Deriving The Dirac Equation

Today I derive the famous Dirac equation. I talk about four-vector notation and contraction, as well as laying out the requirements for the anticommutator relation of gamma matrices.
Dot products of four vectors:
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Deriving the Klein Gordon Equation:
• Deriving The Klein Gor...

[@ShaunRL]

At 3:18 you said that g_{\mu u} p^{ u} = g_{00} p_{0} + g_{11} p_{1 }+ ... when it should be a four vector (g_{00} p_{0} , g_{11} p_{1 }, ...) not a sum

[@AndrewDotsonvideos]

You're right. I was trying to hit home that only the mu=nu terms survive, attached to their basis vectors, but writing it as the sum was not correct. Thanks for the catch!

[@deeptochatterjee532]

Instead of using the "Dotson" you can just do 1 lightsecond/second

[@Wattylearns]

ShaunRL Yes I totally understood that

[@16rumpole]

forgive my extreme math stupidity but where do you get e2-p2-m2 is zero?

[@chaoticstorm8145]

@@16rumpole Rewriting einstein's energy mass equivalence principle but with c=1

[@bryamayvar6590]

level 1 tensor boi tries to understand level 7 tensoir boi gone wrong xd

[@factsverse9957]

I am a Level 0. I understand absolutely nothing but enjoy looking at algebra.

[@maxwellsequation4887]

Andrew is now level ∞ tensor boi now

[@SuperMariocapo]

Even though I didn't understand a thing about this, I still watched the video because you explain everything in such a good way that you make it seem simple maths!

[@anuj7008]

Dirac says "Okay" from heaven. He is a man of few words.

[@user-wi6uf3xy6n]

Dirac was very sure that he was NOT going to heaven.

[@pauldirac808]

Kvothe o Sem-Sangue no I didn’t

[@noether9447]

@@pauldirac808 lmao

[@aniketeuler6443]

😂😂

[@fabiansepulveda7719]

Finally someone can explain this in a simple way! Thanks for all of your videos Andrew!

[@anubhav21dec]

I had loads of fun man. I might sound embarrassing but I'm gonna say it. When you finally arrived at the equation, I had goosebumps!

[@AndrewDotsonvideos]

Anubhav Mishra 😁

[@fatemehpouladrag]

Whenever I'm really desperate to drive a part of the equation somehow magically this channel have a video about it so thank you i wish your thesis going well.

[@Cloclo64]

That was great! I had missed the day when my professor had derived this in my particle physics class, but this clarifies a lot. I salute you sir

[@dcterr1]

Very good explanation! I like how you avoided using the alpha and beta matrices, which is the usual way I've seen it derived.

[@AndrewDotsonvideos]

thanks!

[@McRaylie]

Can we make the Dotson an actual unit, like how a Smoot is a unit.

[@sahhaf1234]

You definitely have to do this "interpretation of dirac equation" video and show us how spin naturally comes out as a consequence of dirac equation!!!

[@seanpaul2562]

I was learning quantum field theory but was having a hard time wraping my head around Dirac equation this video helped a lot keep up the good work

[@AdrienLegendre]

Thanks for video! This video is a common textbook derivation but there is a better first principals approach that is more correct and provides deeper insight. Quantum wave functions are representations for rotational symmetry in space-time; the simplest representation is a scalar, the next simplest is a spinor (spin 1/2), next is the vector (spin 1), etc. Spinors are the building blocks used to make vectors, tensors and other higher order symmetry representations. The chiral spinors (Weyl spinors) are contraspinors (right-handed spinors) or cospinors (left-handed) spinors; contraspinors make contravariant vectors and cospinors make covariant vectors. Parity means invert the spatial components of a vector; this corresponds to switching right-handed and left-handed spinors (right->left and left->right). A single chiral spinor generates a vector (momentum) and pseudovector (spin); the momentum represents a particle with no mass and the representation is not parity invariant. Use both chiral spinors (bispinor or Dirac spinor) and it is possible to generate a vector (momentum) representing a particle with mass and a representation that is parity invariant. This is a correct representation for an electron. Steane (see reference) uses these principals to generate the correct spinors and uses simple algebra to transform this solution to the Dirac equation. Sperança (see reference) generates a Dirac operator from the Dirac equation and shows that the Dirac operator is actually a parity operator; solving the DIrac equation means finding the spinors needed for parity invariance. References: Llohann D. Sperança. International Journal of Modern Physics D, 2018; An introduction to spinors, Andrew Steane, 2013, arxiv.org/abs/1312.3824.

[@syedmraza-ca]

Looked at this video a year ago, didnt understand much, but last winter break i read a book on tensor calculus and recently got into quantum computing and had to do some clifford algebra, now this makes complete sense!!!

[@joshyman221]

Excellent video, I’m fairly comfortable with 4 vectors after having done my general relativity course where I got used to thinking as space time as a pseudoriemannian manifold. However, I hadn’t seen this gamma vector (with matrix components) or seen how one applies ideas of QM in here and I found the way you explained these to be very clear and easy to follow. Thanks!

[@danielfishbein5492]

This was great! Thank you so much for making this! It was fun to watch and you are a great talker/teacher. I loved just listening to you talk and not needing to regurgitate it next week for an exam lol. I find many of my professors try to skip stuff and don't always write out all the smaller steps involved in these kinds of proofs. It was greatly appreciated that you lengthened your video and took the time to write everything out! Keep up the great work man!

[@wrestpinpeace]

Definitely enjoyed this derivation video. Looking forward to its Interpretation!

[@robertschlesinger1342]

Excellent video on the Dirac Equation. Paul A.M. Dirac was a great genius, and like many high functioning geniuses, was rather eccentric. A couple of credible anecdotes follow: A French physicist came to Dirac's home to discuss some cutting edge physics. The physicist was escorted into Dirac's study and he preceded for some time, trying with great difficulty to explain his work in English to Dirac. The physicist was clearly having considerable frustration with his limited spoken English. After quite some time, Dirac's sister, Betty, entered the study with some tea and biscuits, speaking fluent French, and wherein Dirac responded in fluent French. The French physicist who had spent considerable time frustrated in trying to express himself in English inquired of Dirac: Why didn't you tell me you spoke French. Dirac replied: You didn't ask. Another anecdote is from his days at Florida State University. The Physics Department held seminars which Dirac would often attend, sitting near the front row. He appeared to be dozing off throughout the presentations, but during the question & answer period, he would make brilliant comments and ask appropriate questions. He seemed asleep, but was all the while quite lucid.

[@samvelasco9231]

Me, currently doing Physics 1 classical Mechanics and Calculus II: What the fuckshit mickey mouse is this What the shitfuck daffy duck is that

[@jyoung5256]

Sam Velasco I vibe so hard as an Econ and Finance major

[@fedjkhimarouane2105]

this when quantum mecanics and special relativity want to mess around

[@tw5718]

Ive been watching a lot of these series lately. I watched both susskind series on gr and various diff geom series but i think ive made a lot more conceptual connections with this series. Muchas gracias

[@Halo2Trigate]

Man allergy season this year is crazy you were sneezing all video, bless your soul.

[@sanidhyasinha5735]

wow. Thank you very much. I was reading this from David Tong QFT lecture notes and I was little confused. But this video made it crystal clear.

[@JaxzanProditor]

This was really cool to watch and I think I actually understood most of it for once. Glad to have this famous equation finally make a little sense.

[@AndrewDotsonvideos]

Jaxzan Proditor 😁

[@johnchristian5027]

one of your best videos! nicely done!

[@meatballsnacker-sitregald6919]

I have no idea what most of these means, but sitting through the video to see the derivation and after all that approached at an elegant equation feels fking satisfying.

[@Fflurie]

"What's up smart people" Heheh, bold of you to assume that I'm smart

[@nitrozox212]

Derive the standard model lagrangian next please!

[@mappingtheshit]

It's in a fucking book, idiot

[@jeremy2719]

General of your mom aren’t they all in books. All he was requesting was to see him derive it there IS NO NEED TO SCOLD HIM. take a look at what you wrote was it even necessary

[@deeptochatterjee532]

@@jeremy2719 true, but I think more of the point is that the standard model lagrangian is way too fucking long

[@lPlanetarizado]

@@deeptochatterjee532 I read that long lagrangian has "parts": lagrangiano.blogspot.com/2020/01/lagrangian-for-standar-model-physics.html

[@wardippy226]

That is left as an excercise for the viewer

[@ronaldjorgensen6839]

thank you each time for your persistence

[@Devast8r34]

Big thanks for making math avaliable and fun! ❤

[@AndrewDotsonvideos]

😁

[@jerelfontenot1]

and suddenly, many years too late, that class on particle physics makes a whole lot more sense.

[@LordDarkhope]

If you have a formal math training your videos are much more understandable than other phys youtubers that hide the technicalities under the rag (i.e. ty mate really nice video)

[@AndrewDotsonvideos]

LordDarkhope really appreciate it. I just always try to address what use to confuse me

[@LordDarkhope]

@@AndrewDotsonvideos usually, when you try to explain something (that you don't understand) to someone that does not know anything about it, you eventually get it through the process of trying to put it into words. This is my experience with "education" at least.

[@MrGaugeBoson]

Exactly what I needed for my Introduction to particle physics course. Thank you :)))

[@davidsantosromacho2167]

i've just finished watching your video, and it was just amazing, thank you!!

[@jxchtajxbt53]

Excellent presentation. Only 1 typo at the anti-commutator relation for u v. Would have been nice to discuss the solutions. Great stuff.

[@henriquecoratozanarella6006]

I don't know why I'm watching this at 1am, but it's definitely worth it! xD (Also, since i watched this at 1am and understood it, it means you explain things really well. So thanks, and good job! 😁)

[@ChaineYTXF]

I struck gold today. Fantastic channel. Just subscribed. Keep this very interesting lectures coming they are quite useful. I second a suggestion below that the Dotson should be made a unit, but rather a unit of usefulness of a video on KZbin 😁 Thanks again.

[@Schrodinger.cat3]

I can watch you for hours bro please continue the relativistic quantum mechanics

[@MyTBrain]

Sweet! Concise and clear, dude you’re the best!

[@ethannguyen2754]

I understood more than I thought I would. Great job :)

[@kilianklaiber6367]

Great video and such a simple derivation it seems. Why didn't I think of it? Well, this just proves the quality of your explanation! ;-)

[@ゾカリクゾ]

This was simply awesome. Thanks for making this.

[@Amb3rjack]

I suspect Mr Dotson is a magnificent maths tutor because I found all of his explanation utterly fascinating whilst understanding none of it whatsoever . . . .

[@ShubhamBhushanCC]

I have a master's degree in Physics from one of the finest Universities in my country and none of my professors ever, I repeat ever derived Dirac equation. They just handwaved over it. Fucking hate that. Thank you for doing this

[@andygregory2390]

Good work ! Nice derivation with good amplification of some steps for those who have not seen it before. At the end you might have put the spinor indices back in and explained how they work, as this is a point of confusion for the novice. Oh .. and credited Feynman for his 'slash' notation.

[@CuriousPriyanshu]

U make Four vector algebra look easy. Grateful☺️

[@jacobvandijk6525]

Your mean, easy for you.

[@sahhaf1234]

An extremely clear presentation.

[@Wristonfire]

Thanks for making this video Andrew!

[@abdullasulfikkar5282]

Thanks a lot Mr.Dotson. Was in real need of this.

[@unearth3d]

Can't wait to Dirac n roll when i actually understand this better💃💃 great video, Andrew!!

[@Zaustie]

I have my final theoretical physics exam for my masters in a few days and this video helps me see the dirac equation from a different angle, thanks for the good video.

[@liamlau4558]

Keep it up Andrew, the physics majors at the University of Cambridge love your content

[@AndrewDotsonvideos]

Liam Lau 😁😁

[@daveconerly1450]

Oh baby, you left me way back there ....and your still going strong. As flammy would say , WTF ! It seems that your physics vocabulary has gone into the twilight's zone mode. Your education on said subject matter seems to have gone into warp speed and you have gone where you belong....... I haven't heard anyone talk in equation code for a long time. You have stepped up unto your plane of natural existence of being electrified into an extremist of sorts of a real Mr. Physics man . See you when we get to the other side of the universe. With your high knowledge you will definitely get there and back by the time that I begin. Your so speedy fast .

[@lordloneshadow7572]

At 10:30, you substituted alpha with gamma. Now gamma can only be equal to alpha for the same indices yet after your substitution they have different indices. This part irked me a bit, but yes it was very fun to watch. Great content man, and I love your style/

[@yaboylemon9578]

“Dirac and roll” don’t think I didn’t catch that.

[@dangernuke929]

The constant need for physicists to use shorthand notation can be so frustrating! There's already enough to memorize without having to remember what ALL of the shorthand notation stands for. LMAO. Anyway... AWESOME video! It cleared up a lot of things for me!

[@lincolndexter9514]

Dr. Dotson's teaching days of his youth. - 2019 colorized.

[@jkli6031]

I am thinking of a geometric interpretation of the Dirac equation, not sure whether it is right or wrong. Let first talk about some basic ideas of some terminology in math. Consider a space-time M (a manifold, with riemannian metric g on TM). Further consider a copy of a finite dim'l space S on each point of M (we denote the resulting space E), with projection p from E 一>M. We call the entire structure a fiber bundle p: E一>M over M with fiber S. A fiber bundle p: E一>M is of "principal" if S is a Lie group act on M( here, the Lie group is called the structure group). Consider a fiber bundle p:E一>M , with group G acts on the entire bundles, we denote the fiber bundle as an associated fiber bundle associated to a principal bundle where the fiber S is G . Interpretation: Consider the spinor bundle p:E一>M(each point of M is attached to something called a spin space, i.e. the space that characterize the spin of particles) over a space-time M, with maps Psi : M 一> E call the section of the spinor bundle. (Psi is mentioned in the video,it is also a 4-vector in space-time and its fiber spin-space)The derivatives operator is a connection tell you how Psi changes in the tangent space of space-time, and the gamma matrices tells you how the vector psi changes in spin-space (gamma matrices are rotation matrix in spin-space). the combination of the two tell you how the spin and the vector components in space-time of psi changes. It is just a guess... Apologize if I get something wrong. I am just an undergrad student.

[@ameenmahmood5672]

The only words I understood was dot product and vector, so I’m basically a fake grad student 😂

[@legendarylightyagamiimmanu1821]

Ameen Mahmood he didn’t even understand what the fuck he wrote after somewhere around 20:00 it all went to shit

[@BlueGiant69202]

Maybe you would benefit from Geometric Calculus and Spacetime Algebra to derive the Real Dirac Equation. geocalc.clas.asu.edu/html/GAinQM.html geocalc.clas.asu.edu/pdf/Observ-opers.pdf

[@BlueGiant69202]

I would be interested in Mr. Dotson's comments on the criticisms of Dr. David Hestenes regarding the Dirac equation in terms of Geometric Calculus and SpaceTime Algebra. geocalc.clas.asu.edu/html/GAinQM.html geocalc.clas.asu.edu/pdf/Observ-opers.pdf

[@aniketeuler6443]

😂😂

[@dean532]

16:05 it blew mine too. Years after graduation. Lol. TBH this is the most simplistic approach unlike those animated videos (which only print half of the information onto the screen) leaving us no room to think.

[@dean532]

Also when entropy is finally deterministic we can have Dotson as its unit. Are you Derek’s (from Veritasium) cousin?

[@AndrewDotsonvideos]

@@dean532 lol no but I get that a lot. Glad you found the video helpful!

[@lukesaunders4776]

Anyone who is interested in the Gamma matrices, read about Clifford Algebras

[@tomkerruish2982]

Ah yes, Clifford, the big red algebra.😂

[@duncanw9901]

Boutta do a sleep but I caught this boi 5 minutes after upload. Hecc.

[@AndrewDotsonvideos]

gotta watch all of it 3 times now

[@tuele4302]

I love hardcore physics videos! Thank you for making them!

[@69erthx1138]

@15:48 matrices appeal to our senses as generalizations of ordinary numbers. The fact that the cross terms, which rep interference, go to zero is counter intuitive. On the outside, this leads to the idea of spinors not being covariant objects. All the algebra aside, these core ideas are very intriguing.

[@isaiashernandez5265]

Damm i got a lot to look forward to in my physics career

[@FermatWiles]

Man, you're an awesome teacher!

[@evariste1686]

Really elegant in 20 minutes.

[@rc5989]

As an interested learner of physics, I decided to search for a vid on deriving the Dirac Equation. Of course, my dude Andrew Dotson has a derivation vid. Boom!

[@aniketeuler6443]

Physics is always a enjoying thing to watch

[@Jay-jn6ul]

At 15:25 I believe you meant to put the not equals between mu and nu (even though what you wrote turns out to be true). Anywho thanks for the great video! This makes much more sense than when my lecturer explained it

[@Gismho]

Very good explanation of a normally "somewhat difficult to understand" equation's derivation. Thank you.

[@fardeenrafiq]

I'm currently in ist year of my college and didn't even got a word but still enjoyed the whole video ♥️😍

[@jonhouck4981]

I basically understood most of it, I haven't done any work in tensors or four vectors before but I don't think it's much more complex than the 3 dimensional real space. I only have my undergraduate physics degree and so we didn't get Into the Dirac equation to much. Cool video by the way

[@jeannettendjemgang3691]

I fully understood it. It's wonderful for me

[@SR-kd4wi]

Next year when I get into uni, probably I'll understand your videos.

[@haarithio1621]

you wont be doing this for a while

[@Ryan_Perrin]

Yeah maybe when you're learning freshman string theory and the universe as a simulation during your second semester

[@ゾカリクゾ]

@@haarithio1621 i encourage you to learn it on your own. this is great stuff, i'm learning this outside school

[@shivamgakkhar9537]

absolute ripper great video by sir andrew dotson

[@acatisfinetoo3018]

I wanna see you derive the Maxwell equations!

[@darindotson2874]

Great job my cousin!!

[@gerardogutierrez4911]

I like how I have no clue what youre saying, then I take a certain class and im like, oh what andrew is talking about isnt that bad. Cant wait to go Relativity. Just finished 1st semester quantum mechanics and classical and EM. Do more EM videos cause I honeslty didnt get it the first time around.

[@TheZod00]

You lost me at 0:47

[@livedandletdie]

It's not that we get lost, it's that higher order mathematics sounds like someone insane trying to explain to you why there are ghost hiding in the 3rd trashcan on the left at the Science Museum in Lübeck, and that they all like Pineapple Pizza with Sriracha and old socks, but only on Fridays when Angela Merkel stubs her toes twice.

[@user-en5vj6vr2u]

@@livedandletdie indeed

[@stenarsk6877]

@@livedandletdie brilliant

[@Jaijitesh]

thanks for giving such knowledge

[@apolowalker4592]

Do in the next video a derivation of the Einstein field equations from the expected value equations in economics.

[@oldpariah]

Great explanation.

[@مصريملحدللأبد]

You are a genius. Genius

[@frankhedbergkindbom3436]

17:31 ”let’s s make sure this makes sense” Good luck with that

[@samueljohanes5219]

so cool. keep up the good work!!!!

[@dagimyishak6532]

Him: hello smart people Everyone else: not really

[@twistedsector]

I'm pretty sure you meant {gamma^mu, gamma^nu} *=* 0 for mu *=/=* nu

[@angelmendez-rivera351]

DiehardTheTryhard What he wrote is correct. {γ^μ, γ^ν} =/= 0 for μ = ν is correct because 2g_(μ, ν) =/= 0.

[@jacobwyngaard291]

Phenomenal Job!

[@eslamabdelrahman1653]

Man this is an awesome video ❤❤🙏🏼 the best 👌🏽

[@TranquilSeaOfMath]

Nice presentation.

[@quantumleap7964]

such a brilliant vid!

[@MrJdcirbo]

The part I understood the best in all this is setting c equal to 1. Since the speed of light is the max velocity, any velocity less than c is necessarily a percentage c.

[@angelmendez-rivera351]

WhoistheJC? Yes.

[@davidhidary]

Anyone else not know enough math

[@samvelasco9231]

Im doing Calculus 2 and Physics 1. What the fuck is all of this

[@chymoney1]

Sam Velasco tensor calc, Lie groups and algebra, differential equations and Clifford algebra and prob more

[@krishnasimha8097]

I'm in high school

[@robertmorrison1657]

@@krishnasimha8097 8th grade. Checkmate.

[@malawigw]

this is kindergarden math

[@debajyotisg]

Hi there, I really liked this approach towards the derivation. Might I suggest an alternative approach where you arrive at the same set of equation starting from the decomposition of Lorentz group into two commutating SU(2) groups? I find that very elegant.

[@sumsar01]

It's also way more advanced and requires at least some understanding of group theory. Maybe not where I would start if you have to write out tensors.

[@stefanmilicevic5322]

Slash notation, what kind of wizardry is this ? In all seriousness, i liked it how you derived the Dirac Equation.

[@wolframalpha8634]

That's legendary!! 🤘

[@macmos1]

Nice video... good refresher!

[@pancreasman6920]

That was extremely helpful, thanks

[@AK-km5tj]

Hey Mr. Dotson! I win! You’ll do tensor next right? Also can you do a small video on everything wrong with classical physics?

[@AndrewDotsonvideos]

Yes tensor will be next!

[@davidm6624]

@@AndrewDotsonvideos ehhm excuse me but no physics meme review will be next (pls)

[@Finkelthusiast]

Hey Andrew, I love this video! Do you think you can make the video you mentioned at the end about interpreting the Dirac equation? Would love to see how we can get antimatter from this