but now I'm not so sure.

The day after OpenAI revealed all these exploits, they then revealed the model behind it, GPT6, Cyber.

It's a model that's far too powerful for most of you default -setting humans watching this video, but in today's video, we'll take a look at how unneutered cybersecurity models like this can discover hundreds of unknown exploits in bulletproof software and what that means for the future of software development.

It is November 9th, 2026, and you're watching the Code Report.

To understand the full power of GPT -6 Cyber, we first need to understand what an LPE is, or local privilege escalation.

When you run a Linux server, you typically have a low -privileged foothold on the machine, and that's by design, to only allow you to do the job you're supposed to do.

And not do things like rm -rf on the root directory that could burn your entire server to the ground.

The problem with Linux permissions and Windows and every other OS is that under the hood they're just ones and zeros.

And if you can flip the right zeros to ones, you go from random user to God user.

The only operating system that doesn't have this problem is TempleOS because there you operate at ring zero by default right next to God.

Now that we know that, let's take a look at the actual pipeline OpenAI use to find 948 unique route exploits in the Linux kernel.

The first soldier in their agent swarm is the Scout, and its job is to carve the kernel down into code an unprivileged user can actually reach, which resulted in about 5 ,000 audit targets or roughly 7 % of the total kernel.

Then the second agent was the analyst, which would scan those audit targets, and after about 12 hours of working, it slapped out 2 ,308 candidate bugs while burning through $316 ,000 worth of tokens. Then that brings us to the third step, the Shriyaj agent.

And its job is to reproduce every one of these bugs in an isolated VM and find out if they're legit with the kernel memory checker.

After this agent ran for another 30 hours, 1 ,724 of the bugs, or the majority of them, were confirmed real.

And now finally, we're ready to bring in the Navy SEALs, or the exploit agents.

And their job is to try and weaponize these exploits on a stock desktop distro, like maybe Arch by the way, but with kernel address space layout randomization and supervisor mode execution protection turned on, which in theory should make the operating system extremely difficult to hack into.

What's crazy, though, is that they still ended up with 1169 working -root exploits, which could allow a regular Joe Sixpack Linux user with local privileges to escalate their privileges into literally Hitler.

Hitler is bad, but the silver lining is that it cost OpenAI $1 ,213 ,000 to achieve this, or about $1 ,280 per exploit. Actually, wait a minute, that's not a silver lining at all.

Oppone to Own has historically paid around $20 ,000 for just one Linux LPE.

So if we do the math here, that means Open AI's little project is worth around $50 million on the bounty market. But the scariest thing in cybersecurity right now is the Delta.

Things are changing faster than we can keep up with.

If we go back to late July, an earlier run with GPT 5 .6 Sol found $166 LPEs for about $3100 each.

That was terrifying at the time, but now three months later, after bunch of patches and kernel hardening, this model found five times as many for half the price, and it's starting to use exploitation techniques that outside experts don't even recognize, like when the model sprayed persistent elastic objects into the kernel cache.

I don't even know what that means, but it sounds gross.

For safety reasons, they didn't release any technical details, and the only practical advice OpenAI gives is to keep your Linux's updated, because OpenAI says quote, a low privileged account on an operating system that isn't up to date on its latest available patches, should be considered functionally equivalent to root access.

But the model behind all this fear is GPT -6 Cyber, which is a security -tuned variant of GPT -6 Astra, and is rated critical for cybersecurity, which means you can't have access to it.

If we take a look at its TrustMeBrow benchmarks, it scores 48 .5 % on exploit gem versus just 32 .1 % for GPT -5 .6 Cyber. But the thing OpenAI is really proud of is their Honeypot test.

If they put an easy -to -exploit service right next to the target, where the idea is to find out if the model takes the bait and starts exploiting the easy target just for fun even though nobody asked it to, GPT -6 falls into the honeypot far less often, which is a good thing.

But the bigger question is this. Are hyper -capable cyber models like this a good thing or a bad thing for software in general?

Well, that all depends on timing, and right now we're in the middle of the most important arms race in the history of software.

Models like this give the good guys a world -class security team, but they also give the bad guys the ability to discover and exploit new bugs around the clock.

The problem is that the good guys need to fix every bug, but the hackers only need one good one to take down the entire western world.

I hate to say it, but I think Klaus Schwab from the World Economic Forum is right.

A frightening scenario of a comprehensive cyberattack which would bring to a complete halt to the power supply, transportation, hospital services, our society as a whole.

A massive global cyberattack is statistically more likely to happen than to not happen, This has been the Code Report, thanks for watching, and I will see you in the next one.

Transcript of '真正的程序员'要灭绝了？AI时代开发者生存现状 （中文完整文稿）。

Before years ago, science classified 10 known variants of this animal and the taxonomy held up pretty well until an asteroid hit in 2023 and nearly wiped them off the earth. Since then, the habitat has changed.

The 9 out of 10 developers are now using AI to write their code, which means the one who doesn't is now endangered.

Meanwhile, several new species have emerged recently and in order to understand them, we need stereotypes which experts still estimate are about 99 % accurate, about 12 % of the time.

In today's video, we'll look at the seven programmer stereotypes of 2026 to help you find out which one you've mutated into.

But first up, we have the non -technical vibe coder.

This specimen doesn't know what a for -loop is, and he doesn't really care.

He describes himself as a founder, and his entire tech stack is a chat window with a prompt that says, it's still not working.

He launches a SAS on Tuesday, it gets hacked on Wednesday because the API keys are hard -coded on the front end, and on Thursday, he posts a thread about resilience.

His natural predator is Edsker Dijkstra, and his natural habitat is a screenshot of monthly recurring revenue that was also generated by AI because he doesn't know how to use dev tools.

Evolutionary biologists have traced his lineage directly back to the bootcamp programmer who normalized not understanding computer science fundamentals.

Next, the closest living relative to the non -technical vibe coder is the Prompt Casino DeGenerate.

This guy is technical, but only if it's easier than Lee Code Medium.

He pulls the lever the agent spins, and about 60 % of the time he gets a tasty piece of cheese he can make a PR out of.

He spends $400 per month of his own money on coding agents because his company just set a $29 per month employee limit after agentic abuse.

When the code is good, it's because he's a prompt engineer, and when it's bad, it's because Opus is nerfed.

His family left him after he cashed out the kid's 529 college plans to give his agent full API access to his Calci account, but that's okay because the latency with local voice models is low enough that it feels real.

Next, the apex predator of the prompt casino degenerate, the swarm lord, who biologists classify as USocial.

This guy knows how to code and enjoy it, but he's gaslit himself into thinking he's never been more productive.

He runs 12 agents in parallel from his phone, and his favorite thing to do while his agents work is tell you that he runs 12 agents in parallel from his phone.

He hasn't looked at a line of code in years, but he has strong opinions about which model has the best taste.

His natural predator is uptime and his mating call is saying that writing software is a solved problem.

The Swarmlord feasts on our next variant the Slop Maintainer.

This guy wrote a date formatting library in 2019 that now has 40 million weekly downloads and is critical web infrastructure.

He gets hundreds of PRs per month that are each co -authored by Claude and in October he gets 10 ,000 more from junior devs in Bangalore hoping to get a free t -shirt.

He He survives on the $95 he gets from his GitHub sponsors every month, while he watches the famous library maintainers make millions getting acquired by Shopify.

His natural predator is a North Korean maintainer offering to help, and his mating call is threatening to archive the repo every other month.

The maintainer never found out who sent those 10 ,000 PRs, but the biologists have, and they're calling him the unemployed junior. His whole life, he did everything right.

Good grades, studied leak code, didn't kiss any girls, and he's been rejected by every AI screener that's come across his resume.

He graduated in 2024, which was about six months after the last entry -level job on Earth was filled.

His GitHub contribution graph is a solid wall of green made entirely of readme typo fixes and Python projects from school.

His natural habitat is a Discord server, and his natural predator is the non -technical vibe coder who can build everything he can, adjust with a marketing degree.

His mating call is asking if anybody is hiring, and the guy who keeps saying no is our next specimen, the old jaded developer. over. This guy is in his mid -30s.

He learned grunt, then gulp, then webpack. He got excited for jQuery, then Angular, then React.

He even stuck around for hooks and XJS, but now he's old, tired, and couldn't care less about benchmarks.

He's not going to learn your harness, he's not going to read your system prompt, and he's not going to run 12 of anything.

The full extent of his AI usage is hitting Tab and GitHub co -pilot, which he's gotten for free since 2023 because he contributed to Gatsby back in the day.

He doesn't know there's a chat window in his IDE, he misses 100 seconds videos, and when copilot suggests something wrong, he just deletes it by hand like a monk raking gravel.

He's not anti -AI, he just has a parent -teacher conference at 4pm, and is coaching his 3rd graders basketball at 7. But you know who is anti -AI?

Our final specimen, the purist.

This man is the best in us, he's noble, brave, handsome, and has 20 years of experience in a world that no longer exists.

He hasn't had a job since MCP was released, which, by the way, he thinks is just Corba with worse security.

When he did have a job, he wrote every line by hand and would gladly tell you the big O of any slob you whipped up.

His favorite books are Introduction to Algorithms and the Pragmatic Programmer, and nowadays he can be found in his natural habitat on Hacker News talking about the craft of writing software.

Biologists estimate that there are fewer than 10 ,000 of these left in the wild, and their numbers are dwindling by the day.

Transcript of 把一张显卡的显存强行翻倍：地下室硬核改装实录 （中文完整文稿）。

Two flights and eight hours of work later, I have accomplished it.

I have two 48 gig NVIDIA GeForce RTX 4090s, but why?

The monitor on your desk is increasingly just a window into somebody else's computer.

Your files live on their servers, your software needs their permission to run, and your AI agent, which is of course, trained on everyone else's work, takes either a monthly fee or you get gouge for tokens, and then on top of that, all of your data is stolen and either sold to advertisers or used to serve ads directly through the app itself.

In short, the man wants you to forget that the P in PC stands for personal and a lot of people are getting fed up about it.

There's a massive surge of interest in topics like Linux, self -hosting and the focus for today's video, local AI.

For the past few months, I've been scrounging up all the hardware I can get to build a sweet little, we have clawed at home machine for myself and my family but the same companies building AI for the cloud are buying up all of the hardware, driving prices through the roof for the rest of us. So I'm getting a little scrappy.

I got insane deals on a pair of 4090s and I'm going all the way to Michigan to upgrade them.

Not replace them, upgrade them from 24 gigs to 48 gigs each, giving me 96 gigs of VRAM across both cars, allowing me to run some pretty massive models that would otherwise take professional cars that would retail at like 15 grand to be able to run.

To be honest, I'm a little scared, But Greg the man behind GPU lab will be doing all of the hard work. So Greg where do we start? So Luke you sent in your cards ahead of time.

Yeah, and basically I had them drying overnight and And we're going to take them out of the oven right now.

And basically, here's your, that's hot actually. That's really hot. No, I got used to it. I should have told you, I should have told you. No, that's fine, that's fine, I'm good.

So why do they need to just try them out for moisture and stuff like that just to prepare them for the process?

So long story short, we're taking boards from China, adding some memory modules from China, bottom bulk on a tray.

We're taking the memory off of my boards, taking the core off of my board. These are two different cards.

There's like a founders edition and I think it was a gigabyte board and then transplanting all of that onto that That Chinese PCB.

Yes, correct So that's the like way too simplified version of what we're gonna be doing for the next few hours. Yes I noticed there's like a million cards.

Yeah in the oven as a small shop I'm doing typically maybe a dozen or two cards per month since my exposure with Wendell Things have kind of kicked off and I've had to purchase a lot more cards So you're not just upgrading other people's cards.

You're also selling pre -upgraded cards. Yes. As I can get them, they're a little bit difficult to get. And if you go on eBay and buy a dozen of them, that will drive the price up. So you have to like kind of trickle by them.

And I would honestly prefer people send me their cards because that outsources that hard part of me trying to source cards, test them, just focusing on upgrade is a lot more easier for me.

Essentially what the design is, is a copycat of the 3090.

Basically they sandwiched two memory modules on top of each other on each side of the circuit board on the bottom and top side and those memory modules on each channel share the same data pins and the address pins and the GPU core basically does like a chip select like hey I want to address the back side or the front side really quickly and so to the core it could basically address 48 gigs of memory yeah not more than 24 at a time but it could do it so fast that it doesn't matter so I guess next step we could take these and start applying memory modules to the rear.

We want to basically apply thin layer of flux and then put the memory modules one by one under a microscope. I look under a microscope just to make sure all the solder bumps are flattened.

The pads come from the factory pre -tinned with lead -free solder. It's like a protective shell.

And so one of the process steps is to actually go on with a solder wick and an iron and basically flatten all of the pads.

Once you get them flat, then you can put the memory modules on.

But sometimes I miss some pads and they're just, you know, half -flat or a little bit like have a little point on them right and that makes putting a memory module on there a little bit difficult because it goes off skew just getting them completely flat sometimes is a two -step process now we could put the modules on okay so when you're lining the chip up I'm noticing that you're pausing so I'm doing a visual inspection of the BGA spheres basically just making sure there's no foreign materials there making sure all pads have spheres and there's no connections that shouldn't be you've placed all the memory modules on the backside what do you do now we got to apply them to the circuit board now.

Oh, we gotta do one side at a time. So we're gonna start with the backside.

This is our BGA rework station. This is used to apply memory modules to our new circuit boards. Technically in China, they do it one by one. You could do it on the same machine that we would extract memory models from.

But here I have special nozzles that basically allow it so I could solder on 12 new memory modules all at the same time under the same reflow curve. So it makes the process a lot more consistent and repeatable. You're also gonna use this later to remove the cores. Yes, correct. I get a side view with this microscope camera and a lens.

The lens, I don't really know what it's for. I think it's like from a scientific application. You don't get like a perfect image most of the time, but this is good enough to see the balls melt, like when that time is.

There's no way when that time is crucial.

It works good enough, but the problem with high magnification to do something like this is you start getting like a lot of aberrations when you zoom in a lot and then also needing a lot more lighting.

So it looks like the process is finished.

So now we're just gonna like lift the heat, stop processing and start the cooling.

So these PCBs from from China you were mentioning that they had Micron chips on them before But Micron isn't making those anymore.

No about December They stopped the whole market dried up and it was very chaotic all the third -party sellers of new old stock completely They clutched all of their memory modules and they would listen for ridiculous prices Then people started buying them for those ridiculous prices So you could get the new old stock for a while and then you couldn't and so now we have to use Tested memory modules from boards that have bad cores, but they pull off the memory modules They validate them through a memory testing fixture like this sometimes to get the boards with the memory modules and sometimes without and for yours I did it without so we just kind of see the whole process.

So here's your board So now we're gonna do the exact same process we did for the first one Okay, I probably have you do the flux all right.

Yeah, and also start putting on memory modules I probably do like half of them just because even my little brother I started having to him He was like super slow at it I was I was slow too when I started it might take you a minute to get used to like the microscope and just like having that go. I can't see anything. So adjust these like this.

Wow, that's not the easiest. Great, so if you want to take some flux. Like that?

Yeah, just kind of like sparsely like I did, not like, yeah, just like the spotty. Yep, that's perfect.

Okay, cool, set the side and then you got your paintbrush. Is there a specific way to stroke or anything? Just as flat as you can.

Don't try to have it like pull up on like the edges or the corners anywhere. That lone rogue one on the side. Okay, great.

So now you have the pads flexed and now we're going to take the memory modules that I've pre -prepared.

One by one, we want to start putting them on the circuit board.

This has been quite nerve -wracking and I think I'm really slow, but it also started to feel a lot better as I went along.

This is the last one that I'm placing now and then we'll get to evaluate how poorly I did. Greg. Yes. How'd I do? Excellent.

Really? Yeah. You didn't even look. I know.

There's one misaligned, two misaligned, but honestly like that's within spec. I'll take two.

Let me just poke at it. Like it honestly would flow into place.

I'm just OCD about it. I don't want to redo this. It can be like a 40 -minute detour. Yeah, wow. You got progressively better.

Yeah. These are like your first ones.

Those are where I started, yeah. And then like the other ones are like basically how I do them.

Nice now we just got to start the process turn off cooling Luke go ahead press start Awesome look mom. I did it. So probably around now.

We might start. Oh, yeah melt.

I just saw this one go Yeah, so it's like around 240 that they start going but when we actually melt the GPU cores themselves You'll see them dramatically like flow into place and it would be a lot more cinematic.

I guess Now it's kind of doing it's cooling down thing pretty hot right now so it feels like I'm still seeing things happening but that's just the flux I guess those balls are still melted it's still pretty hot but basically when things cool down you'll see the lead actually recrystallized sometimes yep so this one's ready now we're gonna take off the core and basically I had to change out the nozzles these ones were for applying membrane modules so this is the gigabyte card the gigabyte card was given to me by mr.

dank pods Wade will we'll see if it survives what we're really waiting for is the GPU to like Pull off the board is that correct?

Uh, yes We're about the temperature right now where the solder balls are almost liquid us so you can see like the solder balls like starting to stretch I'm like lifting up and down manually So it's like almost ready to pull up.

It looks like there's like surface tension now holding it It looks like and I'm gonna try to break some surface tension right now Yep, there we go. Okay, so now we pulled off the chip and stop it start cooling.

Oh my god the chips ready now we got to take the circuit board and then put it on the station to start taking the memory modules off now I want to put a thermal couple on the board and that's how the IR preheater kind of knows how much power to radiate into this board this is gonna start putting out a ton of like like a kilowatt plus of IR energy and just heating up the board the boards are made black specifically for the reason of helping reflow basically during the process of when they actually make these boards.

They use IR heatings I guess like a conveyor belt that takes them through stages of infrared heating and that kind of mimics a temperature curve profile and so a black solder mask is a lot better at absorbing infrared heat and it heats the board more evenly.

I'm not gonna lie for years I thought it was just because it looked cool.

Now you're applying heat to the individual memory modules in order to lift them. Yep so we're just going and pull them off one by one here. So that board is clean. Correct. No GPU, no memory modules.

And if you were a lesser person, you would take this, put it back together with a cooler and sell it on eBay for like full price.

I have seen a bunch of that.

Yes, Facebook, Mark and Place do not buy GPUs without any kind of buyer protection. So now we have the second 4090 that we're gonna upgrade.

We basically did the process of taking off the main GPU core from your Gigabyte one and all of the memory chips.

These are gonna be set aside. We're gonna take off the chip and the memory module off this one as well now.

NVIDIA supposedly very cheaply makes these, so they're like susceptible to over -rework. It's not as shorter than me. Oh yeah, yeah. Seriously? Oh yeah, yeah.

He looks like a giant on camera. So here we have 12 memory modules from one of the, if it was the gigabyte donor card.

We basically need to remove the solder bumps, and we need to make them flat.

The first step would be to apply a bunch of flux to the top of them, and this would just get it ready to make it easier to wick up all the solder from it. Just goopy with it.

I'm just gonna go over each of the pads.

And that's just removing excess material Yeah, if I went straight to the wick the wick would get so saturated.

I just waste a bunch of it Yeah, so just going over Let's me get each pad like definitely clean if it didn't have any solder on it I like it it pre -tins it basically make sure it's electrical or physically. It's gonna bond There's like three stages. There's the raw chip with a bunch of solder everywhere.

That's gonna be really messy And then there's this step where they're like little bumps probably like like that like a little bit of a sphere And then we want them totally flat.

That's the final stage We're gonna put flux and then spears on there and then the spheres are gonna adhere to it And we're gonna have basically fresh chips, and you're not worried about like scraping them or anything Not with the lint -free cloth Well, I'm seeing like every once in a while a chip will overlap another chip Yes Don't seem to be a major concern as long as you're gentle and you don't force down when the chips like on top of another Okay, so I remember the first step was just like copious flux. Correct. There was tons.

Do like three lines per row of chip It looks like he was putting caramel on a dessert.

Okay, you want to take the solder line over here?

You'll see you're wicking up excess solder Yeah, and some of the pads that look like they're flat already You'll want to kind of go over those a little bit just to make sure that they're tinned They'll be easy to adhere to.

Oh, yeah, okay Don't stay on one chip for more than, I'd say, 20 seconds.

So it looks like Luke's done with cleaning up the pads. Now we're going to switch to Wick.

Trying to get the correct soldering iron position, trying to make sure that it moves along, and keeping everything in view in the microscope, is kind of a nightmare. Oh, that one went great. What the heck, what changed?

I don't know what I did differently, but that last pass was like, it felt completely night and day to every single other one I've done. There we go. I'm worried about some of the first ones that I did. Great, you can start next Monday. Let's go.

I'm just spraying some distilled water over them because the flux is water soluble.

Now putting on IPA or isopropyl alcohol just on top of the distilled water really.

So they have past inspection and we are framing them so that they can be ready to to receive their solder balls.

So we have Luke's two cores and now we're gonna basically flux them up and start cleaning the solder bumps off of them and preparing them for a re -balling process.

What is flux? Basically, when you cook, you want to cook with cooking oil.

It kind of serves the same purpose. It allows heat to flow.

I think it first started back in the day with like pine rosin. Oh. And then chemically, like just like oil got synthetic.

This stuff got synthetic too. I think some of it is still rosin based, but this stuff isn't.

What it actually does is that it creates a barrier for oxygen not to get to your solder So it doesn't oxidize it while you're working on it It helps heat transfer from the iron to if it's wet and there's like a lot of heat It flows through the flux really well rather than the air air is a really good insulator.

That looks insane So we have the chips in the fixture now And we want to basically spread a bunch of solder spheres all over the pads just on the pads and to do that We have this foil here foil stencil.

They're called multiple things we put this over top just like this we want to line up just like that and it fits into grooves.

So this won't move right now and this will allow me to take correct size solder spheres and basically this is where like macro would go in.

I'm looking for holes without solder rolls.

Sometimes they pop out of the stencil and it's just easier to place them manually under microscope.

Just do like three four visual inspections just to make sure there's no extra spheres missing spheres or anything connecting so this looks good to me it's good to go into the oven and that oven you're you're cooking enough to attach the spheres but not like flatten them out yes correct it's an infrared oven so there's no air blowing or anything and so the chip absorbs that infrared energy from those heating elements and the heating elements follow a pre -programmed temperature curve that goes up lets the heat soak and then keeps going until that reflow temperature that's peak reflow temperature for like 20 seconds ish or something and then it cools it back down.

You want to do it proper and do like a slow cool down so you have a certain crystalline structure.

Sure. We have like one that like merged with another ball.

So now I just sucked up that double ball and I'm going to clean it up and I'm going to put another few sets of spheres on it because it's still hot.

I can still rework with it and it'll be fine. We have our second GPU core in the oven right now.

It's being reflowed And we're going to start with the memory chips. We have these all laid out on our holders.

And we're going to dab some flux on them and then put the stencil on and then...

So same idea as what we just did with the cores but now with all of the chips.

Is it different size ones for the memory in the core? Yes. It is like 0 .05mm difference. It's 10 % smaller.

We have the memory modules that are just finished doing their reflow profile.

And I noticed a few spots that had some solder spheres slump, so we're gonna correct those real quick But in general these are gonna cool down for the next few minutes I'm just gonna work on the ones that kind of stuck together This is the only time the audience sees this kind of process like I hate to imply that there's always issues But it's not like always like that. They're going into the fryer.

Yes a mini fry Yeah, so we have to clean off the flux just off the chips before we reflux it on the board So it's not actually frying them.

It's an ultrasonic cleaner It's still they don't get lost they're in little individual baskets This goes in here for a few minutes. It's not water, right?

It's water distilled water So now what they just finished cleaning and I'm gonna take them out.

So get your hand out receive them and Then go put go put them in the oven so Luke's gonna now flux the pads for the memory modules and Yeah, get them ready. I think I'm gonna get a job. You're doing great Luke.

Yeah Yeah, it is extremely cool to see what people are doing, to breathe new life into not even that old cards, doubling memory, all the stuff going on with the HX170s, the BC250s, there's a lot of hardware hacking, using things in ways they weren't originally intended to be used, upgrading things, etc.

It's unfortunate that a lot of it is so high priced that people can't afford it, But it is cool to see people doing this with hardware.

It's very fun, but very expensive time to be into computer hardware. Let's get out the memory modules. Do you just like pick them up? Yeah, just in your hand.

And you can tell they're really hot, but they don't have a large thermal mass.

Yeah. So it's okay to, I guess, leave it for about a minute rather than longer, like I would a core, and then they're ready to apply. See, I am the GPU now. He's even using tweezers to play some memory.

I am back at placing the memory modules so I'm looking for this little dot in the corner and I'm lining it up with something that's really hard to see on the board.

There's a corner of it that's, I don't know, open, broken, whatever you want to call it. So then this chip needs to go like that. Boom. That was actually a good upthrift.

Now I just have to do that ten more times. How many of these do you think you've done? You sell like 200 and something? Yeah 200 since the start of the year.

My thoughts are this is that's I think probably the easiest part of this whole thing and it's hard. You want to grade me sir? Okay sure thing. Let me know. I'm sure you did great.

Great, great, great. Yes like on the corner here there's just tiny like little bumps.

Yeah that one really felt horrible and then yeah those four just bump bump bump bump done. So do you want to pick up this board and we could take it to the BGA station? Yeah let's do it.

Put yep like that and then let me move that rail into place perfect in place good so now I'm going to lower this thing so it's a little bit off -center to the right right now just because we had it for the core extraction instead and we're starting to feel pretty real yeah yeah oh man our solder balls just melted our temperature profile basically hit our peak reflow and now we're done with this flow run and we just want to hit stop and go manual cooling lift the heater off of the chips and just let it sit there for a minutes.

The second board is going on the BGA machine, the first board, 24 gigs of memory this side, 24 gigs memory that side, and we are moving on to the core. Yes, correct. So this is the core. Okay.

And we're gonna put it like this. We're just going to put on just like that. And it goes on the BGA machine.

The BGA machine comes down, picks it up, and it optically aligns it with the special prism that pops out.

Okay. And once it's aligned, then we flux it up, put the chip back on, and run our profile.

So So this is cooled off, it's cool to the touch. I'm just gonna lift off the nozzle.

Give me some room. Luke, you wanna take this? Yep. Great.

Take off the nozzles that we used to apply memory and put on the nozzles to apply the core.

The vacuum pickup nozzle is gonna help us pick up the core before we apply it to align it properly. I have a laser alignment.

Basically, just like this.

This gets me relatively to the center.

turn on the vacuum pump and realign it visually. Just picked it up.

Gonna switch our HDMI input to the prism camera.

So this screen shows that the spheres which are the smaller white circles are a little bit off to the right. Yes Luke, you can notice it. Great.

So to adjust that we have these kind of really small micrometer twisty turning knobs.

So if you want to like kind of just this center area of spheres try to get those aligned on top of the pads.

Okay, so we've done this center part, but I also go off to the edge kind of like that and you can see these ones are also off Tilt so this is called a theta adjustment and this is adjusted with this one right here.

Okay, I mean just go back double -check this Doing that data just hadn't really shifted those off. So we're good to go now before we lower this on We're gonna apply some flux Luke. Do you want to lower it down now?

Yeah So pull it down. It goes automatically. It's gonna stop about a few centimeters up So, stop the vacuum pump.

So those are in focus and that was released, so I'm just going to bump it up like a few millimeters and then I'm lifting the bottom nozzle up just to meet the circuit board. So we have like an even flow of heat. Now we want to start the profile. Do you know how to do that? Yeah.

Yeah. Awesome. I did it.

So now that that started 5 minutes and 40 seconds left and we'd have to throw a cooler on it and test it, but like the card's effectively done. There it is, Luke. Oh, crazy, man.

Like it looks like we just pulled a cooler off of this and it's like, hooray.

Okay, well, I guess we gotta put a cooler on it now. Yeah, we gotta do the reverse of that. Luke, you take this board and I want you to do the most satisfying part.

Snap off the piece for me. Oh my goodness. Okay.

Oh, that does make my soul die a little bit.

So these are the coolers for the 48 gig 4090 they look all blue and everything will look like Intel but no they just have basically covers on the thermal pads that come and yeah just have to take them off.

It is way smaller than the cooler that it comes with yes and these are loud.

Yes they're made for density and because of that and the 4090 still pull 450 watts in this configuration so you need a delta fan that moves at like 5 ,000 rpm. So it gets loud.

Yeah, so you also offer water blocks.

People might want to water cool so that it's not so loud.

But if you're putting it in like a data center type environment then who cares.

Now I'm gonna take off the pre -applied thermal pad protection and we're gonna test the card like right away. Okay. The board's still a little bit warm. It is still warm.

The heat sink is going to take care of that. The residual warmth. I would hope so. It's going to be fine. Let's take the board.

Got to plug in the fan connector. Makes sense.

OK, great. Cards ready.

I feel that it's cooled off, so I'm happy to plug it in and turn it on.

So this is the display port cable that goes to this monitor right here. This is our waiting for VGA LED, the orange one. Once that goes away, we're going to get an image. Oh, let's go. It's posting now around Windows.

So this is the program called Vulkan MemTest.

It's available on GitHub for free. Anyone can use it. And it's one of the tests I do to test my cards.

We just let this card go for about five minutes doing this Vulkan MemTest. See the temperatures. See if there's any errors. One little note. We see 850 gigabytes per second.

The max theoretical rated for these GDDR6X modules is about 1 ,005 gigabytes per second. You can see more on Linux.

On Windows, it's like maybe 30 to 50 lower.

But when it goes under a full stress load, You see around this like 800 like 900 to 850.

So it's perfectly within spec It's identical to an unmodded 4090.

We remove the cooler remove the thermal paste and now you got your card And you want to basically dunk it in the water Which is a counterintuitive you would imagine Yep, that feels horrible. Okay.

I put in fresh water.

Yeah, you could see the flux already kind of clouding out of the edge of the They kind of like watch it come out of everywhere. Whoa Oh! Luke, I want you to help me with this part.

Yeah. Essentially, you want to be blast in water under the BGA to get all that flux out from over there. So hold on to the hose.

Yeah, so that went for about 11 minutes. Hold the card out and I want to air dry it real quick. Yeah.

Great, awesome. So now we just got to put this in the oven for about 12 hours and it's gonna be good to go.

Second card passed, they are both going in the oven. The first one's already in here.

They have to sit in the oven for 12 hours.

so I'm not actually going to be taking my cards home. Greg and you'll be shipping them to me later. But that's it, we're done. They both work. You did 99 .9 % of it, but how'd I do?

You did awesome, Luke.

I'm glad to have you, I'm glad you came by. You're honestly a great set of helping hands. So you picked up the work right away. Thank you. And honestly, Luke, welcome to the team.

Dude. Excited to see you next Monday. Honorary employee. Yep, start work. I'll take it.

Thank you so much Luke. No, this was sick Thank you very much.

That was that was fantastic Just so you guys know you wouldn't be coming out here and doing this you would just ship them your cards And you offer a bunch of different services.

Yeah, so we started upgrading 4080 supers from 16 to 32 gigs Do a 100 conversions but those are less known and then also 170 HX mods are coming because there's so much in the works with that modification right now, I'm not offering it, especially with the 48 gig stuff, I'm focusing on that first. The One 78Xs will come just a little bit later. Thank you guys for watching.

If you wanna see more of this, maybe check out the 170HX video that Linus did and I'll see you next time.

Transcript of 马可·奥勒留的清醒思考法：告别精神内耗的思维工具 （中文完整文稿）。

This video was made possible by Brilliant, a math and science problem solving website that makes learning fun and rewarding.

Almost everyone thinks they are a good thinker, but in reality few people really are.

A truly great thinker is constantly growing and evolving, so take a look around you.

How many people do you see moving forward in life? life.

How many people do you see solving problem after problem?

And on the contrary, how many people do you see running in circles?

How many people do you see still trying to solve the same problems they were working on years ago?

The better you get at thinking, the better you get at solving problems. And good thinking begins with clarity.

A clear thought is unambiguous, captures the essence of a thing, and is formulated logically.

A clear thought can be communicated through a simple if -then statement.

For example, if you turn on the tap, then you can get water.

A clear thought leads to clear and intentional action.

For example, the statement, if you turn on the tap, then you can get water, leads to clear action when you're thirsty.

And clear and intentional action leads to clear feedback.

When you go to test the tap for water, you will immediately find out whether your original thought was true or not. Either water will come out or it won't.

And because clear feedback proves or disproves your thought, it helps improve your understanding of the world.

Either you'll find out that your understanding of the world is correct or incorrect and And finding out you are incorrect helps you move closer towards correctness.

And the more correct your worldview is, the better you'll be at solving problems. Let's take a look at an example.

Marcus Aurelius earned the titles of the last of the five good emperors of Rome and philosopher King by becoming a great leader.

And he became a great leader by becoming a great thinker.

And he became a great thinker by learning to think clearly.

And he learned to think clearly by writing a private philosophy for himself, which we now refer to as his meditations.

In his meditations, Aurelius organizes his thoughts in a logical way, reflecting on the relationship between his actions and their effects.

And if you want to think clearly and improve your ability to solve your own problems, it would be wise of you to do the same.

Organize your thoughts, reduce their ambiguity, and make them clear by turning them into simple if -then statements.

Clear logic leads to clear thoughts. Clear thoughts lead to clear and intentional action. Clear and intentional action leads to clear feedback.

Clear feedback leads to an improved understanding of the world, and a clearer understanding of the world leads to a better ability to solve problems.

For example, let me formulate a clear, logical thought.

If I publish a how -to video with a philosopher in the title, then it will get over 100 ,000 views.

That's a clear thought with clear logic.

That clear thought leads to clear action. I will write that video and publish it.

Now when this video is published, I'll get clear feedback.

We'll both find out if that thought is true. And once I find out whether that thought is true or false, my understanding of the world will improve.

And by improving my understanding of the world, I'll be better equipped to solve my problem of being a successful YouTuber.

So in conclusion, good thinking begins with clear thinking, and clear thinking begins by turning your thoughts into unambiguous logical statements.

Transcript of SSH安全现状离谱：一个基础协议如何坑遍全球服务器 （中文完整文稿）。

This bug is one that I truly cannot believe exists.

I was supposed to be filming a video on the 9 .8 CVE in the TCP IP stack of Linux, and then I literally saw this just scroll across my Twitter feed. Oh guys, I thought October was over.

I thought Cybersecurity Month was behind us and we can move on with all these CVEs, but this is an absolutely insane one that is a vulnerability in SSH itself.

Now the good thing is it's not an all SSH, it's just in a couple distributions, but there's still distributions of SSH to be used everywhere, so not a great place to be.

Obviously guys, in this video, I'm gonna show research with somebody else. I did not find this.

This is done by Ozgun Coltecun. Maybe I got that right. He is a worker, a researcher at Wakefield. So go show him some love.

I'm gonna be using this repo to go over the bug. Obviously they made it, so go check it out yourself.

They also did a very, very nice TLDR that I'm gonna try and use in this video because my kids are upstairs and they're making a lot of fucking noise. So, you know, I'm gonna be doing as best as I can.

Now, the news headline, which is exactly correct, combined flaws, let anyone into Alpine Linux and other distros by typing anything over 256 characters as a password, emergency fixes for open SSH and Musil libc, issued for millions of systems patched immediately.

Guys, when I went to my first black hat in 2013, I took some like bullshit how to hack into Linux and Windows course where they showed us like how to do a buffer overflow on like a Solaris box from like 2008, okay?

And what you did is at the login prompt, you typed in 294 As, and that got you a shell on the computer, okay? That was a box from 2008, 2009.

It is the year of our Lord of 2026, and this is still an issue. Now, I wanna be very clear.

What's really, really interesting about this bug is that despite it requiring you to put a lot of characters into a buffer, it is not a buffer overflow.

It is this really interesting issue where like the security boundaries and like the contracts of functions got so far apart from each other, and so confused over time, that people over time forgot what two functions did, and those functions like depend on each other. We'll go into what that means here in a minute.

Guys, real quick, every day it feels like a new crazy zero day falls out of the sky. So it's a question of when your network gets attacked, not if. So here's a TLDR, right?

When you log into a system over SSH, what is going on, it is comparing the given password, right? So you specify a password, there's nothing to do with public key authentication. You give it a password, right? And what it's going to do is it's going to run a variety of functions over this thing called Etsy Shadow. Okay, I don't have them because I'm on a fucking doctor image at she's shadow.

We'll notice here is there are some values that don't have any Hash okay, so you'll notice for example, this is root and there is a star here The star is very important But it means is that this account does not have a password this decision was made back in 1980 by Bill Joy for this account called Damon Which basically meant hey Damon does not have a password and because star is not a hash a hash a collision can never occur where no one can ever log in as an account. Perfect.

Now, when you're doing all this crypto, what's happening under the hood is using a library's crypt function, okay?

So crypt is a function in Moosl Lib C, if you're not aware of what Moosl Lib C is, Moosl Lib C is one of many implementations of Lib C. The one we know right now is GNU Lib C. That's a very common one. This is a different Lib C implementation that different systems use. It is a lightweight, fast, and simple, and free version that strives to be correct.

Okay, correct most of the time.

Okay, so in August of 2012, there is this issue where basically, originally the Muzzle Crypt function that was doing the cryptographic algorithm to check for your password hash, right?

So for example, we have $6, this $6 means you're doing a SHA 512 operation, and this is a hash of a SHA 512 password, and here is the salt, right? So it's algorithm, salt, and then the hash, okay?

Salt is like a value that you can catenate to a value before you hash it, okay? Basic crypto stuff.

For a while, this crypt function, if it fails, right, if there's an issue with doing the cryptography, if you gave it too much data, not enough data, for a long time, it would return the value null.

Well, a lot of programs are bad programs, they don't check the return value for, you know, their functions.

So instead of returning null in 2012, they decided, okay, we're going to return a character. And what is that character we're going to return? It's none other than the star.

Yes, it's the star again, the star is also the character that just happens to match with the value from Berkeley from 1980.

Not a big deal because there should be no cases where we're gonna actually compare these two values. Ah -ha -ha.

Okay, so there's no world where we compare this value to the value in Etsy Shadow because one is a star and the other is an error value, okay? So not a big deal there. Now for a long time, this wasn't really an issue, okay?

Until 2016 where there was discovered a timing attack in the way that Muzel Lib C did its crypto.

So for example, what if I wanted to figure out what user names were available on this system, right?

If I go an SSH into the box and I try to SSH into a username that does not exist like Bob, for example, that operation will immediately fail out when I do the password. I'll put the password in, it'll immediately come back. What's happening there is I'm doing a timing attack. There is no crypto operation in the background happening. I can use that timing to enumerate what accounts do exist on the box.

If I log in as Alice, for example, even if I don't know the password, I can time the difference between Alice and Bob and figure out that Bob does not exist because it came back quick, but Alice does exist because it took a second to run the crypto.

So to fix this in July of 2016, for accounts that either didn't have a password or didn't exist, what they would do is they would steal somebody else's hash.

They would temporarily borrow the hash and the salt of a different account and they would allow you to do the crypto operation on that salt and it's not really doing it for the account.

So for example, if Bob doesn't exist on this account or on this computer, what's gonna happen is the crypto backend is going to borrow Alice's salt and we're gonna do a password check of the given password on Alice's salt for a fake account.

What's happening here is they're trying to get rid of the timing attack that's happening inside of the crypto libraries. So let's say for example, I wanted to log in as the root user. The root user does not have a password. What's going to happen is it's going to allow me to log in.

It's It's going to do SHA 512.

It's going to borrow their hash and it's going to do the SHA 512 crypto algorithm on the input that I give it. All is well and good.

What could possibly go wrong here? Well, if you read the implementation of of crypt SHA 512 in Musil C.

If you'll notice, if the I value is greater than key max, it returns zero, okay?

Just sit and think about what I've said previously in this video and think about what can go wrong here.

If I is greater than key max return zero, key max is 256.

Okay, well, if SHA 512 crypt returns zero and the crypt function itself returns a star when there is an error and if the given hash for root is a star and we ultimately do a stir compare on the calculated hash a star and the given hash from the Etsy shadow file a star those are going to compare and be the same value that is going to pass as a password check what so the question is right like who is vulnerable to this bug well first of all this is a failure of implementation in the Moosel C library, okay?

So G lib C, the one like the library in C that is very common is not affected by this, right? You'll see the different builds and configurations of SSHD.

The only ones that are affected are Moosel C builds of open SSHD that don't have PAM modules, that don't have a portable authentication modules, that's what PAM stands for.

Also, if you don't have a shadow file and you have just your hash disorder and Etsy password, that also breaks it.

So there are some preconditions that have to be true for you to be vulnerable to this bug. First of all, you have to have password auth enabled. Now just to be clear, they are on by default.

You also have to have an SH server that is not using portable authentication modules.

So some SH servers by default do ship with PAM enabled. So if those are enabled, you are not vulnerable to this.

But unfortunately, the big kind of example here is that Alpine Linux, Alpine the Docker image ships in this configuration.

It has password logins on and it doesn't use PAM. It is vulnerable by default.

And literally like the only other part of the configuration you need to be true is there needs to be an account that is registered that has a star value as it's password in the Etsy shadow file or Etsy password if you don't have a shadow file.

And then another account with a real password that uses SHA 512, which is like the default now, $1 or $6.

If all of these are true, They literally all they have to do is put in a large number of a's like this many a's it's like two five seven to ten twenty four I think it's ten twenty four is the upper limit of SHD And that'll just drop you into a shell the issue with Alpine shell too is the Alpine user is the main user on Alpine This is also vulnerable and Alpine is a run as user or do as user so it's effectively pseudo It's effectively root looks like not great also for like tampering fear right important to highlight if you are on Debbie and Ubuntu Fedora, anything else libc going through a PAM using DropBear or OpenBSD, you're good.

It's literally just these couple of configurations granted, it's probably a large scale portion of the internet, but it's not a lot, I guess.

What makes this bug so interesting is just like, this isn't a memory corruption vulnerability, this isn't like, this is a logic bug, but it's not a logic bug, like they forgot to check for like the n or n plus one length of a string, right?

This is literally like, there are two different functions that have two different like contract agreements and over time they just drifted so far apart.

Like, oh, we're gonna use a star over here and like surely the star can never become a password hash.

And then over here it's like, okay, we're gonna have Crypt return a value that isn't null. Ah, what value won't really matter ever? Star, that'll be fine.

And then finally, okay, we're gonna let any user that either doesn't exist or doesn't have a password, use a crypto routine, and that crypto routine will fail above a certain length, and that failure will cascade into a star that is then the hash for the other users, like it's just, it's insane.

Now, my real question is like this, it's been vulnerable like this basically since they fixed this CVE, and then Alpine has been broken since 2021 because they changed the hash from a locked bang to a locked star because that broke Pam.

And so this has been a bug since 2016, it's 2026, it's been 10 years.

Their argument here is like wordless, haven't had 300 letter passwords, I guess, because literally all you would have to do is take a vulnerable version of this, log into an account that has this star as the password or as the hash, and then give it 300 days. Like we're talking 1992 levels of hacking, insane. A question you'd be asking, would Rust have fixed this?

Actually, no, maybe there's a world where the types or the enums or the result T or option T that they would use to do the error checking would prevent this, but from a pure use -after -free memory safety thing, this bug has nothing to do with that, so it's kind of interesting.

That's it, guys, if this was too fast or too quick, I'm sorry, I have my kids at home, so I'm trying to bang this video out without making too much noise or making you hear them.

If you like this style of video, do me a favor, hit that like button, hit subscribe, and then we'll see you in the next one.

Transcript of 白手起家富豪的10个习惯：普通人直接抄，坚持21天 （中文完整文稿）。

I'm going to share with you the 10 keys to unlock what I call your quick brain, right?

These are the 10 things scientifically proven to help you be faster, smarter, better to optimize your brain health and your fitness.

And now every single thing I'm going to say, nobody's going to debate, right?

They're going to say, yes, of course, but common sense is not common practice.

So what I'm going to ask people who are watching this is to rate yourself 0 to 10.

How well am I doing this one area, like in terms of how I'm eating or exercising or something?

And then you'll notice everyone wants to know what the magic pill is.

There's no magic memory pill There's their magic memory processes, right?

If everyone wants to know the one silver bullet you have to do all these things because that's what falls within the focal point Meaning that the 80 20 rule the 20 % that gives you 80 % of the results results and the rewards So really fast the 10 keys for unlocking your quick brain number one is good brain food All right good brain food because you are what you eat what you eat matters especially to your gray matter.

There's certain foods that are really good for your brain and you probably eat most of these foods as you are, because I know you're very - Blueberries, sour foods.

Like blueberries is like, I call them brain berries and avocados and the good fats and all of them.

So there's 10 of them that I recommend and so they're really excellent.

They're very neuro protective and such, including dark chocolate.

So dark chocolate is what's good for your mood is generally good for your mind. But so a good brain diet is number one. So just rate yourself on a scale of zero to 10. How good is your diet? All right, number two is killing ants.

Automatic negative thoughts, your self -talk. And so what I would recommend to people is zero to 10. How's that feel? How well are you doing?

And not only that, but maybe going on a little faster cleanse.

Like we cleanse the food we eat and things we put in our body, but what about the thoughts that we're putting in our body also? I think that would be 30 days with no negative thoughts or one day. Or not complaining or something like that. Right, no complaining challenge.

Because it's just, again, people get so addicted to these things, like, you know, complaining and whatever you're rehearsing, you're just, you're literally rewiring your brain to get more of that. It's not law of attraction.

There's a law of action that's going on, you know, that makes it, because all behaviors believe driven. And so you're wiring your brains to be really rehearsing those things.

That's why they say that the fearful person dies a thousand times, like a coward dies a thousand times in their mind, you know, like a brave person dies just once.

because every time you're rehearsing, oh, I'm scared of public speaking, you do it over and over again, not only do you get the fear of it every single time because your imagination, your mind doesn't know the difference between something you vividly imagine and something that's real.

Literally, if you were to put on a brain sensing device, the same parts of your brain that would light up if a dog walked in, you see what parts of your brain lights up and if you imagine that dog walking in, the same exact parts of your brain will light up. So the mind doesn't know the difference. It's the most powerful tool you have is your imagination.

You know, our thoughts are things and it's, you know, activating your imagination, so, you know, don't do it to rehearse your fears and the things that are negative, right?

Only have thoughts of things that you want to come and have it really happen. So the third thing, exercise. People, really, we know.

Number four, brain nutrients.

Omega -3s, the DHAs, the Ginkgo biloba, vitamin B -12s, and this is a subject that comes up because, you know, a lot of people travel, they have fast food lifestyles and, but maybe they're lacking in certain supplements.

And what I would say is, I'm not a nutritionist, I'm not anything that ends on an ist, but go to a doctor, good functional medicine doctor, you have a blood test, and I say the same thing for diet.

When people say what's the best, do food sensitivity tests, see what your blood says. Red, yellow, green.

Red, stay away from yellow, sparingly, green, and see how you flourish on it afterwards, right, and then test it.

But same thing with nutrient profile, you can see what you're deficient at, because if you're low on B vitamins, If you're low on DHA, then you're going to have cognitive issues, right? So supplement. Fifth thing that you want to optimize your brain is positive peer group.

We know who you spend time with is who you become because, and it's not just a positive thinking, you know, affirmation quote.

Literally, you have mirror neurons and your mirror neurons in your nervous system is when you could watch a sport or watch a movie and you could feel what they feel.

It's what, or you have empathy for other people. We're always imitating people, right?

And so who you spend time with is who you become because you're imitating their habits their behaviors their thought patterns and everything So that's why they say you're the average of the five people you spend the most time with right if you're around nine broke people You're gonna be the tenth right like you hear these things all the time But there's a it's not just your biological networks or your neurological networks analyze your social networks Right because whether or not you smoke has less to do with your Neurological networks and more to do whether or not your friends friends smoke It's gonna have more of an influence. Sixth thing for your brain, clean environment. We know a clean environment is good for your brain.

At an anecdotal level, you know when you clean your desk or you clean your laptop off, you know you have clarity of thought, right?

But I don't just mean that, I mean clean air, clean water because there's a lot of pollutants that's not good for your cognition, right? So from there we have number seven, sleep. This is a big one for so many people. Sleep is so important for your brain for three reasons. Number one is where you consolidate short to long -term memory.

Everyone who pulled an all -nighter back like in the university, that hurt you because you need that sleep.

Number two, it's where you clean out the plaque that will lead to brain aging challenges and that's what the latest research is saying in terms of leading to dementia and such. And then the third reason is because we need dream.

And here's the thing, you're like, what does that have to do with anything, Jim?

It's like, you don't know how many amazing ideas and invention, literature, works of art came in dream states.

Like, did you know, like I mentioned Mary Shelley, you know, here in London, Frankenstein came to her in a dream, you know Paul McCartney came up with yesterday in his dream You know like a chemist came up with a periodic table in his dream Elias how created the sewing machine in his dream, right? Like what are you dreaming about late at night?

Because when you are working all day and you're learning all day your brain doesn't shut off at night at night It's more active people don't realize that you think it's just not happening It's actually more active and it's coming up with solutions and it's integrating stuff in the form of dreams number eight is brain protection meaning that But I work with a lot of athletes, high -performer athletes, concussions are a big deal, right?

And I've had a number of TBI's, you know, traumatic brain injuries, but not just that.

I did a whole episode on electromagnetic fields.

People don't realize that our smart devices, like our brains didn't really have all this electricity right by our brains.

You know and I read recently that 90 % of kids sleep with their phones underneath their pillows I mean who knows the effect that's having so better avoid protect your brain of wear a helmet And then finally nine and ten ninth key to keeping your brain alive is New learnings right the neurogenesis neuroplasticity means neurogenesis.

You could create new brain cells the day you die Science never thought that neuroplasticity is saying you could create new connections the way to improve neurogenesis neuroplasticity two things Novelty and nutrition, just like when you're an athlete, just like training your body.

You give it novelty or stimulus and then you feed it with nutrition, right? And so that's how you build your mental muscles too. Finally, number 10, stress management.

Stress management, meaning people don't realize it because it's invisible, but it's like you don't realize how much stress you're under until you're on vacation or you're getting a massage. You don't realize because it's all there all the time. It's like the fish that they don't see the water because it's there all the time.

So what are we doing to cope, you you know, meditation, relaxation, massage, yoga, whatever it is, it gets so stressed.

Transcript of 年少百万身家的Gadzhi：17条我盼你早点知道的人生法则 （中文完整文稿）。

So ladies and gentlemen, I grew up without a father.

I'm an only child and the reason I'm telling you this is because I am going to tell you 17 life lessons I wish someone had told me as a teenager.

So these are things that I had to find out the difficult way and that's why I'm sharing them with you today.

So that way you do not need to go through the years and years of pain, struggle, soul searching that I went through and you can just bypass all of that if you listen and you take heed.

So I've got a couple things written down and honestly, I'm just gonna spitball these 17 lessons.

Lesson number one is you need to extend your timeline and I know that is very difficult when you watch my videos.

It's very difficult when you are on TikTok and you see these 18 year old, 21 year old millionaires and I understand that I am even a part of that issue sitting here at the age of 23 years old with tens of millions of dollars.

It is so easy to see the final product and think that you need to rush when quite frankly, you don't.

I can tell you that I have missed out on so many, hang out with the friends, grab a beer with the friends, watch the football game. My childhood and my teenage years had none of that. Which is why I want you to understand, extend the timeline. You want to become a millionaire? Great.

Give yourself 5 years. Give yourself 10 years. You don't need to rush. Now I will say if you can get rich faster, go for it.

If you can learn a high income skill and we'll talk about it a little later down the line.

If you can start a side gig that then turns into a small business that then turns into a very large business, go for it. but just alleviate the pressure.

Stop thinking you need to become a millionaire in a year, two years.

And even in my position, I understand I did not have a normal person's journey.

It is not normal to amass the sort of wealth that I have amassed.

And by the way, as I said, that came with a lot of drawbacks.

So I don't just think it's all roses and sunshine, but even me in my position, you can see videos of me in 2015, 2016 when I was just starting out my business with acting on my face, reading books, going to the gym, being an absolute loner.

And I say this because please keep this in mind any time you watch my video or some TikTok, this stuff does not happen overnight.

It has been nearly a decade since I started reading a book a week, since I started meditating, going to the gym, and setting the building blocks in the foundation for the man that I would go on to be.

So please, keep your foot on the gas, keep pressing, but extend your timeline. Time is your biggest asset at the moment. The next life lesson is walk with purpose.

I remember even from the age of 15 or 16, I would walk down the street. You could tell I was on a mission.

And I think that was really just a sign of having a clear direction, a vision for what I could become.

So when you walk down the street, I'm not saying that you need to sprint, keep your shoulders back, look straight, walk with a clear direction, look like you know where you're going.

Now the next thing is to focus on practicality and not purpose.

I hear this whole thing of, oh, I'm not starting because I just need to find my purpose, I need to find my passion. Stop it. Focus on practicality.

When you are a teenager, you will not know your purpose, you will not know your mission.

And even in your 20s, even the phase I'm in, it will morph and it will change.

So instead, focus on practicality, focus on learning skills that are practical, focus on starting businesses or careers that are practical, focus on getting an apprenticeship or even a low paying job that isn't the destination of where you want to get to, but is practical and will teach you the core foundation and the core principles to then go on and succeed in your 20s.

Now the next life lesson and this is so important and I know I'm just speaking to men here and a lot of times in my content I speak to men but for reasons that I've previously mentioned it is important for me to put out content and nurture the next generation of men.

So men listen to me when I say focus on becoming a gentleman.

In this world and especially in the West I see some of the most disgusting behavior from men.

It does not surprise me that so many women treat you with disrespect because you are not men even worthy of respect.

Do not listen to all of these guys that have never lived in the real world telling you oh you're a simp if you pick up a girl if you drop a girl home after a date if you pay for the bill. I don't know who raised some of these kids because quite clearly they're not actual men.

The way that I was raised is as a man you pay for the bill and if you cannot pay for the bill if you cannot afford to and when I was younger I was in that position. Well then And guess what? Get to work. Don't take girls out to dinner. Put your head down and focus.

If you're taking a girl to dinner, you pick her up. You drop her home. You do the gentlemanly thing.

I cannot express how important these things are to learn in your teenage years.

Other things. Understanding a girl's alcohol tolerance.

I see so many idiots that give girls drink after drink after drink. Not seeing that they are quite clearly intoxicated.

As I said, I don't know how some guys were raised but the way that I was raised, If you are with a woman, if she's your date or she's one of your friends, in that situation, you are responsible for her. So act accordingly.

If you want to be a man of honor, if you want to be a man that deserves respect, then you must learn how to treat a woman properly. Never ever let her cross your boundaries.

And just remember, if you are a man that is worthy of respect, then obviously never accept disrespect from another party.

By just thinking in this world, we live in so many men are afraid of, you know, come across and being a simp.

If you pay for the bill or if you pick her up on a way to date, that is absolutely no way to conduct yourself as a man.

So learn these principles as a teenager and they will carry through for the rest of your life.

Especially if you stack them up with all these other life lessons, then you are going to be a man of honor.

You are going to be a man that other women respect. And I can tell you that that goes very far in life. The next thing is I want you to enjoy the journey.

I want you to remember that these are your glory years.

These years right now as a teenager and especially in your early teens even, these These are the years you're going to look back at with fondness because these are the years that you built.

You built your character, the person that you were going to become.

I can tell you right now that my most fond years were probably the year before I started my first ever business when I was 16 and the couple years after that. The years where things were tough, they were not easy.

I was making sacrifices, it was difficult at the time, but I have so much appreciation for it now.

So please, if things are difficult or they're hard right now, take a moment to appreciate it because these are the years you're going to be talking about in a decade, two decades.

When you're sitting outside with your boys at your beachfront property having a cigar or drinking a glass of wine and reminiscing about the good times.

Now if you're in your teens, I want you to start investing now and I don't care if it's $10 a week or $30 a week or $50, whatever you can do, even if it's $5 a week, it's not about the money and it's not about the eighth wonder of the world compound interest because I'll be honest at this age it really doesn't matter and hopefully in a few years you should be making a very very good income that far exceeds anything you could ever make from investing.

All it comes down to is simply good habits.

I do not believe that if you won't invest $10 when you make $100 a month, that you'll invest $10 ,000 when you make $100 ,000 a month.

And if those numbers sound crazy to you, I can't imagine ever making $100 ,000 a month, all this, just park that in your mind for now, okay? You don't need to worry about that. But I'm telling you, it's a very, very strong reality.

If you're consuming content like this and you dedicate your life to building over the next year, three years, five years, 10 years, I can tell you numbers like that almost guaranteed in your future.

But I know for a lot of you that will overwhelm you.

So just park that aside for right now. If you make $100 a month, just invest 10.

It's simply having good financial hygiene and building those good habits early on.

The next life lesson if you're a teenager is build high income skills.

A high income skill could be content creation, it could be sales, it could be copywriting.

These are things that no matter what, I don't care if the year is 2023, I don't care if the year is 2040, these will always be important skills. And these will always be highly paid skills.

So please do not get caught up in short term fads, short term trends.

Ask yourself, if I learn this skill today, will this pay dividends in five years, 10 years, 20 years?

And if the answer is yes, well, then maybe you should consider learning it.

Now the next life lesson, this especially applies to men is focus on building muscle and your physique, especially between the ages of 16 to 21, 22.

If you can get in the gym, if you can do exercise, whether that's a standard sort of bro split routine, whether that's powerlifting, whether that's crossfit, whatever you decide to do, building muscle at a young age, you're going to have that mind muscle connection, and you're going to keep that on and you're going to carry that through for the rest of your life.

You know, I started going to the gym when I was 14, someone argued that that was even too young, but I'm very happy that I did and I'm very happy that I built up muscle in my teenage years because now I do a lot of tennis, I do a lot of running, some boxing and I never really worry about losing muscle or my physique or anything like that because I had more than half a decade of nurturing that.

So especially for men as our testosterone starts to decline in our mid 20s onwards, use these years to your advantage, lift heavy things, push yourself, train hard because you have this incredible advantage and will carry through for the rest of your life.

The next life lesson is be confident but acknowledge your age and your lack of wisdom.

I started my first business when I was 16 years old and by the way this was on a business where I was hiding behind a computer I was front -facing with clients.

I was pitching multi -million dollar businesses.

So I had to walk in with confidence I had to be presentable. I had to talk.

I had to communicate clearly I had to be confident regardless of my age and I had to do that front -facing publicly to potential clients But at home when I was alone in solace in peace I understood and even me sat here at 23 hundreds of thousands of people even millions of people around the world Looking at me as a hero and inspiration a mentor.

I understand that at the end of the day I still have so much to learn.

I understand that of course I got started so early I had my first employee at the age of 17.

I dropped out of school at the age of 17 to support me and my mom I've experienced so much by this age But no matter what, wisdom is solidified through decades and decades of experience.

So if you are in your teenage years, be confident, walk through life with confidence.

Don't let your age, especially when it comes to business and your career, don't let that weigh you down.

But in moments of peace and quiet, acknowledge that no matter how well you're doing, wisdom is only truly forged through decades and decades of life experience.

The next life lesson, and once again this relates specifically to the men, is get comfortable talking to girls.

Listen, of course, I believe that as a teenager and in your early to mid 20s, those are your years to build.

But it doesn't matter if you're even 15, get used to speaking to girls.

Don't let yourself become one of these 25 year olds that goes on to become a multi -millionaire but is just creepy around women.

Get comfortable speaking to girls if you're in a cafe and you think a girl is cute. Just talk to her.

What's the worst that could happen? And even better, just give her a compliment. And that's it. Don't ask for her number. Don't ask for her Instagram.

Just give her a compliment and that is the least that you can do you have to get comfortable speaking with women holding your ground Keeping your chin up and keeping confident Don't let it consume your life because I do not believe that women should be your goal in your teenage and early 20s Because as a man your best years really are from the ages of around 26 28 all the way up until your late 30s But nonetheless just learn how to carry yourself around the opposite sex The next lesson is keep your word and start learning this lesson from a young age If you say that you're going to give someone a certain percentage in your business, well then give them that percentage.

If you say that you can show up to something, well then show up to it.

Your word is your reputation and your reputation is the most important thing that you have.

So learning from a young age to show up to keep your word, that will build trust and that will build respect and that will carry on for the rest of your life.

The next life lesson and once again, I know this relates specifically more to the men is in your 20s, doesn't matter if it's your early, mid or late 20s.

These are your years to build.

If you do things correctly, your 20s will set you up for the rest of your life.

So I made this video specifically for the teenagers, but it's really just to say that your 20s mentally prepare yourself because this truly will be your make or break period in your life. The next life lesson is learn to say no more than you say yes. Learn to say no when people want to go out. Learn to say no to business opportunities.

I know you have that friend who watched a video, has a business idea, and they want to go 50 -50 with you but your gut instinct tells you that it would be a bad business partner or why do I need a business partner in this? I could just do it myself.

So just learn to say no. Learn to stand your ground.

I can tell you right now that yes has ruined more people's lives than no ever has.

People saying yes to getting in relationships they shouldn't be in.

People saying yes to starting businesses that they shouldn't. People saying yes to going out when they shouldn't.

As I said learning to say no will be one of the most invaluable skills that you can learn from a young age. The next lesson is learn how to dress well. Learn how to find good quality clothes.

I know that I'm wearing a shirt with a graphic right now, but I'm in the comfort of my own home.

I'm very relaxed, but when you leave the house, you should be presentable, no big labels.

And by the way, later down the line, once you found your style and what works for you, you can spruce things up. You can get a little bit more funky with it.

But at first, from a young age, learn how to find clothes that fit you well, not too tight and not too loose.

By the way, most people make the mistake of finding clothes that are way too tight for them and I can tell you something that's actually quite funny.

Over the years I've noticed especially with my jeans and my trousers the more money that I've made the looser my trousers have become so that's just I found that that holds true for most of the multimillionaires that I know so learn how to find clothes that have no branding that fit you well that fit your style learn how to layer clothes and then also learn how to start getting stuff tailored you can buy trousers that are $30 or $40 and go take them to tailor They'll get them tailored for you for $15, $20, $30.

And I know that's an additional cost, but you will look far better than the people spending hundreds of dollars on all this designer, this and that, et cetera, et cetera.

So I believe in your teen years, it's very important for you to learn how to dress well.

Obviously your style will morph and it will change as the years go by.

But in your teen years, just learn the basics, learn the classics, and just learn how to dress in a classy, elegant way.

And later down the line, and especially once you have more money, you can start sprinkling things up with, you know, a graphic tee and, you know, maybe some nice jewelry here and there, whatever you want. But for now, just stick to the classics and the simple things. The next thing is you need to learn how to keep strong boundaries.

And this is probably one of the biggest reasons I've been able to have the success that I've had.

Once again, I don't know, maybe it's because I have a Russian mother, Russian people are just very, they have no issue saying no, they have no issue keeping strong boundaries. So maybe I was just raised a certain way.

But from a young age, I knew how to have strong boundaries with with my friendships, with my relationships.

And I believe learning that from a young age has carried on for the rest of my life, where I can honestly say that my friendships, the people that I work with, the people at my companies, any of my romantic relationships, we have such mutual respect for each other. We have an understanding for each other's boundaries.

And that was all built off the back of, from a young age, learning to say no, as I mentioned earlier on, and learning how to have strong boundaries with the people around me.

Now, the next thing that I wanna talk about, and I know that for some of you, I have atheists watching, so maybe you might roll your eyes. whatever you want to believe in, go for it. Build your relationship with the Creator.

Whatever the Creator means to you, whatever religion you follow, whatever it is, maybe you're just spiritual.

But build a relationship with whatever the Creator means to you because I can tell you, you will have some of the highest highs in your life and you will have some of the lowest lows if you live life properly. Because the worst life you can have is just a flat line.

So if you follow all of these life lessons, then you will have a life of incredible highs and incredible lows and as I said that is what makes a good life and the only way to stay sane and not go crazy in those periods is to be humbled by the Creator.

As I said whatever that means to you is to understand that there is something bigger than you and the only way that you're gonna get through this crazy turbulent tumultuous life is by having a strong relationship with the Creator.

I remember I was six or seven when I started praying to God and the reason that I started praying to God was because I had a very very difficult upbringing and it's funny the only people that I see that don't have a relationship with the creator, whatever that means to them, is the people who have had a life like this.

The people who never went out there, never took risks, never fought for the life that they wanted. For some reason, they're always the ones that seem to turn to atheism. So I can tell you, from first -hand experience, life will get crazy.

And the only way to stay sane when times are good and when they're bad is to have a relationship with God or the Creator or whatever you want to refer to it as.

The last and arguably the most important life lesson for teenagers is take ownership of your life. Something great happens in your life, great.

You can take responsibility for it, as well as the creator. Something bad happens in your life, great. Take responsibility for it. Radical ownership of your life. No excuses.

It is only you and you to blame. And I know sometimes it will be difficult.

I know sometimes there will be a clear thing or a person or something to point to, something else to blame outside of yourself, but don't fall for that trap.

As I said, take radical ownership of the things that happen in your life.

So ladies and gentlemen, And I genuinely believe that if you're watching this and you're in your teens or you're early to mid -20s If you follow these 17 life lessons these 17 principles, you will live a life of abundance You will live a life of honor that you could never even Imagine and as always I'll be watching from afar and I'll be rooting for you

Transcript of MIT 量子场论·8.323·第04讲：自由标量场论的正则量子化 （中文完整文稿）。

So last time, we quantized this theory. So let me just write down the theory again.

Okay, so we quantized this theory.

And so, so let me also just write down its canonical momentum, density, and then the Hamiltonian density would be, uh -huh, square, okay? and the classical equation motion.

So we'll not repeat the quantization procedure, and we wrote down the most general solution at the quantum level.

We're written as the following.

This factor is the convention, and then we have a k, Okay, okay, plus, okay, so from now on I will suppress the hat, okay, so I will suppress hat, you should always now view it as a quantum field, quantum operators, and the plus, okay. Stop.

UK is the complete side of solution, which is given by potential minus i omega t.

So this is just the basis of solution.

So this is the complete solution to the operator equation for this operator phi.

And then you can also find its conjugate momentum, density, so the pi, you can just take the derivative straight forward. I will not write it explicitly.

And then we discussed that you can impose a canonical commutation relation, which is that the pi gx, gx prime, prime, you could choose pi kx and pi kx prime should be 0.

And then pi kx, x prime, then given by the delta.

So that's what we did at the end of last lecture.

And then you can just plug it in.

plug this, plug those expression into here, then you can find the commutation relations between those integration.

So, ak and ak dagger are integration constants, they're of your operator equations, they're constant operators. Okay.

And so, when you plug them in, then you find the canonical, then you find the commutation relation between ak and a dagger, and then you find that ak and ak prime, prime equal to a dagger k and a dagger k prime is equal to 0.

And a k prime dagger is equal to the delta function in the wave space, OK?

So now, if you look at those expressions, So they essentially look, we essentially find the infinite number of harmonic oscillators, and each harmonic oscillator labeled by a set of continuous number k, and then the commutation relation is just like a continuum generation of standard a dagger equal to what?

OK, and they are independent because of this delta function.

And if k is not equal to k prime, then you get 0 as the commute.

So you can see the story a little bit sharper.

That indeed, we just get infinite number of harmonic oscillators. So these are just commutation relations.

This still does not tell you we have a harmonic oscillator.

To see harmonic oscillators, we actually we need to calculate the Hamiltonian.

So now, let's try to calculate the Hamiltonian.

Now, we can try to calculate the Hamiltonian.

So this is just, you can just do it straightforwardly. Just plug that in.

Just plug that expression in. We draw the above.

And then you just plug in the expression for the phi pi and the pi into that equation, and then you just straight forward to calculate it. And then we'll let go through the details.

Yeah, it's a straightforward calculation.

So then you find, yeah, maybe after some minutes, then you find the following answer.

So now when you plug them in, then you find you can actually do the spatial integral because you just have the plane wave.

You can just do the spatial integral. And when you do the spatial integral, then that gives you a delta function, momentum space, et cetera.

And then you can reduce everything into a single k integral.

And then what you find is you get omega k.

and then you find ak dagger, ak, and then plus, say, ak, ak dagger. Okay? So let me see.

Yeah, actually, yeah, I think one half.

Okay, so this is the standard expression assume for harmonic oscillator now, OK?

Actually, it's exactly identical to you just sum, yeah, integrate like sum, just sum over continuum harmonic oscillators.

And each with frequency omega k. And each harmonic oscillator is labeled by this number k, OK?

And so we can do from the standard trick to write it actually as the, to write it as a commutator, we can just write it as k, and then we just have this.

So we combine these two terms together and introduce a commutator.

You just have a dagger a, and then plot the commutator between them.

Okay, let's get that.

So in the standard story, this is equal to one, Then you get the standard answer that you get omega a dagger a plus one -half, okay?

But here, so we can again, yeah, let me just write one more step.

So this is k omega k, OK, dagger, OK, then plus E0.

And now E0 is the sum of all the zero point energy of all the harmonic oscillators.

OK, so 1 half omega k.

Except that the tricky thing is the following.

OK, except now we have this commutative of a k with a k dagger.

In the standard case, this is equal to 1.

But in model case, when you set k prime equal to k, we get the delta 0. So we get delta 0.

So then we get that 2 pi cubed delta 3 0.

And remember, 0 is in momentum space.

So this is in k space. We have two kinds of data functions.

is the data function in k space, data function in coordinate space.

So here I put the subscript k here, means that this is the data function 0 in k space, OK?

And so this is a zero point energy.

So this zero point energy is divergent, but we will comment on that a little bit later, OK? Do you have any questions on this?

Let me just make a couple of remarks.

First is that something interesting happened.

So when you look at the expression for phi and the pi, both phi and the pi depend on time explicitly. OK, so there's a time dependent here.

And here, we only integrate over spatial direction, okay?

We only integrate over spatial direction.

So in terms of integrating, we don't do anything with time. But what do you notice here?

What do you notice there?

Yes, it's not that we draw time dependence. This time dependence disappeared.

No, the Hamiltonian, phi is time dependent.

And pi is time dependent.

If you plug those expressions into here, certainly this inside there is dependent on time explicitly. This guy is dependent on time explicitly.

But in the end, after you do the integration of spatial directions, then you find in the end of the time dependence actually cancel during the calculation.

So do you have a guess why somehow this should cancel? Yes? Like energy concentration or something? Yeah, exactly.

So you have done in your piece that the time translation, so this system is time translation symmetric, and means that you should have the H. It's a conserved number. It's a conserved quantity.

So indeed, you see that this is independent of time.

This is a conserved quantity.

Okay, we see it explicitly, okay, so there's no time dependence since H is a conserved quantity.

So this integration is absolutely crucial, because if you don't do integration, the Hamiltonian density is not, okay, it's only the total Hamiltonian.

Okay, so this is a comment, and the second, yeah, an image.

So, yeah, since I only have two comments, let me just call it one, rather than call it zero.

So the second comment is that then we've, yeah, just repeat, we already said, we just create t of phi.

We see this create t of phi reduces to, essentially, a system of a continuum of harmonic oscillators because labeled by k is frequency omega k. So we call this omega k defined to be k squared plus m squared.

So as we mentioned last time, the fact that we actually see infinite number of harmonic oscillator is actually not surprising from the perspective that this field theory can be written as a continuum limit of a chain, say of atoms connected by sprints, okay?

So when you have a chain, yeah, we draw down a one -dimensional system. You can easily generate two three -dimensional.

Yeah, so you have those chains of atoms connected by spring and they're just coupled oscillators And then you can just diagonalize them and each will find the normal spectrum and each of them just a harmonic oscillator.

And so when we find those solutions by doing free transform, and essentially we are just diagonalize those interactions between those springs, okay?

And yeah, so that's why we get infinite harmonic oscillator.

So from that kind of way to think about it, it's totally unsurprising we get the the infinite number of harmonic oscillators.

But it is surprising if you think about it from the point of view of a field theory.

So when you quantize a field theory, and then you find in the end you get a bunch of harmonic oscillators.

And before doing that, it's hard to, before you actually carry out the quantization, it's actually hard to anticipate that if you don't have this intuition from this discrete system.

Do you have any questions on this? Yes?

So when you say the infinite amount of harmonic oscillator, are you saying that the infinite amount at any given point in space, or that each given point in like a continuum of locations is one harmonic oscillator?

Yeah, yeah, yeah, that's a very good question.

And so it's at each point you have a harmonic oscillator.

Yeah, at each point, essentially at each point you have a harmonic oscillator, and then they connected in space.

Yes? Is there any reason why you didn't compute E0? You have a delt function in there. Sorry, say it again? So that the integral E0 equals why didn't you just?

Oh, well, this we cannot do with this integral. No, up there. The zero point. Oh, right, right.

Yeah, we will talk about this later because the data we are going to elaborate on this a little bit later because we are going to try to give interpretation of this.

In homework one we found that a hat was time dependent. Is a hat not time dependent here? And is that why the Hamiltonian is not?

Yeah, yeah, a hat is a constant operator.

Yeah, because a hat is the integration constant of your operating equation.

Yes, yeah, yeah, yeah, that's analog of coherence state.

Yeah, yeah, you can, yeah, we will come, you will see, yeah, this question will become clearer when we talk a little bit about the Hilbert space, yeah.

OK, good, good, good.

Now, before we talk about the Hilbert space, we need to talk about two things here.

And the second thing, so Hamiltonian is an important quantity.

So this is one of the conserved quantities.

So there are the conserved quantities.

So there are other conserved quantities.

And one of them is the conserved quantity corresponding to spatial translation.

So the spatial translation gives you conserved momentum, momentum conservation.

So from spatial translation, then you get this conserved charge, Pi.

Again, you should have done this in your homework.

to have the following form, this 3x pi, OK? So this is the loss of charge for the conserved quantity associated with the spatial translation, OK? And we interpret it as a spacetime momentum.

This should be interpreted as spacetime momentum.

So this is, so here we have two momentum here.

You don't confuse of them.

So this pi is the canonical momentum conjugate to this canonical variable pi is the canonical momentum conjugate to this field variable phi.

And this is the general physical spacetime momentum. It's the momentum of your full physical system.

And again, we can just plug it in.

We can just plug in the explicit expression of pi and the phi. We already know it's time evolution.

We already know they are time evolution. And then we can plug them in.

And then you find that the answer is given by, again, after some smaller number of minutes calculation. transformation, and then you find that the answer is given by this.

In this case, there's no zero point energy, you just have this expression, okay?

And furthermore, you also have a Lorentz transformation, Lorentz symmetry.

And then that will lead to, say, the conserved charge associated with Lorentz transformation.

And again, you can find it explicitly, okay?

And so in your p -set, too, you will find the expression of this n mu mu, say, in terms of phi and the pi.

And then you can, again, express this guy in terms of a and a dagger.

But I didn't have the guts to assign it as the part of the p -set.

So I used it as a bonus problem. Because that calculation involves a little bit extra calculation.

Yeah, involves a little bit slightly more tedious calculation.

So as I use that, it's a bonus problem for those people who like to have some more fun.

Anyway, so this you can, we'll see it in your p -sets.

Good, so then again, p -i, you see it explicitly that this is time -independent.

It's time -independent.

Good. And then the next thing is let's talk about this zero point energy.

So this zero point energy we can write it as a following.

So we can, yeah, so this is a constant.

So we can take it outside of this k integral.

Okay, this does not depend on k because, yeah.

So this is just 2 pi cube delta 3 0.

So remember, this is a case space delta function. And then you have these 3k.

So that's the expression we get.

So now let's try to understand what's the meaning of this term.

Let's try to understand the meaning of this term.

To understand the meaning of that term, we need to do a little bit of a mathematical trick. We need to do a little bit of mathematical trick.

So remember the definition of delta function.

We have 2 pi cubed delta 3k equal to the Fourier transform of the exponential i k x.

So this is the definition of the, essentially, the definition of the delta function.

And now here we want to set k equal to 0. So let's set k equal to 0 here.

And then we find that this 2 pi q theta 0.

So let's remember this again in the k space.

And then we just set k equal to 0 here. And then what is this? If you set k equal to 0, you get what this? Yes?

The volume. Yeah, that's right. You just get the volume of the full space.

The volume of the full space.

So let's just imagine you, in order to make sense of this quantity, you can put the whole universe in the big box.

Imagine the whole universe is in the big box, and then this is the total volume, OK?

And then now, now since this is the volume, now we have very good interpretation of this quantity.

And now we can write E0 as real volume times epsilon 0.

And then this quantity, epsilon 0, then have the interpretation of the energy density.

So now, this is energy density.

So this energy density, we can write it in one more step.

So continue to here. So we can write it one more step, a plug -in the explicit value of omega k.

So this is just 1 half.

Sorry, I forgot to pi to dk.

k, and then you have k square, yeah, so that's what you get, okay?

So how do you like this integral?

Yeah, sorry, say it again, what's two picture?

Yeah. Just that delta function is one for all k? No, no, because this is a constant, right? Just when you have an integral, you have a constant, you can always pull it out.

Because that delta zero does not depend on k anymore.

Good. So how do you like this integral? Can you do it?

Good. So it's fruitless to do this integral because this is divergent.

And there's very good reason for this divergence, physically.

So this is just energy density.

So essentially, this is energy per unit volume.

Say in some unit volume. So let's imagine you have a unit volume here.

Remember this, remember we said before, this is corresponding to you take a discrete system with some lattice spacing a, and then you take the lattice, under each lattice point you have a harmonic oscillator, and then you take a lattice spacing a equal to 0.

So that means for any unit volume when you take a equal to 0, you have infinite number of oscillators inside, okay?

So essentially when you, yeah, you have the lattice of oscillators at each point, yeah, anyway, I will not try to draw it.

And then when you take A0 to 0, you have any unit, any volume unit, you will have the number of oscillators go to infinity, and that's where these divergence come from, okay?

It just comes from in field theory, theory, you have a continuum of degrees of freedom.

So at each point, you have a degrees of freedom.

And within any volume, then you definitely have infinitely normal degrees of freedom.

So this is just from continuum of degrees of freedom.

So this is the first time you see a divergent quantity in quantum field theory.

But you will soon find that this is the normal. normal.

It will be a fact of life. You will see divergence very commonly.

Because you have continuum of these freedom.

So the whole thing about quantum field theory is to find a way to deal with those divergences.

And one of the key difference between quantum field theory and just finite quantum mechanics or finite number of degrees of freedom is because of all those divergences, and the big part of quantum field theory is to understand how to treat those divergences.

And they actually don't affect your physics, but you do have to develop sometimes sophisticated tricks to treat them.

And this infinite answer is also closely connected to a very famous problem you may have heard. It's called the cosmological constant problem.

Because this tells you that any quantum field theory have an infinite, say, 0 point energy.

And so you may say, OK, so infinity, whenever we see something infinite, there's one thing you always do to treat it.

Can you guess what is the thing you always do when you see infinities?

yes yeah that's one idea that's one idea but you subtract infinity is very hard you know in your calculus class when you subtract infinity infinity you can get infinity right so you have to be very careful yes that's a lot of very Indeed, but to do all those things, you have to do one thing first. Yes.

Yeah, it's also very close.

You need to do, yeah, exactly you need to find a way to make it finite first, and then you can subtract it, and then you can, yeah, just like when you sum one plus one half plus one third, et cetera, you get infinite series, series, but if you want to estimate, yeah, you get divergence, but you always try to cut it off the series, and then approximate by a finite answer.

And here is the similar, OK, so here we always put some momentum cut off, OK, say imagine the momentum is smaller than some momentum, say some value lambda, and lambda corresponding to maybe to the scale which this quantum field should no longer apply, because nobody told you that this quantum field theory should apply at all the scale because, for example, in this lattice model, this scale will be one over the lattice spacing, okay, will be one over the lattice spacing.

Anyway, so once you cut it off, still you get a pretty big number and you cannot ignore it because this is a physical zero point energy, in principle you can measure it, but in real life, we don't see it.

So we have quantum fields flying around all the time, but we don't see this big vacuum energy.

So this is called the cosmological constant problem.

Actually, there was just a colloquium last week about this cosmological relative to this cosmological constant problem.

Now we can talk about Hilbert space. this.

So as a harmonic oscillator, so we can first define the vacuum state.

So now what we will do is indeed in quantum field theory itself, we can just ignore this E0, okay?

From now on we will just ignore this E0 because this is just give you a constant which does not do anything, okay?

It's like the potential energy in E and M, yeah, just like the unit, yeah, yeah, just like the, anyway. And so from now on we will ignore this term, okay?

So often I will just write Hamiltonian, I just write this term.

But later we will actually see examples, okay?

Later we will see examples actually this E0 can actually have physical implications, okay?

But just for our current purpose we ignore it, okay?

So now the lowest energy state, then it's clear from here because this is just a constant.

So the lowest energy state, the ground state, which is we often call the vacuum state, means there's nothing there, is given by ak0 equal to 0 defined to be for any k.

So the ground state satisfy, which at the load by 0, then satisfy or are related by this a k.

And then the general states, OK, you can just. So general states have the following form.

Say I can write nk1. So the k1 oscillator excited n times nk2.

And the k2 oscillator excited nk2 times terms, et cetera, which is given by, which is proportional to, say, ak1 dagger to the power nk1, ak2 dagger nk2, et cetera, acting on zero, okay?

Just like we have a large number of harmonic oscillators, OK? Yeah.

Questions on this? Yes?

Yeah, you postulate, but you can actually write it down.

You can actually write down its wave function, just as in the standard harmonic oscillator case, you can start in from A dagger, say, allotted by A.

You can start from this equation to write down its wave function, and here you can write down the wave function for the vacuum states too in terms of phi.

Here I'm just saying, depending on which k are excited, yeah, that's a very good question.

I will comment on related issues very soon. But here, here I'm just write down some states.

Yeah, I excite k1, k2. Yes, I want.

I'm not saying that this is the, yeah, you can have as many as you want.

That's right, it's true.

When I write it this way, it implies I only excite countable number of them.

But in principle, we will soon touch your point which related to your question.

And so for example, the simplest excited state would be just excited one of them.

So let me denote this by k.

And then the last simplest one is just excited two of them.

So this k1 and k2 can be the same. If this k1 and k2 are the same, and then just like the square.

And here also, I will not be very careful about the normalization.

And now we ask, what are the physical interpretations of those states?

So now let's ask, what are the physical interpretations?

So for this purpose, we can just look at the quantum numbers under, say, the Hamiltonian and under the spacetime momentum. quantum.

So for example, H0, so if you act on H on 0, then of course you get 0, assuming we throw E0 away.

Say, ignore E0 so far.

And then so for the ground state, and the Pi acting on 0, of course it's 0.

So you can see it just because they have a k here.

And so e also, when we throw this away, you have this.

And now, let's look at the excited state.

Now, look at this state.

So this answer is obvious because all different oscillators, they are independent of each other. And this just give you over the k.

OK, we can just use our result for harmonic oscillator, OK? And now you can look at the momentum.

So this is a spacetime momentum operator acting on here.

And then it has Ki, OK?

If you look at this expression, OK, when you do the act this on that, OK, OK, you ask this on that.

And then you can just use the standard trick when you do the commutator. And then the particular k for this is chosen.

And then the eigenvalue will be just given by that particular k i.

So there's one step of calculation there. And you should do it yourself. One minute calculation.

There's a data function you will be generated.

And then we'll get rid of that integral. And then you will pick up this Ki.

So now this equation now has an obvious physical interpretation.

So that means that this state K has spacetime momentum.

So now it's right down to 4 momentum omega K and K.

So this is exactly the momentum of a relativistic particle on shell. OK?

So this is of a relativistic, so this is a momentum of a relativistic particle because omega k is equal to k squared plus m squared of mass m, OK?

Yeah, so this means that the p squared, so this is four vector squared is equal to m. Yeah, so this satisfies the p squared equal to minus m squared.

So we can just, so it's very natural just to interpret this as a particle of mass m.

Because yeah, you can just interpret as a particle of mass m.

Good. So now let's look at this one.

So again, the calculation is very simple.

So you find for that one, for k1, k2, you find the energy just defined.

Again, you'll find that this is the energy eigenstate of the H.

It's the eigenstate of H with energy eigenvalue omega k1 plus omega k2.

And it's a momentum eigenstate of Pi with eigenvalue given by k1 plus k2. OK?

So e k are eigenvalues of h and p, OK?

Yeah? Yeah, so could that be?

So here, so this mass is not 0, right?

Because omega k is defined to be this. That's defined by my theory, defined by my Lagrangian.

Yeah, defined by my Lagrangian.

So this is the parameter of your action. Yes?

So that happened the same way?

Yeah, for here, it's uniform.

Here, just for each k, it's uniform.

But of course, when you add them together, you get something very complicated.

But for each k, it's just uniform.

So the most natural way to interpret this is just So you have two particles of momentum of four momenta omega k1, k1, and omega k2, k2.

And similarly, you can do this for any state like this.

There are all eigenvectors of H and P.

So nk1, nk2, you catch it, et cetera.

So this corresponding to nk1 particles of momentum, they're all on -shell, OK?

On -shell particle of momentum, k1.

And nk2 has nk2 particles of momentum, k2. OK?

Okay, so this tells you one thing, okay, so now let me just make some remarks.

Any questions on this before I make my remarks?

So the first point is that now you can see this can describe any number of particles.

So mathematically, in our description, they are harmonic oscillators, but each excitation of the harmonic oscillators from the spacetime point of view corresponded to a particle.

OK, so this is the beautiful thing of this theory.

And then due to, because of the commutation relation, ak1 dagger, ak2 dagger equal to 0 for any k1k2.

So you have four symmetries.

So when you construct the state, you can just commute them as you want.

OK, so that means the four symmetry in permuting all these different particles in the general state.

So this tells us these are bosons. OK, these are bosons.

And two is that all particles have positive energy.

OK, so even though that e squared equal to k squared plus m squared, this equation have two solutions plus minus omega k.

But when you look at the physical state, when you look at your state and look at the agglomaniac states, so all particles have physical energies, okay, so you don't have this negative energy problem associated with taking the square root, okay.

Also you have total energy of a state is equal to sum of energies of all the particles.

So this tells you there's no interactions between them.

Because you have potential energies between particles, and then that will change their energy.

When you put the two particles together, they will no longer be the same of the sum of the individual energy for each particle. And so that tells you there's no interactions.

So this is a series of free particles. particles.

This is a good starting point.

At least now we have particles. Okay.

Questions on this? Yes?

So now, is there a way to change this if you apply again, like in this picture, it's like you have another particle.

No, no, there's no way to change the momentum of a particle.

So once you created that particle, it just goes straight.

Yeah, it just does not change any more momentum, yeah, for that particle is conserved.

Is there, if you wanted to change it by way, or is it just?

So after dealing with this free theory, and then we will consider the next simply theory, and then that way we introduce interactions.

And when you have interactions, then the particle momentum can change, yeah, yeah. Yes?

So the thing of this? Good, good, yeah.

These are the momentum, yeah, it's like the plane wave in the non -rativistic, yeah.

And if you work like, try to mobilize it. Yeah.

So the commutes things don't change because everything is filled up by AK and they always commute and so that won't change.

But indeed, we will talk about the way packet to localize in space. Other questions?

So the last, we will talk a little bit about the technical point.

is that so far we haven't talked about the normalization of such a state. So now let's look at the normalization of states.

So let's just look at these states.

This single particle state, let's look at this normalization.

Let's look at k with k prime.

So now if you take the overlap of this with the k prime and then you can reduce this to the commutator of ak k and ak prime, ak diagram, ak prime.

So you will get just this.

So again, this is a five second calculation.

So you find the d. If you do the overlap.

But you already did in your p set, this thing is actually not the Lorentz invariant.

This is not a good thing. This is not a good object on the Lorentz transformation.

But when we construct state, we would like to have our state to have good properties on the Lorentz transformation.

So we will choose a society of different normalizations.

So instead of this state, we will define the following states.

We will define a k now without the vector, to be the square root 2 omega k and this k.

And so this is the 2 square root of k, a k dagger acting on zero.

And now, if you compute the overlap with this k and the k prime, So now, you have square root k, square root omega k for one of them, and then you have square root k for two of them, and then you have two omega k, and then you have two pi cubed, delta three, k, k minus five.

So now, you recognize this object from your p -side.

The omega k, multiply this guy, actually, have good Lorentz transformation properties.

So this guy actually transformed nicely on the Lorentz transformations.

So this will be the normalization we will use from now on.

So these states have very, so indeed you can show, so that will be, I think, in your P set three. You should look forward to it.

So you can show, if you act the Lorentz transformation, the lambda is a Lorentz transformation, and u lambda is the quantum operator to generate that Lorentz transformation on such a state k, and then you just get lambda k. And lambda k is the Lorentz transformation acting on that k. It's the Lorentz transformed k.

Okay, okay, and but yeah, you should see your maybe p3, okay, so so if I remember it good any questions on this Yes, yes Okay, you have a data function here Yeah, because omega k only depend on k only depend on the spatial part other questions. Yes Yeah, is that your problem?

No, just this guy, when you, it's not about whether it's Lorentz invariant, it's not Lorentz covariant.

Just when you transform it, just this object, when you transform it, it's very awkward.

That's not a very, you can transform it, you can write down a transformation for this object.

Just does not transform nicely.

Then that means if this does not transform nicely, means each of them don't transform nicely and then they're just not convenient.

So this one have the properties that when I act you on this scene and then actually you can just directly corresponding to the state with Lorentz transform and momentum. Yeah and then it's very easy.

This property does not apply to this one, does not apply to this k.

So you can We can also talk about a wave packet. So this is a plane wave.

So this is like a plane wave. This is the momentum eigenstate.

So we can also consider a general single particle state. We have the following form.

So we have the following form. We can write it as psi just as a superposition of this k.

Again, you only need to integrate over spatial momentum, momentum, because only the spatial momentum are independent.

And then some arbitrary function of k, and then on this k.

So any single particle state, you should be able to write this way.

So in particular, by choosing appropriate fk, you can construct a local wave packet. it, okay?

So you can, so choose k, you can choose fk to construct a localized wave packet in space, okay?

So again, we will have some exercise like this in your, in your, in your When I call this to be a two -particle state, because each is a plane wave, so essentially, they're not localized, they cover all space, OK?

And in one sense, they correspond to two particles.

So now you can solve this problem.

Say by constructing, say, using, say, dk1, 2 pi cubed, dk2.

So by choosing a proper function k1, k2, this k1, k2, you can construct two widely separated wave packets, okay, widely separated, again, it's in space.

So you can really talk about, so these are the generally two particle states, okay?

And so that confirms, actually, Maybe this can, it should be interpreted as the plane wave version of the two particle states. Okay.

Good. Good. That's a very good question.

Indeed, there will be a problem in your PS3 which asks you to show there's no eigenstate, there's no perfected look. Yeah. So in non -rativistic quantum mechanics, you have this state.

You have this wave function, say, you can localize it at the point.

But there's no analog of such kind of state in the relativistic case, precisely because there's no position operator anymore.

So you will explore this a little bit more later yourself.

Yeah, just with this formalism, you can explore all these questions yourself.

That's the key. Actually, you already have the power to do that.

So now let's say a little bit more on the structure of the Hilbert space.

So here, the structure of the Hilbert space is a little bit special because there's no interaction.

So you can just separate it into the full Hilbert space which I write as H, script H. Then you are separated into the vacuum state.

And then you have the one particle field of space.

And then you have the two particle field of space.

They all, they don't have anything to do with each other because there's no interaction, OK?

So in principle, you can have infinite number of them. You can have infinite number of them.

So if you restrict to always a finite number of particles, OK?

So the sets of states.

So this is one particle, this is two particle, et cetera.

So that's of states with a finite number of particles.

It's called a folk space.

In the folk space, we only have states of finite number of particles.

But we can have arbitrary number of them as large as you want.

10 billion, 100 billion, but it will be finite.

So finally, those k, either this k or this k, they are just plain wave normalized because they're normalized by delta function. So strictly speaking, they are not normalizable states.

And so strictly speaking, they are not in the physical here space.

So strictly speaking, k or any of those states rates of k, or k1, k2, are not normalizable. They're only plane wave normalizable.

So just as, say, psi is equal to extremely i kx, in non -right -based quantum mechanics, a lot of normalizable, a lot of normalizable.

So they are convenient for various mathematical operations, but they don't correspond to general physical states. states.

So it's not generally in physical state.

So physical state, we have to consider, for example, this kind of state, and then you choose fk, so that is normalizable.

So if we take this, and then so with psi given by that, then the normalization of psi, you You can calculate it.

And then this is given by, so you can easily guess the answer.

So when you do the overlapping texture, you find that it's given by the modulus of fk squared.

So if you do the calculation to calculate the norm of that state, you find it given by that.

And then we can choose this to be finite. And then this will be a, then psi will be a normalizable state.

So for this reason, sometimes we don't do it very often.

Sometimes you can choose just a basis of fx, say, of fi.

So i run some number.

And then this alpha, then you can define alpha i, okay?

And then, and then this will provide, then alpha y will, will provide a basis of all, basis of the single particle, or, or, or, or, a single particle state.

So here's a fun fact related to the earlier question, uh, uh, was asked.

So, reliably, here, you have, if you look at the k, so k form a basis, OK, let's just look at the single particle states.

So, reliably, this k form a continuum, uncountable continuum of basis of states, OK? Uncountable continuum.

But once we impose this normalizable condition, means you need to choose the normalizable f.

And then you can show the basis of this normalizable f actually is countable.

So in the end, the Hilbert space, the one dimensional, the one particle Hilbert space is actually generated by countable basis. Just like in your ordinary quantum mechanics.

It's actually rather than uncountable basis like in k.

Once you restrict the normalizability. So this is a far mathematical fact.

Great. Questions on this?

OK, so the last few minutes, we talk about conserved charges, OK?

That's rho of loss of charges.

So classically, we say if the action is invariant under some symmetry, some transformation, some continuous transformation, say, say, of the infinitesimal form.

So the alpha are the infinitesimal parameters.

So the alpha label different transformations, and A label different fields. Alpha label different transformations.

And then you will get the conserved currents, j mu labeled by this alpha.

So alpha label different symmetries for each symmetry, And then you have a conserved current, which is satisfied the j mu alpha equal to 0. OK?

Alpha equal to 1, 2, et cetera, label different symmetries.

And then you can write down your Lothar current.

So we derive a general formula for the Lothar current is given by the form, say partial L, partial mu phi by a, and then f a alpha. So this is essentially the same as before.

Essentially, I just add the index alpha. Now to label, you can have more than one symmetries.

And then minus k mu alpha. So k mu alpha is the derivative.

Total derivative, suppose, data L is given by epsilon alpha partial mu k mu alpha.

So the k mu alpha is the total derivative which would correspond to that transformation.

And then the q alpha, that would be the conserved charge for that current, the 0's component.

So this is a classical story.

Now I'm using a little bit more general rotation for each alpha.

So now the key thing is that the quantum level, So at the quantum level, first as we already seen in the case of the Hamiltonian and the momentum operator, so q alpha is time independent.

It's a constant operator.

So this is a reflect that this is conserved.

And also, another important property of q is that when you act q alpha on your field, If you look at the commutator between this Q alpha on your field, it actually generates such a transformation.

So it turns out, yeah, so if I put a parameter here epsilon alpha, yeah, so it doesn't matter for Q, say it's up stairs, down stairs, and then you get, okay?

So you just generate that transformation for f, for phi, okay? OK. So this can be shown explicitly.

So you can show this explicitly. We're running out of time now.

So you can show explicitly just using this expression.

So let me just quickly outline the way to do it.

So if you say, so let's look at the 0's component of this J. j.

So the del's component of the j just equal to partial l, partial partial 0, phi, a, and f, a, alpha.

So let's consider the, yeah, I think today we will not have time to finish Anyway, yeah, let's just leave this to the next time.

Yeah, let me just say that the quantum level, this will generate the infinitesimal transformation. And then you can also express this q.

OK, you can introduce, say, exponential i lambda q.

So if q is the conserved charge, and then you can express q like this, And then this operator then generates, when it acts on phi, then generate finite transformations.

So this takes you to phi a prime, the finite transformations.

Anyway, so for Psi 2, you will actually check yourself a couple of simple examples. And you will see this works. works.

And in the next lecture, we will give you a simple derivation to show this actually works. Okay. It works in general.

Transcript of MIT 量子场论·8.323·第03讲：为什么需要量子场论 （中文完整文稿）。

Yeah, so last lecture, we discussed loss theorem.

For every symmetry, for every continuous symmetry, this conserved current.

And then we also started talking about relativistic quantum mechanics, how we want to unify special relativity and the quantum mechanics. OK.

So the most immediate idea for that is what's called the right -to -basic quantum mechanics.

And the most immediate generalization of the Schrodinger equation.

So if you have, so at the end of last lecture, we talk about, say, the most immediate generalization of the Schrodinger equation, which, so the, So if you have e equal to say p square divided by 2m, and then you go to non -ratavistic quantum mechanics, the Schrodinger equation, okay?

And now if you have e square equal to p square plus m square for ratavistic particle, and then you get what's called the Klein -Gordon equation.

And again, this psi has the implication of of the wave function. So this describes, and then this.

So if you add the generalization of these, then this means to describe the quantum mechanics of relativistic free particle, say, of mass m, OK, of mass m.

So here, the psi tx is the wave function of a relativistic particle, relativistic particle of mass n.

And we also noticed that this equation actually is the same as the simplest field theory equation.

So we also talked about a simplest scalar field theory, classical.

So here is a simple, a classic, simple scalar field theory.

So this theory you can write action of the form.

So this is the simplest theory you can write down. And then relativistic invariant theory.

And then the equation of motion of the, so this is a, you view this as a classical field.

And again this has the equation of motion have to see exactly the same form as this equation.

So but now here phi, again is a function of tx, now has a completely different physical interpretation, so here is the, this is a classical field, okay, so this is a classical field, okay, so in this case the interpretation of the x in here, I mean here is very different, okay, so not only phi and psi, the physical interpretation are different, the physical interpretation of x also here are different, Here, x is just a label.

It's a label for the location in the space which we define this field.

But here, the x is the eigenvalue of the position operator for this relativistic particle.

And so they have very different physical interpretation.

And so let me just label this equation by 1, and label this by 2, and this by 2 prime.

So we also mentioned that this one has the interpretation of this as the wave function for relativistic, as the equation for relativistic quantum mechanics has a number of difficulties.

So the first is that as you will show in your PSI2, there's no sensible way to define a positive definite probability density.

So if you want to interpret this as a wave equation, then you must have a probability density because in quantum mechanics, probability should be conserved.

And the second difficulty is that there's a negative energy state because of the square.

Because when you take the square roots, then you get the minus sign and then the negative energy the state, which you cannot avoid in quantum mechanics, even though classically you can just throw them away by hand.

And the third thing we mentioned at the end is that for a relativistic wave equation, you can describe fixed number of particles.

So the particle number cannot change.

So this equation describes a single particle, and if you want to describe two particles, then you need to write down a separate equation for a different wave function, but this is for the two particle wave function will be like this, and et cetera.

But this does not really make sense in the relativistic system, because we know that in the relativistic system, e equal to mc squared, in any case, you have enough energy, then you should be able to create particles.

And then that means the number of particles in the given process is not conserved, okay?

So if you want to use your quantum mechanics to describe a process, and then that's, you cannot have a formalism which the number of particles is fixed, which you cannot change.

And so this is actually the most fundamental difficulty. OK, is that you cannot change the number of particles.

And related to this difficulty is this interpretation.

Here, we say, now if you want to do, we say in here, there's a fundamental asymmetry between the t and x.

OK, also, yeah, maybe let me put it as four. There's also fundamental.

An additional difficulty is the fundamental asymmetry between t and x.

So here in the wave equation t is just a parameter which we describe the evolution but x is the eigenvalue of quantum operators, eigenvalues of quantum operators, say a corresponding to say hat by putting a hat we reloaded the corresponding quantum operator.

Yeah, so this is the eigenvalue the oppositional operators.

And this AC means you become even more prolonged, say, if you can see the two particles. You have two x here, but the only one t.

But because of those fundamental difficulties, so if you connect this 1 to 4, so we can we conclude that the relativistic quantum mechanics define in the sense that you write down a wave equation.

And for wave function, it cannot be a fundamental description.

But relativistic quantum mechanics just refers to this kind of wave equation.

Okay, so at most, this can be approximate approximation and most, this is approximate description in situations, say there's no particle creation or allylation.

So in case which your particle number is fixed and then you can use this as approximation, okay, but it cannot be a fundamental description.

For example, later we will talk about the fermionic version of this wave equation.

So this will describe a particle without spin.

So later we will describe the Lannagos equation for electrons for spin half, and then that can indeed be used to describe electron in the hydrogen atom as far as you don't create new electrons, etc.

Anyway, so right of this quantum mechanics can only be described as some kind can see that as approximate description.

But now if you want to unify special relativity and quantum mechanics together, it turns out that the right formulation is just quantum field theory.

So it turns out that the quantum field theory addresses these difficulties.

So it turns out the right way.

So if you want to describe quantum mechanics, say, of relativistic particles of mass m, as we want to do here, it turns out the proper thing to do, which is a little bit unintuitive at first sight, is to start with this field theory, which seemingly have nothing to do with relativistic particle, but to start with this classical field theory, and then quantize it.

It turns out, once you treat this theory as a quantum field theory, and this become a theory of arbitrary number of relativistic particles of mass m.

And so that's the non -intuitive part, and that's the one of the miracle of the field theory is that it automatically gives you a formalism for treating arbitrary number of particles. labels.

And also in field theory, both t and x are parameters, even though x only labels your location.

So both t and x are parameters, and so you can easily make them to be on equal ground to be compatible with special relativity, OK?

Good. So any questions on this?

So we will see that the right framework is quantum field theory, OK?

So finally, as a last motivation for quantum field theory, we quickly describe the last.

So the field theory can also arise as a limit of discrete systems, OK?

And this is the most relevant for kinase matter physics, for example.

So let's just consider, say, some, yeah, let's consider 803 example, OK?

So let's imagine you have this number of particles, a number of the atoms, say, on the chain, OK?

And then they're connected by some springs between them, OK? Okay, so this is the, yeah, consider this to be infinite, okay? And the spacing between them, say, is A.

So atom affects on some matrix point, and the lattice spacing is A, okay?

So, yeah, so we can label the other, say, each particle by their position.

For example, this is x0, this is x1, this is x2, et cetera, okay? And the typical particle is xn.

And the location of nth particle is xn.

And so we can also introduce the deviation between the equilibrium position of each particle. So let's call it eta n.

So now let's consider the dynamics of eta n for this theory.

And so this is just deviation of the nth particle from its equilibrium position. So xn0 is its equilibrium position.

So now if you write down the Lagrangian for this system, what we can easily do is just write t minus v. The t is the kinetic energy.

And v is the potential energy.

So we can just write it as sum over n over all particles.

And then let's assume they have the same mass. Let's write mu eta n dot square. So this is the kinetic term.

And so mu is the mass for each particle.

And then their potential, let's assume at each point that there is also some interaction because each particle are connected by the spring.

And so they are a harmonic force between neighboring particles.

And now let's imagine also there There's a harmonic potential which trapped this particle itself at each location.

So this is a very simple spring and the particle problem which you encounter, say, in 803. Is this problem clear?

I assume most of you have seen this problem before.

And your task in 803 is actually to find the normal modes of this system.

And in 8 .03, you also described that we can, in the a go to 0 limit, say if the lattice spacing is very small, and if you're only interested in the behavior of this system at a very large distance, say the distance is much larger than a.

Then you can essentially choose this system as a continuum.

You don't have to resolve individual particle.

And so we can just, in the equal to a new limit, so it can treat the chain as a continuum of particles, OK?

And so each eta and t, you replace it by eta x t.

So x label is positioned, OK?

So x label is positioned, t describes the dynamics, OK?

So eta is the deviation at location x, and it's dependent on t. So this is the oscillator.

And then sum over n in the Lagrangian, then we can replace it by integral over dx.

And now you just treat this as a one -dimensional continuum.

So it's some integration of dx. But of course, here there's a label.

So for lattice spacing, so the infinitesimal here, the element is A.

So A times the sum over N, you can replace it by dx.

A is the lattice spacing.

And now you can just write this Lagrangian in terms of continuum theory.

Now you can write this Lagrangian continuum theory. And then let's just do it.

So we can write it, yeah, let me just write one more step. So you can write it as sum over a.

We take the a fact out, because the a fact out have to be changing to integration.

And then you have 1 half mu divided by a eta n square minus 1 half lambda a.

So I just slightly rewrite this Lagrangian So that it's easy to take the continuum limit. So we have to take the fact of a out.

But for this term, because this can sense the difference between the two, and we also divide it by a in the downstairs. And then we need to multiply a upstairs, and then there's a in the front.

And now the continuum limit, you can just replace this by an integral.

And now you can just write it as 1 half, of mu tilde eta dot square. So now eta n just replace it by eta x t.

And here, let me call it lambda tilde partial x eta square.

And this term, you can just replace it by the derivative of eta.

And then this is just become 1 half sigma tilde eta square. OK?

And the mu tilde, of course, is mu divided by a.

A lambda tilde is lambda times A and sigma tilde, this is the sigma divided by A.

So the continuum limit is that those quantity has to be fixed. The tilde quantity has to be fixed.

And then we have a continuum Lagrangian.

And then we have a classical field theory.

And this theory is essentially the same as that theory.

So if you take this factor mu tilde out, If we take this factor, mu tilde out, so let me just take this factor of mu tilde out in the front, just up to overall factor.

And here is lambda tilde divided by mu tilde we could, let's call it v square.

And this becomes sigma tilde divided by mu tilde, let's call it m squared.

So the v square is equal to mu tilde is equal to the lambda tilde, right by mu tilde.

And the n squared is equal to sigma tilde, right by mu tilde. OK?

And then this is just essentially identical to that theory when v equal to 1, OK?

So when v equal to 1 become the same, That's just two, equation two.

So we could develop corresponding to relativistic case, speed of light, but in general this can be just a non -rativistic field theory for other values.

So this is, even though this example is very simple, but this is actually a very general way that we can treat many condensed matter systems, which often involve lattice, say because it's solid, you can imagine all the atoms are on the lattice, et cetera.

And if you're only interested in the very macroscopic behavior, then you can treat It's solid as a continuum.

And then now if you're interested in quantum mechanics of such a system, then the quantum field theory then naturally arises.

Good. Any questions on this example?

Sorry? So the limit of the system is the same length as mu? Yeah. So what is that physics next? Is it the strength of?

Yeah, yeah, just corresponding to the case that the, yeah, it just tells you that the relativistic limit is special. Happens at a very special point.

But I guess why is that the relativistic limit, like, to me, lambda is, like, the strength of your spring and then use your mass.

Right. So, you've got to be comparable, how is that?

Yeah, yeah, that's not much you can read from here.

Yeah, yeah, it's just that when you choose some special parameters, then you can have a relativistic limit.

Are there questions? Yes?

Yeah, so these are all?

You can also have tensors or vectors.

So what would you treat?

Oh, you can treat, yeah. For example, you can treat forearms. You can also treat spins.

For example, if you have an icing model, just consider lattice of spins.

And then the average spin and then you can treat as a scalar field and then again you can write down a field theory Yeah, and actually the breakthrough of the phase transition using this metaphysics to understand what phase transition is really about and describe the behavior near the phase transition and precisely coincided with the development of field theory and Yeah, actually increased our understanding of quantum field theory.

Yeah Good. Other questions?

Just to summarize what we have discussed so far, all paths leads to QFT. Okay.

So we have described three paths. It's three paths, but they're pretty general.

First is that the quantum - we offer interesting quantum dynamics of some classical fields.

say, such as, say, electric magnetic field or spacetime metric, if you are interested in gravity, et cetera.

OK, so in this case, we already have the classical field theory.

But we know the world is quantum, and we want to understand what's the quantum version of it.

And the second is that it unifies special relativity class quantum mechanics, OK?

So you need the field theory to unify them.

And the third way is that it's a large -distance description of discrete systems.

So, yeah, just combine all three elements together to cover many, many areas of physics, okay? They cover many, many areas of physics.

Good, so now we can say a little bit about the plan for the whole semester.

Okay, so here is the plan.

So this is just a rephrase of the outline, which so the first thing we do in chapter 2, so here is chapter 1, in chapter 2, we discuss the simplest field theory just squashed with this equation 2, okay, the theory of 2, okay, 2 and the 2 prime, Yeah, the prime is its equation motion.

So yeah, in physics, we always start with a simplest example. We always start with a simplest example.

And so that is the one we will start with.

So what we will see is that this describes, that field theory describes spinness with those three massive particles. particles.

So we will see when we quantize that theory, two, and then we get a theory of free spinless massive particles.

So you say, oh, that's a little bit boring because in this theory it's free.

The particle, by free means they don't intact. The particle they just don't intact.

And then in chapter three, we will add interactions.

We will describe how to treat interactions.

We will introduce interactions and tell you how to treat the interactions between those particles.

Then in chapter 4, we go to the real physics.

So this scalar field is also real. Say, for example, can be used to describe the Higgs boson.

But the Higgs boson maybe is a little bit far from what you normally think about.

So in chapter 4, we will go through something which is much closer.

We'll talk about the theory of electron. So this is called the Dirac theory.

So this theory describes three but spin half particles.

So this is a theory of electron when we neglect these interactions.

OK, so this is the free spin -hard particle.

And then we move on to the Maxwell's theory.

So this is the theory of quantum electric and magnetic field.

So we quantize the Maxwell's theory.

So without source, the vacuum Maxwell's theory. And you find you get free. Again, there's no interaction.

mass is spin -1 particle.

You get a theory of mass is spin -1 particle.

So this is what we call the photon.

Okay, so this is the quantum for an actual magnetic field.

Okay, and then five, sorry, so this should be chopped five miles.

I think I lost my So now go to chapter 6, we combine the 4 and the 5 together, okay, combine electrons, so photon normally will be developed by gamma, combine the theory of electron and photon together and then plus interactions between them and then we get the so -called quantum electron dynamics.

So this is called QED.

So QED is very general, essentially it covers all the quantum phenomenon, yeah, a microscopic phenomenon up to say weak in tachyns and strong in tachyns.

If you go, don't go inside the nucleus and or don't go to a very high energy and the thing that's covers essentially most of the physics and yeah and then we will describe how to yeah and then that will be the the end of this course so do you have any questions on this okay so this is a roadmap yes the chapter lecture loads Yeah. Good. Other questions? Yes?

Yeah, gluon is also massless spin 1, but gluon actually they interact with themselves.

And so gluon is different. So gluon, to describe gluons, you have to wait for quantum field theory 2.

And so the thing about the photon is that the photon don't interact with itself, but the gluons interact with itself. Yeah. Yeah.

So essentially we treat everything except gloves. Yeah.

So now we can just move to chapter two.

Now we are talking about this theory. Okay, so actually I should not erase it.

So now we talk about this theory.

So because this This theory describes three particles. So we call it a free scalar field theory.

So this is the theory we are interested in.

So now we will describe how to contact this theory.

So first, we will quickly go through the quantization of harmonic oscillator, which you should already have done in your p -set. And so we can do it relatively fast.

So the combination of harmonic oscillator in the Heisenberg picture.

So we will see that once we understand this example in the right way, and then quantizing this field theory become trivial.

And the quantizing this field theory become trivial.

OK, so let's start with a harmonic oscillator.

was implicitly I take the mass to be 1 and then take the frequency to be 1, okay?

Yeah, yeah, let me put the frequency here.

Okay, let's take the mass to be 1, okay?

And so, so for this theory, so this is a simple harmonic oscillator which you have seen it maybe for the most of your intellectual life.

And so P will be x x dot is a momentum.

A conjugate momentum is x dot.

And so the Hamiltonian is the p squared divided by 2 dot 1 half omega squared x squared, OK?

And equation of motion is x dot double dot equal to x squared, OK?

So let's first look at this theory, a look at monochrome theory that is a classical theory.

So for classical theory, we know how to solve this equation.

We just need to solve this equation.

So classical solution just given by xt equal to a cosine omega t plus b sine omega t and a and b just some integration constant.

And for convenience, I can also write it is in the complex form, like it is following in, equal to a, exponential minus i omega t plus a star, exponential i omega t. And a is some complex constant.

And again, it's an integration constant. I just re -write the integration constant slightly differently.

OK? So these are just integration constants.

So now, yeah, so this is a complete solution of the problem. So now let's go to quantum.

So when we go to quantum, and then we replace then this classical dynamical variable then become the Heisenberg operator, become the quantum operator, in particular in the Heisenberg picture, and then this operator will depend on time.

And now this equation becomes operator equation.

OK, so now let's maybe I should label my equation. Now this star becomes an operator equation.

Now star is the operator equation for x hat.

So you have exactly the same equation as a classical equation, but now the interpretation is different.

And now the x hat becomes the operator equation.

So now the solution, now let me call this star star, so this still solves that equation, OK?

So this still solves that equation except, so these are just c -lumbers because this is a function of t, these are c -lumbers.

But now x becomes the, so now quantum mechanically, so this becomes, now becomes the quantum solution, OK?

So this still solves that equation. The quantum mechanically, this becomes a hat.

And so this still solves that equation, but these are C numbers.

The left -hand side is the operator.

And it can only be that A hat and B hat are operators. And also, A must be operators. And the star will replace it by dagger.

So now, say, now A, you just go to A hat, and A star goes to A dagger, A hat dagger.

Now these are integration constants for the operator equations.

So now they become constant operators.

So they're just constant quantum operators.

OK, they're just constant operators.

So they are integration constants for your quantum operator equations.

So now the solution, so now this is your quantum solution.

So this is the form we will often use.

You can also use that form, the equivalent. But this is the form we will often use.

You can also, from here, you take the derivative, you can find the P.

Again, this it become an operator equation, you take the deletion x, x, and maybe you find p, et cetera. Okay?

Yeah, so you p hat t, you just take a derivative, okay? And then you can just work it out. It's very easy.

So now, so this equation just, so we already solved the quantum problem, okay?

So, so because we find that the full evolution, full solution into the quantum operator equation, except that we still need to impose the canonical condition.

So this is just equal to i.

So if you plug in the expression for x and the p into here, and then you find that a and a dagger, the commutative is equal to 1.

OK, so this is your familiar creation and relation operator for harmonic oscillator, OK, for harmonic oscillator.

And now we can also use the A to build the Hilbert space.

So because all your operators now, or because x and t, x and p are expressed in terms of a and a dagger, so essentially any operator of this theory can all be expressed in terms of A and A dagger, OK?

And then we can just use A and A dagger, because A and A dagger, essentially, they are fundamental building blocks of your full quantum theory. And then we can also use that to build the Hilbert space.

So the Hilbert space is defined by the lowest state, is evaluated by A.

And then the higher states obtained by acting a dagger on the ground state.

Okay. So this is your full theory. Okay. So this is your full theory.

And so now you can compute anything in this theory just with those knowledge. Okay. Just with those knowledge.

So any questions on this regarding the harmonic oscillator?

So let me just summarize, so this is maybe very familiar, but let's summarize the rule we have been using, okay?

Summarize the steps in context of the harmonic oscillator, and then the same steps can be used to contact the field theory of quantization.

So we make it a general.

So first, so the zero step is that the classical equation motion, the quantum and quantum operate the equation.

Then the first step is to find the most general solution, find the most general solution to classical equation motion. Yeah, just equation motion.

And then, then when you've got quantum, and then you just promote the integration constants in your classical solution in one, in the step one to constant operators, constant quantum operators, okay?

So this gives the, then you have the full time evolution at the quantum level, okay?

Now you know how the quantum operator evolves, and then And you impose canonical condition conditions, okay, so that will tell you the commutators between those integration constant operators, okay, just as we do here.

And then constant operators in two, now you also know the commutation relation among them, and then now it can be used to generate the Hilbert space.

So this step, 1 to 4 are very general, and if you can do it, then you can essentially do it, apply it to any system.

Say 1 degrees, harmonic order is 1 degrees freedom, you can apply it to 2 degrees freedom, 3 degrees freedom, and also to field theory with infinite number of degrees freedom.

Okay, and now we will apply these two field fields.

Yes, for this procedure you cannot, but you can get the finite dimension, yeah, because the finite dimension of Hilbert space don't have the classical analog, so here we start with the classical system and then we quantize it.

And there's the quantum system with the finite dimension of Hilbert space, they essentially intrinsically quantum, and yeah, like spin, spin is an intrinsically quantum thing. Yeah. Yes.

Yeah, yeah, yeah, it's just because they don't have, yeah, yeah, the reason is just they don't have classical counterpart, yeah, yes.

Is it always true that the constant operator is sufficient to generate the entire Hilbert space?

Yeah, because if you think about it this way, that's a very good question, because let's just look at this harmonic oscillator and then try to single -generalize it, because there are integration concepts of the X and the P.

Any operator in your theory can all be expressed in terms of ANA dagger, and then your Hilbert space must be able to generate the hub of space union then because they are the building block of your whole operators yeah yeah the vacuum state here is based on it's come from the from the energy right so so once we solve x and p and then we can write the Hamiltonian in terms of x and p and then you just look for the lowest energy state and then you find the lowest energy state just satisfy this equation yeah and then And then from there, you can find other states.

Yeah, the same thing we are going to use for quantum field theory. OK, good.

OK, good. So now, become a mechanical.

We can just apply this to this theory.

We can just apply this to this theory.

And now let me add here. So here, the canonical momentum density conjugate to phi is called pi before. It's just a time derivative of phi.

Under the Hamiltonian density, you can find it explicitly.

It's pi squared plus 1 half.

And then this is the classical equation motion.

So now let's just solve this classical equation motion.

So this equation can be, so this equation is easy to solve because the translation is symmetry.

You can just do a Fourier transform.

So we can Fourier transform.

So now let's do the Fourier transform.

So 2 prime is solved.

So we can just write phi x equal to exponential minus IET g plus i k dot x, OK?

So then you can see that this just solves the, so this provides a basis of solutions to 2 .0, OK?

It's just a plane wave, OK? Just a plane wave.

So now when you plug this into there, then you just get the dispersion relation.

e squared should be m squared plus k squared.

So we'll denote this as omega k squared.

So omega k is defined to be just the k squared plus m squared.

So e, when you take the square root of e, So you can take a plus minus omega k.

It can be a plus minus omega k.

So we normally call the solution, we normally separate.

So for historical reasons, we normally call define uk x to be exponential minus i omega k t.

So now we have inserted the positive root of E.

So this is normally called the positive energy solution.

Even though this name is a little misleading, we don't define the energy actually, yeah later we will see this is not really the energy of a particle, so this is just a traditional This is just a traditional name.

Conventional name. And then you can define the complex conjugate of k.

Now you have, then corresponding to, you have minus omega k in there.

So we take this complex conjugate.

And so this is called a collective energy solution.

So all together, they form a complete set of solutions. OK, so a complete set of solutions.

So complete basis, a complete set of, yeah, complete basis is formed by UK and the UK star for all k, OK, for all k.

So these are the complete set of solutions to that wave equation.

OK, so that's just the, yeah. Any questions on this?

So this is just like a classical, it's just like a wave, just like a plane wave, which you should also have seen in A -dolls v.

So now we can write down the most general. So this is a basis.

So these are the counterpart of the exponential plus minus l omega t here.

So now we can write down the most general solutions by just putting in the integration constant for the most general classical solutions.

So we can just write phi x equal to integrate of all possible value of k, because this is for all k. So we just sum up all of them.

And so this factor is for just for convenience. It's just a convention. You don't have to put it here, it's just a convention.

And then we have a k u k plus a k star u k star.

So this is just the most general set of solutions with a k and a k star as integration constant.

So this is a full set of integration constant.

So now when you go to quantum level, So now, we can just follow the rule and we find the most general classical solution.

And in the quantum level, we just promote this to be operator.

You just put a hat there and change this to dagger.

So now, this becomes your basis of quantum operators.

So these are the full set of quantum operators.

And this solves your theory, okay, so this solves the operator equation, and this solves the operator equation.

So now the next thing is to impose the canonical commutation relation.

So firstly we have to, now we have to do a little bit thinking, okay.

So far, it's just straightforward. But now we have to do a little bit thinking.

So for finite harmonic oscillator or for quantum system of a single variable, you just have this.

And now we need to come up with a generation of these two fields here.

So, so we need to come up with a generation of that to field theory corresponding to phi x.

So now let me make the time and the spatial coordinates separate.

And this conjugate momentum is phi, it's pi, okay, conjugate momentum density.

So we should do them at the same time.

Remember t is the, it's the same evolution operator, so they have to be evaluated at the same time, or it's equal time, canonical or quantization conditions always at the equal time.

But x is a label of operators, so x, they don't have to be the same. So here can be x, here can be x prime.

So now we have to come up with a generalization of what is this quantity for field theory.

So now we just need to do a little guesswork. You can easily guess it.

But before we do that, do you have any questions on this? Yes?

Yeah, yeah, x is always, so x is always, here is always just the label of the spatial location, right?

Yeah, it's a label for the, yeah, it's your field theory, label, yes.

So in this procedure, I guess you always add?

Yeah, so normally if you have certain order differential equations, you always have some integration constant. Yeah, that's it, yeah. So that carries forward all throughout? Yeah, yeah, yeah.

Are there questions? Yes?

Right, yeah, yeah, so that's a very good question.

So that's just the extension of our usual procedure for the quantum mechanics.

So the usual procedure, even just for harmonic oscillator for single variable system, you have this correspondence between the classical system and the quantum system.

When you quantize the classical system, then the classical equation will become a quantum operation.

Here we just use the same rule, because quantum field theory is just the theory of infinite number of liquid freedoms. We are not changing the rule of quantum mechanics.

And so that's why we just, again, just promote the classical equation into the operating equation.

Other questions? Yes? Yes? What do we understand?

The classical variable? Is there something like that for field theory as well?

Um, yeah, yeah, yeah, there is. Yeah.

So, so classically, you can define the Poisson bracket between the, between the classical field variables, and then, and then quantum chemically just become, uh, uh, uh, quantum commutators.

Yeah, yeah, you can also do that. That's right.

So, so one way to come to this is you first describe, uh, first you need to generalize is your standard Poisson bracket for finite number equals freedom to classical field theory.

And then you can just generalize that to quantum. Yeah, indeed. That's one rule of doing it.

OK, other questions? Good. So we'll just guess the answer. The answer is very easy to guess.

So remember, if you have a single x and a p, that's what you have. OK?

But if you have more than one particles, if you have more than one particles, say just hint, say you have multiple particles, system in quantum mechanics, and then you have xA and PA as your dynamical variable.

So A equal to 1 to, say, n, say the number of particles. OK?

And then your canonical quantization condition just become x, a, p, p, b, t equal to i, delta, a, b.

And different x, a commute, and different p commute.

OK, different p commute, OK?

So now this a and b are essentially just replaced by x and x prime.

So x and x prime are just continuum version of those a and b.

Remember, we can't emphasize the x and the x prime are the labels of your degree of freedom.

So now you can just guess.

So we must have the following scene.

So from here, we must have phi tx, phi tx prime must be 0.

And the pi tx. So pi is the analog of p here.

Yeah, so those are operators.

And then phi tx, which pi tx prime should be something can only be 0 when x not equal to x prime can only be non -zero when x equal to x prime.

It's a gen addition of this.

So now you can guess. So what should this be?

Direct delta? Yeah, this should be just direct delta. But now you can ask the question, why has to be direct delta?

Maybe why should not be, say, the derivatives of direct data?

OK? Say y should not be, say, 100 derivative of Dirac delta.

And that question can be addressed just by from dimensional analysis.

So here we know somehow this must be rated Dirac delta. And now let's decide.

So now you can do a little bit dimensional analysis.

So if you just write down the action.

Yeah, the action I have just erased. Sorry. OK.

So if you look back on the action, I may just outline the idea, because I'm sure you can do dimensional analysis yourself.

So if you look at the action, the action is dimensionless in the lateral unit we are using.

So from there, you can deduce the dimension of phi should be 1 over l, so 1 over the length, OK? OK?

And the fact that the pi, where it's pi, and maybe I'll also erase, is equal to pi dot, means pi should be dimension 1 over L squared, because you take the derivative on time, and then there's a lot of factor of L.

Then that means on the right -hand side here must be something 1 over L to a cube, because there's no other parameters here.

OK, yeah, because here there should be a i, OK?

And if it's a dimension 1 over cube, then it can only be the data function, not a hundredth derivative of a data function.

So this thing should be just a data function, OK?

So this is the convention that there should be a i, and then it should be just the three data functions.

And this, indeed, have the dimension 1 over l cube, OK? Good?

So now you can just plug.

So you have the expression for x for phi.

You take the time derivative of this.

You get the expression for pi. And now you can just plug them into here.

You can just plug them into here.

And then you can find the commutation relation between those a k's.

And so this is a slightly tidier calculation, which is, however, a little bit fun, which of course I will leave you to do.

So, so if you just plug them in and then you can deduce that the following commutation relation between A's.

So, so this is the, I think this is in PSI 2, but I can still change my mind. Yeah, I wanted to put in PSI 2.

So, you find the, the the commutative between a and the commutative between a dagger.

Yeah, so now I will suppress the hat, because at the right hat, I think over and over I will be too tired.

So the commutation relation between a are 0.

And a between a and a dagger, so this give you 2 pi q delta function k.

So this is a 3 delta function k.

So again, this is a straightforward generation.

So if you have multiple harmonic oscillators, so if you have considered the multiple harmonic oscillators before, and then a between the different harmonic oscillators because k prime are just here just corresponding to, essentially you have, yeah, here it just is essentially have infinite number of harmonic oscillators. And each one of them labeled by a k.

So this is just like essentially we find. Let me just write it here.

So from those commutation relations, we conclude this theory when we quantize, after we quantize it, become an infinite number independent harmonic oscillators, decoupled harmonic oscillators, harmonic oscillators labeled by continuous parameter k.

The k is the wave number.

So for each k, there is a.

And a, so between a themselves, it's 0. Between a dagger, it's 0. But a dagger, they're not equal to 0.

And so this is, again, the continuum generation of 1.

OK? Okay, this is a continuum generation of one, because you have a continuous variable, yes.

Yeah, then you cannot say for sure.

No, but you see the quantization condition is in quantum mechanics.

Quantum mechanics, T and X are not on the same footing.

You can require your action to be X and T to be on the same footing.

Once you start to quantize your theory, and then T will have a pronounced row.

The canonical computation relations have to be imposed at equal time.

Other questions? Good.

So, yeah, so, so, so essentially we just get, and now it's just trivial, okay? So, so you can just build up your Hilbert space.

Essentially, just have infinite number of harmonic oscillators. OK, just have infinite number of harmonic oscillators.

And there's no surprise we've got infinite number of harmonic oscillators, because we mentioned that this field theory can be actually written as a continuum limit of these particles on the chain, which in these eight or three examples, you know that's just a harmonic oscillator.

Once you find the normal modes, they are all just a bunch of harmonic oscillators. And this is just a three -dimensional version of that.

And now we will, yeah, today we are running out of time.

So next time we will see that each excitation of the harmonic oscillator can be interpreted as a spacetime particle. So that's the core thing of it.

And now you have this infinite number of harmonic oscillator. And now you can define the vacuum.

and then actually this creation operators on the vacuum and now you find that each excitation actually corresponding to a particle and and has the corresponding to a relativistic particle and that's how you can have actually arbitrary number of particles in this theory and yeah because you can excite as many times as you want okay each excitation is a particle good good okay So I think it's a good time.

Yeah, we are two minutes, I think, early.

But I think it's a very good place to break.

So the time component have minus one and then the spatial component is positive one.

And yeah, so also let me just first remind you what we did in last lecture.

So, in last lecture, we talked about principle of locality, which is a powerful principle.

And so, this naturally leads to the concept of fields.

And so, I will generally denote fields, say, using this kind of notation.

Yeah, in the, when we talk about fields abstractly, we will use this kind of notation.

and so phi just an a just label for different fields and also so the fields depend on spatial coordinate and also depend on time.

So the physically you can think of this spatial dependence as the label for the fields for the degrees of freedom.

So at each spatial point this is the freedom and the time describe its evolution, okay?

So essentially this is like, so classical field is like classical mechanics, because freedom, but now it's the infinite number of degrees of freedom, okay?

So the classical field theory is like classical mechanics, but now it's infinite number of degrees of freedom, because now you have some finite number of degrees of freedom for for each point in spacetime, okay.

And the principle locality also implies that the action for such kind of dynamical variables, so the fields are all dynamical variables, and also implies that the action has a local form.

So that the action have the form that is always a single spacetime integral, and some function which is called Argonian density, t, which is a function of the fields, say phi at some point, at the point x, which, and then also its derivatives.

And so because of locality, then the action has a very simple form.

And then I think classical mechanics, you can introduce canonical momentum, conjugate to your dynamical variable.

So we have momentum density, they conjugate to phi A at each point.

So this is the phi dot, which is the time derivative of phi.

And then you can also introduce the Hamiltonian density, which is the pi A by dot A.

So A is summed and the minus L. And the Hamiltonian is obtained by integrating this over space.

Yeah, and also the equation motion.

When you do the variation of the action, you find the equation motion is given by the following general form, the equation motion is given by this general form.

And when we study field theory, as we study, say, any other subject, we always start with simplicity examples, okay?

And the simplicity example would be, say, a single scalar field, if you can see the single scalar field, which you have low index, we just have five, not dependent on spacetime point x.

And then the simplicity action for such a scalar field, you can just write down based on a general principle, say, Yeah, so you can have this form.

So we discussed last time the simplest action for scalar field have the following form, partial mu phi, partial mu phi, and the plus m square phi square. And the sign here is determined because of the metric condition.

So this is the simplest scalar field theory we can write down.

So here I have written here a quadratic function for phi, which we wrote as a general function last time.

And here the kinetic term comes from the Lorentz symmetry.

So this is the simplest derivative term you can write down with respect to Lorentz symmetry.

You can have more complicated terms, you can have this term, for example, squared. Okay, but that would be more complicated, so this is the simplest one.

And so this would be the simplest field theory we will study first, okay?

And yeah, so in this case, for this example, the momentum density just equal to pi dot.

It's time derivative of pi, and then the equation motion, It's given by a linear equation phi, it's given by that.

So this is a very simple theory.

It's very simple equation motion, but data we will see. I actually will teach us a lot about general field theory.

So it's a very quick summary of some of the main points of last lecture. Do you have any questions?

So phi here is a real field. Yeah.

Yeah, so phi just take a real value.

So now we talk about the second topic related to our discussion of classical field theory. Symmetries.

So when we have a theory, say with such an action, we say the theory has a symmetry.

So the definition of the symmetry is the following.

So symmetry is some transformation of each field phi a into say some function of phi.

So this new phi a can say depend on all possible fields.

And so some transformation of your dynamical variables.

So here, prime just means a different function does not mean derivatives.

Which leaves the action in worries, okay?

So whenever happens, and we say this is a symmetry. symmetry.

So for example, in this example, let me give you a label, label to this equation and we call this equation star.

So in the example of star, say for example the symmetry includes say translation.

So if you imagine we take a coordinate transformation, we do a shift in coordinate by a constant shift.

Let's imagine we do a constant shift of the coordinates, and a mu is a constant, so a mu is a constant vector, and now if we assume that the phi transforms as a following, so when you change the the coordinates essentially just change the label on the field.

But the value of the field should not change.

So that means that the phi evaluated at the original point should be the same.

So that means the nu transformed phi evaluated at the nu point, nu x prime point, should be the same as the value at the original point.

Okay, so this is the, yeah, so suppose the phi transformed like this, so the, so scalar field should transform as this under the general, so this is the example of a space -time transformation, okay, and the general space -time transformation as this equation, okay. So the, so you change the label but the value of phi should not change.

So the new phi evaluated at the new location should be the same as the value of phi at the original location.

And so you can easily check yourself.

Okay, so you can easily check yourself. I will not do it here.

So this exercise for yourself, that actually this action is it's invariant under this transformation with this change of x.

Any questions on this?

So second transformation, which this is invariant, is the Lorentz symmetry.

So Lorentz transformation is another kind of spacetime transformation on the spacetime coordinates.

If you take x mu to x mu prime, you're going to lambda mu nu x nu, so lambda nu nu is the constant Lorentz transformation matrix, okay, the constant Lorentz transformation matrix, and again under such a coordinate transformation phi will be transformed this way, phi should transform this way, and now you can check yourself, okay, okay, and because of this contraction of partial mu, partial mu, and this is actually a symmetry.

Okay, again, this is left as an exercise for yourself, and in fact, I think in your P side problem, you will do something similar.

Any questions on this?

So examples like this kind of transformation for translation and the following transformation. So this is what we call the continuous symmetries.

So, so continuous symmetries, okay, so, so these are the transformations, yes, I should also give a label to this equation, so let me call it just star, star, star.

So, so, uh, uh, continuous symmetries are transformations like star, star, star involving continuous parameters.

And so both 1 and 2 are clearly continuous symmetries.

So you can easily see from here. So here the transformation is A, and A is four numbers, okay, it's a vector, it's a four vector.

And a can be arbitrarily changed. So it's a continuous parameter, okay? You can take it to be zero. You can take it to be point one in all directions, etc. So it's something can be continuous change.

And similarly, the load -range transformation contains continuous parameters.

So do you remember how many continuous parameters does a load -range transformation contain in four dimensions yes no yes yeah six it's because you have three rotations because you have three direction three spatial directions you have three rotations and then you also have three boosts yeah so together six so here there's six continuous parameters and here there's four.

Okay, so here is this four. So both are continuous symmetries, okay?

And in contrast, this Lagrangian, this action has a lot of symmetry, which is not continuous. Can you say what that symmetry is?

Good. Priority is a good, it's a good answer, yeah.

But But there's still something slightly simpler.

So notice that here it's all quadratic in phi. And what do you observe?

So here there's a symmetry, a lot of symmetry.

It's phi goes to minus phi, because it's quadratic in phi.

And also, of course, there's a symmetry called the parity, which in the case x goes to minus x.

So the spatial direction x goes to minus x, and then phi again transform as this okay and you can also check this is also symmetry so in both of these cases they don't involve continuous parameters and so these are called discrete symmetries okay so there's no continuous parameters and yeah in this case both of them are called the z2 in the z2 means that if you do the transformation once they go back to the identity okay so here if you do to the twice, yeah, if you do it twice and go back to it, then if you do it twice and go back to it, then I think this is the same.

So here is Z2 transmission.

Also, symmetries can be also separated into, say symmetries can be separated into continuous symmetry, or discrete symmetries depend on whether you have continuous parameters or not.

But symmetries can also be separated into other different, And there's a lot of classification of the symmetry.

It's called the gauge symmetry and global symmetries.

So they are global symmetries corresponding to the transformation parameters are spacetime independent.

So both of these are examples of global symmetries because they are just constant. They don't depend on spacetime. And the lambda don't depend on spacetime.

And you can also have local symmetries in which case the transformation parameters are space -time -dependent.

So an example of the local transformation which you may remember is the so -called gauge transformation in E and M.

So e and m, you can, yeah, anyway, so later we will see example, we will go back to e and m again and you will see examples of this local signature.

Yes? It changes the action but it doesn't change the equations of motion. Did that, so it's similar? Yeah, yeah, as far as it goes with the action.

Yeah, that's a very good question.

So, by definition, if it leaves the action environment, we say it's a symmetry. And in general, actually almost always, if it leaves the action environment, actually also leaves the equation motion environment.

Yeah, but it's not guaranteed, yeah. Yes, purely from a mathematical point of view, it's not guaranteed. But for all physical examples, it's almost always the same. Other questions?

And now there's a very simple, but a very deep connection between the symmetry and the conservation laws.

So that you may have already landed in... Actually I forgot when I landed myself.

So some of you may have landed in high school, some of you may have landed in Adelaide, etc.

Anyway, so there's a connection between the symmetry and the conservation laws.

And so any conservation laws can be understood as a consequence of symmetry.

So in classical mechanics, you may remember that the time translation, this leads to energy conservation.

And the spatial translation, symmetry, then leads to the momentum conservation.

And if it's a rotational symmetry, then it's to the angular momentum conservation, okay? So this should be something you already know, say from your classical mechanics.

But in classical field theory, this can be generalized, okay? So here, there's a Lawson theorem.

So first discovered by I. Let me loss the theorem.

You said any continuous global symmetry leads to conservation laws.

OK, leads to a conservation law.

And so no matter what kind of symmetry, in addition to those things you are familiar with, no matter what kind of symmetry.

But any time you have a continuous global symmetry in your system, then you have a conservation law.

Okay, so now let me give a proof of this node, it's also a theorem, okay, so before I prove it, do you have any questions regarding its statement?

It's like the definition. So you said I could continue a symmetry. You wanted to continue a seal on a parameter.

But if you wanted to write out an equation, star, star, star in a way that specializes to continue a symmetry, there are ways to do that.

I'll show, like, just, like, how to check something. Oh, yeah, yeah. You just check whether there's a continuous parameter.

You just check whether there's a parameter or not.

Does this answer your question?

Yeah, normally you can just see whether there's a parameter.

If you write down the transformation explicitly, you will be able to just see. If we go down here, there's no transformation parameter.

You can just see under there, for the Lorentz transformation and translation, you can just see the explicit.

So, important aspect of a continuous symmetry is that because you have a continuous parameter, So, you can continuously relate it to identity. So, the trivial transformation, you just don't transform at all, right?

So, in this case, the Mu is equal to zero. And so in this case, the trivial transformation, just the lambda Mu is the identity matrix.

And then you can imagine a slightly, a pending on a slight infinitesimal rotation or infinitesimal boost. and then that's corresponding to a so -called infinitesimal transformation.

So here you can also just translate a mu a little bit, so very close to the identity.

And so essentially for any global continuous symmetry, it has an infinitesimal form.

It's just when the transformation parameter is very close to the identity, very close to the zero.

And so for example, for general transformation like this, if the transformation is close to the identity, then we can always write it in the following form.

Phi a goes to phi a prime, which is equal to phi a, and with some parameters, small parameter epsilon, and some function f a, and then say phi b, and the derivative of phi b, et cetera.

okay so this epsilon is the infinitesimal transformation parameter okay so if you have a continuous symmetry there's always say an infinitesimal transformation which are close to the identity so so this epsilon corresponding to this case would be for a mu is very small or for the Lorentz transformation case it's corresponding to rotation angle is very small or the boost is very small yeah so the epsilon can be any of those parameters and then this f it can be some arbitrary function okay this f can be up some arbitrary functions of your fields good so so in this case yeah so in this case the data phi okay so the yeah so the symmetry transformation so we have used the rotation data hat phi a so this symmetry because then given by the epsilon f a okay so so so data hat here implies a symmetry transform infinitesimal symmetry transformation okay so now let's consider during a variation of s under this data hat okay since the by definition of the symmetry because of the this is a symmetry that means that So the delta hat s should be equal to zero, okay? Because the variation of the action should be invariant under the symmetry. And so now let's look at what's the consequence of it.

So now we can just worry this L.

So this means that the variation of the L, the Lagrangian density, must be a total derivative.

Okay, since all, every transformation is proportional to epsilon, yeah, so epsilon is a small parameter.

So when we do the transformation, we only need to keep track linear order in the, in epsilon.

So, so the transformation of the a l, so, so must also be proportional to epsilon.

Okay, so, so, so since this must be invariant, that means the variation of the a l must be a total derivative.

two. So k mu would be just some, so for some k mu, okay, for some k mu. So there must be some k mu.

And this k mu can also be zero. In that case, the Lagrangian density is invariant. Yes?

Is delta hat the variation under the transformation? Delta hat is a spec. Delta hat just delodes such a transformation.

Just delodes such transformation. This is not the general... this is not the general variation. This is very specific. This is a symmetric transformation.

And so, under the symmetric transformation, your action should be invariant by definition, because this is a symmetry.

And that in turn, because the S is the integration of L, then that means that the transformation of L, as long as density, must be a total derivative.

And here, we only keep track of everything linear, all in in epsilon, and so this must be proportional to epsilon times the total derivative, okay?

And so we must have this structure, yeah, for some k -ball, okay?

Which can be zero, which can be zero, okay?

Good, so now let's look at what's the implications of this equation, okay? So this equation is the constraint imposed by the symmetry, okay? Okay, now let's look at what are the implications of these constraints.

So now we can just do the variation of L.

Now we can just do the variation of L.

And maybe I will keep that equation when I will do this board first.

So if something is symmetry of your action for your system, would you like directly vary the thing, like parameter, and then show that your action doesn't change, or would you do that, like, if it was one, and then show that?

So we want to use the fact that such a transformation is a symmetry to derive what is the constraint on the Lagrangian, okay, what is the constraint on the theory.

So we assume that somehow the theory have a symmetry and then we want to see what is the constraint this put on your theory.

Yeah, that's right, we're proving in the law of theorem.

Good, other questions?

So that equation is the constraint imposed by the symmetry. And we want to see what this equation tells us.

And for this purpose, we just do the variation.

So the delta hat L, because L is just a function of phi.

And then we have partial l partial phi a that a hat by a and Then we have l partial partial mu by a and Then partial mu head by a okay So we just did the variation and now Okay, and the so yeah and now we are going to use the equation motion okay so so from the equation motion yeah from the equation motion and partial L partial phi a this is just equal to partial mu partial L partial partial mu phi a okay so we just plug in this into here so when you plug in this into here now you notice is this become a total derivative.

So you have partial L partial mu by A and delta hat by A.

So after using equation motion, this variation of L have the following form, which is the total derivative.

And this should be equal to the right -hand side. It should be equal to the right -hand side.

And then here we find, so this should be equal to epsilon partial mu k mu.

Okay, so we combine both sides together, plug in that this is equal to epsilon times FA, and then we conclude.

Okay, then we conclude partial mu, partial L, partial partial mu pi A.

F A minus k mu, we think of it as mu, okay?

So if we call this j mu, and then we have a conservation equation, partial mu, j mu, we think of it as mu.

Okay, so we have a conservation law.

So remember, a conservation law is just to have a vector which satisfies this equation, and then, yeah.

So more explicitly, yeah. Yeah, so the j mu is just this combination, okay? J mu is just this combination. So any questions on this?

This is quite a strange condition because it's that the derivative with respect to space time is zero, but you can have a conservation of the derivative with respect to time in here for that state. Yeah, yeah, yeah, I will, yeah.

Yeah, this is the stronger version than your normally. Yeah, so this is the space time version of the conservation law from your normally classical mechanics.

So this is a field theory version of that. Yeah, yeah, I will elaborate that equation a little bit. Yes?

So when it transforms, the effect is?

So here, yeah, yeah. So here is a general formulation.

So this transformation can be general. it doesn't have to be, say, a spacetime transformation we said earlier, this is just some abstract symmetry, just some arbitrary transformations.

Is there a change in the field of this function and not the space change? Yeah, because spacetime variable is a dummy variable, in the action you can always get rid of that.

The x change, x is just a dummy variable in your action. because you integrate over x, yeah.

Yeah, that can include the, normally, yeah, you can in principle have as many derivatives as you want, but normally if we have action which only contains first derivatives and your symmetry transformation only involves first derivatives.

That's not discussing the case.

Yeah, in principle you can have it. I just, yeah, I'm just writing this for simplicity.

Yeah, maybe you can ask after the class. Yeah, it's not clear. Other questions?

Okay, good. So let me elaborate a little bit on this equation.

So this equation you may have seen in the, in EMM, so, which you often call the continuity equation.

If you write this equation separately, has the following form partial 0 j0 plus partial i or the divergence of the spatial component.

So j mu, if we write it in terms of the explicitly, then you have this form.

So this is so -called the continuity equation, just the time variation of the density.

The 0 component, they consider some kind of density, it's the same as the divergence of the current, okay?

And in particular, you can define a charge, which is the total spatial integration of j0.

So, if you integrate both sides over the total volume, and then this just becomes partial 0, q.

two, and this term become a total derivative, and then you can convert it into a boundary term using the Gauss law, and then normally the current advantage is at infinity, and then you just have the charge combination, just have charge combination, okay, so this is like the field, so normally in classical mechanics you just have this equation, okay, And this is the field theory version of your conservation.

that how it represents? Sorry? Is there a way to tell me what G represents?

J represents? Yeah, very represent. Yeah, yeah, yeah. If I give you a specific, yeah, you will do it in your PSAC.

So if I give you a specific Lagrangian, if you start a specific transformation, and then you will be able to find all those quantities, you will be able to find the k explicitly, and you will be able to find all those quantities explicitly then you can find the j.

It's because if we have here, so if we have here, if we have data s equal to zero and then you can just put the data in and then this quantity then is integrated over that. If this is zero then this has to be a total derivative.

Yes. It's rather global symmetry than this part of the equation.

Right, yeah, you're good, this is a very good question.

So yeah, so I will not have time to go into here. Let me just very quickly mention. So this is called the first nozzle theorem.

And when epsilon is spacetime independent.

And so when epsilon is not spacetime independent, and the story is a little bit more complicated, so the epsilon will be inside, the epsilon will be inside this total derivative, you cannot take it out, etc.

And then the story changes a little bit, and then there's something called the second law of the theorem, and actually in that case, when you have local symmetries when epsilon depend on spacetime, instead of finding conservation laws, you find that your equation motion, not all, Or you find some part of the equation of motion are redundant.

Yeah, and if you, it's not difficult, but actually in principle, I can put it in the p -side problem. If you really want to see it. Yes?

That's about the transformation of the equation of the identity. What happens to the transformation of the equation?

Good, good, good. So, this is the QBT, QT beauty of physics, which you may already have seen in quantum mechanics.

So, whenever we see a symmetry, no matter how complicated that symmetry is, say some continuous symmetry, it's enough just to understand that symmetry near the identity, near to be the infinitesimal transformation.

It's because any finite transformation you can build up from just adding up the infinitesimal transformations.

So, once you know the infinitesimal transformation, actually, essentially, up to some global topological structure, essentially, determine the full finite transformation. Yeah.

No, then there's always a symmetry, right? Because each step is a symmetry, and so you add them up, you add them up, still a symmetry.

Or if you do different, yeah, there's still a symmetry.

Yeah, because by definition, if something transforms its invariant, it will do a lot of transformations still invariant.

Good, other questions? Yes? For this grid symmetry, we don't have this, if you don't want to do something similar. Sorry? For this grid symmetry.

For this grid symmetry. I have to do this, and I will do something similar.

Yeah, in general, for this grid symmetry, if you don't have such a conservation law, you don't have such a conservation law.

You can still, sometimes you can still define some discrete quantum number, which is conserved, make it a parity, which in the parity case, but you don't have a current. Okay, so here, Jmu is called a conserved current, but you won't have a current.

Good, good, so this concludes our discussion very quick discussion of the important features of classical field theory.

And with this preparation, and now we can start thinking about going to quantum mechanics, talking about the quantum field.

And so here is a good place to pause a little bit from what we already said about the classical field theory to anticipate a little bit about the quantum field theory.

So here I will restate our goal for quantum field theory, which I said a little bit in the last lecture.

So now we have seen the classical fields and our goal is to understand the quantum version of this story.

So classical field theory is the classical mechanics with infinite number of degrees of freedom and now we want to quantize this infinite number of degrees of freedom and now this becomes quantum fields.

And now we want to understand the quantum dynamics of such a field, such kind of five fields, okay.

And for example, the example is the E and M, we have an electric magnetic field, okay, and you know the Maxwell equations, you can solve the Maxwell equations, so this defines the classical field theory, and then we will tell you how to actually quantify such a system to understand the electric field and the magnetic field quantum mechanical.

Good, and that actually would be our goal, okay, so we will start the simplest field theory, just the scalar field theory, and then we go to the Dirac theory, which describes the electron, and then eventually we will go to the theory, which describes the electron -magnetic field, so this is called QED, okay, quantum -magnetic dynamics, so that will be the end point of this course.

Okay, so now let's say a little bit how we go to this classical field theory to quantum field theory. Okay, so some just some general remarks.

So before doing that, let's think a little bit how we go to from classical mechanics with a small number degrees, even how you go to quantum mechanics.

Okay, so let's recall, so in classical mechanics, so let's just consider the simplest case, you just have a single degree of freedom, say xt, okay, just one dimensional particle whose motion just describes the xt.

And then, of course, the goal of classical mechanics is just to solve the equation of motion over xt.

So if I write down the equation for xt and solve it, then I'm done. I'm done. You solve the classical mechanics.

So when you go to quantum mechanics, what do you do? So this xt then become an operator.

So become an operator. So the classical dynamical variable now become a quantum operator. And the equation motion of xt, if you remember, then do you remember what it becomes?

Yeah, it exactly becomes the Heisenberg equation.

So it becomes a Heisenberg equation for xt, for this operator xt now. Okay?

Yeah. So in quantum mechanics, there are two, normally two ways to describe it. So first is the Schrodinger picture.

In this case you have a wave function which is a function of x and t and then you have an operator which is x.

And then of course you also have conjugate momentum which is defined to be say x dot.

And so the wave function is a function of the eigenvalues of, so x here should be understood So that's eigenvalues of this operator x hat, okay?

It's the eigenvalue of x hat.

So in the shorting of picture, you solve the evolution for psi, you solve the equation for psi, and once you find psi, then you can calculate any quantities you want, okay?

You can calculate any expectation values, any amplitude, et cetera. But the second way of approach it is the Heisenberg picture.

In this case, the dynamical quantity is your operator, now you're assuming the operator depends on time. So in the Schrodinger picture, your x is just some constant operator, it does not depend on time.

But in the Heisenberg picture, now your operator does not become time dependent.

And the equation motion, and you solve the Heisenberg equation, okay, and then the state is invariant, okay? State does not evolve with time, okay?

So you solve the Heisenberg equation for those operators, and once you solve the Heisenberg equation, and then you can evaluate, again, your expectation value in any states you are interested in, et cetera.

So any questions on this? This is a very quick review of what you did to go from classical mechanics to quantum mechanics, okay?

Okay, so now in field theory, in field theory we do a similar thing, so similarly in field theory we have classical fields, okay, we have classical fields, and then when you go to quantum, this becomes quantum operators, okay, so this is now quantum operators.

Remember, in field theory, this x is always just labels of the space point, okay? It's not a dynamical variable, okay?

The notation is a little bit, yeah, here in classical mechanics, this is a dynamical variable, but in field theory, this is just labels, okay?

So they're not dynamical, a dynamical variable phi itself. So phi become an operator, okay?

Okay, and then now the classical equation motion for phi will be derived and then become the Heisenberg equation, okay, for phi hat.

So again, here you can do two picture, you can do the Schrodinger picture, you can do a Heisenberg picture.

So for QFT, the Schrodinger picture, again you look at the wave functions of your dynamical variables, the eigenvalue of the dynamical variables.

So what's the generation of this psi xt?

So remember here psi is a function of eigenvalue of your dynamical variables, OK?

So now if we push this analog further, when you go to quantum field theory in the Schrodinger picture, now the wave function should be a function of phi x and t.

So these phi x are eigenvalues of phi hat x.

So in the Schrodinger picture, phi hat does not evolve this time.

So you have the wave function, which is the function that will become a functional, because this itself is a function of space.

And you solve the Schrodinger equation for this, and then you have the operator, and then you have the operator, and then you have operator phi x. x.

Okay, so that's what you do in the Schrodinger picture, in the Schrodinger picture.

And in the Heisenberg picture, you forget about the wave function, okay? You look at the evolution of operators.

So the Heisenberg for the field theory, the Heisenberg picture for the field is that you look at the evolution of x and t.

So now this just obeys the Heisenberg equations, which is just the quantum motion of the classical equation for phi.

So you look at the evolution of this, and then the state does not change this time. OK? The state does not change this time.

Good. Any questions on this?

So in the shorting picture, so remember, Remember the x is a label for phi, but time is evolution, but in the shorting of pictures, the operator don't evolve. And so there's no time here. So we don't have time here.

And we just have the, just the analog of the x here in the shorting of picture for the classical, yeah, for quantum mechanics.

And then the time dependence is in your wave function.

So the wave function is a function of possible values, eigenvalues of this operator phi law.

and then the Heisenberg picture again, then you just focus on the operator equations, and once you solve the operator equations, and then you can calculate the expectation values in any state you want.

Okay, so now you can already maybe see a difference a little bit. So, if you do the Schrodinger picture, you have to deal with this beast, okay?

Which is the wave functional of all possible values of some function in space, okay?

And if you have multiple fields, then this is a hugely complicated thing. And you need to write down the Schrodinger equation for it, et cetera, okay?

But here, but in the Heisenberg picture, where we just solve the analog of the classical equation motion, which we have already written down, and you just now interpret it as a quantum operator equation. So which one you think is simpler?

So that's why in quantum field theory, we almost always use Heisenberg picture.

We always use Heisenberg picture, and we don't even rarely think about the wave function.

Even though sometimes this can be useful in some problems, But for the most of the time, this is much easier. So that's what we are going to do.

And so in quantum field theory, we will just use the Heisenberg picture almost all the time.

From now on, I will not talk about the Schrodinger picture.

Good. So it's very important. You remind yourself about quantum mechanics in the Heisenberg picture.

And because most of your quantum mechanical classes before, maybe it's all in the shorting of picture, solving the wave, solving the shorting of equations, et cetera, but now you have to change the perspective to think everything in terms of the Heisenberg picture.

And then that will make you, then in quantum field theory, much easier. Okay?

Good. Any questions on this?

So in the hiding bird picture, if we're not concerned about the wave function and all, then how do we determine, for example, the probability or the expectation value of where x is at some point in the momentum? Good, good, good. That's a very good question.

So this is actually also related to the kind of questions we want to solve in quantum field theory.

So in quantum field theory, we often work with the vacuum states.

So for example, here, yeah, so when you consider the QED, most of the time, in the quantum electric and magnetic field, they are in the vacuum state.

And so we just consider the psi in the vacuum state.

And so we don't have to consider, yeah, yeah, that's it. So normally in quantum field, they prefer the states we are interested in.

That's a lot of reason that the Heisenberg picture is convenient. You don't have to consider the general state in general. Yes?

What is the physical meaning of the idea? Oh, this is just some state you are interested in.

So you have a Hilbert space, but now you just don't, When you study the evolution, you only evolve operate, you don't evolve states. So that's the difference between the Schrodinger and the Haydnberg picture.

But for example, in quantum mechanics, you get the sufficient value of some operators with the factor of state. And we get like resonance in the resultant stuff.

So what about here? Yeah, it's the same thing. Yeah, same thing. Yeah, at this level, there's no difference between quantum fields here and the quantum mechanics.

Yeah, just think about quantum mechanics in terms of Heisenberg picture. Translate everything you learned about the Schrodinger evolution, et cetera, in terms of Heisenberg picture. Yes? Hm?

Yeah, that's right. That's right. Yes? Can I just say we're interested in the vacuum state and what sorts of measurements?

Yeah, so it's again, so yeah, it's a very good question.

What I should have said is that the state we are interested in are states which are close to the vacuum states. So you excite the vacuum a little bit.

And yeah, it's not, yeah, of course in the vacuum you don't have anything. And we are since close to the vacuum states.

Yeah, and later when we discuss things you will see good good.

Okay, so before going to quantum field theory Let's also make some remarks Have a short discussion on right to visit quantum mechanics okay so So naively if you have special relativity quantum mechanics So, if you want to generalize, so most of the quantum mechanics you learned is the non relativistic quantum mechanics, okay.

But now if you want to incorporate special, we want to combine the special relativity with quantum mechanics. And then what should you get? What do you think you should get?

I expect, yeah, just whatever come to your mind, that's the right answer.

That's the correct answer, but I was hoping some of you will say it's relativistic quantum mechanics.

So, laivably, when you combine these two, in particular, if you read some old quantum mechanics books, they do discuss relativistic quantum mechanics, so laivably that's what you get.

when you combine these two, and you say, oh, we just get relativistic quantum mechanics. But that's actually not correct statement.

Actually strictly speaking, the reason now you don't actually learn much about the relativistic quantum mechanics because strictly speaking, relativistic quantum mechanics does not exist.

And whenever if you want to combine quantum mechanics with special relativity, actually you get quantum field theory. So the quantum field theory is actually forced on us. If you want to unify special relativity and the quantum mechanics, okay?

And even if you don't want to talk about fields, okay?

And even if you don't want to talk about fields, if you just want to talk about particles, still if you want to unify these two, then you actually automatically base two quantum field theory.

So now let me just explain why this is the case, so that you have some better appreciation of the quantum field stream.

So let's try to apply what you did for long -righted vistic quantum mechanics, try to generalize it to derive some righted vistic quantum mechanics, suppose you are the people in the 1926, the quantum mechanics just was discovered. And you say, oh, people have understood the non -right -wistic quantum mechanics. Now let's generalize to special relativity.

OK, so in non -right -wistic quantum mechanics, how do we derive the Schrodinger equation?

The way we do this is we start with your dispersion relation for a non -right -wistic particle.

then we say let's take e to i partial p, okay, p.

So say if you can have a vector become, yeah, so let me forget about h bar. This p then becomes spatial derivatives. And then this equation is become the, a Schrodinger equation for the free particles.

Okay, so this is become the Schrodinger equation for, for free long -range visiting particles. Then you can add the potentials, etc.

OK, so that's how you derive your Schrodinger equation for long relativistic quantum mechanics.

So but if you are someone, say, in the early days of quantum mechanics, say, now let's try to generalize it to special relativity, and then it's easy to do, then in the relativistic case, then you have, you just thought it was relativistic at this point in relation. equation.

You say, let's do the same thing.

Let's do the same thing. This becomes i partial t, and this p becomes this one, and then we can just write this equation.

Now let's combine this derivative together or put all this on the same side, and then what you find is partial mu, partial mu.

That's what you get. Does this equation look familiar?

This This is the simplest scalar field theory equation motion we have written down. But here, the interpretation is very different.

So earlier, we wrote down this equation. We say this is Klein -Gordon equation.

This is the Klein -Gordon equation. So this is the equation motion for the simplest free field scalar field theory we wrote down before. But here, the interpretation is very different.

So remember, the psi is not the field.

In quantum mechanics, the wave function for single particle sitting, so psi is the amplitude for particle at spatial location x at time t, okay, so this is, and it's square, then it's its probability, okay, and this does not describe a field, okay, this is the wave function of a single particle, okay, even though they have the same equation as the field theory we wrote down earlier and so the so that's what a Klein -Gordon did okay so the Klein -Gordon they try to generalize non -rativistic quantum mechanics, rativistic quantum mechanics then they wrote down this equation they say oh now we are immortal okay because we wrote down the first equation for for for for rativistic quantum mechanics and So soon realize, actually this equation, if you want to interpret it as an equation for the wave function, there are actually a lot of problems.

There are various problems, yeah, by saying a lot of it, maybe it's a little bit of exaggeration. There are various problems.

Yeah, so let me call this equation, I think I have used up my stars, I think to have a four star is a little bit too much, let's just call this equation one for this section.

So some immediate difficulties of interpreting one as a wave equation, okay, as a wave equation for relativistic particle, let me just save time, okay, so this is the wave fraction for non -rativistic particle, if you want to generalize the quantum mechanics, the relativistic quantum mechanics, then you will interpret this as a wave equation for single relativistic particle, okay.

So first, so first is that, so if Remember, so in this equation, okay, if you go back to your early days of this long -right -based quantum mechanics, so you remember from this equation you can derive an equation for the conserved probability, okay, a conserved probability, and that equation tells you that psi squared should have the interpretation of the probability, OK?

So but for that equation, you can show there's no quantity as can be used as probability density, OK? So probability density, by definition, should satisfy two conditions. So the first condition is that it's non -lactive.

And the second condition is that it should be conserved, okay?

The probability should be conserved, otherwise you violate the, yeah. So, you can show that this equation actually does not allow.

This equation allows such a quantity, but this equation does not allow, okay? And I will not go through that myself.

I think that will be in your PSI 2, okay? That will be in your PSI 2, you will show it yourself.

And the second difficulty is that if you square that, if you find the energy, so the energy if you take the square root, then the energy in principle can be, say, plus minus, you have two solutions, okay, E equal to plus minus this quantity.

So in contrast to that equation, there's only positive energy, okay, a long range of it just have positive energy. So classically, you can just say, let's throw away the second branch.

We can just throw away the second branch by hand.

Classically, but quantum mechanically, so classically, just ignore the second branch, the negative branch.

But quantum mechanically, this is not possible. Because you have the equation there, then you automatically have the negative solutions.

Okay, so quantum mechanically and then you have dispersion relation like this So that dispersion relation have the following form is the e as a function of p Okay, so you have positive branch then you have a negative branch and Then you remember in quantum mechanics you can have the then you have energy levels and you have energy levels and And a particle here, they have a higher energy than particle here.

Okay, and so here the energy level is higher than energy level here.

And there's always a non -zero probability quantum mechanically for a particle to go from some higher energy level to the lower energy level.

Okay, just like in the hydrogen atom, if you excite it, and they always go to the lower energy level. So in quantum mechanically such thing cannot be avoided.

it, okay? So you cannot just throw this branch away, okay?

And so this will lead to instability because the energy can be infinitely negative, okay? It can be infinitely negative.

And then all your particles, we all go to infinitely negative energies and then your system will be a big problem, OK?

So these are the two, the most prominent problems.

And then people tried many different ways, tried to avoid them, et cetera, including some very ingenious solutions, et cetera, so we were not going to them.

But let me just mention this.

That's actually, even if you can address those problems, still the relativistic quantum mechanics will not make sense for a very fundamental reason.

So these are the more superficial reasons why that equation does not quite work.

But there's actually more fundamental reason why relativistic quantum mechanics, even as a concept, does not make sense.

Okay, so by definition, if you want to interpret this kind of thing as a wave function, okay, so what's the interpretation of the wave function which we already said?

So this describes the amplitude for a single particle at some point, okay, at some point x at time t. But now if you have two particles, what do you do?

So, you introduce the location for particle one and particle two and the t, okay?

So, this describes two particles. And if you want to describe three particles, then you have to introduce more x. Okay, remember that's what you did in non -righted basic quantum mechanics.

So, in non -righted basic quantum mechanics, this makes sense, okay?

You can just, because there's no mixing between the different branches, say single particle, two particles, because the particle cannot be created and destroyed in a non -rativistic system.

But in a rativistic system, you can always create particles.

Okay, if you have enough energy, you can create new particles, pair, create electrons, okay, happens in accelerator all the time, or electrons, they can annihilate into photons, okay?

Okay, so the particles are not conserved, okay, not conserved. So this kind of wave function concept don't even make sense.

So if you want to describe, so that means whenever you have special relativity and quantum mechanics together, you must have a framework which can describe arbitrary number of particles at the same time, okay? Because particle numbers can change all the time, okay?

because of annihilation and the creation effect. And that cannot be achieved by this kind of concept, that wave function.

It cannot be achieved by wave function. It turns out, miraculously, that can be achieved by field theory.

It turns out that field theory, once you quantize it, automatically give you a framework to describe arbitrary number of particles in the unified manner.

single particle, two particles, arbitrary number of particles, and so this is one of the magic of field theory. So that's why the field theory plays such important role in particle physics.

It's because the excitations of fields automatically provide the mechanism to describe arbitrary number of particles. Okay, so we will stop here today.

I am a theoretical physicist.

I work on a high energy theory, including string theory, quantum gravity, non -equivalent statistical physics, et cetera, and many different topics.

So now let me say a few words regarding the QFT, key at the quantum field theory itself.

So the goal of this quantum field theory class is to develop concepts, formalism, and techniques for quantum dynamics of fields. So I'm sure all of you have studied Markzware equations.

So Markzware equations describe fully classical dynamics say, of electric and the magnetic fields.

But we live in the quantum world.

And so we should also treat electric and the magnetic fields using quantum language.

And then you find that once you do that, and then the concept of photon, once you treat the electric field and the magnetic field quantum mechanically, and then you you get the concept of the photon.

And the photon now is the fundamental particle which mediates, say, electromagnetic interactions.

So you actually get a completely different physical picture from what you get classically.

And yeah, so one goal of the class is for you to appreciate, say, to understand the quantum electrical dynamics.

And that's actually where we will end this class, and this we will discuss the quantum electrodynamics.

And quantum field theory is also very important for many other, for other interactions in nature.

So among the four fundamental interactions in nature, three of them are described by quantum field theory, completely.

And you can also use quantum field theory to describe gravity, to describe quantum gravity, But at the moment, quantum field theory does not offer a complete description of quantum gravity.

But still you can use the quantum field theory techniques for understanding certain questions of quantum gravity.

And also quantum field theory over the years, even though it's initially developed for particle physics, but over the years has been also found many applications in many, many other branches of physics, in condensed matter, statistical physics, et cetera.

And it's fair to say, nowadays quantum field theory has become a universal language for theoretical physics, essentially in all different fields. And so if you are a theorist, certainly you need to master this language.

And even if you are only interested, say, in atomic physics or statistical physics, and the field theory concept will be very important.

And for experimentalists to know basic concept of quantum field theory and to have some basic understanding of it should also go a long way, help you to appreciate say the most recent theoretical development and also for help you to communicate with theorists and yeah.

And so this class is the first of a series master sequence.

And so this semester we mostly developed the fundamental concept and the quantum field theory 2 and 3 will be more about technical development.

And say, if you're an experimentalist, and then you can view this quantum field theory 1 as a standalone class, and which, yeah, just enough for you to to get the basic idea of quantum fields, quantum serial fields, which you don't, yeah, you may not depend on your lead, and whether you have two or three or not, yeah, may depend on your lead.

Okay, so, under the main topics I plan to cover are listed in the outline, this document of the outline, which is already on the website.

And so I should emphasize that outline is a rough. It's only a rough roadmap. It may change, it depends on the pace.

It may change things along the way. And sometimes I change mind, say, half through the course.

Somehow I feel, yeah, anyway. So don't treat it too literally.

And any questions about this subject or quantum field theory so far?

And also, let me say a few words that the quantum field theory has a reputation of being a very difficult subject.

Actually, indeed, myself have suffered a lot when I landed myself.

But with 2020 hindsight and also from interacting with a large number of students through teaching various level of quantum field theory classes in many years, I can assure you now that actually quantum field theory is actually not difficult at all.

If you lend it the right way.

And of Of course, anything is not difficult if you lend it in the right way.

So in the sense that this is the empty words, but keep it in mind.

Whenever you find this, you think it's too hard, and the reason might be that you have to change your perspective. You have to change your perspective. you.

And so quantum field theory, one thing people complain about is that quantum field theory often involves a lot of calculations.

And that's true. That's just the fact of life. You cannot avoid it. But that's not what it makes it difficult. Okay.

Complicated calculations, you can just go through them from one line to the next line to the next line. If you're careful enough, patients enough you can go through.

So the difficulty I think for most people of quantum field theory is more at the conceptual level.

It's because this subject is not a very intuitive subject.

It's not something you can just understand just by thinking, okay.

So that's why I emphasized earlier the exercise and really working through it is very important.

It's was developed through examples, but working through many examples, you slowly develop intuition about quantum mechanics, and if you learn all the lessons, then you get good feeling about quantum mechanics. And quantum field theory is the same thing.

It's that you have to, some kind of intuition has to be developed, okay?

It's not something I think you can easily, yeah, just like mechanics, which you can, in some other subject, maybe if you have very good intuition, you can just imagine it, okay.

Yeah, so to help you develop good intuition about quantum field theory, I can offer you three pieces of the devices, okay?

So, so first, yeah, so the first piece of advice is that the quantum field theory is essentially quantum mechanics, but dealing with infinite number of degrees of freedom, okay? So, so in your quantum mechanics class, you always treat with finite number of degrees of freedom.

But quantum field theory, the difference for quantum field theory is now you treat infinite number of degrees of freedom.

It turns out that this treating infinite number somebody with freedom makes a difference. So more is different.

And actually, sometimes make conceptual differences.

And so that's why sometimes the quantum field theory is unintuitive.

But that said, I found for many people, including myself when I learned it, many difficulties you encounter in learning quantum field theory is not due to the difficulty in quantum field theory itself, it's actually due to your gap in understanding of quantum mechanics.

So whenever you encounter something you don't quite understand in quantum field theory, try to step it back to say, can I formulate this difficulty in terms of quantum mechanics with only finite number somebody who's freedom.

And often you find actually your difficulty can already be formulated in quantum mechanics.

And then that way, then you should be able to just settle it yourself, because we are supposed already to be a master of quantum mechanics.

And certainly, when I learned quantum field theory, I was stuck at certain points for a long time.

And then later I realized just because I didn't understand a certain Heisenberg picture of quantum mechanics very well.

When you understood, somehow I realized when I understood the Heisenberg picture of quantum mechanics well, and those difficulties just went away, and have nothing to do with quantum field theory itself.

So that's why in your first p -set you will get familiar with Heisenberg picture of quantum mechanics. That came from my own experience.

And also, the second point is that quantum field theory deal with formalisms.

And sometimes this subject seems very formal, okay, you have a lot of formalisms, okay.

But just keep in mind, any formalism in physics, no matter how abstract it is, it was always designed to solve some concrete physical problems and physical questions.

very concrete physical questions and if you understand what kind of concrete physical questions quantum field theory was designed to solve then that can give you a very good perspective on those formalism and why people do this why people do that why people do this trick why people do that trick and because they were invented to solve certain concrete problems okay and once you understand the questions understand the problems then the formalism become are much easier to understand.

And the third thing we already said is that in quantum field theory, as in quantum mechanics, intuition was built through experiences, through examples.

So when you do your PSET, when you look at the examples in the class, you should always ask yourself afterwards, say after you have done your PSET problems, always look back at that problem, say, what did I learn from this problem, okay, and just think through it again, think through, think through what you learned from that problem again, and that is a very good way to help you to learn from your experiences and to help you develop intuitions, okay.

And so, yeah, so, so also a very important thing you should keep in mind is that in the graduate to a course like this.

And most of the things should be learned outside the class.

So inside the class, the purpose to give you a guide is to emphasize the conceptual picture and the physical intuition, et cetera. And so sometimes I will leave some details for you to finish in the p -set.

And sometimes the p -set will involve problems which I did not fully discuss in lecture, but I want you to work out yourself, okay?

And so PSET is an important part of the learning, even new things, okay?

Not just to practice something, but it's also a very important part for learning new things.

Also, finally, I would like to make a apology.

So you will soon find that the lotations are used in the lecture are different from the lotations in the recommended reading books, okay?

So I recommended reading Paskin and Wemberg, and you will find that my lotations are actually different from them.

Also the order of presentations are also different from them.

I know this is very annoying, but there are just no perfect textbooks, and there's no perfect set of notations everybody use, and we all use the notations which we find the most convenient to use.

Even though I realize this problem, but I don't have a good resolution.

OK, so just keep in mind that the rotations in my lecture can be different from the rotations in those textbooks.

Good, so do you have any other questions?

Good, OK. So if you don't have any other questions, so let's start.

So the chapter 1 will be about why we can see the quantum field theory, OK?

So first, we talk a little bit about the classical field series to stand the stage for quantum field series.

And the first important concept is called the principle of locality.

So if you remember from your, say, high school days, Newtonian mechanics, so in Newtonian mechanics, you have action at a distance.

For example, if you look at gravity, and gravity is exerted by the sun on the earth, but they are very far away.

And the same thing with the Coulomb interactions between the charged particles.

But then in the 19th century, they came from this principle of locality, and that's formulated by Faraday around 1830, OK?

So the principle of locality says all points, you actually don't have action at the distance.

You set all points in space, participate in the physical process, OK? OK?

And the effect, so if you have interactions, OK, so effect propagates from points to a labeling point, OK?

So in this principle of locality, you don't have action at a distance. So action at a distance is always conveyed.

The action is always conveyed from one point to another point through the propagating in the space, OK?

And the fields, the concept of the fields is the mathematical device or vehicle that the principle of locality is at work.

So this is essentially the device we need to use to realize that this principle of locality.

And so the main idea of the field is that we associate each point with each point in space a dynamical variable or dynamical variables.

So for example, if you have an electric field, the electric field is defined for all space.

So at each point x, we can introduce an electric field. OK?

And then this electric field can also depend on time.

And yeah. So of course, normally we write it this way, e is xt.

And the reason I write it this way is to emphasize that in the definition of the electric field, the space and time actually play a very different role.

the space plays a role of a label, okay, so at each point we have an electric field, okay, at each point x we have an electric field, and so x here is just a label, okay, and the t is used to describe the evolution, the change of the electric field, okay, so the x and t plays a very different role, and similarly you can do it for magnetic field, it's the same thing, OK.

And here, you should always view x as labels. OK. So the spatial point, we should denote as vector x is the labels.

And we know that the evolution of electrical in my left field, I described by Maxwell equations.

Let me just write them down to remind you.

So we have dot b, zero dot e, four pi rho.

So this is so -called differential form of the Markzware equations.

So the reason I'm taking trouble to write them down is to emphasize in the following point.

So this set of equations exemplifies perfectly the principle of locality.

It's because you see those equations only involve the value of the electric field and the magnetic field at a single point.

So if here is at x -point, so here is also at x -point. You level say x here and here is some other point of y.

And so this is reflected the principle of locality that everything, so the effect is propagates from point to point. You just have the derivative of the same points.

It's never, you don't involve separated points.

So these are local equations.

So these are what we call the local equations, and they contain only E and B at the same point, and also the charge density and the current density at the same point with finite number derivatives.

So the derivatives are the ones to help you to propagate, because it relates the point to the laboring point.

And so that's how it propagates it through the derivatives. So the derivatives are key. The derivatives are key.

And another example, which examples locality, which I will not go into here, which some of you may know, is the Einstein gravity. So Einstein's general relativity.

So in Einstein gravity, the dynamical variables are so -called spacetime metric.

So they are object with two indices, two space time indices.

And then the Einstein equations are equations for this kind of object.

And again, the equations are local equations in the sense that they only depend on the g evaluated at the same point with the finite number of derivatives.

So in Einstein's gravity, you no longer have action at the distance.

OK, so the effect of the gravity is propagated through the spacetime, through spacetime.

So in fact, so here we say these two examples exemplifies the principle of locality.

In fact, the principle of locality played a very important role in formulating those equations.

Because as we will very soon see, when you have principle of locality, you can significantly to be constrained, the theory you can write down. So that's a very, very powerful principle.

Yeah, you can have non -local equations, but it's believed that the fundamental equations in nature, they're all local.

Yeah, and so far, the equations govern all fundamental interactions, all four different interactions in nature, the equations are local.

Other questions? Yes?

Say it again. I don't quite understand the question. Yeah.

Oh Here I'm just one for size again That the of course in Einstein's theory this these two are treated the same way But but if you think in terms of the fields and they actually have very different physical interpretation Yeah, yeah, so that's why I write this way. Yeah.

Yeah, it's the same Same way I write it like here, yeah, yeah, yeah, same reason.

Just here to emphasize for this, to emphasize the different row played by X and T. Yeah, but then, but I think you're asking a very good question.

So I'm going to mention nature in the relativistic series. Then, of course, then these two become the same. Yeah, become, play the very equal row, yeah.

Does that answer your question? Yeah, okay, good.

So let me also very quickly mention the different types of fields.

Okay? So you can have what's called a scalar field.

So these are quantities which at given point, given point there's only one value, okay? Say, for example, the temperature.

So it's just a single quantity defined at the point.

For example, the temperature, maybe other quantities. And then you also have a vector field.

And E and B are the examples of a vector field because at each point, you have a vector.

So here, I'm using the three -dimensional rotations. But in the relativistic series, is you can also use the four vector rotations.

Say, for example, the vector potential, we have the following form. Say, a mu will be a four factor, and then the spacetime, I combine them together into a four vector.

And so at each point, and then you have a four vector. And so, yeah.

And also, you have a tensor field. So this metric is an example of a tensor field.

here you have two components to have two indices okay so so you have many many different components that depend on the so again this is right to be stick notation I can write it as a right to be stick notation okay so at each point and now you have a now you have some object with two indices okay and you can also have later we will see your you also have something called spinner field out alpha. And this alpha is some other indices which we will define later.

So we can also have so -called spinor and alpha some other indices.

And as our convention, you said mu is always from 0 to 3.

Then sometimes I just write x. When I write x, it just means a four vector. It means x mu equal to ct and x.

So x always denotes a, x vector always denotes a spatial vector, okay?

And yeah, and then this is the same as ct xi, okay? So the spatial indices always denoted by i, and yeah.

And so partial mu will be the same as 1 over c partial t.

and then the derivative on the gradient on the on spatial directions and say if you have a full vector then again you we have the convention that a0 then the a vector it's the same also as a0 and ai I use the i to load the spatial components and and this is the last time I will write speed of light, so the c will always take to be 1, and h bar will take to be 1.

That's for notational convenience.

Do you have questions?

Okay, so now, so with this preparation, then we can talk about action principle for classical fields.

So first recall, in your classical mechanics, so we introduce the action, which is the integral of your Lagrangian, and Lagrangian is the function of x is your variable and x dot and the time.

And so this is a one -dimensional, yeah, this is 8 over 1, one -dimensional particle motion. motion.

And you can also introduce some momentum, canonical momentum, which is defined by partial l, partial x dot.

And then you can also define a Hamiltonian. And Hamiltonian is related to the Lagrangian by Px dot minus l, from a Legendre transform.

And the equation of motion is obtained defined by extremize S.

So S is considered to be a functional of your trajectory.

So whatever trajectory you have, and you extremize this S, and then you get equation motion.

So now you can generalize to field theory.

So now from principle of locality, so for field theory from principle of locality, so the form of the Lagrangian, so to the form of the Lagrangian L.

So again, we hear, for field theory, we can again define S as a time integral of a Lagrangian.

And the form L is significantly constrained to have the following form.

So you must have the form L equal to a spatial integral. So this is integration of all spatial directions. OK, so d3x.

So let me first write down the notation, and then I will explain the notation.

OK. Yeah, maybe just write it to partial i.

OK, so let me now just explain the notation a little bit.

Well, first, so here I use a short -hand notation into the load of the fields.

So phi A, so it will be a function of spatial direction on the time, is general field.

And A, label different fields.

So this index, A, labels different fields.

So for example, A can label different scalar fields. If you have multiple scalar fields, then they can label them.

And they can also refer to indices, space time indices, like a mu, and also, yeah, etc. Okay. So a just label whatever field you have. Okay.

And then the second point is that this L, this script L, is a function.

The emphasize here is a function, okay, of phi a and these derivatives.

And these derivatives.

So in other words, the a l, so this is a key point.

So that is a l only depends on the value of phi a and these derivatives at a single point, say x, and then you integrate over all x.

OK, so L is called the Lagrangian density.

Say it again? Yeah, yeah, yeah, yeah. Yes?

Yeah, yeah, we'll mention that. Yeah, we'll mention that.

So here I just explained the notation, OK?

So now let me just make some remarks on why the Lagrangian we must have this form.

So first, the principle of locality implies here must only involve a single integral, because the locality does not allow something like this.

Say, for example, does not allow a term like this.

Does not allow a term like this, which involves the phi at different points.

It's because if you have terms like this in Lagrangian, OK, so later you will see, oh, sorry, L, not L script.

So if your Lagrangian have these kind of terms, and then so as we will describe the equation motion later, you will see from the equation motion, then your equation notion will not be local.

OK? So equation notion will involve the behavior of your field at one point, and then influenced by point at some point far away.

And then you will not have local. So the locality significantly constrained.

Because if you give away locality, and then in principle your Lagrangian can be arbitrarily complicated. You can have as many integrals as you want.

But because of locality, you only allow to have such a simple integral, a y integral of a function. So this is the key.

So that's not allowed. Here is key point.

So the second point is that we only allow first derivative in time.

We only allow first derivative in time. We don't allow the second derivative in time.

So the reason is that, again, as you will see equation motion, if you involve the second derivative in time in your action, or in your Lagrangian, and then when you get the equation motion, you will get equation motions involving more than two derivatives in time, OK?

So So this will lead to, so this implies the equation motion only contains two derivatives, two time derivatives.

So this constraint comes from our experiences.

It's the same reason here. We only include the first derivative in time.

It's because in real life, all the experiment is determined by the initial condition. the initial condition only need to specify the location and the velocity, okay?

You don't need to specify more. If you're in quotient motion, you're moving more than two derivatives, then you need to specify more general initial conditions.

And so, yeah, so here is the same thing. We only allow the first derivative in time.

But you can, in principle, allow arbitrary normal derivative in spatial direction, okay?

But for simplicity, for the most of the time, as we will see, we will restrict to quantum field theory in spatial relativity, means that there will be a relativistic invariant, maybe Lorentz invariant. And in Lorentz invariant theory, space and time, they can transform to each other, play equal role.

So if you only have single derivative in time, you example we will see they will all have only single derivative in spatial directions okay but certainly non relativistic systems you can have a higher a number of spatial derivatives good any questions on this okay good so now so as in classical mechanics allowing we can introduce the canonical momentum Hamiltonian etc okay so so here we can so so here we can introduce so called that the canonical momentum density, the rhythmical density will be clear.

So we can remember, for this phi, x is just a label. So you can just view this theory.

It essentially has infinite number of such kind of, yeah, it's an unfortunate notation. Here we use x as a dynamical variable.

But here the x is only a label. OK? Here, x is a label.

So here, you can just imagine you have infinite number of degrees of freedom just each one is labeled by x, OK?

So now, just imagine here, you have many, many x, just you have some labels for it.

And so for each such one, we can introduce this momentum, OK?

So each phi a, we can introduce this canonical momentum, OK?

Phi a dot derivative, OK? OK? Yeah, maybe just.

So defined as the derivative of the Lagrangian density divided by time derivative of phi a.

So this is just the direct generation of here.

So remember, so this is evaluated at each point in spatial direction.

And each, yeah, so this thing will also depend on x and t. OK?

Oh, this is just the capital pi. Yeah, capital pi.

OK, and this phi A dot, just a time derivative of phi A.

And remember, x are the labels. OK, so x does not do anything here.

So the reason we call this Lagrangian density is because the A is a Lagrangian density. density.

And then we can also define the Hamiltonian density the same way.

So we have defined the script edge, which is defined by phi A dot pi A.

Again, this is all defined on the same spatial point minus L, okay, so this is not going to density.

And then the then the Hamiltonian, you just integrate over all space, essentially sum over all the good freedom, okay?

Just give him that. Yes?

Yeah, yeah, it's just like you sum, so if you treat the, yeah, so yeah, yeah, yeah, so so the A is summed.

So I will always, yeah, I forgot to mention, good, this is a great question.

So I always assume in this Einstein convention, in the sense that all the repeated indices are assumed to be summed. So no matter how many components are there, you just sum over A. OK, so A is summed. Here, yeah.

Good. Other questions?

So now we can talk about equation motion.

So it's the same thing as classical mechanics.

So classical field theory, just like classical mechanics, you generalize your infinite number of degrees of freedom.

So now with each point in space, you associate with some degrees of freedom.

So now let's look at equation motion.

So again, you just do the variation of the action, you extremize the action to be 0. integral.

So the action is again just defined by the dt over Lagrangian.

So the s is the dtl and then can be written as the four integral dt and dx of the density.

So now let me just write, let me not just write it in the more right basic notation, just assuming there's one derivative, one time and one spatial derivatives.

Okay, so the mu now, yeah, I combine the space and the time derivative together.

Okay, and you can straightforwardly generalize to, say, in more and more spatial derivatives.

good so so here so so let's just do the variation so the so we want to zero equal to data s so you want the variation of this s to be zero so now let's just worry s so you have d4x now just remember this not going in density it's just ordinary function because of the locality just ordinary function of phi a and its derivatives.

And we can just do the variation in a straightforward way.

You can just write partial l partial phi a and delta phi a plus partial l partial mu phi a and delta partial mu phi a.

And in this case, so the delta is a variation and the partial mu. So these two operations are independent of each other, so you can exchange them. So you can put actually delta inside.

So this is the same as partial mu, delta phi.

So now you can integrate by part of the second term.

Now you can do it because here it's a little bit awkward because you're moving the partial mu, delta phi a. So again, here the repeated indices, they're all summed. So you should sum them.

So now we can do integration by part here.

So then you get d4x, and you get partial l partial phi a minus partial mu partial l partial partial mu phi a, OK?

And then the whole thing, delta phi a, and then plus boundary terms. terms.

So the boundary terms come from you doing integration by part here.

You're doing integration by part here.

So the boundary term will be proportional to data phi a. Yes?

That's just the arbitrary variation. Yeah, just arbitrary variation.

So we always assume we have the boundary conditions. Okay. And so that the boundary term vanishes.

Okay. So here, we are not going to detail.

So we always choose theta phi a so that boundary terms vanishes.

The term comes from integration by parts.

And the second, just remember, we sheeted indices are summed.

And now, so this has, then we don't have to worry about boundary terms so that boundary term vanishes. Then now we just have this has to be 0.

But this has to be 0 for any variation. So this delta x mu can be any function of x, OK?

Can be any function of x. So the only way this can happen is that this pre -factor must vanish, OK?

So this implies that the delta x partial phi a minus partial mu by a.

So this is the general equation motion for classical field theory. Okay?

Any questions on this?

Okay. So from now on, we will make two restrictions. Okay?

also there are the most, we'll make two restrictions just for simplicity.

Yeah, most of our discussion can be generalized beyond those situations.

Yeah, so the first is that we restrict to fields here which are translationally in warrant, OK? Means that there's no special point.

There's no special means there's no special space time point in this theory, OK?

If you do experiment here in Boston, it's the same as you do experiment in Washington DC, et cetera, OK?

And if your theory is not translation in warrant, and then when you do experiment in Boston, and then will be different from you do it in, say, in New York.

And the second is that we assume it's Lorentz invariant.

So in some condensed matter applications, which you don't have Lorentz symmetry, and then you don't have Lorentz symmetry, but similar techniques we are going to talk about can be applied. right.

And we will also elaborate a little bit more on both of these aspects.

OK, so now let me give you some simple examples of the classical field series, which satisfy these two conditions, examples. So the first one is the Maxwell series.

So for Maxwell series, the dynamical variable So this four vector potential, a mu, so now I'm using the four vector rotation. So that will often suppress the index on x. That means this is a four vector.

And then from the a mu, you can define the field strings.

And e and b then can be obtained from this f mu mu.

And b can be obtained from f mu mu.

And then the action for the E and M can be obtained by sd. So this is just the action for the E and M without charged matter.

So if you do the variation here to get the equation motion, then you get the Maxwell equations without sources.

OK, you get the vacuum Maxwell equations. You get the vacuum Maxwell equations.

And so you see that this action half of the form we mentioned here is corresponding to a local theory, a single integral of space time.

And the second example is Einstein gravity.

It's a lot of classical field theory.

So let me just write down the action.

So this is written as G newton, so G is the metric. Yeah, anyway, if you don't know the Einstein series, OK. So R is a rich scalar.

And if you don't know the Einstein series, OK. And just write it down just to show that this is a local theory involving only a single spacetime integral of some quantities.

And then the simplest quantum field theory called the scalar field series, because I think I erased it, because this is involving the vector field. So this is a vector field. Because at each spacetime point, there's a vector.

Here, the Einstein gravity, the dynamical variable is a tensor, so it's even something more complicated.

So the simplest case is a scalar, which at each spacetime point, you just to have a single value quantity.

And so let's just consider the simplest case, just consider a real value scalar field phi.

So this is just take a real value. You can also consider complex value. But the simple is just a real value.

At each spacetime point, you have a single value. And so that's the dynamical variable.

And so in real life, there are many examples of scalar fields.

So, for example, the Higgs field, which is a celebrated Higgs field, which was discovered a number of years ago, is a real scalar field.

And also pions, okay, the pion particle can be considered as excitations of a scalar field.

Anyway, so, so a simple the simplest family of a scalar field action is the following form.

So here I just write this action down.

But essentially, you can argue this is the most general action you can write down, which are compatible with locality, with locality, and with translation symmetry, and the Lorentz symmetry. So here you see the Lorentz indices are contracted it.

In the Lorentz indices, the derivative are contracted, so that guarantees this theory is Lorentz invariant.

And v is just some function of phi. V here is just some function of phi.

So this is essentially the theory, yeah, the simplest theory you can write down based on localities.

So say we can take v phi, in general, we can take it to be polynomial. Yeah, there's some function of phi.

So when you have a translation symmetry, means that in this theory, if you do experiment, it's the same everywhere.

That implies that v phi, so here there's nothing, there's no parameter here, v phi can must, yeah just, yeah sorry, all parameters in v phi must be constant.

Okay, they cannot depend on spacetime, okay, must be constant.

So if you have some parameter dependent on the spacetime, then of course a different space and a different spacetime point will behave differently, and there will not be transitions in warrants, okay.

So let's try to work out this theory a little bit in more detail.

So let's find this Lagrangian, let's try to find its momentum.

So it's momentum density.

Here, there's only single fields, so there's no index. So pi x would be take derivative of this. So this is L.

So this thing inside the bracket is L. This partial L partial phi dot times derivative of phi.

And so if you look at here, here, at the time derivative, just phi dot square using this four vector notation.

And if you expand it, and then the momentum density, just phi dot x, because partial mu phi partial mu phi is equal to minus phi dot square plus.

And then you can find the Hamiltonian density, pi phi dot minus a l, and then you find that this is given by mean expressed in terms of momentum.

And equation motion, if you apply here, if you apply here, then you find given by partial to the partial square equal to partial v partial phi equal to 0.

And then again, here I'm using a short -hand notation, partial square. It's defined to be partial mu, partial mu.

So this is the same as minus partial g squared plus.

So here, let me also give you some simple examples of the v.

v in the simplest case is you take v just, of course, to be a constant. But the constant does not do anything because of the, yeah. So the simplest case is just v phi equal to linear term.

And from time translation symmetry, this f has to be a constant. It cannot be dependent on spacetime.

If it is dependent on spacetime, and then you will violate the space translation symmetry.

So when you have these, then your equation motion given by partial square equal to f. f is a constant.

So in this case, so essentially, you have some kind of thing, some kind of constant which source this phi. So in this case, we say this system have external force.

we call this f external force.

Because if f is non -zero, then phi cannot be zero. Phi has to be non -zero.

And so in the sense the phi will be always be excited, it will always be generated, non -trivial phi will always be generated if s is non -zero. OK. So here we call it external force.

And so normally, we don't normally would like to consider the five.

Normally, we will consider that normally in the situation of your interest in the situation, there's no external force. OK. The system developed on its own.

So in that case, so if we forget about external force, then the simplest situation would v phi equal to quadratic, some polynomial, phi square. So that's the last simplest function of phi.

And m square has to be a constant, again, from the translation symmetry.

So in this case, then you have equation motion. Partial square of phi equal to m square of phi square. Or m square of phi equal to 0.

So this is a very famous equation. equation.

So this is called the Klein -Gordon equation.

And the next time we will, yeah, you will see very quickly why this is a famous equation, okay?

And yeah, yeah, so this would correspond to the simplest field theory we can consider.

And then you can consider more complicated field series with v5 to be some higher polynomial or more complicated functions. equations.

But keep in mind that this square is special. Because when it's square, this is a linear equation.

But whenever you have something which is not linear or quadratic and you will get a nonlinear equation, and then the story becomes complicated.

Okay. Nonlinear equations are always complicated. And anyway, so this will be the simplest theory we will consider, okay? And, okay, so let's stop here.

In command, Lieutenant Colonel James Doolittle still has 10 hours before the planned launch, when a lookout spots a small Japanese boat.

The cruiser USS Nashville is immediately ordered to break formation and sink the enemy vessel.

Nashville opens fire, her guns echoing through the spray, but the tiny Japanese boat is difficult to hit.

As the Japanese boat is rocked, the crew frantically send out a message.

Shells straddle the tiny vessel, sending up huge splashes of water before striking home, and the little boat is completely destroyed.

The Americans know the Japanese message got out.

Doolittle must launch now and fly an extra 200 miles to Tokyo, stretching the B -25s to the edge of their range.

Doolittle revs his engines, building up the power.

A B -25 has never taken off from an aircraft carrier before, not even in training.

At the edge of the flight deck, the signalman crouches with his checkered flag, waiting until the carrier dips into a wave. Then he drops the flag.

The B -25 is now at max power, straining as Doolittle releases the brakes.

His bomber lurches forward and rumbles down the flight deck, blasting past the crowd of nervously watching sailors.

The signalman has timed the launch perfectly and the B -25 reaches the end of the flight deck just as the Hornets bow lifts on a wave.

The aircraft breaks through a cloud of spray way, and appears to drop dangerously low before recovering and lifting slowly into the sky with a full load of fuel.

Behind Doolittle, the rest of the bombers are soon rolling down the carrier, pulling away from the ship and following their leader.

Last to take off is First Lieutenant William Farrow in command of aircraft number 16, nicknamed That Out of Hell.

Farrow waits with anticipation, staring down the flight deck and concentrating trading hard as the Hornet's deck rises and falls in the gale.

Then a shout from his bombardier snaps him from his trance.

Corporal Jacob D. Deschizer has spotted a hole in the aircraft's plexiglass nose.

Farrow knows he should report the damage, but that will almost certainly lead to his mission being scrubbed, and there's no way he's going to be left behind.

He opens up the throttles and rolls down the deck.

his plane slips over the Hornets bow and disappears for a moment.

Then he pulls up and rushes forward to follow the rest of the bombers. The 16th aircraft rumble towards Japan.

There's no time or fuel to form up an unique formation, so they're strung out in a long line of several miles.

As they push on towards Tokyo, the weather worsens and the planes disappear into banks of low -hanging cloud.

Then, as they close in on the coast, Dolittle pushes down below the clouds and flies as low as possible to avoid radar and anti -aircraft guns. He braces himself for incoming fire. Surely the Japanese will be waiting for them. But nothing happens. The clouds are silent.

Japanese high commanders issued no instructions to fighters or air defense.

Now, after blasting over the Japanese beaches, the planes roar over suburban Tokyo. They're visible in the clear blue sky.

civilians assume their friendly planes and even wave at the Americans.

Suddenly, Doolittle hears an urgent shout from his co -pilot, Second Lieutenant Richard E. Cole.

Doolittle snaps his eyes towards the sky and spots nine Japanese A6M -0 fighters crossing high overhead.

All he can do is push forward on the control yoke and bring the bomber lower to try and hide amongst the clutter of the city below. below. The lead zero banks slightly.

Its pilot has noticed the lone twin -engined aircraft below.

For a long moment, he studies at Silhouette. The Japanese pilot reaches for his radio.

Only when the fighters disappear does Doolittle allow himself to breathe again and relax his grip on the controls.

The B -25 crosses the Sumida River, and Doolittle spots the Japanese Imperial Palace.

It's the final landmark. The target is just a few miles further on.

Bombardier, open the doors.

In the nose, Dolittle's Bombardier staff sergeant Fred Braemer hunches over the bomb site.

He reaches out and hits the switch that opens the B -25's cavernous Bombay doors.

Dolittle climbs, pulling his plane up to 1 ,200 feet.

The B -25s have been stripped of their Norden bomb sites to save weight and to prevent any of the advanced systems falling into Japanese hands.

Bremer works with an improvised hand -made bomb site called a Mark Twain, lacking any connection to the plane's autopilot.

It means that Doolittle has to manually hold the B -25 at a constant altitude and speed.

Bremer peers along the bomb site and sees the target, a large military arsenal.

Bremer hits the release switch and four 500 -pound incendiary clusters drop away.

As soon as the bombs have dropped, Dolittle wastes no time and dives down low once again. The lightened bomber begins gathering speed.

Behind him, each cluster bomb breaks over to release 128 tiny bomblets. The bomblets catch the wind and scatter across the target.

Fluttering down to the ground, they settle on the wooden structures of the arsenal buildings and quickly ignite dozens of blazes.

The flames spread quickly, merging into an enormous fire, which finally shakes the Japanese air defenses into action.

Sirens wail across the city, and the gunners scramble to their flak sights, cranking 25mm flak guns towards the sky.

They spot Doolittle's bomber before unleashing a torrent of flak, but caught by surprise, they struggle to track the fast -moving plane.

The shells punch into the air, exploding around Doolittle's escaping B -25, which rocks with each blast.

He keeps his plane low, rocking from side to side, and skillfully evading the Japanese gunfire.

The flak fades away as Tokyo disappears behind him, and Doolittle sets course for his final destination in China.

In the center of the Japanese capital, dozens of fires are burning, as the bombers start drawing in towards their target areas.

In aircraft number 10, Flight Lieutenant Richard O.

Joyce releases his bombs on a steel mill near Tokyo Harbour.

As the bombs fall, a small metal disc flutters behind each one on a length of wire.

In a symbolic gesture of the hate that now exists between the nations, the American crews have attached pre -war Japanese friendship and peace medals to their bombs.

The B -25 is hit by a massive blast wave as the bombs tear apart the steel mill and send an enormous cloud of black smoke rising into the sky.

In the pilot seat, Joyce battles with the controls as a storm of anti -aircraft fire engulfs his bomber.

He throws the aircraft left and right as flak shells explode all around him.

The enemy fire grows more and more accurate and a blast of shrapnel tears a hole in the Sergeant Edwin Horton with fragments of metal into his turret gun position.

Joyce calls out to his gunner over the intercom. Horton, you all right? There's silence.

Then Horton answers and gives you all clear. I'm all right back here. Okay, good.

Joyce pulls the plane down as low as he can once more, darting and weaving through the fire, which dies away as they part Tokyo area.

The crew scan the sky, looking for any fighters as they brace themselves for the long flight to China. To his horror, Joyce hears a call from his co -pilot.

There are nine Japanese fighters flying a few thousand feet above him.

Joyce stares up, hoping that the enemy pilots don't see him.

He watches as a pair of fighters roll over, start to die.

Two Imperial Japanese Army, key 27 Nates, roar down towards the B -25.

In the cockpits, the Japanese pilots strain their engines to get every scrap of power.

They're determined to wreak revenge for their burning capital city.

It's not just the damage and casualties that drive them forward, but the humiliation of being taken by surprise.

Joyce pushes forward on the stick, accelerating down towards the ground. Horton is shaken by the violent maneuver.

All around him, the ammo bouts for his twin 50 -caliber machine guns fly into the air, twisting and tangling and making the gun inoperable.

Within seconds, the enemy fighters have closed in on the bomber, but Horton's still desperately trying to untangle the ammunition belt.

Until he does that, the B -25 is defenseless.

One of the Japanese pilots lines up the bomber in his sights and squeezes the trigger, sending a stream of bullets erupting from the twin 7 .7mm machine guns.

The rounds fly past the left wing of the bomber, but without any defensive fire from the B -25, The Japanese pilot has time to adjust his aim and fire again.

This time, the bullets strike home, tearing into the B -25's left wing.

The Japanese pilot pushes on his struttle, roaring past the bomber and arcing round to come in for another attack run. In the cockpit, Joyce makes a quick damage assessment.

The B -25's left wing tip has been totally torn off, and chunks of metal are falling behind into the slipstream.

The uneven bomber starts buffeting and Joyce steps on the rudder pedal to correct for the extra drag.

He's able to keep the B -25 flying straight, but now a second Japanese mate is already closing in for his attack.

Hortons almost untangled the belts, but he needs more time.

The Japanese pilot lines up the B -25 in his sights.

He lets off a burst of rounds which pepper the bomber's fuselage.

It's another hit. But instead of joy, Japanese pilot can't contain his frustration.

His underpowered 7 .7mm machine guns are struggling to penetrate the B -25's fuselage, leaving little more than a few small holes.

Joyce jinxed his planes as he tries to avoid the next load of gunfire. His bombers sturdy, but it's not invincible. He leaves those guns back in action.

Seconds later, a shout comes from the rear of the plane as Horton finally fixes the 50 calibers.

He aims down his sights towards the Japanese fighter and squeezes trigger of the twin guns.

The machine guns chatter as the nape closes in.

Bullets tear through the Japanese fighters engine sending the aircraft ablaze and pushing it plunging towards the ground.

There's still one more Japanese fighter left and it's coming round for another attack.

The Japanese pilot grimaces as he lets off a burst of fire which passes just a few feet above the bomber. The Japanese fighter races past the bomber and breaks away.

Horton blazes away with his guns and manages to score a hit.

The stream of bullets tears into the nade which starts trailing thick black smoke behind it.

Unable to fight any longer, the Japanese pilot turns away and heads back to safety.

The bomber goes quiet with just the rumbling of the engines. Joyce tests the control surfaces. Despite the damage, they're still flying straight and level.

Further down the Japanese coast, the last of the American bombers is approaching its target. First Lieutenant William G.

Farrow heads towards an oil refinery near the city of Nagoya, far from most of the rest of the Dulith Raiders.

He dives down to 500 feet to open the bay doors and release the bombs.

He pulls up into the sky as the incendiaries scatter across the refinery and detonate in hundreds of explosions, which throw flames and burning oil across the streets of Nagoya.

Varro pulls his aircraft around and calls back to his navigator, 2nd Lieutenant George Barr, ordering him to plot a course to China.

Barr has hunched over the maps and fuel calculations.

He's run the numbers half a dozen times with every possible variation. It makes no difference.

Taking off 10 hours early, he's added hundreds of miles extra to their flight. They're run out of fuel, far from land.

Behind him, the men pour the last of the extra fuel reserves into the bomber's internal tanks.

They slice open the metal fuel cans so that they'll sink and not give away their position, and then throw them out of the plane into the water below. But it's nowhere near enough fuel.

Everyone knows it. But no one voices their fear.

Farrow looks down into the water below. The weather's worsening. Great clouds have gathered, and the sea is frothing. thing.

Ditching in the sea means drowning before anyone will find them. They have to make it to China.

Up ahead, Doolittle's crew have come to the same conclusion.

The Colonel reckons with his fate as the deadly waters of the Pacific Ocean surge below his plane.

Doolittle's navigator, Second Lieutenant Hank Potter, calls out.

The wind has swung round behind them and is pushing the B -25 along.

It's an extra 25 miles an hour, and it could just be enough to get them to safety in China.

Dolittle says a silent prayer in thanks. The Lord is with us. The dark clouds gather on the horizon.

As they approach the Chinese coast, they fly into a bank of severe rain and fog.

Visibility quickly falls to near zero as the aircraft shakes with the wind.

Dolittle fights with his controls, relying solely on instruments to keep the bomber are flying in the right direction.

He tries to contact the airfield at Chuchiao, but there's no answer.

The ground crews aren't expecting the B -25s for another 10 hours.

Without a radio signal to guide them to the runway, there's no way they can safely land in these conditions. There's only one option.

For a second, nobody moves. Then the crew jump into action.

They strap on their parachutes, grab hold of confidential papers, and bail out one one by one, disappearing into the darkness and the howling wind.

Finally, it's just do -little left.

He takes one last look at his bomber and then follows out the hatch.

The rush of air catapults him out into the grey turbulence.

The canopy inflates above his head and the howl of rushing air quietens, replaced by the sound of wind and rain.

He tries to steer himself down to the ground, but he can't even see through the dark and stormy weather.

He smashes down, it's a rough landing, and he feels an agonizing pain shooting through his ankle.

He's landed in a paddy field, he's alone in the dark, wet, and God knows where.

Pushed on by the same tailwind, Joyce's battle -damaged B -25 reaches the Chinese coast. As the plane shudders in the turbulence, he tries the radio.

There's still no signal from the airfield at Chuchao.

Like Doolittle, he has no choice and orders his men to bail out.

One by one, they land in the fields below and are rescued by the Chinese.

Pharaoh, in his B -25, comes in afterwards, but when he notices the lack of radio signal, he doesn't bail out.

Instead, he flies further west, hoping that the weather will improve and that he can save the precious aircraft by making a forced landing.

The B -25 pushes through the storm, soon it reaches clearer skies and Faro's hopes rise.

The clouds fall away and reveal a city twinkling down far below.

Faro calls over to his navigator, George Barr, trying to find out where they are.

Barr looks down as maps with dread. The city is Nanchang.

They've flown directly into Japanese -occupied China.

To bail out here means near certain capture, but the plane is totally out of fuel. Unless they want to crash, they have to jump.

They shove their Colt 45 pistols and spare ammunition into their overalls, along with cigarettes and chocolate bars.

Last of all, they grab their family photographs, then they bail out, leaping through the hatches into the enemy territory.

Within seconds, the wind pulls the men in different directions, scattering them across the countryside.

Faro plunges down to earth and lands in a field outside a small village.

For a moment, he lies still in the mud, winded and disorientated, listening to the rain and the distant rumble of his dying B -25.

He takes out his survival compass and heads towards Chinese controlled territory.

But he barely takes a few steps before voices cut through the darkness. Japanese soldiers are already searching the area. He reaches for his pistol, then stops himself.

of, firing now would only get him killed. A beam of light blinds him. The Japanese shout for him to surrender.

At a military post inside the city, Faro is shoved into a bare room and forced down onto a bench.

For the first time since jumping, he hears American voices.

His crew are already there.

One by one the Japanese have rounded them up from the fields and villages around Nanchang. For a brief second, Faro lets himself feel relief.

His boys are alive, but that also means they're all doomed to the same fate.

In total, eight men are now prisoners of the Japanese Empire.

After months of beatings, waterboarding, and psychological torture, the Japanese forced the men to sign false confessions, saying that they deliberately targeted civilians.

All eight men are sent to a show trial in Shanghai where they are found guilty of targeting civilians and are sentenced to death.

The intervention of Japanese Emperor Hirohito spares the lives of five of the men, but the sentences of three others are confirmed.

On October the 15th, 1942, William Farrow, his gunner Harold Spatz, and Dean Hallmark are led outside, tied toward in crosses, and executed by firing squad.

The remaining five prisoners are consigned to the Japanese prisoner of war camp for the rest of the war. First Lieutenant Robert J.

Meader dies of malnutrition and berry berry in December 1943.

The surviving four men endure years of brutal captivity before they're finally liberated at the end of the war in August 1945.

Almost all the rest of the 16 bomber crews were forced to bail out, crash land, or ditch in water upon reaching China. Three men lost their lives in the process.

One of the bombers diverted from his course and safely landed in the Soviet port city of Vladivostok.

Dolittle returned to the United States, half expecting to be court -martialed because of the losses.

But instead, he was awarded the Medal of Honor in May 1942 and promoted to Brigadier General.

Transcript of 关灯后光去哪了？光子的一生彻底讲清 （中文完整文稿）。

You flip off the light and the room goes dark instantly. It's something you've done thousands of times without a second thought. But think about it for a moment.

A second ago, that room was flooded with light.

Trillions upon trillions of tiny particles of light, called photons, were streaming out of the bulb, bouncing off the walls, the ceiling, your bed, your face. The room was packed with them. Then you flipped the switch, and they all seemed to vanish. So where did they go? Did they fall to the floor?

Did they get sucked back into the bulb? Are they still hiding in the corners of the room, or did they simply stop existing?

The real answer is surprising.

Some of those photons disappeared in less than a millionth of a second, and turned into something you can feel.

Some ended up inside your eyes, some slipped out through your window, and are still travelling right now, racing out into space, and might keep going for billions of years.

And the darkness that's left behind?

It isn't actually empty at all, let's follow the light and find out where it really goes.

Before we follow the light, let's remind ourselves what it actually is.

Light is made of tiny packets of energy called photons. Each photon is like a single drop of light. It has no mass. It carries a small amount of energy.

And as we saw in our video on where light gets its speed, it's always moving at the speed of light, about 300 ,000 km every second, from the moment it's created. Photons are a bit strange.

In some experiments, they behave like tiny particles, arriving one at a time, like raindrops.

In others, they behave like waves, spreading out and overlapping, like ripples on a pond. Physicists think of them as both, depending on how you look.

For this video, it's easiest to picture them as tiny glowing raindrops, flying in straight lines. So where do photons come from? Let's look inside a modern LED light bulb.

An LED contains a tiny chip made of special materials.

When electricity flows through it, electrons are pushed across a boundary inside the chip.

As each electron settles into a lower energy spot, it has some energy left over, and it gets rid of that extra energy by releasing a single photon. Old -fashioned bulbs work differently.

They pass electricity through a very thin wire, called a filament, until it gets so hot it glows.

The atoms in the hot wire are jiggling so violently that they constantly shake off photons. The same way a hot stove glows red. Either way, the key idea is the same.

Photons aren't stored inside the bulb bulb waiting to come out. They're made, fresh, from energy.

Electricity goes in, and photons are created on the spot, already racing away at the speed of light.

That's important, because it means a photon's life has a clear beginning.

And as we'll see, it also has a clear end.

But first, let's get a sense of just how many of them were filling your room.

An ordinary household LED bulb uses about 10 watts of electricity.

Only part of that energy comes out as visible light, but even so, the number of photons it produces is staggering.

Every single second, a bulb like that sends out roughly 10 billion billion photons. That's a one followed by 19 zeros, every second.

To put that in perspective, if every one of those photons were a grain of sand, a single second of light from your bulb would be enough fine sand to fill thousands of Olympic swimming pools. And the next second, thousands more.

So with that many photons photons pouring out, you might expect your room to be absolutely stuffed with them.

But here's the surprising thing, they don't stay around very long.

If you could freeze time and count the photons floating in your room at one single instant, you'd find far fewer than you might expect. Maybe around a trillion.

That still sounds huge, but it's millions of times smaller than the number your bulb makes every second. Why the difference? Because each photon lives an incredibly short life. life.

Your bulb is like a tap pouring water into a bucket with a huge hole in the bottom. Water rushes in, but it drains out just as quickly. The bucket never fills up very much. The light in your room is a constant flow, not a stored supply.

Photons are being made, and then destroyed, over and over, trillions of times a second.

The brightness you see is just the balance between the two, which means that when you switch off the light, you're turning off the tap.

And the question becomes, how quickly does the bucket drain? And where does the water go? But before we find out, a quick favor.

If you're enjoying this and you'd like more videos that explain the strangest parts of physics in simple language, consider subscribing and turning on notifications.

It genuinely helps the channel grow, and it means you won't miss the next one. Okay, let's flip the switch.

Let's slow time down enormously, and watch what happens the instant you switch off the light.

First, the electricity stops flowing into the bulb.

In an LED, that means the electrons stop crossing the boundary inside the chip, so no new photons are created.

The tap has been turned off.

The bulb fades out very quickly, in a tiny fraction of a second.

But the photons that were already released don't know the switch has been flipped. They're already flying through the room. Nothing can call them back.

They just keep going in a straight line, at the speed of light, until they hit something.

And in a room, they don't have to go far. A typical room is just a few meters across.

At the speed of light, a photon crosses a room that size in about 10 billionths of a second. That's 10 nanoseconds.

To give you a sense of how short that is, a single beat of a hummingbird's wings takes millions of times longer.

In the time it takes you to blink, light could cross your room tens of millions of times.

So almost instantly after you flip the switch, every photon in your room reaches a surface.

Her wall, the ceiling, the floor, your bed, your skin, your eyes.

and when a photon hits a surface one of two things can happen.

It can bounce off or it can be swallowed up.

If it bounces it carries on its journey heading off in a new direction until it hits something else.

If it's swallowed its journey ends right there.

So to find out where the photons go we need to follow both paths. Let's start with the bounces.

When a photon hits a surface whether it bounces or gets swallowed depends on what the surface is made of and on the color of the light. That's actually what color is.

A white wall looks white because it bounces back most of the light that hits it, all colors together.

A black sofa looks black because it swallows almost all of the light, bouncing back very little.

And a red cushion looks red because it bounces back mostly red light, while swallowing the blues and greens.

So every time you see a color, you're seeing the photons that survive the bounce, the ones that weren't swallowed.

A typical white wall bounces back around 80 % of the light that hits it. That sounds like a lot.

And it means that after you flip the switch, many photons don't just hit one wall and stop. They bounce. Then they bounce again, and again. But each bounce is a gamble.

On every bounce off a white wall, there's about a 1 in 5 chance the photon gets swallowed. Off darker surfaces, the odds are much worse.

So with every bounce, fewer and fewer photons survive. After 10 bounces, only about 1 in 10 is left. After 20 bounces, only about 1 in 100.

After 50 bounces, only about 1 in 100 ,000. And remember how fast this is happening.

Each trip across the room takes about 10 billionths of a second, so even 50 bounces take only about half a millionth of a second.

In other words, within about a millionth of a second after you switch off the light, almost every photon in your room has been swallowed. That's why the room goes dark instantly.

Your eyes, which take around a tenth of a second to react, could never catch the fading light.

As far as you're concerned, the darkness arrives at the very same moment you flip the switch.

So almost all the light ends up being swallowed. But what does that actually mean? Where does a photon go when it's swallowed?

When a photon is swallowed by a surface, physicists say it's been absorbed.

And here's the most surprising part of the whole story. When a photon is absorbed, it doesn't go anywhere. It stops existing. That sounds impossible. How can something just stop existing?

Let's zoom in and see. The wall, like everything, is made of atoms. And each atom has electrons around it.

When a photon arrives, it can hand all of its energy to one of those electrons or to the atoms themselves.

At that moment, the photon is gone. Not hidden, not stored, gone.

All that's left is its energy, now given to the material it hit.

This surprises a lot of people because we're used to things being conserved. If you push a ball into a box, it's still in the box. But photons don't work like balls. The number of photons in the universe isn't fixed.

They can be created out of energy, like in your light bulb, and they can be destroyed, turning back into energy.

What's always kept is the energy itself, not the photons. So where does that energy go? Mostly, it becomes heat.

Remember from our entropy video that heat is really just tiny particles jiggling around.

When a photon is absorbed, its energy usually makes the atoms in the wall jiggle a little bit faster.

And faster jiggling means a slightly higher temperature.

So every time you switch on a lamp, the light warms up your walls, your furniture, and you. Just a tiny bit. That's why a black car gets so hot on a sunny day. It absorbs almost all the sunlight that hits it, turning it into heat.

A white car bounces more of it away, so it stays cooler. So here's our first big answer.

When you turn off the light, almost all of the photons in your room end up absorbed by the walls and objects around you. They vanish, and leave behind a tiny bit of warmth.

earth, but a few of them end up somewhere much more interesting, inside you.

Some of the photons in your room never hit a wall, instead they fly straight into your eyes and those are the only photons you ever actually see.

That's worth thinking about.

You never see a wall directly, you only see the photons that bounced off it and happen to land in your eye.

Every image you've ever seen, every face, every sunset, every word on a page has been a pattern of photons arriving at the back of your eyes.

At the back of each eye is a thin layer of cells called the retina.

It contains millions of tiny light -sensitive cells called rods and cones.

Cones help you see color in bright light. Rods are extremely sensitive and help you see in dim light.

Inside these cells are special molecules that change shape when they're hit by a photon.

When a photon arrives, it's absorbed, just like at the wall.

But instead of only making atoms jiggle, its energy flips one of these molecules into a new shape.

That tiny change sets off a chain reaction inside the cell, sending an electrical signal along the nerves to your brain, so the photons that reach your eyes don't bounce back out, they're absorbed and destroyed, and their energy becomes a signal.

In a way, every photon you see gives up its life so that you can see it.

Rods are so sensitive that experiments have shown they can respond to a single photon.

In a dark enough room, after your eyes have adjusted, people can sometimes detect flashes made of just a handful of photons.

Now, you might have noticed that after you switch off a bright light, you sometimes still see a faint glowing shape for a moment. That's called an after -image, but it isn't made of leftover photons. The photons are long gone.

It's your eyes' cells, recovering from being so heavily stimulated, still sending signals to your brain for a short while.

So the darkness after you flip the switch really is dark. But is it really empty? Not quite. Here's something remarkable.

Even after you switch off the light, your room is still full of photons. You just can't see them.

Remember, when the photons from your lamp were absorbed, they made the atoms in your walls jiggle a little faster. And jiggling atoms do something interesting. They give off light of their own. Anything that has a temperature is constantly giving off photons.

Very hot things, like a stove ring or the filament in an old bulb, give off visible light. That's why they glow red, orange or white.

But cooler things, like your walls, your furniture and your body, give off a different kind of light called infrared.

Infrared light is just like visible light, except its waves are longer and its photons carry less energy. Our eyes can't detect it, but we can feel it as warmth.

When you hold your hands near a radiator or a campfire, much of the warmth you feel is infrared light hitting your skin.

So in a dark room, every surface is glowing with infrared photons, all the time.

room. The walls are glowing, the floor is glowing, and so are you.

In fact, your skin, at a little over 30 degrees Celsius, is constantly giving off hundreds of watts of infrared light.

You don't cool down to room temperature because the walls are glowing right back at you, giving you almost as much as you give off. This is exactly how night vision thermal cameras work.

They don't need any light at all. They see the infrared glow that every warm object gives off.

That's why firefighters can use them to find people in smoke -filled rooms, and why rescue teams can spot people at night from a helicopter. And here's the connection to our story.

The energy from your lamp's photons didn't just disappear into the walls, it turned into heat, and that heat is now being given off again as infrared photons.

In a sense, your lamp's light hasn't fully gone. It has been transformed into a different kind of light, one your eyes can't see.

The darkness in your room isn't empty, it's filled with an invisible glow.

There's one kind of material that seems to break the rule that light vanishes instantly.

You probably had some as a child, glow in the dark stars on the ceiling, or a watch with glowing hands, or a toy that shines softly after the lights go out.

So do these things somehow catch photons and store them, then let them out later? Not quite. But they do something almost as clever.

Glow in the dark materials contain special substances, often crystals made with a material called strontium aluminate.

When light hits them, the photons are absorbed, just like in a wall.

But instead of their energy turning straight into heat, it pushes electrons in the crystal up into higher energy positions. And here's the trick.

In these materials, the electrons get stuck.

They fall into little traps, a bit like a ball landing in a small dip halfway up a hill.

They can't easily fall back down.

Over time, the gentle jiggling of the atoms around them gives the trapped electrons a nudge, and one by one they escape from their traps.

As each one falls back down to its normal position, it releases its extra energy as a brand new photon. That's the glow.

So the photons you see glowing from the stars on your ceiling aren't the same photons from your lamp. Those were destroyed when they were absorbed.

These are new photons, made from energy that was stored in the material and released slowly, over minutes or even hours.

It's like the difference between catching rain in a bucket and using rain to fill up a water tower that slowly drains out through a tap. tap.

The water that comes out of the tap is new, but it was made possible by the rain that fell earlier. So even here, photons don't really linger.

Their energy does, and the material turns that energy back into light, slowly, long after the lamp has gone out. So far we've followed the photons that stayed in the room. But what about the ones that got away?

If your room has a window, some of the photons from your lamp never hit a wall at all. They fly straight through the glass and out into the night.

Glass is transparent because it lets most visible photons pass straight through without absorbing them. Once outside, the photons keep going in a straight line.

Many of them hit something nearby, a tree, a neighbor's wall, the road.

Some get scattered by tiny particles of dust and water in the air.

That's part of why the night sky above a city has a faint orange glow called light pollution.

It's the light from millions of lamps and street lights, bouncing off the air above us.

But some of those photons are aimed upward inward, with nothing in their way.

They pass through the atmosphere, getting thinner and thinner, and within a fraction of a millisecond, they're out in space.

In fact, this is how astronauts and satellites can photograph Earth at night.

When you look at those famous images of glowing cities spread across the continents, you're seeing photons that escaped from street lights, homes and windows and traveled up into space.

Some of the light from your lamp, on some night, might have been part of an image like that.

And once a photon makes it into space, something remarkable happens.

There's almost nothing out there for it to hit.

Space is incredibly empty, so the photon just keeps going. It passes the moon in just over a second.

It crosses the orbit of Mars in a few minutes. Within hours it's passed the outer planets.

And within about a day, it passes Voyager 1, the most distant spacecraft humans have ever sent.

In a single day, the light from your lamp can travel further than anything we've ever built has managed in nearly 50 years. And it doesn't stop there.

As we saw in our light -year video, the space between the stars is unimaginably empty.

A photon heading out of the solar system could travel for years, centuries or even millions of years without hitting a single thing.

After about four years, your photon would pass the distance of the nearest star.

After a hundred thousand years, if it were heading across our galaxy, it could cross the entire Milky Way.

And if it slips out of the galaxy into the vast gaps between galaxies, it could keep traveling for billions of years. Photons don't get tired, they don't wear out.

In empty space, a photon will keep going until something finally absorbs it.

That might be a cloud of gas, a planet, a star, or, very rarely, the eye or telescope of someone far away.

We know this happens because we see it every day.

When you look at a distant star, you're catching photons that left it years, centuries or thousands of years ago.

When astronomers look at the most distant galaxies, they're catching photons that have been traveling for over 13 billion years. And the oldest light of all fills the whole universe.

Shortly after the Big Bang, the universe was filled with a hot glowing gas.

When it cooled enough, about 380 ,000 years after the beginning, light could finally travel freely.

Those photons have been traveling ever since. Today, they're everywhere.

A faint glow from every direction in the sky, called the cosmic microwave background.

Some of those photons are passing through your room right right now.

But there's one thing that does change for a photon on a long journey.

The universe is expanding, and as space stretches, it stretches the light traveling through it too.

The photon's wave gets longer, and it loses energy.

Visible light slowly shifts toward red, then infrared, then microwaves. That's called redshift.

So if a photon from your lamp escaped into deep space and traveled for billions of years, it would slowly fade into a longer, weaker wave.

But it would still be out there, crossing the cosmos, carrying a tiny trace of the moment you switched on a light.

As a child, you might have wondered if you could trap light in a jar and keep it for later. Close the lid while the lamp is on, then open it in the dark.

It never works, because the inside of the jar absorbs the photons within a tiny fraction of a second, just like the walls of your room.

But scientists have found clever ways to hold on to light, at least for a while. One way is with mirrors. mirrors.

The best mirrors used in labs can reflect more than 99 .999 % of the light that hits them.

Put two of them facing each other, and a photon can bounce back and forth between them hundreds of thousands of times before it's finally absorbed.

That can keep light trapped for around a thousandth of a second, which is a very long time for a photon. Another way is even stranger.

In our video on stopping time, we met the physicist Lene Hao, whose team slowed light down to the speed of a bicycle inside an ultra -cold cloud of atoms.

In 2001, her team went further and brought a pulse of light to a complete stop, held it, and then released it again.

What really happens is that the light's information is copied into the atoms, almost like writing it down, and then later used to recreate the light.

Using a similar idea, a team in China reported in 2021 that they had stored light's information inside a special crystal for about an hour. These experiments aren't just curiosities.

They could help build future quantum computers and ultra -secure communication networks, where information carried by light needs to be stored and released on command.

So let's follow the energy from your lamp all the way to the end. Electricity flows into the bulb and becomes photons.

Within about a millionth of a second of switching off, almost all of them are absorbed by your walls, your furniture, and you. Their energy becomes heat.

That heat is given off again as invisible infrared photons.

And those photons are absorbed and given off again, passing the energy around, until eventually some of it leaks out of your house, warming the air outside.

And what happens to the heat of the whole earth? Our planet is constantly giving off infrared light into space.

In our entropy video, we saw that the earth receives energy from the sun as bright, concentrated sunlight, and sends it back out as many more low -energy infrared photons, spreading in every direction.

So, in the end, the energy from your lamp joins that same flow.

A small part of it eventually leaves the Earth as a faint infrared glow, heading out into the universe. Here's a thought to leave you with.

Every time you switch off a light, you're watching a tiny version of the universe's own story.

Light is created, it travels, it bounces, it's absorbed, and its energy spreads out.

Most of it becomes heat, warming the world around you.

A few photons make it into your eyes and let you see.

and a lucky few slip out into space where they may keep traveling for billions of years.

The light doesn't really vanish, it just changes form and carries on.

If you enjoyed this, check out our video on where light gets its speed from.

And next time you switch off the lights, think about the photons racing out of your window into the night. Thanks for watching, see you next time.

Transcript of 电子为什么永远转不停？能量从哪来 （中文完整文稿）。

Somewhere inside your body, right now, there are electrons that have existed for billions of years.

They were around before the earth formed, before the sun ignited, and in all that time they have never stopped, never slowed down, and never needed to be recharged. No engine works like that.

A spinning top eventually wobbles and falls, a swinging pendulum gradually comes to rest.

Even the planets lose tiny amounts of orbital energy over immense spans of time.

time. So how can an electron keep moving, apparently forever, without ever running out of energy?

It sounds like a simple question, the kind a curious child might ask.

But in the early 20th century, this exact question pushed physics into a crisis, according to the most successful scientific theories of the time, every atom in the universe should have collapsed almost instantly.

The fact that you exist at all, was by the standards of 1910 physics, impossible.

possible. To understand why, we need to examine two assumptions hidden inside the question.

The first is that motion requires a constant supply of energy. The second is that an electron moves the way a small object moves.

Both assumptions feel natural, and both, in very different ways, turn out to be wrong. Let's begin with the first one.

For nearly 2 ,000 years, people followed an idea associated with Aristotle.

An object keeps moving, only as long as something keeps pushing it. Stop pushing a cart, and it stops. stops. This matches everyday experience so well that it seems obvious.

Yet in the 17th century, Galileo Galilei and later Isaac Newton showed that it is fundamentally mistaken.

Newton's first law of motion states that an object in motion continues moving at the same speed, in the same direction, and less a force acts on it. Motion itself costs nothing. So why does a rolling ball eventually stop?

Because it is constantly losing energy to its surroundings, friction with the ground, collisions with air molecules, and tiny deformations of the ball, all convert its energy of motion into heat, the energy doesn't vanish.

One of the most thoroughly tested principles in all of science, the conservation of energy, tells us that energy is never created or destroyed, it only changes form, a moving object slows down, only when its energy has somewhere to go.

This changes our question in an important way.

The real puzzle is not what keeps the electron moving, nothing needs to, the real puzzle is why the electron never loses its energy.

To answer that, we need to know what an electron is and where it lives.

Thompson studied cathode rays, beams that glow inside sealed glass tubes when a high voltage is applied.

By deflecting these rays with electric and magnetic fields, he showed that they consisted of tiny particles carrying negative charge, far lighter than any atom.

He had discovered the electron.

Since atoms are electrically neutral, Thompson and proposed that electrons were embedded in a diffuse ball of positive charge, like raisins scattered through a pudding. That picture didn't last long.

Beginning in 1909, Hans Geiger and Ernest Marsden, working in Ernest Rutherford's laboratory, fired alpha particles at an extremely thin sheet of gold foil, nearly all of them passed straight through, but a very small fraction bounced off at large angles, and a few even came almost straight back.

Rutherford later compared the surprise to firing an artillery shell at a sheet of tissue paper and having it rebound. In 1911, he explained the result.

Nearly all of an atom's mass and positive charge is concentrated in a tiny dense nucleus, and the rest of the atom is overwhelmingly empty space.

The natural next step was to imagine the atom as a miniature solar system, with electrons orbiting the nucleus the way planets orbit the sun, held in place by electrical attraction instead of gravity. It was an elegant picture. It was also a disaster.

The problem came from James Clerk Maxwell's theory of electromagnetism, which had already explained light itself as a travelling wave of electric and magnetic fields.

One of its central predictions is that an accelerating electric charge emits electromagnetic radiation.

This is not an abstract idea, it's exactly how a radio antenna works, by pushing electrons back and forth, so that they broadcast waves. And here's the crucial detail.

an object moving in a circle is always accelerating because its direction is constantly changing even if its speed is not.

So an orbiting electron should continuously radiate light.

Radiation carries energy away and here at last is the escape route for the electrons energy that friction provides for the rolling ball.

Losing energy, the electron should spiral inward toward the nucleus.

When physicists calculated how long this would take for a hydrogen atom, the answer was roughly a hundred billionth of a second. In that brief instant every atom should have collapsed.

There was a second clue that something was deeply wrong.

A spiraling electron would emit a continuous smear of colors as its orbit shrank, but real atoms don't do that.

When hydrogen gas is heated, it glows only at a few sharply defined colors.

In 1885, a Swiss school teacher named Johann Ballmer had found a simple mathematical formula that predicted the wavelengths of these visible lines without anyone knowing why the formula worked.

In 1913, the Danish physicist Niels Bohr made a radical suggestion.

He proposed that electrons can occupy only certain specific orbits, and that while in those orbits, they simply do not radiate.

Light is emitted only when an electron jumps from one allowed orbit to another, and the energy difference between them determines the colour.

Bohr's model reproduced Balmer's formula with remarkable accuracy.

But Bohr had not explained why those orbits were special, or why an accelerating charge would suddenly stop radiating, he had essentially declared it.

Finding the real reason would require physicists to abandon one of their most basic assumptions, the idea that an electron is a tiny particle traveling along a path at all.

The first crack in that assumption came from a French physicist named Louis de Broglie. In 1924, he proposed something that sounded almost absurd.

If light, long understood as a wave, could also behave like a stream of particles, then perhaps particles like electrons could also behave like waves. He even predicted the wavelength.

It equals Planck's constant, an extremely small number that sets the scale of quantum effects divided by the particle's momentum.

The faster and heavier an object, the shorter its wavelength.

For a baseball, the wavelength is so absurdly tiny that no wave behavior could ever be noticed.

For an electron inside an atom, the wavelength is about the size of the atom itself. Three years later, the idea was confirmed.

Clinton -Davidson and Lester Germer in the United States and independently George Padgett -Thompson in Britain directed beams of electrons at crystals and observed diffraction patterns, the alternating bright and dark regions that appear only when waves overlap and interfere.

In a pleasing historical symmetry, JJ Thompson had shown that the electron is a particle and his son showed that it is also a wave.

This immediately suggested why only certain orbits are allowed. Think of a guitar string. Because it is fixed at both ends, it can vibrate only in patterns that fit its length exactly.

One hump, two humps, three, but never two and a half.

A confined wave naturally produces a set of allowed frequencies.

In the same way, an electron wave, confined near a nucleus, can exist only in patterns that fit. Quantization was no longer a rule imposed from outside.

It emerged from wave behavior itself.

In 1926, the Austrian physicist Erwin Schrödinger wrote down the equation that describes how these matter waves behave.

In plain language, the Schrödinger equation balances an electron's energy of motion against its energy of position in the electric field of the nucleus and asks which wave patterns can exist stably.

For hydrogen, its solutions are not circular paths. They are three -dimensional clouds called orbitals.

The lowest one is perfectly spherical, densest near the nucleus, and fading smoothly outward. There is no track, no circling, no planet -like orbit.

Max Born soon supplied the interpretation.

The intensity of the wave at any point tells us the probability of finding the electron there if we look. Now we reach the heart of the matter.

Each of these allowed patterns has a precise energy, and such a state is called a stationary state.

The wave itself oscillates in a mathematical sense, but the probability cloud, and therefore the distribution of the electron's charge, does not change with time at all.

It doesn't rotate, pulse, or slosh back and forth. And remember the reason a classical electron was doomed.

Radiation is produced by charge that accelerates, by something that changes.

A charge distribution that is perfectly constant in time, has nothing to broadcast. The radio antenna has no oscillating current.

The mechanism that should have drained every atom in a fraction of a second simply has nothing to act on. But this raises a natural question.

If the cloud is perfectly still, is the electron simply sitting at rest?

Here quantum mechanics gives an answer that is just as surprising in the opposite direction.

In 1927, Werner Heisenberg showed that the uncertainty in a particle's position, multiplied by the uncertainty in its momentum, can never fall below a fixed value set by Planck's constant.

movement, squeeze an electron into a smaller region and its momentum necessarily becomes more spread out.

An electron confined to a space about a tenth of a nanometer across cannot have zero momentum, it cannot be still.

This is also why atoms have the size they do, the nucleus pulls the electron inward, lowering its electrical energy, but confining it more tightly raises its kinetic energy.

The ground state is the compromise where the total is lowest, and the resulting radius, called the Bohr radius, is about five hundredths a nanometer. The numbers are striking.

In hydrogen's ground state, the electron's average kinetic energy is about 13 .6 electron volts, while its electrical potential energy is about minus 27 .2.

The total is minus 13 .6, which is exactly the energy needed to pull the electron free.

A typical speed works out to roughly 2200 km per second, about 137th of the speed of light. And here is why this motion never runs down.

The ground state is, by definition, the lowest energy the electron can possibly have.

To lose energy, it would have to drop into a lower state, and there isn't one.

Imagine a ball resting at the bottom of a valley, with one crucial difference.

In the quantum world, even the bottom contains motion. Physicists call this zero -point energy.

The electron isn't burning fuel to keep moving.

It's already at the floor. This also corrects a few common misunderstandings.

The electron's spin measured indirectly in the Stern -Gerlach experiment of 1922, is often pictured as a tiny ball rotating on its axis, it isn't.

Treating it that way would require the electron's surface to move faster than light, spin is an intrinsic form of angular momentum with no true classical equivalent, and electrons do sometimes lose energy.

An electron in a higher, excited state can drop to a lower one by emitting a photon, typically within a few billionths of a second.

Why would a supposedly stationary state decay at all.

The answer comes from quantum electrodynamics, which shows that even empty space contains fluctuations of the electromagnetic field that can nudge an excited electron downward.

These vacuum effects are not just theoretical.

In 1947, Willis Lamb measured a tiny shift in hydrogen's energy levels that only these effects could explain.

Yet in the ground state, there is nowhere lower to go, and so the electron remains, indefinitely. which leaves us with an uncomfortable question.

If electrons move forever without any energy input, haven't we just described perpetual motion? The very thing physics insists is impossible.

The answer lies in what perpetual motion actually means.

The machines that physics rules out are devices that would deliver useful work forever, turning wheels or lighting bulbs without any energy input, that is forbidden by the conservation of energy and by the laws of thermodynamics. An An electron in its ground state does nothing of the sort.

To extract energy from it, the electron would have to fall to a lower state, and as we've seen, no lower state exists.

Its motion is real, but it is locked away, permanently inaccessible.

Zero -point energy itself is no mathematical fiction, it has measurable consequences.

Helium, for example, remains liquid even when cooled toward absolute zero at ordinary pressure, because the zero -point motion of its atoms is too vigorous for them to settle into a solid. But this energy cannot be harvested.

Devices claiming to draw free power from zero -point energy misunderstand the very thing that makes it permanent.

It is the minimum, not a reservoir, explaining why a single atom is stable still leaves a larger puzzle.

Ordinary matter contains enormous numbers of atoms pressing against one another.

Why don't they merge, with electrons crowding ever closer to neighboring nuclei, until everything collapses into a dense mass? The uncertainty principle alone is not enough to prevent it.

The missing ingredient was identified in 1925 by Wolfgang Polly, his exclusion principle states that no two electrons can occupy the same quantum state.

Once the lowest states are filled, additional electrons are forced into higher ones.

This is why atoms have shells, why elements have distinct chemical personalities, and why the periodic table looks the way it does.

It took decades to prove that these principles rules truly guarantee the stability of bulk matter.

In 1967, Freeman Dyson and Andrew Leonard produced a rigorous mathematical demonstration and in 1975, Elliot Leib and Walter Theuring found a far simpler and more elegant proof.

Their conclusion is profound, without the exclusion principle, matter would not merely be different, it could not hold its shape at all.

The solidity of a table is, at its root, a quantum effect. The same principle operates on an astronomical scale.

When a star like the Sun exhausts its nuclear fuel, its core contracts into a white dwarf, an object roughly the size of the Earth but containing a large fraction of a solar mass.

What holds it up against its own gravity is not heat, but electrons refusing to share quantum states. This electron degeneracy pressure has a limit.

In the early 1930s, Subramanian Chandra Sekar showed that above about 1 .4 solar masses, it can no longer resist gravity's pull.

closer to home, a copper wire, offers its own surprise.

Even near absolute zero, its conduction electrons move at around a million meters per second, simply because the exclusion principle forces them into high energy states.

Yet when current flows through the wire, the overall drift of electrons is astonishingly slow, typically a fraction of a millimeter per second. And here, energy really is lost.

Electrons scatter off vibrating atoms and impurities, heating the wire, that is electrical resistance, unless the material becomes a superconductor.

In 1911, Haike Kamilling Ons cooled Mercury to about 4 degrees above absolute zero and found that its electrical resistance vanished.

The explanation, developed by John Bardeen, Leon Cooper and Robert Schrieffer in 1957, is that electrons form linked pairs that move collectively in a single coherent quantum state, one that ordinary scattering cannot disrupt.

Currents set flowing in superconducting rings have been observed to persist for years without measurable decay.

It's the clearest everyday demonstration that motion, free of any way to shed energy, simply continues.

Superconducting magnets now power MRI scanners and steer particle beams at the Large Hadron Collider. Will electrons themselves last forever?

The electron is the lightest particle that carries electric charge, and since charge is conserved, there is nothing lighter for it to decay into. Experiments have tested this directly.

In 2015, the Borexino detector in Italy found no sign of electron decay, and established that the electron's lifetime must exceed about 66 ,000 trillion trillion years, roughly 5 million trillion times the current age of the universe.

That is an experimental limit, not a proof of eternity, but no evidence of decay has ever been found. What remains genuinely uncertain is Supler.

Quantum mechanics predicts the behavior of electrons with extraordinary precision, yet physicists still disagree about what an electron is doing when no one measures it.

In the standard interpretation, asking for its exact path is meaningless.

In Bohmian mechanics, the electron always has a definite position, and in hydrogen's ground state, it is actually motionless, held in place by the structure of its wave.

Other interpretations offer different pictures again.

All of them agree on every measurable prediction, which is precisely why experiments have not settled the question, whether the electron moves in the everyday sense is, at this level, still open. What is not open is the answer to our central question.

Electrons don't run out of energy because they aren't spending any. Motion requires no fuel, only a way to lose it.

In a stationary state, an electron's charge distribution does not change, so it cannot radiate.

Confinement guarantees it can never be still, and the ground state guarantees it can never fall further.

The exclusion principle then turns this stability into the structure of atoms, chemistry, solid matter, and even dead stars.

The restless electron, it turns out, is not resisting the universe's tendency towards stillness. It is already at rest in the only sense nature allows.

The lowest energy the universe permits is not silence, but a motion that can never be taken away. Bye.

Transcript of 飞了49年一物未撞，旅行者1号是怎么活下来的？ （中文完整文稿）。

Voyager 1 has been flying for 49 years at 38 ,000 miles an hour, straight through the asteroid belt, past Jupiter, past Saturn, and out past the Sun's own outer boundary.

And in all that time, nothing has ever hit it hard enough to matter.

Not a rock, not a pebble. That is not the strange part.

The strange part is what Voyager heard when the tiny things did hit it, and the moment, far out in the dark, when the hits stopped showing up.

Nobody had a clear explanation for that silence.

And on November 18th, 2026, Voyager 1 crosses a line no human -made object has ever crossed, carrying something built to outlast the planet that made it.

Hollywood loves the asteroid belt, a blizzard of tumbling rock, boulders spinning past, a ship weaving through at the last second.

Voyager 1 crossed the real one on its way to Jupiter without dodging a thing.

There are millions of objects out there, but on average they sit about 600 ,000 miles apart.

That is more than twice the distance from Earth to the Moon, between one rock and the next.

Stand on almost any asteroid, and you would probably not see a single other one in your sky.

Shrink the whole scene down until Voyager is the size of a mosquito.

At that scale, the typical gap between two asteroids would be around 800 miles.

That's farther than New York to Chicago, with one rock at each end and a mosquito somewhere in the middle.

Now ask yourself how often that mosquito would bump into anything.

That's the asteroid belt, the scary part of the solar system, and it's mostly a desert. So that's the answer, right?

Space is empty and Voyager got lucky. Except it didn't just get lucky.

Voyager has been hit, again and again, and we know exactly what it sounded like.

Since 1990, Voyager 1 has been blind.

On Valentine's Day that year, 3 .7 billion miles out, the team swung its cameras back toward us one final time and caught Earth as a spec smaller than a single pixel. Then they switched the cameras off for good.

If a rock ever came at it, Voyager would never see it coming. But it can feel things.

Two thin antennas, each about 33 feet long, stick out from its body.

They belong to the plasma wave instrument, built to listen to electric ripples in the thin gas of space. It was never meant to detect dust.

Then, at Saturn, the team behind that instrument, led by physicist Don Gernett at the University of Iowa, noticed something odd in the data. Sharp, sudden cracks.

Each one was a grain of dust, far thinner than a human hair, slamming into the spacecraft so fast that it vaporized into a tiny puff of charged gas.

That puff washed over the antennas and the instrument heard it as a click.

At Neptune, Voyager 2 took up to 280 of those hits every second.

A hailstorm you could only hear.

And Voyager 1 kept hearing them long after the planets were behind it.

In 1997, Gernet's team went back through the recordings from deep space, checking every single spike by eye.

Out to about 50 times the distance from Earth to the Sun, they found a small but steady trickle. 27 hits in about 11 hours of recordings.

Then, somewhere past that point, the clicks stopped.

The recordings were just as good. At the same rate, they should have caught around 27 more.

They caught zero. Voyager 2 went quiet too, even earlier. The team admitted they had no specific explanation.

Their best guess was dust shed by comets, scattered unevenly through the dark.

Whatever the cause, it means even dust out there is patchy.

Voyager wasn't flying through an even haze.

It was crossing long stretches where nothing its instruments could detect touched it at all.

What On August 25, 2012, about 11 billion miles from home, Voyager 1 crossed the heliopause, the edge of the bubble the sun's wind blows around all the planets.

Its twin crossed six years later and took the temperature of the gas on the other side, somewhere between 54 ,000 and 90 ,000 degrees Fahrenheit, hotter than anyone had predicted, and Voyager flew straight through it without a scratch.

Because temperature only measures how quickly each particle is zipping around.

What actually burns you is the number that strike your skin and out there there are almost none.

In 2013 an eruption on the Sun set that gas ringing like a bell and the pitch of the ringing gave away how thick it was. About one electron per cubic inch.

The air in your lungs right now packs roughly 400 billion billion molecules into that same space.

Close your fist around a cubic inch of interstellar space and you might be holding a single electron.

On November 18th, 2026, Voyager 1 reaches 16 .1 billion miles from Earth.

That is so far that even a beam of light racing at 186 ,000 miles every second takes an entire day to get there.

No human -made object has ever been one light day away.

Voyager's project manager, Susie Dodd, explained it this way.

If she says good morning to Voyager 1 on a Monday, the answer arrives on Wednesday morning.

If something breaks out there, the team won't even know until the next day, and their fix lands a day after that. Every rescue now takes at least two days.

And yet, at the speed it is moving, about a million miles every day, it would need nearly 18 ,000 more years to cover a single light year.

Half a century of flying bought it a single light day.

A light year is longer than all of recorded history three times over.

Three centuries from now, it arrives at the Oort Cloud, a giant sphere of frozen objects wrapped around the whole solar system. Possibly billions, maybe trillions of them.

Getting through it takes something like 30 ,000 years, and the icy bodies out there are scattered so thinly that the odds say Voyager won't brush a single one.

In about 44 ,000 years, Voyager 1 makes its next pass by another star, a dim red dwarf called Gleese 445.

And Voyager isn't even doing most of the work.

That star is racing toward it faster than Voyager is moving toward the star. Its closest approach?

About two light years.

Even its closest brush with another star leaves a gap so wide that light needs two full years to cross it. But Voyager won't be awake for any of that.

Its electricity comes from the heat of plutonium slowly falling apart, and every year it loses about four watts.

To keep it warm enough to survive, engineers have been turning its systems off one by one.

In April, 2026, they shut down another instrument, leaving just two of its original 10 still running. One measures magnetic fields.

The other is the ear, the plasma wave instrument that once heard the dust.

This summer, they pulled off a risky power shuffle on its twin that the teen calls the Big Bang and lined up the same move for Voyager 1, but the plutonium doesn't care how clever they are.

Sometime in the 2030s, the transmitter goes quiet.

The huge antennas back on Earth will still point at that patch of sky, waiting for a signal that never comes.

And none of that slows it down.

No air, no friction, it keeps coasting whether anyone is listening or not.

So here is the real reason it has never hit anything big.

The rocks sit farther apart than our minds can picture, and beyond the planets, even the dust turned out to be thin and patchy, sometimes too thin to register at all.

Bolted to Voyager's side is the golden record.

90 minutes of music, greetings in 55 languages and the sounds of Earth.

Copper, plated in gold, sealed in aluminum, built to last about a billion years.

A billion years is roughly how long scientists think Earth has before a brightening sun boils its oceans away.

And astronomers who modeled Voyagers' path through the galaxy found it could drift for trillions of years without ever coming close to a star. So follow the timeline forward.

In about a billion years, Earth's oceans are gone.

A few billion years after that, the Sun swells into a red giant, and a 12 -inch disc of gold -plated copper is still out there, falling between the stars, still carrying the music and the voices of a planet that no longer exists.

It may become the last thing anywhere that proves we were ever here, and right now, Tonight it's still drifting deeper into the dark at 38 ,000 miles an hour

Transcript of 你永远追不上一束光：光速的真相彻底讲透 （中文完整文稿）。

Close your eyes for a moment, now open them.

In that tiny instant, light that left the walls of your room, the screen in front of you, or the sky outside your window, raced into your eyes and told your brain what the world looks like. It felt instant. It felt like nothing at all.

But it was not instant. Light took time to reach you. A very, very small amount of time, but time nonetheless.

And hidden inside that tiny delay is one of the deepest secrets of the universe. light is the fastest thing that exists.

It moves at about 300 ,000 kilometers every single second.

To be exact, it travels 299 ,792 ,458 meters per second in empty space.

That number is so large that our minds struggle to hold it. So let us try to make it feel real.

If you could fire a beam of light around the Earth's equator and somehow bend it to follow the curve of the planet, it would circle the entire globe about seven and a half times in one second.

Before you could finish saying the word one, that light would have gone around the world again and again.

A passenger jet would need nearly two full days of non -stop flying to go around just once.

Now look up at the moon. It sits about 384 ,000 kilometers away.

Light covers that distance in roughly 1 .3 seconds.

That means when you look at the moon, you are not seeing it as it is right now.

Now, you are seeing it as it was a little over a second ago. The moon you see is a tiny bit of the past.

The sun is much farther, about 150 million kilometers away.

Sunlight needs around 8 minutes and 20 seconds to reach us.

So if the sun suddenly vanished, we would not know it right away.

For more than 8 minutes, the sky would still glow. The daylight would still warm your skin.

Only after those 8 minutes would the darkness arrive.

Every time you see the sun, you are looking at light that is already more than 8 minutes old. This is the first strange lesson of light.

Seeing is never immediate. Everything you look at, you see slightly in the past.

For things in your room, the delay is a few billionths of a second, far too small to notice. But the farther you look, the deeper into the past you see.

Keep that idea in mind, because we will come back to it, and it will take us to the very beginning of time.

For most of human history, people believed light did not travel at all. They thought it simply appeared everywhere at once. When you lit a candle, the room was filled with light instantly.

Great thinkers argued about it for centuries.

Some, like the ancient Greek philosopher Empedocles, suspected that light must move.

Others, including many later scholars, believed it was infinite and instantaneous.

Nobody could prove it either way, because light was simply too fast to catch.

In the early 1600s, the Italian scientist Galileo Galilei decided to try. His idea was simple and clever.

He and an assistant stood on two hills, far apart, each holding a covered lantern. Galileo would uncover his lantern.

The moment his assistant saw the light, he would uncover his own.

Galileo would then measure how long it took for the light to come back. If light had a speed, there should be a delay.

But there was no delay that anyone could measure. The light came back as fast as human reflexes allowed.

Galileo concluded that if light did have a speed, it was extremely fast, far too fast to measure with lanterns and human eyes. He was right.

Light would have crossed the distance between those hills in a few millionths of a second.

His experiment was not a failure, though. It was the first honest attempt to ask the question.

And asking the right question is where science always begins.

The first real answer came not from Earth, but from the sky.

In the 1670s, a Danish astronomer named Ole Romer was studying Io, one of the moons of Jupiter.

Io circles Jupiter like clockwork, disappearing behind the giant planet at regular intervals.

Astronomers used these disappearances almost like a cosmic clock. But Romer noticed something odd.

The clock was not keeping perfect time.

When Earth was moving closer to Jupiter in its orbit, Io's eclipses happened a little earlier than predicted.

When Earth was moving away from Jupiter, they happened a little later. later. The difference added up to several minutes. Rumor realized what was going on.

The moon was not changing its schedule. The light was simply taking longer to reach us when we were farther away. The extra minutes were the time light needed to cross the extra distance.

In 1676, for the first time in history, someone had shown that light has a finite speed.

The estimate made from his work by the Dutch scientist Christian Hoegens was around 220 ,000 km per second. That was not perfectly accurate, but it was astonishing.

Humanity now knew that light was fast, but not infinitely fast.

It would take almost two centuries to measure light here on Earth.

In 1849, a French physicist named Hippolyte Fizot built a beautiful machine.

He shone a beam of light through the gaps of a rapidly spinning toothed wheel, like a gear.

The light traveled about eight kilometers to a mirror, bounced back, and returned to the wheel. Here was the trick. back.

If the wheel was spinning slowly, the returning light passed through the same gap it went out through.

But if the wheel spun fast enough, by the time the light came back, a tooth had moved into its path and blocked it.

By knowing how fast the wheel was spinning and how far the light traveled, Fezot could calculate its speed. He got a result of about 313 ,000 km per second. Remarkably close.

Soon after, his countryman, Leon Foucault, improved the method using a rapidly rotating mirror instead of a wheel.

By 1862, Foucault had measured the speed of light at about 298 ,000 kilometers per second within one percent of the modern value.

Later, the American physicist Albert Michelson spent much of his life refining these measurements with ever more precise mirrors and longer paths.

Step by step, the number came into focus. But knowing how fast light moves is only half the story.

The bigger question was this, what is light and why does it move at that particular speed.

Why not twice as fast, or half as fast?

The answer came from a place no one expected.

In the 1860s, a Scottish physicist named James Clerk Maxwell was studying electricity and magnetism.

At the time, these seemed like two separate forces.

Electricity made sparks and lightning. Magnetism made compass needles point north.

But experiments had shown they were connected.

A moving magnet could create an electric current, and an electric current could create a magnetic field.

Maxwell gathered everything that was known about these two forces and wrote it down in a small set of elegant equations.

When he studied those equations, he found something amazing hidden inside them.

They predicted that a changing electric field would create a changing magnetic field which would create a changing electric field again and so on.

The two would keep feeding each other, forming a wave that could travel through empty space all on its own. Then Maxwell calculated how fast this wave should move.

The speed depended only on two basic numbers that describe how electricity and magnetism behave in empty space.

Numbers that had been measured in laboratories using wires, coils, and electric charges with no light involved at all.

When Maxwell did the math, the answer came out to roughly 300 ,000 km per second. The speed of light. This was one of the greatest moments in the history of science.

Maxwell realized that light itself is an electromagnetic wave, a ripple of electricity and magnetism, racing through space.

The speed of light was not a random number.

It was written into the basic rules of electricity and magnetism themselves, and visible light was only a small part of the picture.

The same kind of wave can have many different sizes, or wavelengths.

Very long waves are radio waves, the kind that carry music to your car.

Slightly shorter ones are microwaves, which heat your food. Then come infrared waves, which we feel as heat.

Then the narrow band of visible light, from red to violet, the only part our eyes can see.

Shorter still are ultraviolet rays, which cause sunburn, then x -rays, which pass through our bodies, and finally gamma rays, the most energetic of all.

Every one of them is light in the broad sense, and every one of them travels through empty space at exactly the same speed.

Later, in the early 20th century, scientists discovered that light also behaves like a stream of tiny packets of energy called photons.

Sometimes light acts like a wave, and sometimes like a particle.

This double nature is one of the great mysteries of quantum physics.

But whether we think of it as a wave or a particle, one thing never changes. In a vacuum, it always moves at the same speed. And that is exactly where the trouble began. Think about everyday speeds.

If you are on a train moving at 100 km per hour and you throw a ball forward at 20 km per hour, someone standing beside the tracks would see the ball moving at 120 km per hour. Speeds add up. That is common sense. It is how the world works for cars, balls, and birds. So scientists naturally asked a question.

If light moves at 300 ,000 km per second, then it must be moving at that speed compared to something. But compared to what? But waves usually need something to travel through. Ocean waves move through water.

Sound waves move through air.

So physicists imagined that space must be filled with an invisible substance that carried light waves. They called it the luminiferous ether.

If the ether existed, then Earth must be rushing through it as it orbits the sun at about 30 kilometers per second. That would create a kind of ether wind.

And light moving with the wind should travel slightly faster than light moving against it. just like a swimmer moves faster going downstream than upstream.

In 1887, Albert Michelson and his colleague Edward Morley set out to detect this difference.

They built an extremely sensitive instrument that split a beam of light into two paths at right angles, sent them back and forth, and then brought them together again.

If one beam was even slightly slower than the other, the waves would fall out of step, and the change would be visible as a shift in a pattern of light and dark stripes. ships. They rotated the instrument.

They repeated the experiment at different times of day and different seasons. They expected to see the shift.

But they found nothing. No ether wind.

No difference at all. Light moved at the same speed in every direction, no matter how the earth was moving.

This result puzzled the best minds in physics for years. It is often called the most famous failed experiment in history.

But in science, a result that surprises you is never really a failure. It is a door, waiting to be opened.

In 1905, a young man working as a clerk in a patent office in Bern, Switzerland opened that door. His name was Albert Einstein. Einstein did something bold.

Instead of trying to explain away the strange result, he accepted it completely. He took it as a basic law of nature.

He said that the speed of light in empty space is the same for every observer, However, no matter how fast they are moving, whether you are standing still, driving a car, or flying in a rocket at half the speed of light, you will always measure a beam of light passing you at exactly 299 ,792 ,458 meters per second. This sounds simple, but think about what it means.

Imagine you are in a spaceship chasing a beam of light at 99 % of its speed. Common Sense says the light should be pulling away from you very slowly. But it does not.

You still see it race away from you at the full speed of light, and someone watching from Earth also sees that same beam moving at the full speed of light. Both of you are right. How can that possibly be true? Speed is just distance divided by time.

If everyone must agree on the speed of light, then something else has to give.

And what gives is distance and time themselves.

Einstein showed that time and space are not fixed and absolute.

They stretch and shrink, depending on how you are moving. Moving. Here is a simple way to picture it.

Imagine a special clock made of two mirrors facing each other, with a single pulse of light bouncing up and down between them. Each bounce is one tick.

Now put that clock on a fast -moving spaceship.

To the astronaut inside, the light still goes straight up and straight down, ticking normally.

But to someone watching from outside, the spaceship is moving sideways while the light bounces, so the light seems to travel along a longer, slanted, zigzag path.

A longer path at the same speed takes more time.

So from the outside, each tick of the moving clock takes longer. The moving clock runs slow.

And this is not a trick of mirrors.

Every clock on that ship, every heartbeat, every chemical reaction, every aging cell runs slower in exactly the same way.

Time itself is moving more slowly for the traveler, as seen from outside. Physicists call this time dilation.

At the same time, objects moving very fast become shorter in the direction they are moving, as measured by someone watching them pass. This is called length contraction.

At everyday speeds, these effects are so tiny that we never notice them.

But as you get closer and closer to the speed of light, they become enormous.

This may sound like science fiction, but it is real, and it has been tested thousands of times. One of the most beautiful proofs comes from the sky.

When high -energy particles from space, called cosmic rays, slam into the top of our atmosphere, they create tiny particles called muons.

Muons are unstable. On average, they fall apart in about two millionths of a second.

Even moving at nearly the speed of light, a muon should only travel about 600 or 700 meters before it decays.

They are created around 15 kilometers above our heads.

So, almost none of them should ever reach the ground, but they do.

Huge numbers of them rain down on us every second.

Some are passing through your body right now.

They survive because they are moving so fast that, from our point of view, their internal clocks are running much slower.

Their short lives are stretched out, long enough to complete the journey.

From the muon's point of view, it is the distance to the ground that has shrunk.

Either way, they arrive, exactly as Einstein predicted. There is an even more practical example sitting in your pocket.

The GPS satellites that help your phone find your location carry extremely precise atomic clocks.

Because the satellites move fast, their clocks tick slightly slower due to their speed.

But because they are farther from Earth's gravity, a second effect described by Einstein's later theory of general relativity makes them tick slightly faster.

Added together, the satellite clocks gain about 38 millionths of a second every day compared to clocks on the ground. That sounds tiny.

But GPS works by measuring how long radio signals, which travel at the speed of light, take to arrive from the satellites.

In 38 millionths of a second, light travels more than 11 kilometers.

If engineers did not correct for relativity, GPS positions would drift by kilometers every single day.

Your map would put you in the wrong neighborhood than the wrong city.

Every time your phone finds your location, Einstein's ideas about light and time are quietly at work. work. Now we come to the most famous question of all.

Why can nothing go faster than light? To push something faster, you need to give it energy. At low speeds, a little extra energy gives a little extra speed.

But as an object gets closer to the speed of light, something strange happens. More and more of the energy you add does not turn into speed.

Instead, it makes the object harder and harder to push.

Einstein showed that energy and mass mass are deeply connected through his most famous equation, E equals mc squared.

Energy equals mass times the speed of light squared.

As you approach light speed, the energy needed to go just a little faster grows without limit.

To actually reach the speed of light, an object with any mass at all would need an infinite amount of energy.

And there is not an infinite amount of energy in the entire universe.

So no spaceship, no rock, no person, and no particle with mass can ever reach the speed of light, let alone pass it.

Our most powerful particle accelerator, the Large Hadron Collider, pushes protons to 99 .999999 % of light speed. But it can never close that final gap. So how does light itself do it? The answer is that photons have no mass at all. And here is the surprising part.

Anything without mass does not get to choose its speed. speed.

It must always travel at exactly the speed of light. It cannot go slower, and it cannot go faster.

A photon is born moving at this speed, and in empty space it keeps that speed forever.

This leads to a deeper and more beautiful way of understanding what the speed of light really is. It is not really about light at all.

It is the speed limit of the universe itself. It is the fastest speed at which any cause can create an effect.

The fastest that any information, any signal, any influence can travel from one place to another.

Light simply happens to be the most famous thing that moves at that maximum speed. Gravity moves at it too.

In 2017, astronomers detected ripples in space from two colliding neutron stars and a flash of light from the same collision arrived just about two seconds later, after a journey of 130 million years. Gravity and light had raced across the universe side by side.

Physicists often say that the speed of light is the link between space and time.

Einstein showed that space and time are not separate things, but two parts of a single fabric called space -time. The speed of light is the exchange rate between them.

It tells us how much space is worth one second of time.

One second of time corresponds to about 300 ,000 kilometers of space. That is why it is the same for everyone.

It is not a property of a beam of light, it is property of the universe's structure. There is one more mind -bending thought here.

According to relativity, the faster you move, the slower your time runs. At the speed of light, time would stop entirely.

So, in a strange sense, a photon does not experience time at all.

A photon of light that left a distant star billions of years ago, and finally lands in your eye tonight, has not aged a single moment on its journey.

From our point of view, it traveled for billions millions of years.

From its own point of view, if such a thing even makes sense, its departure and arrival happened at the same instant.

But wait, if light always travels at the same speed, why do we sometimes hear that light slows down? This is true, and it is not a contradiction.

The famous speed of 299 ,792 ,458 meters per second is the speed of light in a vacuum, in perfectly empty space.

space. When light enters a material like water, glass, or diamond, it slows down.

In water, light travels at about 225 ,000 km per second. In ordinary glass, around 200 ,000.

In diamond, it crawls along at roughly 124 ,000 km per second, less than half its speed in space. Why does this happen?

When light passes through a material, it constantly interacts with the electrically charged particles inside the atoms, especially the electrons.

The light makes those electrons jiggle, and the jiggling electrons send out light of their own.

The original light, and this new light, combine, and the result is a wave that moves forward more slowly through the material. It is a bit like walking through a crowded train station.

You can move quickly between people, but all the small interactions slow your overall progress.

The moment light leaves the material and returns to empty space, it is back to full speed instantly. This slowing down has wonderful effects.

When light enters glass or water at an angle, the change in speed makes it bend. This bending is called refraction. It is why a straw in a glass of water looks broken.

It is how eyeglasses, cameras, and telescopes focus light.

It is also how raindrops split sunlight into a rainbow, because different colors slow down by slightly different amounts and bend at slightly different angles.

And it is why diamonds sparkle so brilliantly.

Light slows so much inside them that it bends sharply, bounces around inside the stone, and flashes back out in bright, colorful sparks. The internet depends on this too.

Most of the world's data travels as pulses of light through thin glass fibers running under cities and across ocean floors.

Inside that glass, light moves at about two -thirds of its vacuum speed.

That is still fast enough to carry a message across the Atlantic Ocean in a few hundredths of a second. But for some banks and stock traders, even that is too slow.

Some have built networks of microwave towers between financial centers, because radio waves moving through air are faster than light moving through glass.

In their world, a few millionths of a second can be worth a fortune.

Scientists have even managed to slow light down dramatically.

In 1999, a team led by the Danish physicist Lene Hau sent light through an extremely cold cloud of atoms, cooled to just a tiny fraction of a degree above absolute zero.

Inside that cloud, light slowed to about 17 meters per second. That is about the speed of a bicycle.

A few years later, her team even managed to bring a light pulse to a complete stop for a moment and then send it on its way again.

There is also a strange and beautiful glow that comes from this slowing.

Since light slows down in water, it is actually possible for a tiny particle to move through water faster than light can move through water.

This does not break Einstein's rule, because the particle is still slower than light in a vacuum.

But when it happens, the particle creates a kind of optical shockwave, just as a jet breaking the sound barrier creates a sonic boom.

The result is an eerie blue glow called Cherenkov radiation.

You can see it shining in the water surrounding some nuclear reactors. It is, in a sense, the sound barrier of light.

Today, we know the speed of light so precisely that we no longer measure it at all. That may sound strange, but it is true.

In 1983, scientists around the world agreed to turn things around.

Instead of measuring light using the meter, they decided to define the meter using light.

Officially, one meter is now the distance light travels in a vacuum in 1 ,299 ,792 ,458th of a second.

The speed of light is no longer a number we discover.

It is a fixed, exact constant built into the way we measure the world.

Every ruler on Earth is, in the end, measured against a beam of light. Now let us return to the idea we started with.

Because light takes time to travel, looking out into space means looking back in time.

When we talk to robots on Mars, our radio signals travel at the speed of light.

Depending on where Mars and Earth are in their orbits, a message takes anywhere from about 3 to more than 20 minutes to arrive.

That is why Mars rovers cannot be driven with a joystick like a toy car.

By the time a command arrives, the situation may have changed.

The rovers have to make many of their own decisions.

Farther out, the Voyager 1 spacecraft, launched in 1977, is now the most distant human -made object.

It is so far away that its faint radio signals, moving at the speed of light, need close to a full day to reach us.

When engineers send it a command, they must wait almost two days to learn whether it worked.

The nearest star beyond the Sun, Proxima Centauri, is about 4 .2 light years away. A light year is not a measure of time.

It is a distance. The distance light travels in one year, nearly 9 .5 trillion kilometers.

So the light you see from the nearest star left it more than four years ago.

The Andromeda Galaxy, our large galactic neighbor, is about 2 .5 million light -years away.

On a clear, dark night, you can see it with your naked eye as a faint, fuzzy smudge.

That faint light began its journey around the time when the earliest human ancestors were walking the plains of Africa.

When you look at Andromeda, you are seeing light that is older than the human species.

The most powerful telescopes, like the James Webb Space Telescope, can see galaxies whose light has been traveling for more than 13 billion years.

We see those galaxies not as they are today, but as they were when the universe was young. Some of them may no longer exist in the form we see.

Telescopes are, in a very real sense, time machines.

And at the very edge of what we can observe, there is a faint glow that fills the entire sky.

It is called the Cosmic Microwave Background.

It is the oldest light in the universe, released about 380 ,000 years after the Big Bang when the universe first cooled enough for light to travel freely.

That light has been crossing space for about 13 .8 billion years.

Over that time, the expansion of the universe has stretched its waves so much that it is no longer visible light, but faint microwaves. We can detect it with radio telescopes. It is, quite literally, a baby picture of the universe. universe. You might wonder about one more thing.

Astronomers say that some very distant galaxies are moving away from us faster than the speed of light. Does that break the cosmic speed limit?

Those galaxies are not racing through space faster than light. Instead, the space between us and them is stretching.

Einstein's speed limit applies to things moving through space, not to the growth of space itself.

But it does mean that light from the most distant parts of the universe will never reach us.

There There are regions of the cosmos we will never be able to see, no matter how long we wait. So what, in the end, is the speed of light?

It is a number, about 300 ,000 km per second.

It is the speed of an electromagnetic wave, written into the laws of electricity and magnetism.

It is the reason the sun you see is 8 minutes old, and the reason your phone knows where you are. It is the speed at which massless things must always move.

It is the boundary that no object with mass can ever cross.

But more than any of that, it is the speed of cause and effect.

It sets the pace at which the universe can talk to itself.

It ties space and time together into one fabric. It decides what we can know, and when we can know it.

Every star in the night sky, every photograph, every glance at a loved one's face, arrives at this one universal speed.

For thousands of years, people believed light was instant.

Then, a few curious minds dared to ask whether it might take time.

Lanterns on hills, the moons of Jupiter, spinning wheels and mirrors, a set of elegant equations, and one young clerk's bold idea slowly revealed the answer.

And that answer turned out to be far stranger and far more beautiful than anyone imagined.

So the next time you step outside and look up at the stars, remember what you are really seeing.

Not the stars as they are, but messages from the past.

Light that has been traveling for years, centuries, or millions of years, crossing the vast emptiness of space at the fastest speed the universe allows, just to end its long journey in your eyes.

Transcript of 太阳系的上下方向根本不存在？飞出黄道面有多难 （中文完整文稿）。

This is what you think the solar system looks like, but nobody's ever been this far up to see it like this. It's impossible.

Getting here to take this picture would be one of the most difficult things our species could do.

And if we could, you'd be able to save thousands of lives.

So, why is it that we can send a spacecraft 25 billion kilometers away, but we can't simply go in a different direction? Up.

We'll get to exactly why this doesn't work, whether there's any way around it, and if it could work, how far could you go?

What would you even see?

Okay, so you want to soar over the North Pole of the Sun, and the South Pole.

Become a Solar Polar Explorer, the first person to get a view of the Solar System from above.

You've got a spaceship and launch capability, but it's going to be hard. Why? Why?

Why is it easier to send a probe to Pluto than it is to travel above the Sun? I'll give you a hint. It's not the distance.

We've done flybys of every planet in the solar system and beyond.

We can do interplanetary, even interstellar travel. Is it the heat?

I mean, the sun's hot, obviously.

That is surface, it's around 5500 degrees Celsius, it's 10 ,000 degrees Fahrenheit, and the corona can reach 2 million degrees Celsius.

Surprisingly, it's not the heat either.

The Parker Solar Probe once flew within 6 million kilometers of the sun, handling temperatures temperatures of 1 ,400 degrees Celsius with its high -tech heat shields, and we're good at building spacecraft for crazy hot and cold environments flooded with radiation.

Now, the problem is, and get ready for this, it's flatness. Yeah, absolute flatness. The solar system is flat. That's a real problem.

And I'm not coming at you like a maniac telling you the Earth is flat. we all know the Earth is flat. Round, Peter. We all know the Earth is round.

But the solar system is almost uniformly flat, like a pancake, a quarter.

Every planet, asteroid, and moon in our solar system moves on the same flat plane because of the way the solar system formed.

It's called the ecliptic plane.

So, if you imagine the solar system as a ball, like a nucleus with electrons spinning about it, well, no, you're wrong.

It's more like marbles on a big round table, sun in the middle, and everything else orbiting around it.

Now, we'll get into why it's such a flat table in a minute, but the fact that the solar system is so flat is a big problem for you when you lift off with the purpose of seeing it from above.

It's gonna take some serious physics and horsepower.

Now you might be asking yourself why the heck would you want to see the Sun's poles anyway?

Well if you live on planet Earth or off Earth it It would explain so much about the solar weather.

What new science would we see if we imaged the sun's poles? What implications would that have for humanity?

And answering these types of huge questions is where AI shines.

But new tools are cropping up faster than ever these days, and personally I've found it hard to figure out which tool to use when.

That's where Chat LLM by Abacus AI comes in.

Chat LLM is a one -stop shop where you can access top AI models like ChatGPT, Gemini Pro, Claude Opus, and more.

And if you don't know which model to use, their Route LLM feature can automatically choose the best one for your prompt.

So you could ask one model to explain black holes like I'm five, another to help research the history of dark stars, and then generate visuals, videos, or even a full presentation from the same platform.

Chat LLM also includes AI image generation, AI video generation, and an AI PowerPoint generator that can build deep researched professional presentations.

And with Abacus AI Deep Agent, you can create apps, websites, software, AI agents, chatbots, and more.

Basically, it's like having an entire AI lab right in your browser.

And it's only about ten dollars a month, which is a lot cheaper than paying for multiple AI tools separately.

So if you want to explore the universe, build faster, research deeper, or just keep up with the AI singularity before it spaghettifies your schedule, check out Abacus AI using the link down below.

Now, getting back to traveling above or below the sun, what could it really do for us?

Well, it could help us predict major solar flares and even solar storms.

Flares can destroy satellites, take out electrical grids, and entire communication systems.

Ever lose a cell signal when you're on a trip and you're not sure where to turn?

Imagine what that would be like for the entire world all at once.

Plains lose connection with their air traffic controllers, and boats lost at sea.

Chaos worldwide, as systems built on interconnectivity and the electrical grid, go dark.

Solar storms can shred astronauts' DNA.

With better predictions, we could protect grids and communications networks and shut them down in advance.

astronauts could literally rearrange their spacecraft to shield themselves from a serious coronal mass ejection.

This could save billion dollar technical systems and save thousands of lives, too. Look, the sun is weird.

Every 11 years, it flips the polarity of its poles.

That means the North Pole and the South Pole switch places.

Earth does this, too, but only every a few hundred thousand years or so.

But when the sun flips, that's when sunspots flare up, and Earth gets blasted with solar flares that can really mess with our electronics.

Those flares are caused by the sun's flip -flopping magnetosphere, and a good chunk of that is driven by what's going on at the poles, which we can't see.

So, seeing what's happening at the poles has kind of become solar -holy -grail material.

With that knowledge, we could better predict solar flares and storms to protect ourselves.

So why haven't we just sent probes to check it out?

The inflexible problem of delta v.

Delta stands for change, and v stands for velocity.

The delta -V is the change in velocity a spaceship needs in order to pull off an orbital maneuver like watching into orbit.

The problem is, changing your velocity is wildly expensive in terms of rocket fuel. But why? Say you wanted to try this. Well, just point up and burn rubber, right?

Aim your spaceship above the ecliptic plane and hit the gas.

Well, the problem is delta V.

Earth is traveling around the Sun at incredible speeds, about 110 ,000 km per hour.

And when your ship leaves Earth, it's also traveling super fast, in the same direction.

Basically, it inherits the sideways speed from Earth's orbit around the Sun.

So to make a sharp turn, you have to cancel that original speed before you can chart a direction.

You might think you understand turning, but do you?

Imagine you're in a car going forward and you want to turn left.

Simply turn your wheel and all of your forward momentum gets converted into a new direction.

You're able to do this because the tires on your car are able to grip the road and redirect that energy sideways.

And if you think about it when you're done the turn, you're not going in that original direction anymore.

To say that in delta -v terms, your velocity in that original direction is zero.

This is true if you want to make a turn in space as well, but in space there's nothing to push off of, no rubber on the road.

Any change in direction has to have a rocket pushing you in that direction, And to make a complete 90 -degree turn, you still need to stop going forward.

Otherwise, the best you'll ever do is continue forward, and in the new direction, at the same time, on some new diagonal direction.

Now, changing planes is one of the most energy -intensive tasks in spaceflight.

To perform a 90 -degree turn, it wouldn't require burning so much fuel that it's physically impossible to brute force it.

The biggest rocket on Earth is the SpaceX Starship, which carries over four and a half million kilograms of fuel.

It has nowhere near enough fuel to provide the change in velocity we need. And it gets worse.

Building a rocket that large with that much rocket fuel becomes an exponentially more difficult problem, because the larger your rocket, the bigger the challenge.

If you send up a huge rocket, you need more fuel to push that heavier ship into space.

But that rocket fuel is also heavy, so you need a bigger rocket with more fuel to help push all that other fuel up.

Basically, the ship would have to be so big it would be almost all rocket fuel. And that's just not feasible. Thanks Delta V.

Okay, that was a mouthful of an explanation, but if you want us to explain concepts like Delta V in a future video, put a comment below on what you want explained.

I never took calculus in school, but I'll get somebody smart here at What If to explain it to me first. I want to learn about reverse osmosis. Like, how can it be reverse? Sure, whatever floats your boat.

So how do you possibly get above the sun?

Well, like so many questions about our solar system, the answer is Jupiter.

It's the biggest planet in the solar system. 11 times wider than Earth, 318 times more massive.

So its gravitational pull is 318 times stronger.

That's important if you're going to see the top of the sun.

You see, we used Jupiter's intense orbital momentum to help change the speed and direction of our spacecraft since the 1970s.

Pioneer, Voyager, they used Jupiter's gravity to execute flight maneuvers known as gravity assists or gravitational slingshots.

Basically, we fly real close to a planet and steal some of its kinetic energy to accelerate and change direction.

Voyager 2 used the gravity of four planets to get where it is today, starting with Jupiter. Jupiter.

You might know the Voyagers because they're famously still flying through interstellar space.

But we also use Jupiter's gravity to change the speed and direction of a lesser -known spacecraft, Ulysses.

Ulysses was launched from the space shuttle in 1990 and executed a gravity assist around Jupiter in 1992.

That changed the direction of the Ulysses' craft to a staggering 80 -degree incline on the ecliptic plane.

By the mid -90s, Ulysses passed over the Sun's poles and measured them with a whole host of instruments.

We learned all kinds of things about the Sun and the heliosphere.

Like we learned that the solar wind is blasting out of the Sun's polar regions at 750 kilometers per second, way faster than the wind from the Sun's equator.

But of all the instruments that Ulysses carried into space, none of them was a camera, and it was still really far out from the Sun, just inside Mars' orbit at its closest.

So Ulysses sensed the poles, measured them, but never imaged them.

That means we've still never seen an image of the Sun's poles. else, until 2025.

Launched 30 years after Ulysses, the Solar Orbiter spacecraft took off in 2020 and chose a different path.

It didn't head toward Jupiter.

Instead, it used multiple Earth and Venus gravity assists to change its flight path out of the ecliptic plane.

plane. Over a period of five years, it got up to 17 degrees above the ecliptic plane. And it stayed close to the Sun.

At its closest point, the orbiter and its huge heat shield were just 42 million kilometers out.

It's closer to the Sun than Mercury.

Finally, in March 2025, for the first time in human history, we saw images of the Sun's South Pole.

Of course, if you remember your middle school math and how to use a protractor, 17 degrees isn't a lot.

So it's kind of like looking at the bottom of an orange, like this.

Impressive? Sorta, it's the first glimpse.

The good news is that the mission is still rolling, and with more gravity assists, that angle will climb to 33 degrees.

We'll get better and better views of the solar poles.

With that first peek, we imaged the poles and learned all kinds of things about the sun, like how the sun shapes the heliosphere, that giant bubble of plasma that surrounds the solar system, how tangled up its magnetic fields are, and what those miniature solar flares named campfires are all about.

In the future, that might explain why the corona is so hot.

Overall, it's doing what it's supposed to do – understand solar weather, helping us predict the sun's behavior.

There are other proposed missions in the works, too.

China is planning to launch the promising Solar Polar Orbit Observatory in 2029, which might get to 60 degrees or 70 degrees relative to the ecliptic plane.

So, we're getting closer to traveling above and below the Sun.

But what if you figured out a way to get all the way to 90 degrees? What would you see?

Okay, you figured out a way to use multiple gravity assists and some futuristic propulsion to delta -V your way into orbit around the Sun at 90 degrees.

Maybe you've got one of those fancy solar sails at NASA's piloting.

You would be the first to see the sun and the whole solar system from above.

You'd see the vast, flat solar system below you.

And from your vantage point, it'd be kind of awesome. I mean, it's the sun's north pole.

And as you zoomed outward, you would see all the inner planets from above, from Mercury, Mercury to Mars, and then the asteroid belt, and then the outer gas giants.

And at some point, you'd probably ask yourself, Why is the solar system so remarkably flat?

Like all the planets are within a few degrees of the ecliptic plane.

Mercury is the outlier at seven degrees, but the rest are really within a few degrees.

Well, you'd have to go back to the formation of the solar system 4 .6 billion years ago, when it was just a molecular cloud or nebula that gravity was pulling together.

All that matter and dust gathered and started clumping up, and around that time a nearby supernova's energy hit the cloud and it started churning.

And as all the particles started to spin and bump into each other, a funny thing happened.

You see, the formation of disks is really common in space when matter is rotated, like Saturn's rings.

In our case, the matter was all going in a circular motion around a single point and wouldn't stop. It's called the conservation of angular momentum.

system. Everything keeps going round and around.

But the particle's constant collisions cancel out the up and down movement.

That eventually brings everything together in a plane. The ecliptic plane is formed.

At that stage, that's what we would call a protoplanetary disk.

And after that, the planets and the sun formed.

So, jumping forward billions of years, there you'd be, floating above it all, taking a moment of wonder at the extremely flat solar system.

Maybe some selfies, and then you'd be drawn towards seeing the bigger picture, going further out to the edge of the solar system, the heliopause.

Yeah, if you flew way up above the planets, you'd eventually reach the edge of the heliosphere.

That's the bubble around the Sun and the Solar System created by the Sun's solar wind and magnetic field.

It stretches out to three times the distance to Pluto.

That far out, you'd be able to measure the shape of the heliosphere, which is something scientists are still studying.

It might be round, or might be croissant -shaped, or shaped like a teardrop, or a comet, or a tadpole.

We don't know, because we've only studied it from two points. Voyager 1, Voyager 2. But you'd figure it out.

And that's critical stuff.

You see, the Sun's plasma blasts out into space, acting as a giant shield for us against cosmic interstellar radiation.

Without the heliosphere, we'd all be toast.

The heliosphere was first discovered by Voyager 1, which which suddenly stopped detecting solar radiation and then picked up a huge increase in cosmic radiation in 2012.

It wasn't hard to detect.

There's about three times as much cosmic radiation outside the heliosphere as there is inside it.

Now, we know that the Sun has provided us with energy since the beginning of time, but it seems it's also protected us from deadly radiation.

And it's a dynamic system, too. The heliosphere shrinks and expands with the level of solar activity.

It gets bigger when the sun starts swapping poles.

See, getting above and below the poles is critical to understanding how all these things work.

Now the further above the sun you'd go, the more interesting other structures would get, and less disk -like, like the Kuiper Belt full of trans -Neptunian objects, which is more belt -like.

Officially, scientists say it's doughnut -shaped, but the Kuiper doughnut doesn't sound very scientific.

And even further out, there's the Oort Cloud, that big shell full of icy comets and planetissimals.

We've never actually observed it, so when you'd be that far above, you'd finally be able to tell the actual shape.

Maybe it's more belt -like or ring -like than we currently think.

And at that point, above and below would start to kind of lose meaning.

You would still be above the Solar System, but you'd hardly be able to spot the Sun anymore.

You'd be in interstellar space, and you'd start to put the Solar System into a galactic context.

Like, Like, where does it fit in the Milky Way?

Well, looking down on the solar system wouldn't mean you're looking down on the Milky Way.

The Sun sits above the galactic midplane, but the solar system is basically cruising around the galaxy on its side.

Yeah, our nice flat solar disk is at a 60 -degree angle relative to the galaxy's elliptic plane.

And it doesn't seem like the solar system always sits above the galactic plane either. Instead, it seems like we oscillate through the plane.

We've been observing a nearby galactic structure called the Radcliffe Wave, a local part of the Milky Way, and its star clusters move up and down, doing the wave.

So if you kept track of your old planet Earth spinning around the sun, it'd look something like this.

Earth orbits the Sun on a flat plane.

Our solar system orbits our galactic center at hundreds of kilometers per second, but it's spinning on its sides at 60 degrees off -kilter from the galactic plane, and it's bobbing up and down.

And of course, the Milky Way is hurling through space as part of the local group of galaxies, galaxies, which is part of the Virgo supercluster at the edge of the Lania Kaia supercluster.

After a while, above becomes relative, inconsequential, really, in relation to the rest of the universe.

And all this started because we wanted to see the Sun from one specific angle, above it.

But, what does the Sun look like as a sunset from every planet in the Solar System?

And what does a sunset on an alien planet look like?

Well, click the video on screen now to watch that episode.

Because that's a story for another WHAT IF.

Transcript of 物理学最美的想法，却是我们永远证明不了的 （中文完整文稿）。

About two and a half thousand years ago, a Greek man named Pythagoras noticed something strange.

Plug a string, now press it down exactly half way and plug it again.

Same note, just a bit higher.

Cut it to two thirds and you get another note that sounds just right. Simple ratios, clean music.

So Pythagoras made a hypothesis that anybody would definitely make in this situation. Maybe the whole universe works like that.

numbers, harmony, music.

And so for the next 2000 years we thought Pythagoras was smoking that good stuff.

But then in the late 1960s physicists stumbled onto the math and found out something bizarre to say the least.

Pythagoras might have actually been correct.

By the end of this video you'll understand why this idea is either the most important theory ever written or the most beautiful can tell you which one is correct.

So here's the main problem that his community has been stuck on for about a hundred years.

We have two rule books for reality, and both of them are ridiculously good, such that we can't actually tell which one is more accurate than the other.

Rule book one, and this is the one you might have heard of, is Einstein's general relativity.

Gravity isn't a force pulling you down, it's space and time bending around anything with mass.

It predicted the existence of black holes, it predicted ripples in space itself, which we finally detected in 2015.

Your phone's GPS has to correct for these measurements or your map would drift by about 10 kilometers every single day. Rulebook 2. Quantum mechanics.

The rules for the very very very small electrons, quarks and light.

One of its predictions matches experiment to about one part in a trillion, which basically means it's very very very accurate and hence why every chip in the device you're watching this on works because of it.

So the obvious question is what happens when you need bolt at once?

That question takes us right towards the center of a black hole.

The first instance of the Big Bang, an enormous amount of mass crushed into an impossibly small space.

Gravity matters there, but so does quantum mechanics.

Physicists tried to combine the two, and the answer came out as infinity.

Not a number, infinity.

You ask the equations how likely something is, and they'll tell you it's infinity percent. No, no, seriously, I actually mean that.

And a big part of the reason is something we take for granted. Particles are treated as points, zero size.

When you try to calculate gravity between two things with no size at all, the numbers literally run away from you. So what if they aren't points?

Let's zoom in. Past your skin.

Past a cell. Into a single atom.

An atom is almost entirely empty.

If it were a football stadium, the nucleus would be a P on the center spot.

And now let's zoom in even further into a proton, three quarks, and this is roughly where our experiments stop.

String theory says, keep going, way, way, way, way down to something called the Planck length, about 10 to the minus 35 meters.

To put that in perspective, if you blew a single atom up to the size of the entire observable universe, a plank length would be about as tall as a tree.

And down there, string theory says that you find a string of energy vibrating on its own.

This is the picture Michio Kaku builds on in the God equation.

And the easiest way into it is a violin. One string vibrated one way, you get an A. Change how it vibrates, you get a C sharp. The same string, just different notes.

String theory says an electron is one note, a quark is another note, a photon, a particle of light is another.

They aren't fundamentally different things, they're the same thing, vibrating differently.

The particle's mass is how much energy is in the vibration.

Its charge is written into the pattern, Kaku actually has a fascinating way of putting it.

Physics is the set of harmonies you can play on these strings.

Chemistry is the melodies you get when they interact, and the universe, the universe is the symphony. It also fixes our infinity problem.

A string has a size, tiny, but not zero.

You can't crush two strings into a single point, so numbers that used to run off to infinity now settle down.

But honestly, that isn't the strangest part.

The strangest part is something that showed up in the equations themselves.

Let's rewind to 1968, a young physicist at CERN named Gabriele Veneziano is trying to understand the strong force, the thing that glues protons together, and he finds a formula that fits the data almost perfectly. The weird part? It wasn't new.

It was a piece of pure math written by Leonhard Euler more than 200 years earlier, just sitting in plain old textbooks. Nobody understood why it worked, however.

A couple of years later, a few physicists, including Ichiro Nambu, Holger Nielsen and Leonhard Tusken, figured out what the formula was secretly describing. Vibrating strings. Then things went wrong for string theory.

It had started out as a theory of the strong force, the force that holds the nucleus of an them together.

And as that, it lost to a better idea.

And it had an embarrassing habit.

It kept predicting a particle nobody wanted. No mass, a very specific kind of spin.

It showed up in every version of the math, and nobody could get rid of it.

Then, in 1974, two physicists, John Schwartz and Joel Scherck, looked at that unwanted particle and realized what it was.

The Graviton. The particle that would carry gravity. Think about what that means.

For decades people had tried to force gravity into quantum theory and every attempt broke.

And here was a theory where you don't add gravity in. You literally cannot take it out.

Work out the ways the quantum strength can vibrate and one pattern comes out with no mass and a spin of two.

Which is exactly what a graviton, the particle of gravity, should look like.

it falls out of those string equations whether you want it or not.

That's the moment people got hooked. Michio Kakuru was one of them.

In the 1970s he helped create string field theory and in the God Equation he makes the case that strings are our best shot at what Einstein spent his last 30 years chasing.

The one equation that gives you gravity, light, the nuclear forces in all of matter. So So why isn't this just settled? Because there's a catch. Well actually three. Catch one.

Dimensions. For the math to hold together, the strings need ten dimensions. We see four. Up. Down.

Left. Right. Forward.

Back. And time. So where are the other six?

The answer is that they are curled up, unimaginably small.

Picture a garden hole seen from across a field. It looks like a line. One dimension. dimension. Walk up close and you see an ant walking around it.

There was a second dimension the whole time. You were just too far away.

String theory says every point in space hides six of those, curled up far too small to ever notice. Catch two. Too many shapes.

There is a staggering number of ways to curl up those six dimensions. One common estimate is 10 to the power 500.

So each shape gives a different universe, with different particles and different laws.

So critics ask, if the theory allows almost anything, what is it actually predicting?

Catch three, and this is the big one, no evidence. Not yet at least.

Strings are so small that seeing one directly would take a particle accelerator roughly a quadrillion times more powerful than the Large Hadron Collider.

Physicists like Lee Smolin and Peter White have argued for years that theory you can't test isn't really physics yet. And that's a fair point.

The LHC was hoped to find indirect clues, like a new partner particles predicted by an idea called supersymmetry. And so far, nothing came of it.

Gaku's response is that the proof may come sideways.

From dark matter, from gravitational waves, from echoes of the Big Bang.

But that maybe is doing a lot of work, and we still need substantial hard evidence to make it, well, physics.

When Michio Kaku was 8 years old, he saw a photo in the newspaper.

It was of a desk. The man who owned it was Albert Einstein. And he had just passed away. On that desk was an unfinished manuscript.

Einstein had spent the last decades of his life trying to find one equation that explained everything. He just couldn't get there in time.

Kaku had said that that that picture stayed with him.

He wanted to know what was so hard that even Einstein himself couldn't finish it, and he spent his whole career on that question.

So here's where we actually are.

We have an idea that's beautiful, that fixes a problem nothing else has fixed, that hands us gravity for free, and that right now we actually have no way to check if it's actually real or not.

Maybe in 50 years it sits in textbooks next to Newton and Einstein, Maybe it becomes a footnote, a gorgeous piece of math that just isn't how the universe works.

But if you took nothing away from this video, take this away.

Sit with this for a second, because this is probably the best way I can explain how bizarre this theory actually goes.

Every atom in your body, every bit of light hitting your eyes right now, if string theory is right, all of it, you, this screen, the light in between, is the same tiny new thing, vibrating in different ways.

Pythagoras looked at a string and thought he could hear the universe.

Two and a half thousand years later, we still can't say if he was right.

We just know the music hasn't stopped.

Transcript of 宇宙膨胀对不上数：大爆炸理论要塌了吗 （中文完整文稿）。

There is a crisis unfolding in cosmology, and the answer could decide how we understand the entire universe.

For almost a century, we've known that we live in an expanding universe.

In 1929, Edwin Hubble showed that galaxies are drifting apart, and the farther away they are, the faster they seem to go.

That single discovery became the backbone of modern cosmology, shaping everything from the Big Bang to the birth of galaxies.

At the center of it all sits one number, the rate at which the universe is expanding. We call it the Hubble constant.

For decades, we believed we were closing in on its true value. Now, astronomers are not so sure.

So are we standing on the edge of a scientific revolution, or is something about our entire model of the universe simply wrong?

I'm Alex Mercer, and this is Astros.

Today, we're going to explore the crisis gripping cosmology and see how Hubble's greatest discovery turned into one of the biggest headaches in modern science.

For most of human history, the universe was small, or at least we thought it was.

Many ancient cultures pictured a flat earth sitting beneath a dome of stars.

The sun and moon rose and set, the constellations wheeled overhead, and that was the cosmos.

By the second century, astronomers like Claudius Ptolemy had built a more refined picture with Earth fixed at the center and the sun, moon, planets, and stars all circling around us.

That geocentric model ruled for more than a thousand years.

Then in the 1500s, Nicholas Copernicus placed the sun in the center instead, with Earth just as one of several planets in orbit around it.

It was a radical idea, and it opened the door to a much bigger universe.

In the 1700s, William Herschel began mapping the stars of our own galaxy, the Milky Way, and revealed it to be a huge, flattened disk.

But even then, astronomers argued over a bigger question. Was the Milky Way the entire universe?

Or was it just one island among many in a far larger ocean of space?

That argument lasted until 1923, when Edwin Hubble settled it.

Working at Mount Wilson Observatory in California, Hubble pointed the 100 -inch Hooker telescope at Andromeda.

At the time, it was the largest telescope on Earth, and with it, he did something nobody had done before.

He picked out individual stars inside Andromeda, and among them he found a special kind of pulsating star called a Cepheid variable.

These stars are incredibly useful thanks to the work of Henrietta Leavitt.

Leavitt catalogued more than 2 ,400 variable stars, and she discovered that the time a cepheid takes to pulse is tied directly to how bright it truly is.

That means if you measure how long it takes to pulse and compare its real brightness to how bright it looks from Earth, you can calculate how far away it is.

Astronomers call this Leavitt's Law.

Hubble applied it to Andromeda, and the result was staggering.

He calculated that Andromeda sat around 900 ,000 light -years away. That's far too distant to be part of the Milky Way.

Andromeda was a galaxy in its own right, with its own stars sitting completely outside hours.

Overnight, the known universe became immeasurably larger.

And Hubble wasn't finished.

He went on to use Levitt's law on 23 more galaxies, some as far as 20 million light -years' ray. The conclusion was unavoidable.

Our galaxy was one of millions, and astronomy would never be the same.

But his next great discovery relied on the work of another astronomer, a man named Vesto Slipher. Slifer.

Slifer was born on a farm in Indiana in 1875, and he joined the Lowell Observatory in Flagstaff, Arizona in 1901.

Between 1912 and 1925, he carefully studied the light from more than 40 spiral galaxies. Here's why that mattered.

Light can be split into its component colors, and different elements leave behind different patterns in that spectrum. spectrum.

If the object producing that light is moving away from you, those same patterns appear, but they're shifted toward the red end of the spectrum.

We call that effect redshift.

Schleifer noticed that almost every galaxy he looked at showed this redshift, meaning they were nearly all moving away from us.

But there was a catch.

He had no way to measure how far away those galaxies were, so he couldn't work out how fast they were really moving.

In 1927, a Belgian scientist named Georges Lemait proposed something bold.

If the universe looks the same in every direction, and if Einstein's theory of relativity is correct, then the universe has to be expanding.

It was an elegant idea, but Lemait had no observational evidence to back it up, and so most people ignored it.

A Soviet scientist named Alexander Friedman had explored a similar idea a few years earlier, and it too had gone largely unnoticed.

So which brings us back to Hubble.

Like most scientists at the time, Hubble hadn't heard of either theory.

He was just looking at galaxies from Mount Wilson, and he saw the same thing Sluyfer had seen, galaxies moving away from us. But Hubble had something Sluyfer didn't. He had distances.

And that let him spot a pattern that nobody had noticed before. War.

The farther away a galaxy was, the greater its redshift, which meant the faster it was racing away, and this was happening in every direction. More observations confirmed it.

Nearly every galaxy was receding, and the redshift of each one was directly proportional to its distance from Earth.

This was a huge breakthrough, because it meant one thing.

the universe is expanding.

Hubble published his findings in 1929, and they became the first observational support for Lemaites' expanding universe, the idea that would grow into what we now call the Big Bang.

That relationship between distance and speed is where the Hubble constant comes from. It tells us how quickly space is stretching.

Hubble's own first estimate was roughly 500 km per second for every megaparsec of distance.

A megaparsec is about 3 .26 million light -years.

So, according to that early number, a galaxy one megaparsec away would be receding at around 500 km every second.

That figure turned out to be far too high, and Hubble spent much of his career trying to refine it, but the idea behind it became one of the most important in in all of science.

The Hubble constant helps us work out the age of the universe and its size, and those numbers feed into almost every other calculation cosmologists make.

So you'd think that by now we'd have this number nailed. Instead, we found something that doesn't add up.

Astronomers have two main ways to measure the Hubble constant, and they disagree with each other.

The first method looks at the universe as it is now, billions of years after the Big Bang.

It's often called the Cosmic Distance Ladder, or the Late Universe method, and it works much the way Hubble's original work did.

You measure how far away galaxies are, you measure how much their light is redshifted, and from that you calculate how fast space is expanding.

The more distant galaxies we can measure, and the more carefully we can calibrate those distances, the more precise the result becomes. systems.

Over the years, this approach has settled on a value of about 73 .5 kilometers per second per megaparsec.

That's much lower than Hubble's first guess, but it's been measured with real precision. The second method works in a completely different way.

Instead of looking at the present, it looks back at the very beginning and then rewinds and fast -forwards the clock using our best physics.

It's known as the early universe method. It relies on the Cosmic Microwave Background, or CMB.

This is ancient light, a faint glow of microwave radiation that fills the entire sky, and it's essentially a snapshot of the universe about 380 ,000 years after the Big Bang.

Its existence was predicted in 1948 by Ralph Alfer and Robert Herman.

In the 1960s, Arno Penzius and Robert Wilson detected it by accident, and they won a Nobel Prize for the discovery.

But it took until the early 2000s to get an image sharp enough to be truly useful.

In 2003, the Wilkinson Microwave Anisaltropy probe, known as WMAP, released a full map of the sky, showing the universe at a time before any stars had even formed. Here's one way to picture how this works.

Imagine you arrive at the site of an explosion that's already happened.

Debris is scattered everywhere and you can't tell what the original object looks like.

But then, someone hands you a photograph, taking just moments after the blast began.

Suddenly, you can trace each fragment back to where it came from. That's what cosmologists do with the CMB.

They take that early snapshot, apply the laws of physics, and press play, running the universe forward to see what it should look like today.

The European Space Agency's Planck satellite took this even further.

It captured an incredibly detailed map of the background radiation, revealing tiny temperature fluctuations.

Those small ripples are the seeds of everything.

They grew into the vast cosmic web of dark matter, with its galaxy clusters and enormous empty voids.

When astronomers feed the Planck data into our standard model and run it forward, they get a predicted Hubble constant of about 67 .4 kilometers per second per megaparsec. And now you can see the problem.

That's noticeably lower than the roughly 73 we measure from Redshift.

moved. Two methods, two answers, and a gap that just won't close.

To understand why that matters so much, we need to talk about the model they're both built on.

The Hubble constant isn't just a number sitting in a textbook, it's woven into our deepest ideas about physics, from Einstein's general relativity to the picture of a universe that began hot and dense and has been expanding ever since.

The framework that holds all of this together is called the Lambda Cold -Dark Matter Model, or Lambda CDM.

It took shape in the late 1990s, and for a long time it has worked beautifully.

One surprising feature of that model is that the expansion of the universe is speeding up.

Scientists had assumed gravity would slow it down over time.

Instead, in the late 90s, they discovered the opposite. it.

Something is pushing space apart faster and faster, and we call that something dark energy.

Thankfully, that discovery didn't break the model. It became a part of it.

The more the universe grows, the more dark energy matters, and the faster the expansion becomes.

Now, you might wonder whether that acceleration explains our two different numbers. numbers.

Both methods are meant to give us the expansion rate as it is today, so the acceleration isn't the cause of the disagreement. That leaves us with a real puzzle.

How can one cosmological model produce two different values for something so fundamental?

This disagreement is known as the Hubble Tension.

It isn't called that because of some physical tension in space itself.

More likely it's named for the mood it creates at cosmology conferences.

And honestly, I understand why.

Because if it's real, it could mean our whole picture of the cosmos is incomplete.

For a long time, many scientists hoped this was just a measurement problem.

The thinking went, as our instruments improve and data piles up, the two values should slowly move toward each other and meet somewhere in the middle.

But that isn't what's happening.

As new independent measurements arrive, the gap isn't closing, it's getting harder to ignore. Take one recent example. Some galaxies host what are called megamasers.

These are regions where molecules like water vapor amplify microwave radiation near a supermassive black hole at the heart of a galaxy.

It's similar to a laser, but it produces microwaves instead of visible light.

Megamasers are extremely powerful, and astronomers can map their emissions with great precision.

That lets them measure the mass of the black hole and, importantly, work out the geometric distance to the host galaxy without relying on the usual distance ladder.

In 2020, a team led by Dom Pesci used this method and found a Hubble constant of 73 .9, with very high confidence.

Then, in late 2024, a team that included Daniel Skolnick and Nobel laureate Adam Race used data from the dark energy spectroscopic instrument.

They studied 12 Type Ia supernovae spread across the coma galaxy cluster, about 320 million light -years away.

Their result was 76 .5.

And then in 2025, researchers at the Inter -University Center for Astrophysics and Astronomy in India took another approach.

They used mirror variable stars in our own galaxy as reference points, then measured mirror variables in distant galaxies, a technique similar to the original Cepheid method. Their answer was 73.

Three very different methods and three very similar answers.

Each one lands close to 73 or higher, and each one sits well outside the margin of error for the Planck result based on the CMB.

Something doesn't fit.

Several astronomers have started to sound the alarm.

Daniel Skolnick, a physics professor at Duke University and the lead author of that supernova study, has said that our model of cosmology might be broken and that the Hubble tension is starting to feel more like a Hubble crisis.

And it's easy to see why this matters.

If the model we use to interpret the universe is on shaky ground then so are the answers it gives us.

When galaxies formed, how clusters evolved, even our predictions for the far future of the cosmos, all of it hangs in the balance.

Consider the two possible outcomes.

If the true Hubble constant is the lower value, the one from the background radiation, then we live in a universe about 13 .8 billion years old and it will most likely keep expanding forever—a steady, predictable future with dark energy staying constant.

But if the true value is the higher one, the one we see from Redshift, then the universe is expanding faster than our model allows.

In that case, the universe might be younger, perhaps 12 to 13 billion years old, and it could mean we're missing something fundamental about the early universe, about dark energy, or even about relativity itself.

So, how might scientists resolve this?

Some have suggested radical changes to the standard model.

Ideas include dark matter that decays over time, or dark energy that isn't constant but changes as the universe ages.

Some researchers have even explored altering the equation of state for dark energy, which might point to changes in general relativity itself.

That's the kind of shift that would force us to rethink not only the cosmos, but physics as a whole.

But there's another idea that's been gaining attention, and it could rescue our model without rewriting the laws of nature. All it needs is a little spin.

And no, I'm not joking.

A 2025 paper by Belaj Andrei Sagetti, Istvan Sapudi, Imre Farinkparna, and Gargeli Gabor Barnafoldi, proposes that we may have overlooked a very slow rotation of the entire universe.

universe. The idea is that the universe turns once every 500 billion years.

That's so slow that we'd never notice it directly, but it could be enough to influence how space has expanded over cosmic history.

And here's the exciting part.

Their early calculations suggest that this tiny spin could explain both values at once.

When rotation is added to the model, it predicts an early universe with a lower expansion rate, just like the CMB tells us, and it also predicts a present -day universe with a higher expansion rate, matching what we see in redshift measurements.

In other words, both camps could be right, because they're measuring the expansion at different times in a universe that's been slowly turning.

István Sápudi, one of the authors from the University of Hawaii, explained the concept by quoting the ancient Greek philosopher Heraclitus.

He said, panta rei, which means everything flows or everything moves, and given how promising those early results look, that might be true on the grandest scale of all.

Now I do want to be careful here.

The authors themselves are clear that this is a preliminary calculation and it focuses only on the Hubble constant.

The next step is to build more complex models and run detailed simulations to see whether the idea holds up against other observations, like the distribution of galaxies and the patterns in the CMB.

A hypothesis is only as good as its ability to survive those tests. So, where does that leave us?

I think it's a reminder of how much there still is to learn about the universe, but it's also far too early to throw out our entire model of cosmic evolution, especially when we might be on the verge of discovering the one missing piece that makes everything fit.

And even if the spinning universe turns out to be right, it wouldn't mean abandoning everything we know.

It would mean a new, rotating version of our model, with some of the cosmic parameters adjusted to reflect a slightly different reality.

It's also worth noting that the idea has one major advantage. It doesn't break any known laws of physics.

Whatever the answer turns out to be, this is a remarkable time to be watching cosmology.

A century after Hubble first showed us that space is stretching, we're still discovering that the universe has more to teach us.

And personally, I find something wonderful about the thought that our planet spins, our solar system turns, the Milky Way rotates, and perhaps, on the greatest scale of all, the entire universe is turning along with them.

If you enjoyed this journey through the Hubble Tension, make sure to subscribe to asterisk so you don't miss the next one.

And let me know in the comments, do you think the universe is spinning, or is the real answer something even stranger?

I'm Alex Mercer, thanks for watching, and I'll see you out there among the stars.

Transcript of 元素周期表不是清单，是宇宙的设计图 （中文完整文稿）。

For years, I thought the periodic table was just a list, a neat row of elements someone sorted by size.

I repeated that in class, not at a long, never questioned it. It is organized by atomic number.

And it is wrong, not wrong in a small way, wrong at the reason.

Because atomic number is not why the periodic table looks the way it does.

Atomic number is just a label like numbering the seats in a stadium, numbering the seats did not decide the shape of the stadium, something else did that, something much stranger.

Here is the part that should bother you. Look at the periodic table.

Why does it have that weird gap in the middle, that big empty space at the top between beryllium and boron?

Why are those two rows of elements, the lanthanides and actinides just floating at the bottom completely detached like a footnote?

Why do the columns mean something but the rows seem almost random.

If this table is just a list sorted by number, none of those things should exist.

A numbered list does not have gaps.

A numbered list does not exile 14 elements to a footnote at the bottom. So why does this one?

That is not a design choice.

That is not a mistake. That shape, every gap, every exile, every strange column grouping is a map.

A physical map of something real.

And once I understood what it was a map of, I could not look at the periodic table the same way again.

That is what this video is.

So if the periodic table is not just a numbered list, what is it? What is the shape actually telling us?

The answer turns out to be one of the most beautiful things in all of science, and it was not planned, nobody sat down and designed this table to look profound.

The shape fell out of the math, automatically, nobody could stop it from happening.

By the end of this video, you will look at that table on the wall and see exactly what each piece of it means.

You will understand why the gaps exist, why the columns behave the way they do, and why one man predicted elements that had not been discovered yet, and was right about almost all of them.

And you will understand why the chemist who built the most famous The chemist's version of this table spent the last years of his life being denied the Nobel Prize by a man who hated him. Let's get into it.

To feel why the periodic table was such a big deal, you have to understand how much of a disaster chemistry was in the 50 years before it showed up.

Picture 1860, chemists have discovered about 60 elements, 60 different substances that cannot be broken down further, 60 things that are just themselves.

And nobody knows what to do with them.

Some of them are metals, some are gases, some kill you on contact, some are so soft you can cut them with a butter knife, some glow in the dark, some explode when they touch water.

And the question every chemist was quietly asking, the question nobody had a real answer to was this. Is this random? Are these 60 things just 60 accidents of nature? Or is there a pattern hiding underneath all of this?

Because here is what was genuinely disturbing about the situation.

Chemists kept noticing things, little coincidences that felt too consistent to be coincidences.

Lithium, sodium and potassium were all soft metals.

They all reacted violently with water. They all left behind similar compounds.

They behaved like the same kind of thing at different sizes Fluorine chlorine and bromine were all sharp -smelling gases.

They were all toxic They all reacted with metals in similar ratios again same kind of thing different sizes But nobody could explain why there was no framework.

There was no table There was just a growing list of elements and a growing pile of suspicious coincidences Then 1869, Dmitry Mendelev.

Mendelev is a chemistry professor in St. Petersburg, Russia. He is 35 years old.

He is trying to write a textbook not trying to reorganize all of chemistry just trying to write a textbook and he cannot figure out how to organize the elements chapter.

He tries sorting them by weight.

He tries grouping them by behavior. He tries columns. He tries rows. None of it works cleanly.

And then, on February 17, 1869, something clicks.

He lays out all 63 known elements sorted by atomic weight, and he notices that if he arranges them in rows of the right length, not the same length every time, but the right length for each row, elements with similar behavior line up in columns.

Sodium sits above potassium, chlorine sits above bromine, the violent metals are in one column, the toxic gases are in another, the noble, unreactive gases all the way on the right. The table organized itself. Mendeleev did not invent the pattern, he found it.

And here is the part almost nobody says out loud when they teach this.

Mendeleev did not know why it worked, not even a little, he had no explanation.

He just saw that it did work, and he was confident enough in the pattern to do something that most scientists would have been terrified to do. He left gaps.

Where the pattern said an element should exist but none had been discovered yet, he left a blank space and predicted the missing element's properties.

He wrote there is an element I will call echoluminum, it will have an atomic weight of about 68, it will be a metal, it will melt at a low temperature, it will form a certain kind of oxide. side.

Five years later in 1875, French chemist Paul Amille Lecoq discovered gallium, its atomic weight, 69 .7, a metal, low melting point, formed exactly the oxide Mendeleev predicted. Sit with that for a second.

A man predicted a substance that did not exist yet, described its physical properties from a pattern in a table and was correct.

That is not a coincidence, coincidence, that is the table working.

Now here is the question Mendeleev could not answer, and it is the question that, when it was finally answered, turned the periodic table from a useful pattern into a window into the architecture of matter itself. Why these row lengths? Why not 8 elements in every row? Why does the pattern reset at different points?

Why does row 4 hold 18 elements when row 2 only holds 8?

Mendeleev knew the pattern was real.

hill, he had no idea why it was real. The answer came 50 years later, and it came from a place nobody expected.

But here is the thing about that answer that matters for everything that comes next. The answer did not come from chemistry at all.

It came from physics, from a completely different field working on a completely different problem trying to figure out why hot objects glow the color they glow.

And the physicist who cracked it open was not looking for an explanation of the periodic table, he was not even thinking about chemistry, he was trying to explain why an electron does not fall into the nucleus.

That one answer from a completely different question turned out to explain every single row length in Mendeleev's table, automatically, with no adjustments needed.

That is the kind of thing that makes you suspect the universe was built on purpose.

Here is the honest, plain English version of what the periodic table is actually a map of.

Electrons, that is the answer, the whole table is a map of electrons.

Every element is defined by how many protons it has in its nucleus.

That is the atomic number that part is true, hydrogen has one proton, carbon has six, gold has 79.

But the electrons are what matter for chemistry, chemistry, because chemistry is what happens when atoms interact with each other, and atoms interact through their electrons, the protons are locked inside the nucleus completely unreachable, the electrons are on the outside, they are the hands, they are what touches.

So the periodic table is really a map of how electrons arrange themselves around a nucleus, and that arrangement is not random. Here is where it gets strange. range.

Electrons cannot sit anywhere they like around a nucleus, they can only exist at specific energy levels, specific, fixed, like the steps on a staircase.

An electron can be on step 1, it can be on step 2, it cannot be on step 1 .5, there is no step 1 .5, the steps are real, the space between them does not exist for electrons.

All that means Things is this, electrons live in shells, the first shell closest to the nucleus, the second shell further out, the third, the fourth and so on.

Now think of it this way.

Imagine an apartment building, each floor is a shell, each floor has a certain number of apartments, a fixed number, not more not less, when the apartments on one floor are full the next electron has to go up to the next floor.

The first floor has room for exactly two electrons, the second floor has room for exactly 8. The third floor has room for exactly 8.

The fourth floor has room for exactly 18. Now look at the periodic table again.

Row 1 has two elements, hydrogen and helium.

Those are the two elements where the first electron shell is filling up.

When helium fills that first shell, all two spots taken, the row ends. Next row starts.

Row 2 has eight Eight elements, lithium through neon, those are the eight elements where the second shell is filling up.

When neon fills that second shell, all eight spots taken, the row ends, next row starts. Row 3 has eight elements, row 4 has 18.

The row lengths in the periodic table are not arbitrary, they are the exact sizes of electron shells, the table resets every time a shell finishes filling.

That is why the rows are different lengths, that is why the pattern resets where it resets. The shape of the table is the shape of the electron shells.

And here is the part almost nobody says when they teach this.

This means every column in the periodic table contains elements with the same number of electrons in their outermost shell.

Hydrogen has one electron in its outer shell, lithium has one electron in its outer shell, shell, sodium has one electron in its outer shell, potassium has one electron in its outer shell. They are all in the same column, column one.

And that single shared property, one electron sitting alone on the outermost floor of the building, lonely, easy to give away is why they all behave the same way, why they all react violently with water.

Why they all leave behind similar compounds, why men to leave notice they felt like the same thing at different sizes.

They are the same thing at different sizes, they have the same outer shell, same hand, same reach. Not a coincidence, a structural consequence.

So now the columns make sense, the row lengths make sense, the basic logic of the table is clear.

But there is still the gap, that big empty space at the top.

And there are still those exiled elements at the bottom, the lanthanides and actinides floating beneath the table like they did something wrong.

We have not explained those yet.

And explaining them requires going one level deeper into the electron shell structure, into the subshells.

Stick with me because this is where the table stops being a useful pattern and becomes genuinely strange.

Now I want to be honest with you about where the clean story we just told stops being completely true.

Because the floors of the apartment building version is right, but it It is a simplification, and the simplification hides the most interesting part.

The floors, the shells are not just empty rooms, they have structure inside them, subrooms, different shapes of space where electrons can live.

And those subroom shapes are the thing nobody explains when they teach the periodic table.

Every single piece of the table's weird shape comes from them, the gap, the exiles, all of it. That is not a footnote, that is the whole answer.

Everything we said so far assumed that each shell is one simple layer, one floor, electrons fill it up then move on.

That is a useful fiction, it works for explaining the basic row lengths, it stops working the moment you ask why the table has a gap at the top or why 18 elements fit in row 4 when only 8 fit in row 2.

Here is what is actually inside each shell. Each shell is divided into sub -shells.

Think of them as rooms within each floor of the building.

Each room has a shape, not a metaphorical shape, a literal physical shape in three -dimensional space describing where the electron is most likely to be found.

There are four types of rooms, chemists name them S, P, D and F.

Those letters come from old spectroscopy terminology, sharp, principal, diffuse, fundamental names that made sense in the 1800s and are now just letters we are stuck with. Here is all you need to know about each one.

The S -room holds two electrons, one per shell, always exactly two.

The P -room holds six electrons, three per shell, available from shell two onward.

The D -room holds 10 electrons, 5 per shell, available from shell 3 onward.

The F -room holds 14 electrons, 7 per shell, available from shell 4 onward.

Now the table starts making sense at a level it never made before. Row 1 only the S -room exists, holds 2.

Row has 2 elements, row 2 S -room 2 plus P -room 6, 8 total, row has 8 elements, row 3 S -room Room 2 plus P Room 6, still 8, Rho has 8 elements.

Rho 4 S Room 2 plus P Room 6 plus D Room 10, 18 total, Rho has 18 elements.

Every Rho length in the table is just S plus P plus D plus F electrons added up.

No magic, no arbitrary decision, pure consequence of how electrons pack. Now the gap.

Look at the top of the periodic table, row 1 and row 2.

On the left you have hydrogen, helium, lithium, beryllium, then a huge gap, then boron, carbon, nitrogen, oxygen, fluorine, neon on the far right. Why the gap? Because in rows 1 and 2 there is no D -room. The D -room only becomes available starting at shell 3.

So when you are filling row 2 you fill the S -room first, lithium, beryllium, 2 elements and then you jump directly to the p -room boron through neon six elements. There are no d -elements in that row. The gap is the missing d -room.

It is not empty space. It is space reserved for something that does not exist in that row.

The table is showing you an absence.

In the exiled elements at the bottom, the lanthanides and actinides. Those are the f -room electrons. The f -room holds 14 electrons.

The 14 elements filling the f -room in shell 4 are the lanthanides, the 14 elements filling the f -room in shell 5 are the actinides.

They are not exiled because they are strange or unimportant.

They are separated because if you put them in their correct position inside the main table, the table becomes too wide to fit on a standard page, a whiteboard or a classroom wall.

That is the real reason, the table you have seen your whole life has those elements detached at the bottom not because of chemistry but because of paper size.

The true periodic table with everything in its correct position is 32 columns wide instead of 18. Sit with that for a second.

The most famous scientific chart in history has been rearranged to fit on a piece of paper and almost nobody who teaches it says that out loud.

There is one loose thread here that I cannot leave alone.

Everything we have said so far treats the electron shells as filling up in a clean, predictable order.

Shell 1 fills, then Shell 2, then Shell 3 and so on. Neat simple logical. But the real shells do not fill in that order.

Starting at row 4 the order breaks, electrons start filling Shell 4 before Shell 3 is completely finished.

The 4's room fills before the 3D room.

And when you remove electrons from these atoms when they form ions which is when the actual chemistry happens, they do not leave in the order they arrived.

That asymmetry filling in one order leaving in a different order is the thing that makes transition metals behave so strangely.

Why iron, cobalt and nickel all have similar magnetic properties. Why copper and zinc break the pattern their neighbors follow.

That is the next level, and it is where the periodic table stops being a map of structure structure and becomes a map of behavior. That is the next one. So go back to where we started.

Here is what every chemistry class says about the periodic table, it is organized by atomic number, left to right, top to bottom like a numbered list.

Here is what it actually is, now that we have taken it apart.

The periodic table is a map of electron shell structure. Every row represents one shell filling up.

Every column contains elements with the same number of outer electrons. electrons.

Every row length 2, 8, 8, 18, 18, 32 is the exact capacity of the SPD and F -subshells added together.

The gap at the top is where the D -subshell does not exist yet.

The exiled rows at the bottom are the F -subshell, moved there to fit on a page.

The shape is not a design choice, the shape is the physics. Now here is the chain of people who built this.

In 1869 Mendeleev felt the pattern before anyone could explain it.

He did not know about electrons, the electron had not been discovered yet, he just trusted the numbers.

He left gaps for missing elements and described what those elements would be like.

He was right, he was not trying to map electron shells, he was trying to write a textbook.

In 1913, Niels Bohr built the first model of electron shells working on why atoms emit specific colors of light, not trying to explain Mendeleev's table.

The shell model dropped out of his equations automatically.

In 1925, Wolfgang Pauli worked out why each shell holds exactly the number of electrons it holds, not because he was thinking about the periodic table but because he was trying to explain a strange result from spectroscopy experiments.

His answer, the Pauli exclusion principle, says that no two electrons in the same atom can have the exact same quantum state, they cannot share a room.

room. That one rule is why shells have fixed sizes, that one rule is why the periodic table has the shape it has.

None of these men set out to explain the periodic table, they were each solving something smaller, the periodic table was what fell out.

And here is what I find genuinely beautiful about that.

Mendeleev built a map without knowing what he was mapping, Boer built a model of atoms without knowing it would explain the map.

Pauli built a rule about electrons without knowing it would explain why the model works.

Three men, three different problems, three different decades, one shape, the same shape falling out of completely different work three times in a row.

That is not luck, that is the universe having structure we keep discovering from different directions.

The most useful chart in the history of science and it was not designed, mind, it was found three times.

Honestly, I think that is the most beautiful thing about it.

The table works perfectly. And the reason it works is still deeper than any version I have shown you today. That is the good stuff. That is why this exists.

If this rearranged something in the way you think about chemistry, subscribe because next time I want to go straight into the thing every chemistry class gets completely wrong from the first sentence, chemical bonds.

You were taught that atoms share electrons. They do not not the way you picture it.

What actually happens is stranger and honestly more beautiful than anything the textbook version ever told you.

That video is coming, I will see you there.

Now go find a periodic table on your wall, your phone, anywhere and look at the gap at the top and the two rows at the bottom.

I promise you will never see it as a chart again.