Electromagnetism as a Gauge Theory 

Hey everyone, thanks for checking out the video! :) Timestamps: 0:00 Intro - "Why is Electromagnetism a Thing?" 14:24 Dirac Zero-Momentum Eigenstates 39:10 Local Phase Symmetry 52:07 A Curious Lagrangian 1:11:43 Bringing A to Life, in Six Ways 1:27:40 The Homogeneous Maxwell's Equations 1:39:25 The Faraday Tensor 1:47:49 F_munuF^munu 1:53:05 The Lagrangian of Quantum Electrodynamics 1:58:14 Inhomogeneous Maxwell's Equations, Part 1 2:11:07 ... Part 2, Solving Euler-Lagrange 2:31:10 ... Part 3, Unpacking the Inhomogeneous Maxwell's Equation(s) 2:37:55 Local Charge Conservation 2:42:20 Deriving the Lorentz Force Law 3:01:25 Miscellaneous Stuff & Mysteries

"Sorry babe, I cant go to sleep right now Richard Behiel just uploaded a 3 hour video on electromagnetism"

Came for the physics. Stayed for the "pew pew pew" noises.

Best 3h spent this year. Thanks for the shout-out, Richard. The most digestible introduction to Abelian gauge theory I've seen.

Usuallyy when I'm this excited for a 3 hour youtube video it's an unhinged video essay about a game I've never played.

Do you know what you're doing? You are taking education to a whole new level, you are making a breakthrough in the methodology of education

Excellent! There is not enough rigorous and mathematical physics on youtube. Thank you!

The fact that educational content of this calibre is free in today's world is the most mind-blowing of all

When I was newer to gauge theory and first heard that "U(1) symmetry gives rise to E&M", it took me many painful hours to get to the bottom of it. This video does an amazing job organizing the story, and the visuals give me something to grab a hold of.

I feel like this video is a triumph for humanity. You’re making this topic accessible for way more people than could previously grasp it. And it’s so beautiful!

The “words of encouragement” section at 13:49 actually made me cry. I didn’t realize how much I needed to hear that. Thank you 💗

Legend is back with straight 3 hours quality essay .... GREAT MATE

The only time i can be awakend from my eternal slumber is whenever this guy uploads.

Having made a 2 hour video on theoretical physics with stock video/photos I paid for and text that Powerpoint animated as I recorded live in a sitting, I know for certain that what you achieved here is far beyond anything my patience is ready to produce. This is just absolutely fantastic. If anyone wants clarity on sections I covered regarding electromagnetism, this is the video I will send them to from now on. There's no question about it. This is the best current summary I know of on RU-vid.

why do you drop this the same day i finish my electrodynamics course😭😭😭😭

This channel is easily one of the best educational channels I've seen. The quality is off the charts, seriously. There are channels with 100x as many subs that put out worse content. Thank you for this feast

"every pseudovector is secretly a plane" *geometric algebra foreshadowing!!!*

Laying out exactly how you get the E and B fields from the potential/ E-M field tensor, whilst seemingly obvious just minutes after you laid it out, was really impressive - which i think is the sign of a great teacher. Also im annoyed at how i think i will forever remember the way of the 6. Very clear and elucidating all around. Great video so far, thanks for the awesome content.

All I did over 4th of July weekend was sit inside by myself and try to unpack these equations. Best summer holiday. Thank you RB

I think I make this comment on just about every one of your videos, but I want to continue to express my thanks that someone is making videos that lean into the technical/math aspects of these topics.

This answered a LOT of questions for me even though I don't have the mathematical background to follow 90% of it. It really just helps put things in order while contemplating the nature of reality. Thank you sir!

02:36:50 - it's beautiful. The longest lecture I've ever had, but interesting enough to keep me viewing to the end. I paused the video on each statement wich I have to checkout. And the part with derivatives is the easiest part. Man, this video is a lot of work.

I am a second year engineering student who's looking to specialize in physics. Your videos on Schrodinger's equation ignited my interest in quantum physics, and since then I have been spending much of my free time trying to teach myself. Now, just as I've finished Griffith's Introduction to ED and have been searching for a good follow up to learn about gauge theories, you release this. Moments of serendipity like this never cease to amaze me.

Probably the most anticipated/happiest youtube upload of my life, I have been researching all about EM as a gauge theory since the spinor video (I was even reading about it yesterday night!). Can't wait to watch this 3 times! Your content is invaluable, thank you so much

You made and uploaded a 3h video, which is basically a full lecture. Just that shows how amazing this channel is (also your videos are amazing lol)

During my studies of Electrical Engineering, I was never under the impression that Maxwell's equations have such deeper meaning! Excellent video!!

Great video. Dark background is a huge plus makes it easier to learn

I stumbled across your video on spinors a few months back, leading me to watch your full series on quantum physics. After doing some more research and developing a better intuition, I came back to your spinors video and suddenly everything just clicked. This has reinvigorated a curiosity for physics that I haven't felt in years! Thank you for creating that pedagogical masterpiece. Now I'll have to take notes on this one for the next couple days as well.

As someone who has a bachelor's in physics and math but didn't go further, this video is perfect for me. It makes a complex graduate topic that is tough to self study a lot more accessible. Thanks for the effort you make to create these videos. There is something truly beautiful about this topic.

Once again, Mr. Behiel, you've absolutely knocked it out of the park. I am consistently stunned and blown away by the sheer quality and care of your presentation. I say this as someone who has passionately studied physics on my own for all of my life: This video, and all of your other works, are utterly invaluable. I have been studying the topic of "the U(1) gauge theory of electromagnetism" for at least 4 years now, even to the point that I am currently diving deep into connections on principal bundles and covariant derivatives on vector bundles just so I can finally wrap my head around "just how all of this machinery really fits together", so it really says something that this video still actually taught me a lot. It's given me a lot to think about and ponder. I love the way you give intuition-focused breakdowns for the different "moving parts" in both the equations, and the physical situations. The animations are incredible too. The way you blend the rigorous, algebraic expressions, with reference back to the "what is really going on in the physical universe, what does this mean", is truly invaluable for really extracting insight from these often opaque, yet inherently beautiful processes. The way you illustrate the journey... the usual narrative from "demanding local phase invariance of the Dirac field", to "well, local phase transformations result in the addition of a spurious term to the Lagrangian", to "what if we added something that canceled that out?", and so on... this journey is one that has been presented many times before by great educators, but you show the path in its entirety, in a very lucid and well-composed way, and I am deeply grateful for that. More amazingly insightful to me, is how you showed the way that the four-potential is "brought to life" by, apparently, the fact that that antisymmetric quantity (the Faraday tensor) is apparently "immune to" or "untouched by" local phase transformations. And thus, these "wigglings" are allowed to "scrunch up freely". I don't believe I have ever actually thought about it that way before. It will be very interesting for me to investigate later how this figures into my understanding of F as the curvature 2-form of the connection 1-form A on the principal U(1) bundle. In short, this video is nothing short of a masterpiece. I wish I could go part-by-part and really get into just exactly what I love about it. For now, it will suffice to say that this video is an invaluable resource for people like me. Your lucid, detailed-yet very well-paced and friendly-leadings-through of these physical and mathematical notions, vividly exposing the beauty inherent in the "mathematical machinery" of physics, are truly a gift. Thank you for this kindness. Your work is helping people. I hope you have a wonderful night and/or day.

Literally about to write an essay on how gauge fields describe interactions - ABSOLUTE LEGEND 💯

Seated and ready! That you would educate us this deeply is amazing. Thank you.

For those people that are thinking about watching this video but dont wanna commit because its 3 hours long. It is super worth it. Really great video, it doesnt even feel like 3 hours dont worry :)

Coming here as a rising senior studying Electrical Engineering, I was not expecting to be out of my depth within the first 10 minutes 😂 Much love! I can’t wait to revisit this after reading up on that textbook you mentioned and watching your other videos!

Love the relatable “you can do it” style of presentation and the amazing graphics. Doing the algebra in public is rare for RU-vid but combining it with the geometry from the graphics is just what students need to connect the dots and develop intuition. While you’re driving the station wagon through the Grand Canyon of U(1) gauge invariance, it might be nice to take a side jaunt into Diracs magnetic monopole solution, with Wilson Loops and homotopy classes. Another fun side jaunt might be an intro to the path integral formulation of QM ala Feynman’s QED the A Strange Theory of Light and matter. As with most summer vacations it’s hard to say if the station wagon will be roadworthy next year. Anyways, thanks for your content. 😊

Wow, this is amazing. I’m a pure math guy, and there’s a reason I skipped E&M but took classical, quantum, and statistical mechanics at university. This was an excellent derivation, thank you. The one thing I still have to wrap my head around is the fact that the “coupling” of fields seems to be asymmetrical when we minimize the action? I hope that this was a presentation choice for the video rather than a mathematical choice. Meaning if you look at the full QED Lagrangian without a priori knowing what psi is, but having the same initial condition, then one can first solve for psi as an electron and then solve for the photon field with psi fixed as was done in the video. Edit: I am definitely being dumb. I see the derivation is independent of psi and the dependence is the current J. It is hard to be locked in for 3 hours straight! Again, this was amazing. Thank you!

Halfway through this video and I have to stop it because my mind is blown. Omg. This is crazy. Reminds me of a really good prof who explained an entire confusing course to me in 2 hours and suddenly it all made sense. You have a gift for making complex things understandable. Please keep making videos! (I would love to see you cover the component concepts in more detail too as I bet it would greatly add to my intuitive understanding.) Can't thank you enough...for explaining the entire universe to me.

I've been looking forward to this. Will save it for later, but this series has been very helpful in the technical side of my journey to understand QM/QFT.

Imagine never hearing about electromagnetism and this is how you're introduced to it

this kind of stuff should be put in universities

Just watched it once, loved every minute of being reminded of grad school. Going to watch it again to take it all in better :D Really really amazing presentation of such beautiful concepts and mechanisms.

Having learned EM from Feynman's lectures on physics, I have been over this ground from early days, but you have added a level of detail that is both new to me and very satisfying as well. It was also delightful to see professor Dirac on video, whom I had the good fortune to meet in life. The impression I get from your presentation is that "requiring" local gauge symmetry directly(!) gives rise to how (Dirac) charged particles interact. By the way, I have always been inordinately fond of the Dirac delta function, and I notice that you repeatedly took advantage of the Kronecker delta function (without explicitly mentioning it). Very elegant how it crops up, nicht wahr? ~~~~Arthur Ogawa

Damn, and I thought that the ~1 hour spin video was long. You're making extremely hard concepts intuitive and understandable, and I firmly believe that this is the best resource on the internet for both graduate and undergraduate physicists to fit what they've learned into intuition. I hope these videos keep coming, honestly your channel has become one of the biggest sources of my physics learning for me. I'll go become a "patron" now, and I suggest anyone who benefits from these videos to do the same, because the audience for a 200-minute Gauge QFT video is pretty limited for Richard to self-finance from, I suppose.

I love this channel, it's both incredibly insightful and amazingly playful in the explanation of very profound physics and math. Like the whole bit about the **Six Ways** and "2B or not 2B" had me grinning from both the humor and the excellent presentation of these ideas.

This is definitely the most informative RU-vid video I've ever seen! Love your style how you go through complex calculations with colorful animations :) Wish my physics profs at university would have teached in a similar style. Keep going with these videos, I'll watch every of them!

> Over three hours EM vid comes out right as I develop a bunch of questions about it. Hold my algebra, I'm going in.

This is simply awesome. Finally, after years of looking for a complete dissertation, there it is! Love this

Thank you Richard. Great contribution! Also, love seeing your excitement for the math and the connections to our physical world

When I took a QFT class we derived F in class and then got deriving the maxwell equations as homework. It felt really weird. At this point in a physics degree euler lagrange and index raising are second nature. So you do the symbolic manipulation dance you're so used to, not really thinking about anything and pop! there is your entire theory of classical electromagnetism. It's like magic! There are 2 distinct other moments I remember as feeling the same: 1 - When you first do rot rot B and see a wave pop out 2 - Taking dirac or klein-gordon in rindler or schwarzschild metric, expanding to get the form of a schrödinger equation and finding + m*g*h + O(1/c2) at the end of the potential. Btw this video was amazing! I didn't expect to stay for the full 3h, but you've really managed to make this fun even for people who are reliving rather than experiencing the intense joy of seeing all of this play out. My only comment would be that it'd be nice if you wrote out the metric tensor and sums explicitly. I think a lot of "magic" regarding the co and contravariant things has happened here that make being very comfortable with SR a requirement here where more explicit notation would probably make things easier for people who only have some familiarity rather than full immersion in SR. Also, writing out the gs explicitly makes it much easier to see how gravity will end up effecting things. The sum symbol has less of a pedagogical justification and is just something I've seen many people get confused by where a simple symbol would clear things up.

I'm rewatching your entire catalogue so I can fully contextualise this video. I'm a PhD student in physics. I can feel a level of intuition building with QM fundamentals because of your videos that I've only ever felt with classical and statistical physics until now. I'm HYPED!!!

Omgg let's goo! I'm soo exited to watch this video with my mom (she was also a science student and has done electrical engineering) right after my exams finishes! I know this will be a really great video! This man is literally the best person in the world❤️✨️✨️ Love your channel Sir 🤍

It's so weird how as I'm now getting interested with gauge theory, you upload this banger

I love how you seemed to have a great time putting this together! This was the best overview of advanced EM I've seen and helped me understand what my graduate EM courses tried to impart (thanks, Jackson! not)

Richard this is unbelievable as always - I can't imagine the time commitment it must take for you to produce something of this quality, not to mention your explanations are always excellent which makes the topics easier to understand. Thank you so much for making this video, along with all the others you have made!!! 3 HOURS!!!

This is probably the best breakdown of this topic I've seen. I'd love to see you cover SU2 Electroweak theory someday, and how the weak force manifests from it. I always find things like neutrino scattering through the weak force a bit confusing, as well as what weak isospin and hypercharge actually DO

Awesome video, had to rewatch all the earlier videos to refresh my memory and I have to say it really helped me digest this 3 hour journey (or whatever I was able to internalize of it :D). One question that popped to my mind is how would magnetic monopoles change things? The way that Maxwell's homogenous equations were derived seemed to clearly rule out any possibility of magnetic monopoles, whereas in my studies, where Maxwell's equations are just given and not derived one could easily formulate them again to add the existance of magnetic monopoles if they were ever found. And Paul Dirac himself proved that if magnetic monopoles do exist, electric charge is quantized. But now it seems that Dirac's own equation rules out even the possibility of magnetic monopoles?

Watching this masterpiece while solving problems on random walker and statistical mechanics and i rarely comment. BTW Your videos also helped me in advance quantum mechanics.

FINALLY. I've always wanted a in depth explanation on this after so many years. It seemed like no one would ever give an example that would flesh out such a profound discover this is. As a chemistry undergrad it seems that nearly the entire field is ultimately dictated by electric charge, and I wanted to know on a deeper level where that came from. I am so glad you acknowledge that accepting something like space/time/matter is far different from accepting electromagnetism, I have always thought the exact same thing. Its almost felt like something "on top" of spacetime, and it needed more justification. Also, it's never touched on how this is so fundamentally different from many other applications of math within physics. Here the physical reality almost seems as just a manifestation of pure math. Incredible video!

Richard, Congratulations. This is the best presentation I’ve seen on the topic. The graphic animations really add a lot. Thanks. Bill

Having watched most of this now, I can say that this is a great introduction to many concepts. I think this is the first time I'm starting to understand what a Lagrangian is. Which I may not have tried to understand explicitly before, but whenever I was looking up other more specific topics, people would keep bringing it up without explaining what it is and it made things very difficult. Having this introduction here is very great and I feel like it's a bit of a primer for jumping into even more complex topics that use a lot of these concepts however (Lagrangian, gauge symmetries, vanishing nudge factors). Before this, QFT was this daunting topic that I really had no way of probing since I had no idea what the mathematics were (and there's really no quick way to look them up). This video isn't about QFT and yet on top of its actual topic, I feel like I finally have a path towards it since the vague interaction terms or mass terms that people bring up every now and then now make sense in the context of the Lagrangian. I still don't really know much more about QFT than I did before, but I know what I need to do to start getting there and that's applying second quantisation (however that works).

Best way to start off the week. Bless you!

Man, just discovered your channel, it's amazing.

been preparing for this video by watching and re-watching all of your other quantum physics videos!

I would be grateful if you could make videos of this type that are really complete and wonderful and in terms of time, the person can communicate with the subject. It is a wonderful method. Such works are permanent.

This video is absolutely incredible. After an undergraduate particle phys class this is an incredible explanation of what I’ve seen

Wow! A video tuned just right to my level. Really appreciate videos at this level. Most are too simple and just words or for advanced grad students in quantum general relativistic theory

This is the first time I've given someone a Super Thanks, but for this video, a like simply wasn't enough. Thank you for spending countless hours creating this. I'm only 19 and have wanted to explore quantum mechanics beyond Schrödinger for a while now. This was incredibly insightful and "easy" to follow. Thanks a lot!

OMG, so excited to work thru the whole video. Thanks for the upload

Word for word, what a bliss of a content! Instantly subscribed!

Hello, I just found your channel and 30 seconds in has already earned a sub. The quality is astounding!

You might not really see this, but I’m a massive fan of all of your videos. I just love how rigorous they can be and how passionate you are about the topics you deal with, as well as the way you deliver everything in such compact detail and animation. They remind me of 3Blue1Brown but I honestly prefer them because they deal with mainly physics rather than strictly mathematics and they give such a beautiful visual intuitive understanding for the topics you talk about. They also just make me want to learn so much more to be able to understand them to the full extent so that the effort you put in has complete value. As a person who really enjoys mathematics and physics but wants to get into biochemistry, I’m quite conflicted on how much time I should dedicate to mathematics and certain branches of it (I would argue physics is a branch of mathematics in a way). Do you or anybody else have advice on this ? There is just so much damn knowledge out there and not nearly enough time to be separated for it all to get a comprehensive understanding. Is it worth that extra mile effort, or is it really just impossible to fully understand mathematics on a deeper level if careers conflict ? I have been slowly doing self study for various topics regardless, but I often question the ultimate value of what I’m doing because all of them are very hard (quantum mechanics is hard, go figure) and not really part of my curriculum or range at all outside a surface level, and wonder if there is something I can focus on that will consistently be in line. TL;DR : I honestly hope to one day be able to fully comprehend your videos, not just intuitively which anybody can get if they have encountered enough background material before watching your videos, but truly understand how elegant these videos actually are. Regardless of wether or not this is directly involved within with a certain career or not, it is just too beautiful to not learn about and contemplate continuously.

THREE HOURS!!?? YOURE THE GOAT! THANK YOU

Amazing 10/10 ✅✅✅. Every classical field theory student should watch

Beautiful. Two thumbs up for this trend of making animation source codes available on patreon or elsewhere. Someone should update some Wikipedia pages with these animations.

all of physics is coming together before my eyes!! this is such a fantastic video

Richard, this video is a masterpiece!! Love how you continue to improve your performance, i sincerely wish you best of luck and that your enthusiasm will never cease. Btw it would be cool if you cover the rest if fundamental interactions, like weak or strong forces. Pretty please...)))

Incredible video. You gave an explanation that appealed to basic intuition but you were not afraid to get into the math. Thank you so much for making this.

Just wanted you to know your are getting through. I have limited aptitude and work as a programmer, musician, and entrepreneur. I respond well to effective teaching and not so well to the rest. This enriches my understanding of the world and rejuvenates my admiration for rigor and tradition in science. I have not made this comment to any other presenter. I am sure that I speak for others when I say that this channel is appreciated.

I'll be waiting for that future electroweak video, even tho I won't understand a thing about it!

You are my favorite youtuber. Even though ive only finished my first year of college ever and theres a lot of academic jargon i dont understand, i still find physics and your videos incredibly fascinating!! Thank you for being inspirational

Yeah, this is a masterclass for introducing particle physics. I have a BS in physics, and this is just the right amount of challenging and doable that I like. Thank you for making this.

Thank you for this! I agree with you: this is just stunningly beautiful; I can't get over it. I learned this stuff a million years ago in graduate school and, having left physics, haven't thought of it since. This detailed derivation, with animations, was much easier to follow.

YESSS finally three hours of beautiful work, thank you Richard Behiel!!!!!!

Awesome job! Makes me studying all of this again before deep diving into the QFT rabbithole

My heart sank when you got to "Contravariant Aμ, covariant Aμ". I have tried to read books that use those superscripts and subscripts and gotten absolutely nowhere. Would love a followable intro to that. (Which I suppose amounts to asking, "teach me relativity." Apologies for that.) EDIT: OK, wishlisted the book you recommended after that. Thanks. With any luck, I'll be up to speed in 2 years!

I am about to get a Master's degree and am applying for a PhD in physics. I have studied QFT for a few years now. While none of the information here was new, of course, the explanation is tiptop. It seems like the perfect thing for someone getting a bachelor who is thinking of getting into QFT for their masters, a perfect balance of handwaving and rigorousness. 10/10 no comments

this is gonna take me a month to understand all of this wow 3 hours is insane amount of work with your quality of editing

This helped me understand some derivations much better. Thank you for making it.

Really excellent explanations, I've spent many years off and on learning this material and this was a beautifully unifying exposition. Taking physics classes this material is often spread out over multiple courses and its hard to see the big picture.

1:26:00 the geometry of the “6 ways” is actually really pleasing and powerful, one of the many treasures discovered in your learning adventure 🤩

43:45 ART!

This is wonderful. Thank you for your time and effort.

Amazing video. Great job at putting all this together and making it easy and accessible. Just as a couple of details, the reason for the homogeneous equations has nothing to do with the lagrangian structure on the kinetic EM side. Even if you include higher order terms you'll recover the homogeneous equations, as they come from the geometric structure of the Faraday tensor as a closed 2-form. Essentially, the homogeneous equations are the Bianchi identity of the Faraday tensor. Check, for example, the Born-Infel theory of EM. It is a classical modification at high energies to avoid singularities on the fields due to point charges and it is related to some determinant of the metric plus the Faraday tensor that, in perturbative approximations include the extra term that you mentioned. This theory still holds the homogeneous equations and only adds some constraints to the intensity of fields at high energies. On the other hand, you're mentioning constantly that this is the QED lagrangian, which is true but imprecise. QED uses this lagrangian, but the quantization comes after. Without quantization, even the Dirac spinors are a classic theory. Still, asside from this two comments, amazing video 😊

Thank you for making these coherent videos!

Oh wow! 3 hour video! You should join SoME! You'll obliterate the competition edit: AND IT IS FOR GRAD STUDENTS??!! HURRAAAYY

As someone with only highschool level of math knowledge, I was able to follow along and engage with many of the concepts you described in this video, thanks to your methods of presentation and visual aid. Thank you!

Really excited to be able to understand this video in a couple years

The way you present the Six Ways is EXACTLY the vibe in my head I had when reading Griffiths' explanation of The Eightfold Way I love it; I hope that was also your inspiration

Bravo! This is absolutely fabulous! Sometimes animations are worth 1000 equations! It's been decades since grad school, but I don't think I ever really "grokked" this stuff, even back then.

OMFG. Amazing. Mind blowing. You just rocked my world.