Deriving the Dirac Equation

Поделиться
HTML-код
  • Опубликовано: 13 май 2024
  • In this video, we'll derive the Dirac equation, and see where it comes from! :)
    Recommended reading: Introduction to Elementary Particles, by David Griffiths, Chapter 7.
    Equations from the videos are available as downloadable PDFs on my Patreon. I'll also be on there to answer any questions you might have.
    / richardbehiel
    Chapters:
    0:00 Intro
    0:38 Three Principles for the Dirac Equation
    3:12 Square Root of the Mass Shell
    7:30 Anticommutation Relations
    9:50 The Dirac Matrices
    10:58 The Dirac Equation
    13:58 Spinors
    #physics #quantum #relativity

Комментарии • 290

  • @RichBehiel
    @RichBehiel  5 месяцев назад +39

    Hey all, thanks for checking out this video! :)
    While reviewing the video just now, I realized I probably should have been more clear about covariance and contravariance. This affects the sign of the momentum terms (more generally, the space-like terms), and is a perennial source of dropped minus signs 😅 The plus signs at 13:20 are due to using the contravariant Dirac matrices, three of which which absorb the minus signs, see the Wikipedia article “gamma matrices” for more info.
    Anyway, the main idea I hope you’ll take away from this video is that the anticommutation relations for the Dirac matrices arise from taking the square root of the mass shell.

    • @piradian8367
      @piradian8367 5 месяцев назад

      I guess it'd be better to pronounce that plus sign before mc at 12:44 is due to conjugation of an i at the denominator.

    • @helicalactual
      @helicalactual 4 месяца назад +2

      ok, so are spinors, 720 degrees to return to its "start", due to, 360 degrees of spin in the electric and then 360 degrees of spin in the magnetic fields, which would then return to its start?

    • @BiswajitBhattacharjee-up8vv
      @BiswajitBhattacharjee-up8vv Месяц назад

      The 1st order space-time variables as you pointed as breathing motion is non commutative of matrices makes space-time an illusion. And the spinors as real .
      Is it ?

    • @SpotterVideo
      @SpotterVideo Месяц назад

      What do the Twistors of Roger Penrose and the Hopf Fibrations of Eric Weinstein and the "Belt Trick" of Paul Dirac have in common?
      It takes two complete turns to get down the "rabbit hole" (Alpha Funnel 3D--->4D) to produce one twist cycle (Quantum unit).
      Can both Matter and Energy be described as "Quanta" of Spatial Curvature? (A string is revealed to be a twisted cord when viewed up close.) Mass= 1/Length

  • @SirTravelMuffin
    @SirTravelMuffin 5 месяцев назад +129

    The "I'm letting the squares breathe..." was genius for geometrically visualizing variables!

    • @bencrossley647
      @bencrossley647 5 месяцев назад +4

      I'm a maths teacher, I've hand waved this idea so many times. I wish I could make animations like this XD

    • @gaHuJIa_Macmep
      @gaHuJIa_Macmep 4 месяца назад +2

      Well, I find breathing squares neat but don't quite get what this breathing is intended to convey...

    • @RichBehiel
      @RichBehiel  4 месяца назад +8

      It’s just there so we’re not tempted to try to pick a specific arrangement of aspect ratios, then try to cancel two rectangles with one rectangle, for example. The breathing (variability) shows that each rectangle has to cancel out with its own symmetric counterpart, since that’s the only way the solution will hold in general.

  • @jim2376
    @jim2376 5 месяцев назад +134

    Dirac was known for both the brevity and precision of his speech. At a conference he was lecturing and writing at a blackboard. A member of the audience said at the conclusion of Dirac's presentation, "I don't understand your equation in the upper right." Dirac said nothing and returned to his seat at the table with other lecturers. After an uncomfortable silence, the conference moderator asked, "Mr. Dirac, are you going to answer the man's question?" Dirac: "It wasn't a question. It was a statement."

    • @RichBehiel
      @RichBehiel  5 месяцев назад +33

      Classic Dirac 😂

    • @veronicanoordzee6440
      @veronicanoordzee6440 5 месяцев назад +11

      Yes, he was a special man. Meeting Werner Heisenberg for the first time, his first question was: "Do you have an equation too?"

    • @sleepycritical6950
      @sleepycritical6950 5 месяцев назад +11

      @@veronicanoordzee6440wasn’t that Feynman?

    • @veronicanoordzee6440
      @veronicanoordzee6440 5 месяцев назад +4

      @@sleepycritical6950 Yes, could well be.

    • @gaHuJIa_Macmep
      @gaHuJIa_Macmep 4 месяца назад +7

      @@sleepycritical6950 No-no, Dirac proper. Feynman wasn't so pedantic, he played drums, picked up girls... and all that...

  • @JakeFace0
    @JakeFace0 5 месяцев назад +36

    A thousand jargon-filled wikipedia articles could not have given me the clarity I now posess thanks to this video. Thanks so much

  • @ralphclark
    @ralphclark 5 месяцев назад +9

    When the math is shouting “PARTICLES AREN’T FUNDAMENTAL, THEY’RE EMERGENT, YOU’RE JUST DOING EPICYCLES AGAIN”

  • @eg5731
    @eg5731 5 месяцев назад +26

    Straight-up the best physics videos on youtube

    • @GregorShapiro
      @GregorShapiro 4 месяца назад +1

      At least the best on Dirac and mentioning spinors!

  • @bastiaanvanhoorn6306
    @bastiaanvanhoorn6306 Месяц назад +3

    The teacher I had for QFT was one of the best teachers i ever met, and yet, this is so much more insightful then what he ever managed on a blackboard. The visualisation is genius.

  • @FunkyDexter
    @FunkyDexter 5 месяцев назад +41

    The reason the coefficients don't commute (so the opposite breathing squares cancel out) and why the coefficient squared are negative numbers, is because a spinor is a unit quaternion! In fact, the nature of spin is a bit less mysterious if instead of using Dirac matrices we use quaternions, for rotations on a 4D hypersurface :)

    • @philipashmore
      @philipashmore 5 месяцев назад +5

      You beat me to it by 5 hours :)

    • @HA7DN
      @HA7DN 5 месяцев назад +7

      When I saw the constraints I just jumped up from my chair and shouted "quaternions!". My head is now spinning, trying to decide the difference between physicists and mathematicians...

    • @JerkoFlapdoodle
      @JerkoFlapdoodle 5 месяцев назад +10

      even better, a Real 3D Clifford Algebra, where the BiVectors e1e2 = i, e2e3 = -j, and e3e1 = k
      All spinors are rotors, and all rotors are double covers of their respective rotational range.

    • @TheBlindfischLP
      @TheBlindfischLP 5 месяцев назад +5

      It's much less mysterious then, because quaternions are known for how easy they are to understand.

    • @FunkyDexter
      @FunkyDexter 5 месяцев назад +5

      @@TheBlindfischLP the basics of quaternions are not hard. Their more advanced uses get a tad bit more complicated, but so do complex numbers. The important thing is that once you use quaternions you know you're dealing with a rotation.

  • @Sol-En
    @Sol-En 5 месяцев назад +13

    Wow this lesson is much better than in Oxford or Harvard. Very interesting to see visualisation of solution of Dirac equation to compare with Klein-Gordon and Schrodinger equations. Also very interesting to see how do the wave functions in the bispinor responsible for spin up, spin down, and antimatter affect each other

  • @evilotis01
    @evilotis01 5 месяцев назад +5

    eigenchris's "Spinors for Beginners" series is a really good introduction to, well, spinors for beginners!

  • @natecoad2258
    @natecoad2258 5 месяцев назад +7

    Wow. I have devoted all of my time to getting into GR and have only had a basic rundown of QM but have wanted to get into the more interesting QM stuff for a while. You did a great job and I love hearing your excitement for introducing new ideas. That was perfect and felt so clean. Great stuff!!!!

    • @RichBehiel
      @RichBehiel  5 месяцев назад +1

      Thanks for the kind comment, and I’m glad you enjoyed the video! :)

  • @ThomasGutierrez
    @ThomasGutierrez 5 месяцев назад +7

    Brilliant content! Looking forward to the next installment on spinors!

  • @pelegsap
    @pelegsap 5 месяцев назад +12

    Absolutely terrific :)
    btw, here's a tip for everyone who sees this: geometric (Clifford) algebra really helps understanding these spinors better (and also space-time in general).

    • @JerkoFlapdoodle
      @JerkoFlapdoodle 5 месяцев назад +3

      the spatial spinors are quaternions, or the 3d real geometric algebra where e1e2 = i, e2e3 = -j, and e3e1 = k. The time-spinor gamma_0 is more mysterious, as it is not the trivector e1e2e3, because that squares to -1 also, not +1.

    • @comradejesus3022
      @comradejesus3022 3 месяца назад +2

      I second this! In particular, the gamma matrices have a one to one correspondence with the basis vectors of spacetime algebra (signature Cl(1, 3)), which are also often written as γ0, γ1, γ2 and γ3, making the conversion even more convenient!

  • @dialgapalkia
    @dialgapalkia 5 месяцев назад +8

    This sort of accessible education will prove to be revolutionary.

    • @RichBehiel
      @RichBehiel  5 месяцев назад +3

      Let’s hope so! :)

    • @derickd6150
      @derickd6150 5 месяцев назад +2

      Yeah I did physics and so I had seen the reasoning he describes in the video. However, watching it, I crave going back and reading those notes now. At the time they seemed so insane. Now I think it would be a joy to read and I think all people going into physics in the future are going to be so grateful for the existance of videos like this :)

  • @plexiglasscorn
    @plexiglasscorn 5 месяцев назад +16

    I think this is the clearest picture of Dirac equation and I have minimal training in math and physics 😊

    • @CHp-up9tx
      @CHp-up9tx 5 месяцев назад +2

      so true, im just at my first semester and i was able to understand most of the things, this man should be proud of himself

  • @Phantores
    @Phantores 5 месяцев назад +13

    I never heard of this way of deriving the equation and now the matrices seem so much more understandable... Congrats dude, you just did the impossible of explaining all of this stuff I tried to understand for months

    • @RichBehiel
      @RichBehiel  5 месяцев назад +4

      That’s the best possible comment I could get on a video. Thank you! :)

    • @mayatrash
      @mayatrash 5 месяцев назад

      You should read “quantum field theory for the gifted amateur”, it’s a texture newish book, but it has a great way of teaching these concepts, they also start with first deriving the KG equation and then introducing “taking the square root of a differential operator”

  • @mahapeyuw5946
    @mahapeyuw5946 4 месяца назад +1

    I was watching Alexander Fufaev video on the Schrodinger equation (He gave an excellent explanation), and he mentioned Diracs equation as the next thing for relativistic SE.
    I'm happy to have found this. It's been years since I finished.
    So here I am learning new things and brushing up the old, to enable drawing out something new if I persist just enough.
    This is excellent and widened my understanding but raised further questions too.

  • @alepheia
    @alepheia 5 месяцев назад +6

    I recently started my postgraduate physics studies and one of my modules (which is by far the hardest one for me) covers the Klein-Gordon equation and (now) Dirac equation. The reading material and textbooks for most of the part pretty much assumed the reader knows how to derive certain things and/or where it comes it or what it represents etc, which is why I'm finding them incredibly difficult. Your videos have literally saved my life by providing the much needed intuition for these topics. Keep up the amazing work!

    • @RichBehiel
      @RichBehiel  5 месяцев назад +1

      I’m glad to hear that! :)

    • @aatrox5559
      @aatrox5559 5 месяцев назад

      I ve began studing physics for past 5 months from ground up, and this one got me for reall this time difficult bad boy

  • @renscience
    @renscience 5 месяцев назад +2

    This is fantastic. A secret closely guarded by academics has been simplified so that even I can understand it with only a BS in Engineering. Thank you
    I need to watch more on spin as it too is a kept secret usually presented and discussed in baloney classical ways. Just a glimpse here helped already.
    I would love for someone to simplify Hilbert space as it too is slung around a lot without explanation.

  • @peterhall6656
    @peterhall6656 4 месяца назад +3

    Fun fact. Back in 1972 I was taught some mathematical physics by the late Joe Moyal. Joe was invited by Dirac to Cambridge during the World War 2 to discuss Joe's work on statistical foundatiions of quantim mechanics (there is a 1947 paper with Bartlett and Kendall which also deals with the issues). Joe met Dirac and I can say that I have shaken hands with someone who has shaken hands with Dirac.

    • @RichBehiel
      @RichBehiel  4 месяца назад +1

      Very cool! I’m jealous!

  • @michaeljin101
    @michaeljin101 Месяц назад +1

    Thank you so much! I was reading the book “The strangest man” by Graham Farmelo, and your video gives an in depth looking into Dirac equation. This let me give the following parallel from a basic math concept to understand Dirac equation.
    While Cartesian coordinates, introduced by René Descartes, provide a structured way to map points in space using two or more axes, the Dirac matrices, part of the Dirac equation in quantum mechanics, offer a profound mathematical framework for describing the behavior of particles in spacetime.
    Just as Cartesian coordinates allow us to pinpoint locations in space with precision, the Dirac matrices enable us to describe fundamental properties of particles, such as spin and angular momentum, within the framework of quantum mechanics. Both systems provide a means to understand and navigate complex phenomena: Cartesian coordinates in the realm of classical mechanics and geometry, and Dirac matrices in the intricate domain of quantum mechanics.
    In essence, while Descartes paved the way for understanding space in terms of ordered pairs of numbers, Dirac expanded this notion to encompass the intricate fabric of spacetime at the quantum level, offering a beautifully intertwined perspective on the mathematical underpinnings of the universe.

  • @Raspberry_aim
    @Raspberry_aim 5 месяцев назад +7

    Been waiting for this one since your last video! Awesome work!

    • @RichBehiel
      @RichBehiel  5 месяцев назад +1

      Thanks Raspberry! :)

  • @DrMcCrady
    @DrMcCrady Месяц назад +1

    You do a really nice job explaining these difficult concepts both formally and casually. And your Manim skills are impressive!

  • @thabomsiza2502
    @thabomsiza2502 5 месяцев назад +2

    I know the Dirac equation but never really saw it being derived before. Either I was incredibly blind in my education or my educators hand waved everything and I struggled from there. These videos are opening my eyes more than my entire journey through undergrad and honors.

  • @movax20h
    @movax20h Месяц назад +1

    So beautiful. You didn't even get to the half of the video and already all the mystery unveiled. Why nobody told me about this when I was studying theoretical physics is beyond me. It is both obvious and simple, and derivation is logical and could not be otherwise. To me the commutation relations were always introduced without any explanation or big reason other than this makes thing work, but not really. All the implications later are just simple logical conclusions (they might be hard to derive, but there is nothing mystical about them).

  • @gigaprofisi
    @gigaprofisi 5 месяцев назад +1

    I said out loud "Ooh!" excitedly when i saw this video as i was working on visualizing the Schrödinger equation for the Hydrogen atom!

  • @sanador2826
    @sanador2826 5 месяцев назад +4

    Fantastic! Looking forward to spinors video :)

    • @____uncompetative
      @____uncompetative 5 месяцев назад

      A spinor is best thought of as a Rank ½ Tensor.

  • @18SV
    @18SV 14 дней назад +3

    Thanks for such a clear explanation. Can you also elaborate on why A,B,C,D matrices should be 4x4. Like what should be motivation behind that to choose 4x4 instead of any higher like 16x16 or so?

  • @KaliFissure
    @KaliFissure День назад

    Great introduction. Thank you.
    In playing with manifold I came across the form below. If you'll notice it is toroid like but it has only ONE surface. A single rotation covers the forever and then, after passage through zero/maxima it is reoriented, inverted.
    The one rotation on one side of temporal manifold, the other rotation on the other.
    Surface(cos(u/2)còs(v/2),cos(u/2)sin(v/2),sin(u)/2),u,0,2pi,v,0,4pi
    I have some videos exploring is topology. But it seems like the inversion version of spinor

  • @brandonprescott5525
    @brandonprescott5525 5 месяцев назад +2

    You are a treasure! You just put so much into perspective for me wrt spin states, what the equations represent, how they come about. Great video! Thanks

    • @RichBehiel
      @RichBehiel  5 месяцев назад

      Glad to hear that, thanks for watching! :)

  • @tim40gabby25
    @tim40gabby25 5 месяцев назад

    The requirement to complete 2 360 degree rotations in order to reset reminds me of mobius strips.. I'll need to watch this lovely video a few times. Subscribed.

  • @monkerud2108
    @monkerud2108 4 месяца назад +1

    was a very nice video btw :). and the quaternion modification, i just figured it is easier to call the non commutative root of plus 1 "u" u*u=1 , u*i=i i*u=-i ect.. then it works out, cleaner than special rules for minus one or one, where you have to basically write the same thing anyway.

  • @0xTJ
    @0xTJ 5 месяцев назад +2

    This is an excellent video! It does a great job making the solution feel intuitive!

  • @MattHudsonAtx
    @MattHudsonAtx 5 месяцев назад +2

    I found the squares (and the whole derivation) breathtaking

  • @nickrr5234
    @nickrr5234 5 месяцев назад +1

    Excellent presentation. I've seen other videos on the Dirac equation but this is the first one that makes me feel like I (just about) understand it!

  • @ashrafhossain9044
    @ashrafhossain9044 Месяц назад +1

    Thanks for clearing everything up!❤

  • @ChasSimpson
    @ChasSimpson 5 месяцев назад +2

    Fantastic explanation! Loved the visualization of the breathing matrix and the colour coding of the terms to make it immediately obvious that the coefficients don't commute and the need for Dirac matrices. And then adding the flag to illustrate the double rotation.. Fantastic! Thank you for your time effort on this video

    • @RichBehiel
      @RichBehiel  5 месяцев назад

      Thanks for your kind comment! I’m glad you enjoyed the video :)

  • @AspartameBoy
    @AspartameBoy 5 месяцев назад +1

    So beautiful tears came to my eyes. Please update to a framework based on Clifford Algebra or Geometric Algebra rather than the most commonly used framework. Thanks for your service!

  • @mathnerd97
    @mathnerd97 5 месяцев назад +8

    Well, Diract went a different direction. When I saw those constraints, my brain immediately went "quaternion"

    • @stanleydodds9
      @stanleydodds9 5 месяцев назад +1

      I'm not understanding how this system of equations could be solved with quaternions. Perhaps I'm missing something, but I don't think the topology or geometry of the quaternions is the same as that of the space needed to solve this system of equations.
      You need 4 numbers, all of which are pairwise anticommutative, where one of them squares to 1, and the other 3 square to -1. In the quaternions, we can certainly get 3 of these - for instance, i j and k are all pairwise anticommutative, and all square to -1. However, I can't figure out what the 4th one would be (the one with the opposite sign when squared).
      1 and -1 both square to 1, as needed, but these both commute with all of i, j and k, and everything in the quaternions (because they are central elements of the quaternions, and really all number systems). These are the only square roots of 1 in the quaternions, because the factorisation x^2 - 1 = (x-1)(x+1) = 0 is true in the quaternions (it is a division ring, where -1 and 1 are central). So there is no square root of 1 which is anticommutative with anything, which it needs to be for this system of equations.

    • @FunkyDexter
      @FunkyDexter 5 месяцев назад +2

      ​@@stanleydodds9 The Dirac equation describes BISPINORS. These are isomorphic to biquaternions, also called complex quaternions. You can do Weyl decomposition to get spinors, which are isomorphic to quaternions. So yes, technically you can't use quaternions in the Dirac equation, because they only are "half" of the solution. If you look at the gamma matrices you will see that they contain the Pauli matrices, which are unit quaternions, and some of them are multiplied by i (=complex quaternions) :)

    • @stanleydodds9
      @stanleydodds9 5 месяцев назад +1

      @@FunkyDexter of course, yes, some degree 2 extension of the quaternions is going to have enough freedom to get this extra number. And naturally it would turn out to be the octonians, given that it's the "nicest" (or at least most interesting/useful) 8 dimensional algebra, in some sense.
      Just saying that it's quite clear that purely in the quaternions themselves, it's not possible to satisfy this system of equations. And of course it's related to the fact that quaternions rotations don't form a double cover.

    • @FunkyDexter
      @FunkyDexter 5 месяцев назад +1

      ​@@stanleydodds9 unit quaternions do form a double cover to real rotations. It's the whole reason they are interesting in the context of spin, and why they can be used in computer graphics. The jump to octonions is useful in the context of a unified description of the wavefunction for both the electron and the positron, but it has more to do with the inherent chirality of spinors rather than extra rotational degrees of freedom.

  • @Joao456Zamper
    @Joao456Zamper 5 месяцев назад +2

    Just thank you, as clear as it can be

  • @fluo9576
    @fluo9576 5 месяцев назад +1

    I'm astonished. This is beatiful

  • @edbuckser3569
    @edbuckser3569 5 месяцев назад +2

    God tier explanation, can’t wait for more stuff

  • @DamianHarriis
    @DamianHarriis 17 дней назад +1

    Informative and concise, great video 🤍

    • @RichBehiel
      @RichBehiel  16 дней назад

      Thanks, I’m glad you enjoyed the video! :)

  • @UJ-nt5oo
    @UJ-nt5oo 5 месяцев назад +3

    Richard: 10:39
    My physics prof: I'm gonna pretend i didn't hear that.

  • @KipIngram
    @KipIngram 29 дней назад +2

    It's also worth noting that the matrix approach is entirely equivalent, mathematically, to the other ways. It's not just that matrices happen to solve this particular problem - matrices form a group, and that group is isomorphic with any group that obeys the same multiplication rules. So not only is this "a way to do it" - you also don't lose ANYTHING by doing it this way.

    • @RichBehiel
      @RichBehiel  29 дней назад

      You’re right to point out the isomorphism between the various approaches, and I agree.
      I also use the matrix approach much, much more than the STA approach, because it lends itself so well to python codes, as well as just to the kind of math I’m familiar with. And it’s much easier to check your work when you’re using the same language as most of the textbooks on the subject.
      But in defense of STA, I do think it’s a slightly more direct and elegant way to visualize the math. It appeals more to the intuition, but IMO is harder to actually use (that’s probably my fault since I’m not great at geometric algebra).
      At the end of the day, it’s helpful to consider various perspectives, even if they’re isomorphic. Studying the different points of view can make a lot of things click, that otherwise wouldn’t have. Honestly I didn’t really understand what the Dirac matrices were, until learning STA. They seemed very mysterious to me, even though I understood the algebra. But the STA approach taught me not to over-mystify these, since they’re “just” the basis vectors for CL1,3(R). I missed out on that insight for many years, even though I had been taught the Dirac equation and had been using it in various simulations.
      en.m.wikipedia.org/wiki/Spacetime_algebra

  • @sumairahmad9464
    @sumairahmad9464 5 месяцев назад +1

    Okay , so I thought Dirac already knew about spinors before combing SR and QM. Thanks to you , now I understand that he might have went for them after finding these relations of A , B , C and D.
    That's nice because sometimes I feel like I should go for a masters in mathematics rather than physics. I am gonna go and be happy with physics for sure now because of this video.

  • @jacobyen9551
    @jacobyen9551 5 месяцев назад +3

    Soooo superbly made😍😍

  • @patb3845
    @patb3845 5 месяцев назад +1

    Great explanation of complex subject.

  • @DiffractionLimited
    @DiffractionLimited 5 месяцев назад +2

    Thank you for another great video :) Cant wait to learn more about the topic.

    • @RichBehiel
      @RichBehiel  5 месяцев назад

      Thanks for watching! :)

  • @ColeCoug
    @ColeCoug 5 месяцев назад +2

    I love these videos! Great explanation and visualization!

    • @RichBehiel
      @RichBehiel  5 месяцев назад

      Thanks, glad you enjoyed the video! :)

  • @bantix9902
    @bantix9902 5 месяцев назад +1

    I'm an undergrad physics student and you made this look incredibly easy. I'm sure I don't really understand the geometry of it but I have a way to imagine the dirac equation now, thank you.

    • @RichBehiel
      @RichBehiel  5 месяцев назад

      Glad to hear that! :)

  • @HunsterMonter
    @HunsterMonter 5 месяцев назад +2

    This was awfully well timed, we just saw Dirac's equation last week in our quantum class 😅

  • @WaluigiisthekingASmith
    @WaluigiisthekingASmith 5 месяцев назад +5

    The way the commutation relations came out from x^2 = E^2-p^2 was so cute. Also the anticommutativity is very reminiscent of geometric algebra and specifically space time algebra.

    • @RichBehiel
      @RichBehiel  5 месяцев назад +2

      If you haven’t read it yet, you might enjoy Hestenes’s papers on the Dirac equation in STA. In my opinion it’s the most elegant way of formulating the equation. Hasn’t totally caught on in the mainstream physics community, but it seems to be gaining popularity over time. Good ideas always win in the end.

    • @gaHuJIa_Macmep
      @gaHuJIa_Macmep 4 месяца назад +1

      @@RichBehiel could you give specific links to the papers?

    • @RichBehiel
      @RichBehiel  4 месяца назад +1

      @gaHuJIa_Macmep yes, let me look for those later this evening. I’m about to get on a boat for a day of scuba diving, but I’ll follow up when I get back. Please remind me if I forget!

    • @gaHuJIa_Macmep
      @gaHuJIa_Macmep 4 месяца назад +1

      @@RichBehiel Oh! Scuba diving! Where are you situated then? Somewhere in southern hemisphere? It's winter in the northern one now...

    • @RichBehiel
      @RichBehiel  4 месяца назад

      @gaHuJIa_Macmep well I live in Santa Barbara, but I’m currently on vacation in Hawaii! :) Went diving today near Honolulu, just east of Pearl Harbor. Saw a bunch of huge sea turtles. Good times.

  • @davidmurphy563
    @davidmurphy563 4 месяца назад +1

    I have zero background in physics and was just listening out of curiosity. But I know linear algebra pretty well and when you said "non-commutative multiplication" I yelled, "they're matrices!" Ha!

  • @GregorShapiro
    @GregorShapiro 4 месяца назад +1

    Wow though I have just studied 1 year of math at the college level I understood the gist of this well narrated video. Good job!

    • @RichBehiel
      @RichBehiel  4 месяца назад

      Thanks, I’m glad you enjoyed the video! :)

  • @realdarthplagueis
    @realdarthplagueis Месяц назад +1

    Once again, THANK YOU!

  • @anjikrish5774
    @anjikrish5774 3 месяца назад +1

    Wow!! Really amazing explanation! Thankyou very much

    • @RichBehiel
      @RichBehiel  3 месяца назад

      Thanks, I’m glad you enjoyed the video! :)

  • @tomyamartino
    @tomyamartino 5 месяцев назад +2

    Perfect, thanks!

  • @tavomama3289
    @tavomama3289 5 месяцев назад +2

    Keep up the good work.

  • @JesseHersch
    @JesseHersch 5 месяцев назад +2

    great explanation and diagrams!

  • @adamtaylor2142
    @adamtaylor2142 4 месяца назад +1

    VERY cool. Thanks for this!

    • @RichBehiel
      @RichBehiel  4 месяца назад

      Thanks, I’m glad you enjoyed the video! :)

  • @zemm9003
    @zemm9003 4 месяца назад +1

    This is completely ad-hoc reasoning that it is amazing that it works.

    • @RichBehiel
      @RichBehiel  4 месяца назад +1

      The physicist’s approach to mathematics! 😂

  • @francescopiazza4882
    @francescopiazza4882 5 месяцев назад +1

    Simply a great video !

    • @RichBehiel
      @RichBehiel  5 месяцев назад

      Thanks, I’m glad you enjoyed it! :)

  • @Drbob369
    @Drbob369 5 месяцев назад +3

    Good work

  • @TheJara123
    @TheJara123 5 месяцев назад +2

    Wonderful!!

  • @raywhitehead730
    @raywhitehead730 25 дней назад +1

    Brilliant!

  • @antoniocotarodriguez5732
    @antoniocotarodriguez5732 5 месяцев назад +2

    Amazing video! Thanks!

    • @RichBehiel
      @RichBehiel  5 месяцев назад

      Thanks for watching! :)

  • @evilotis01
    @evilotis01 5 месяцев назад +1

    it's here! hurrah

  • @diemme568
    @diemme568 2 месяца назад +1

    brilliant !

  • @SeverSpanulescu
    @SeverSpanulescu 5 месяцев назад +1

    Yes, the pleasure of dealing with "highly non-trivial objects"!

  • @ethanostwald8765
    @ethanostwald8765 5 месяцев назад +2

    This is beautiful.

  • @ShyamkantAnwane
    @ShyamkantAnwane 5 месяцев назад +2

    I loved it! Awesome!

    • @RichBehiel
      @RichBehiel  5 месяцев назад

      Glad you enjoyed it! :)

  • @YossiSirote
    @YossiSirote 5 месяцев назад +1

    First, excellent video. Second, one technical point at 14:50 or so, Psi 3 actually corresponds to spin DOWN positron and Psi 4 to spin UP.

    • @RichBehiel
      @RichBehiel  5 месяцев назад

      Thanks! :)
      I’m confused, but I definitely appreciate your feedback! In “Introduction to Elementary Particles” by Griffiths, equation 7.30 and the following sentence state that psi3 is spin up positron, and psi4 is spin down. But then later he converts the u3 and u4 spinors into v2 and v1, respectively, such that now u1 and u2 are spin up and down electrons respectively, while v1 and v2 are spin down and up positrons respectively. So the “up” and “down” are reversed when looking at the indices on the u’s and v’s, but when writing the bispinor as a column, it should still go up, down, up, down, right? Or am I missing something?

  • @xelaxander
    @xelaxander 5 месяцев назад +1

    That was surprisingly quick. And you have indeed done a decent job ;).

    • @RichBehiel
      @RichBehiel  5 месяцев назад +1

      Thanks, I’m glad to hear that! :)

  • @godinhos7797
    @godinhos7797 5 месяцев назад +1

    man,this equation is a kind of awesome awesome art! paul dirac is a genius and one math artist! awesome!

    • @RichBehiel
      @RichBehiel  5 месяцев назад +1

      Dirac was enamored by the beauty of math and physics. You can see why!

  • @AlkezkSH
    @AlkezkSH Месяц назад +1

    Great job dude

  • @monkerud2108
    @monkerud2108 4 месяца назад +1

    i leave it as an exercise for the content creator to look at the differences between the gamma matrices and these types of quaternions with a non commutative real unit.

  • @tayranates8279
    @tayranates8279 Месяц назад

    At the time 12:13 , this equation can only hold as an operator because left side is a matrix and right side is a constant. This means that actually what we are find is only can be satisfied in quantum mechanics and does not any analog representation in classical mechanics.

  • @user-yb5cn3np5q
    @user-yb5cn3np5q 5 месяцев назад +2

    Such a good video. I only wish it was formulated in terms of geometric algebra. One of the major reasons spinors are trippy is because they're usually all about exact coordinates, while for the purposes of explanation it's possible to split it into two parts: "thing where squares are -1, and AB = -BA, and their algebra" and "how to represent numbers of that algebra as 4x4 matrices to do an actual simulation on a computer". You can't neither draw nor get geometric meaning out of numbers of 3D spacetime algebra anyway.

    • @RichBehiel
      @RichBehiel  5 месяцев назад

      We’ll get into Hestenes at some point :)

  • @kukim3285
    @kukim3285 2 месяца назад

    Thank you!

    • @RichBehiel
      @RichBehiel  2 месяца назад

      Thanks for watching! :)

  • @Roxor128
    @Roxor128 5 месяцев назад

    When you hit the bit with the anti-commutativity and the squaring of A,B,C,D to 1,-1,-1,-1, I thought "Hey, isn't that a rotor?". A would be the scalar part, and B,C,D the bivector parts.

  • @pacolibre5411
    @pacolibre5411 5 месяцев назад +3

    Amazing video! I do have a question about the final form of the equation. Since you are multiplying bispinors by 4x4 matrices, does that mean that the “=0” in the dirac equation is the “zero bispinor” or is it literally the number 0?

    • @RichBehiel
      @RichBehiel  5 месяцев назад +2

      Great question! Yes, it’s the zero bispinor. One way of thinking about the Dirac equation is that it’s four coupled differential equations, and the Dirac matrices intertwine the various components of the four-momentum across those equations, and each equation has a zero on the right side.

  • @khuti007
    @khuti007 4 месяца назад +1

    Thank you

    • @RichBehiel
      @RichBehiel  4 месяца назад

      You’re welcome, thanks for watching! :)

  • @alphalunamare
    @alphalunamare 5 месяцев назад +1

    Far out Man, that 'was' real trippy! :-)

  • @alecbooker1368
    @alecbooker1368 5 месяцев назад +1

    Beautiful

  • @ntesla66
    @ntesla66 4 месяца назад +1

    You did good!

  • @GeoffryGifari
    @GeoffryGifari 5 месяцев назад +1

    The constraint of 1st order space and time dependence eventually lead to the Dirac field/"wavefunction" being bispinors, signifying spin-1/2 (fermion). Is it possible to come up with an equation 1st order in space and time, for integer spin fields (bosons)? Is the Klein-Gordon equation the most "elementary" equation for bosons?

  • @angelamusiemangela
    @angelamusiemangela 5 месяцев назад

    Anche qua la Fisica ,( ci sono le numeriche o variabili della Fisica ,No!)non c'entra nulla , è una ricerca che è già stata esposta, dalla Dirac Converse , riguardo gli andamenti del ciclo di vita del prodotto,nelle sue varie fasi, durante una linea di produttività. Ciao ciccino un bacione! Può servire la successione di equazione per il riscontro di un andamento o processo di distanziamento. Questo si è vero! Bene.

  • @maxsamarin9002
    @maxsamarin9002 5 месяцев назад +1

    Amazing explanation and easy to follow, something that I was not expecting when considering how to derive the great Dirac equation. Question (I'm not familiar with Klein-Gordon equation): what is the motivation to get a first-order equation? A second-order equation incidentally does not predict spin?

    • @RichBehiel
      @RichBehiel  5 месяцев назад

      Good question! :) I talk about this in the second half of my KG video. Long story short, the second-order makes probability densities and currents weird, such that sometimes you have negative probability density. This has been contextualized over time, as QFT has developed, but back in the day it was one of Dirac’s main motivations to look for a first-order equation.

    • @maxsamarin9002
      @maxsamarin9002 5 месяцев назад

      @@RichBehiel Thanks! I'll check that video as well!

  • @angelamusiemangela
    @angelamusiemangela 5 месяцев назад +1

    🙂🙂 , Complimenti!

  • @reinerwilhelms-tricarico344
    @reinerwilhelms-tricarico344 Месяц назад +1

    Very nice. I almost understood this, but I don't get what the deal is with the minus sign of the momentum operator. It's usually in QM p_x = - i \hbar \partial_x .
    But you used initially p_x = i \hbar \partial_x with the plus sign, but write in the same line in the beginning at time stamp 1:35 \hat{p} = -i \hbar (\partial_x, \partial_y, \partial_z). Then, in the end, in the Dirac equation it's always p_u = i \hbar \partial_u . Is the trick somehow that the imaginary i should be explained from taking the square root of the metric tensor, or signature (+---) or (1,-1,-1,-1) which makes (1,i,i,i) ? I dunno. This is a bit confusing to me right now, I must have missed something.

  • @kristoferkrus
    @kristoferkrus 2 месяца назад

    Great explanation, and helpful visualizations! However, I think a motivation for the third principle of the Dirac equation would have been nice. Now it was just stated, for no obvious reason. (I know the motivation if it myself since before, but other people may not know it.)
    A thing that I'm wondering-at 7:31 when you write out the equations for A, B, C and D, this strikes me as equivalent to the equations for the basis elements of the quaternions, so would it be possible to write the Dirac equation with the quaterion basis elements instead of the gamma matrices, and letting the wave function be quaterion-valued instead of a vector field?

  • @PillarArt
    @PillarArt 5 месяцев назад +1

    even the way you have the visuals in motion = NB ... ty
    that 4x4 matrix breathing is important for understanding reality.

    • @PillarArt
      @PillarArt 5 месяцев назад +1

      @6:55 lol case in point

  • @ppmealing
    @ppmealing 4 месяца назад +1

    Very good exposition. I was surprised I could follow it as well as I did. It even allows one to play with the matrices to confirm the results for oneself.
    I never understood it before.
    The Wikipedia exposition, by contrast, I find very difficult to follow.

    • @RichBehiel
      @RichBehiel  4 месяца назад

      Thanks, I’m glad you enjoyed the video! :)

  • @____uncompetative
    @____uncompetative 3 месяца назад

    That was excellent.
    Eric Weinstein is using this strateg where his thinks the Modified Yang-Mills Equation Analog has a Dirac Square Root in a Mutant Einstein-Chern-Simons like Equation.

  • @monkerud2108
    @monkerud2108 4 месяца назад +1

    if you modify the quaternions slightly, that is you fuck around with 1 and find out, then its all good as well. what you need is simply to modify the real unit multiplication such that if you multiply the real unit by a non real unit A=+-1 B=i,j,k such that if A*B, A=1 and if B*A, A=-1, then it satisfies the same constraints, basically you make 1 non commutative and you are good to go.

  • @patmichel4724
    @patmichel4724 5 месяцев назад +2

    I have one thing to say : "thank you"!!!!

    • @RichBehiel
      @RichBehiel  5 месяцев назад

      You’re welcome, and thanks for watching! :)

  • @victordelmastro8264
    @victordelmastro8264 Месяц назад +1

    I like my least paths solution to causality: e^(M*{L/lambda}^2)=cos(Lambda*M)+lambda*sin(Lambda*M)!!!, "The Temple Equation for Causality."

  • @JakeFace0
    @JakeFace0 5 месяцев назад +1

    Is it possible to "Dirac-ify" other differential equations in physics? The wave equation is to the second power; are there any mathematical objects we can use to make the wave equation first-order in time?

  • @monkerud2108
    @monkerud2108 4 месяца назад +1

    basically if you replace 1 in the quaternions by a non commutative root of 1, that is what i said by a multiplication rule that picks out either 1 or -1 then we are all good.