Relativistic Quantum Waves (Klein-Gordon Equation)

In this video, we'll unify special relativity and quantum mechanics, to derive the beautiful Klein-Gordon equation! Then we'll explore some of its properties, to see what it can (and can't) tell us about the nature of things. This is part of a video series that's building toward an exploration of the Dirac equation, and then a triumphant return to the hydrogen atom.
Thanks for checking out this video, and if you are so inclined, please consider joining my Patreon, for full access to the slides from these videos, and additional notes and links to supplemental materials. I'll also be on there to answer any questions you might have about physics, math, and life, and your input will help guide the future direction of this channel. All support is greatly appreciated.
/ richardbehiel
Chapters:
0:00 Intro
1:40 Deriving the KG Equation
7:24 Four-Momentum Eigenstates
12:38 Superposition
14:36 KG vs Schrödinger
16:48 Group Velocity & c Speed Limit
23:16 Fourier Transforms & Antimatter
31:01 The 2nd-Order-in-Time Problem
33:15 Probability Density & Current
39:55 The Mystery of Spin
42:27 Concluding Remarks by Paul Dirac
#physics #math #quantum #relativity

[@TheoriesofEverything]

Resplendent. This outpaces any single chapter in a book and any single lecture I've seen on KG.

[@RichBehiel]

Thanks Curt, that really means a lot! :)

[@rv706]

This video is actually conceptually wrong. According to the modern understanding, the KG equation is _not_ a quantum equation: it's a CLASSICAL equation. The ψ appearing in KG is _not_ a quantum wave function: it's a classical field. Also, the Dirac equation is _not_ a quantum equation: it's a CLASSICAL equation. The ψ appearing in the Dirac equation is _not_ a quantum wave function, it's a classical field describing a (classical) spinor i.e. a section of the spin vector bundle that satisfies the Dirac equation. To get quantum in the usual way (i.e. following the Heisenberg picture) in QFT you have to consider "field observables" (that would be certain operator-valued distributions on spacetime). But you can also follow the less common (for QFT) Schroedinger picture for spinors: you'd then have to consider wave _functionals_ on the space of solutions of the Dirac equation. _Those_ functionals would be the quantum wave functions for a spin 1/2 particle, and they have nothing to do with the solutions ψ of the Dirac equation (it's a notational coincidence, or rather a historical misunderstanding, that those are denoted ψ like wave functions for the Schroedinger equation). Cf. "Mathematical quantum field theory" by Urs Schreiber.

[@sonarbangla8711]

So, if phi is classical in all cases, KG, Schrodinger and Dirac, then what you are indicating is QM is still without any theories, in other words INCOMPLETE, as Einstein would say. Does Urs Schreiber provide a complete theory? If not, is QC functions, without any infinite axiom, error correcting algorithm, is the complete theory?@@rv706

[@ShonMardani]

The reason you guys have more conflicting explanations is because they are all FALSE. Hi, I am Shon Mardani, this is my Unifying Theory Of Everything. I have more to fill in between the lines regarding the FORMATION of the Atoms and other Chemical and Physical Interactions once I validated the fundamentals of my hypothesis. Please let me know if you see any conflict with any Observed Facts or Agreed upon Scientific Knowledge, Thank you. [GOD] Created NOTHING, a Void Point in Space. NOTHING Attracts [neighboring] Space as the Only Law of The Nature which gave NOTHING its Property to be the GRAVITATIONAL PARTICLE (GP). Fast Moving Space into GP, Creates its own GP at the [Vacated] Space which Attracts the Surrounding Space which is a Negative Pressure or PULLING/SUCKING IN of a GP on Neighboring Space. There are 3 Pairs of 2 Directional Possible Movement Axis, this Creates Magic Numbers of the Nature, Numbers 2, 3 and their Sum 5. Propagation of the GPs in a Closed Cyclic Patterns / Locked Loops of GPs Create Collection of Virtual Positions in Space known as Atom, Starting with Hydrogen to EVERYTHING else. Atoms are Connected by Overlapping/Common/Shared (single, double bonds ...) GPs to Create Molecules. Hydrogen Atoms Virtually/Positionally Collect to Form Nitrogen and Oxygen Atoms and Form the Atmosphere, in a 4 Nitrogen to 1 Oxygen ratio which I call one ATMOSPHERIC UNIT (AU). Movement of GP toward the Center of the Gravity Transforms 2 AUs (2(N4O1)) into 2 x 3 Carbon atoms with H2O as Hydrogen Transformer and CO2 as State Transformer, the Collection of this Cyclic Process is called LIFE. LIFE Synthesizes the Heavier Organic Elements to Create Species of Independent Life Cycles. Overlapping Fundamental Atoms Create Heavier Atoms/Elements which are collected in the Periodic Table. The [Virtual] Movement/Propagation of GPs in a Circular Patterns within the Connected Atoms has Frequency and Direction which Constitute and are Observed as its Weight/Mass/Gravity, Force, Polarity, Magnetism, Electricity, Heat, Light, Color and ALL other Physical Properties and they Move and Interact by Connectivity. The Gravity Force we Experience on the Earth, Is the Flow of GPs through Matter Moving Perpendicular to the Surface of the Earth towards the Center of the Gravity of the Earth and all Matter Connected to it, like being under the rain from a Firehose. We feel the Flow to the CG as Push and from the CG as Pull. Imagine there is a room full of Marble balls and you remove/disappear ONE Marble from the middle. One of 6 possible adjacent Marbles can move in and fill the empty space, the one on the left, right, front, back, above or below. If the Marbles move into the empty spaces at the Ultimate Speed in Nature, the traces of the MOVING/PROPAGATING GPs (empty spaces) will look like a Spiral. Hydrogen [Matter] is OVERLAPPING GPs. Imagine the Light Sources like Sun, Light Bulb and LEDs are NOT Ejecting Photons but they SUCK IN the GPs which we see and call them Photons. Turbines convert the Gravity Force to Magnetic > Electrical > Heat > Light and everything in between, all different manifestations of the GP [Virtual] Moving in the Matter. Space comes to existence when GPs are created, in other words Space is the Moving GP. Like a battery whose positive (+) side exists only when Negative (-) side is created.

[@ShonMardani]

In the formula F=ma, a has a time variable which means F = 0 when time = 0 and Force can not be calculated unless you provide time duration. The parameter a Acceleration is due to the Gravity and Gravity has a set vector / direction, 0 to 180 degree, in other words there is NO Acceleration if there is NO Gravity, and in Gravity direction and Time needed. Force is 0 (zero) unless you give a non zero time.This is what scientists have been hiding and the space time scam is to explain a flawed formula using the gang leader einstein. E=mc2 is also False since you can change the constant 2 to 3 or any other number, it has no effect because E has no Unit. Also all the formulas mentioned as the successor to E=mc2, have the Time variable with the value 0 (zero) which makes all the calculated Forces, Energies and Vectors Zero. Additionally einstein never wrote and published any of these formulas and no other scientist wanted to claim Fallacy.

[@thekey429]

Thank you for giving a clear concise explanation of these equations that doesn’t require 2 years of obnoxious jargon training.

[@RichBehiel]

You’re welcome! :) Glad you enjoyed the video.

[@gaHuJIa_Macmep]

Well, let me notice, professional jargon knowledge is very useful sometimes... :)

[@thekey429]

@@gaHuJIa_Macmep I agree it can be helpful, but a lot of mathematics and physics is steeped in obscure jargon without transparent explanation in order to create unnecessary barrier to entry to those who don't pay $50,000k per year at a US university.

[@gaHuJIa_Macmep]

@@thekey429 No, I can't agree. You see, professional jargons exist in almost every trade, incl., for instance, pipemen. It's just how this life is designed: people devise jargons to simplify their everyday lives, they tend to economize in their language. And you need to understand it if you want to be able to read professional literature and be part of the Movement. This jargon is explained during regular education in universities.

[@thekey429]

@@gaHuJIa_Macmep This video provides a crystal clear explanation of the Dirac Equation without resorting to reams of technical Jargon, and it maintains strong formalism; this is a real rarity in the United States, where this video publisher is from. He is teaching the Dirac Equation, not giving a lecture to his peers. Physics Academia, particularly in the United States, purposefully obfuscates concepts through tons and tons of unnecessary jargon in their educational system, and require students to go $70K-200K into debt just to obtain an undergraduate level, of which about 30-40% is jargon training. Whether you agree or not, it is a fact that the politics of the big money US education system are designed to create barriers to understanding so that gatekeepers must be paid; its a fact of US Academia, and one I avoided by being trained in Europe; I found it easier to learn an entire new language than to owe the US system 100K to teach me how to understand what a Non-Abelian Gauge Transformation is after 3 years of jargon training. Just my experience, and I appreciate this person for giving such clear explanations that don't play gate keeper with the language.

[@enotdetcelfer]

Absolute clarity, wow. I'll never look at the mass shell, position and momentum, and the speed of light the same again. Between this and having watched eigenchris' spinor videos on a loop, all my childhood discomfort with traditional metaphors completely disappeared and I think I finally understand. I can't thank you enough, it all finally clicked. And they knew this so back in the 20s huh?! You turned the formulas into sheet music and those visuals were so intuitive. Everything as degrees of freedom in this pure math. I gotta watch this like 10 more times and let it really sink in. Thank you so much.

[@RichBehiel]

Thanks for writing such a thoughtful comment! I’m glad you enjoyed the video :)

[@ProfessorBeautiful]

As a physics fanboy with moderate math, I greatly appreciate this video, with many gems of insight tying things together I knew only from snippets. And... to hear DIRAC speak! What a treat--the gentlest possible use of the word 'intolerable'!

[@lawrence-1]

loving the conformal transitions!

[@diana-pestana]

Really liked how you easily presented the topic, adding historical and philosophical context. Great video!

[@RichBehiel]

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

[@trucid2]

14:39 That transition was so awesome I had to watch it multiple times.

[@LuisAldamiz]

I must admit I only grasp the maths somewhat but the explanation and the graphics are awesome.

[@RichBehiel]

Thanks! :)

[@henrybottomsworth]

I have only one word to say for this video: awe some.

[@RichBehiel]

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

[@artlover7770]

But that is a 2, you? Toyu...

[@michaelblankenau6598]

Even though I can't understand any of this ( and that's 100% on me ) I still give you a like because intuitively I know you have done a great job .

[@danielackles4265]

I'm considering going back to school for physics and this playlist is awesome! Thank you for such a clear explanation and visual presentation!

[@RichBehiel]

That’s awesome, studying physics is a great thing to do! I’m glad you enjoyed the video :)

[@purplenanite]

It looks amazing! I loved the closing remarks by Dirac.

[@calummoncrieff1789]

Love the Superposition section, these videos are beautifully put together. Down to earth explanations on complex topics. 👍👍

[@Phantores]

One of my favourite channels uploaded another amazing mathy video again

[@gianlaager1662]

I just started college and I did as my "high school thesis" (not a really good translation) a approximation of the Schroedinger equation because I wanted to see what these waves looked like. Thank you for giving us those animations along side a quite in depth explanation of the math. Keep it going!

[@gavinwince]

When I saw you start this off with + --- I knew this was going to be good!

[@Handelsbilanzdefizit]

This was superb! 🤩

[@RichBehiel]

Thanks, I’m glad you enjoyed it! :)

[@Study-fz5ik]

The concluding remarks by Dirac were awesome! Amazing video!

[@RichBehiel]

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

[@AlkezkSH]

Box operator or unrecognized character. Dryest shit I've ever heard. Love it.

[@Raspberry_aim]

Great content, like always! I really appreciate you sharing these!

[@RichBehiel]

Thanks Raspberry! :)

[@alverpo9935]

Cant wait to see the Dirac equation video. I totally trust you to finally get me to understand the spinors.

[@RichBehiel]

Glad to hear that! :) I’ll try my best. Honestly I’m a bit worried about how to present all the algebraic stuff in a way that’s visually intuitive and appealing, but I think it’s doable.

[@trucid2]

Just wanted to say that your video is a masterpiece. I'm not just saying that. Usually when I see equations in physics videos my eyes just glaze over and I stop paying attention. You made it so simple to understand that apart from the latter part where you started talking about eigenvectors I followed all of it.

[@RichBehiel]

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

[@edd.]

That is an epic ending!! Well done!

[@RichBehiel]

Thanks, I’m glad you enjoyed it! :)

[@nickst2797]

Thank you very much for all your videos, demystifying QFT!

[@RichBehiel]

Thanks for watching! :)

[@DiffractionLimited]

Another great video ! Thank you for the effort you put into these. They helped me to get some understanding of topics I previously found hard to grasp. The great visualizations also help to get a more intitive understanding of what the formulars describe. Looking forward for the continuation :)

[@RichBehiel]

Thanks for the kind comment, and I’m glad you’re enjoying these videos! :)

[@piradian8367]

Fantastic! Ok, you got me intrigued!

[@piradian8367]

I've been watching this video series kzbin.info/aero/PLJHszsWbB6hoOo_wMb0b6T44KM_ABZtBs. At the same time I watched your video about Schroedinger's equation. But I wasn't able to put puzzle together, the question immediatly raised: how spin states are included in Schroedinger's equation?

[@sirnate9065]

I found the phase velocity vs. group velocity graphics to be super helpful. Definitely put a clear visual to the fuzzy, vague image I had in my mind.

[@RichBehiel]

I’m glad to hear that! :)

[@tayranates8279]

I did not know that klein gordon equation and dirac equation has this much beauty in it.

[@Jaylooker]

The linearity of the solutions to the Klein-Gordon equation, the d'Alembert operator in the Klein-Gordon equation factoring algebraically like a^2 - b^2 = (a + b)(a - b), and the justifications mentioned here for trying to derive a first order Klein-Gordon equation suggests trying to derive what is the square root of the Laplacian. The Dirac operator is the formal square root of the Laplacian and leads to spinors.

[@vb6database]

I don't have much, bu your videos are priceless please keep making them.

[@RichBehiel]

Thanks, that means a lot! :)

[@yashmohangupta5379]

Wowww…!! Absolute beauty!

[@ARBB1]

Great video! Specially your visualization of the momentum eigenstates with progressively higher modes in the mass shell, I hadn't thought of that picture. On your Fourier point, I've made a video on the subject and briefly mentioned it, but Feynman's proof of the case is very good and enlightening to showcase why antimatter must arise, using basically the same ideas. Cheers.

[@RichBehiel]

Thanks! :) I’ll check out your vids, and will look up Feynman’s proof. Always love a good Feynman proof!

[@jdbrinton]

damn, dude. This is quality! bravo!

[@RichBehiel]

Thanks! :)

[@aieousavren]

Absolutely incredible work! I don't mean to simply blindly add more praise to the pile of already glowing comments, but I genuinely really appreciate this series and your hard work that you clearly put into it. I enjoy the style of your thinking and your presentation, the visuals are of course gorgeous, and the mathematical details really fill everything out. I've learned a ton from your approach to thinking about quantum mechanics, and I continue to watch eagerly and wait patiently for this great work to continue. Every time you upload, it is a joy. Thank you so much! ❤

[@RichBehiel]

Wow, thanks for taking the time to write such a thoughtful and kind comment! :) It’s feedback like this that makes it all worthwhile, and knowing that there are people out there looking forward to these videos, makes it a lot easier to stay motivated.

[@aieousavren]

@@RichBehiel I'm happy that you enjoyed my comment! It's people like you who have really refined my passion for theoretical physics, and the beauty one can find in seemingly obtuse equations if only you know the right way to look at them to get to the core of what's going on. And again, I really enjoy your narration, and the casual yet excited tone of it. It's clear you have a passion for this stuff, and I really admire that.

[@beamshooter]

Great closing remarks. I look forward to your video on the Dirac equation. Another motivation for searching past KGE is how do we introduce a potential field, as we do in Schrödinger. We necessarily require not Ê_rel^2, but purely Ê_rel, so as to have [Ê_rel + V] (psi) = [Ê_total] (psi)

[@RichBehiel]

Very good point! Now I wish I had included that in the video 😅

[@gaHuJIa_Macmep]

Huh! Good point! I've never thought about it this way! Thanks!

[@FunkyDexter]

Very, very interesting and well put together. I also find very interesting the recent fermilab experiment where they finally proved antimatter falls down like normal matter, thus proving it has positive energy (and making it fall on the positive mass shell). I think we really need to reinterpret the meaning of the mass shell itself to get out of this uncomfortable situation, and I think the key is reevaluating the importance of the phase of the wavefunction: we could for example substitute E with ihd/dt, suddenly it's not a matter of positive and negative energies but of relative positive and negative phases. Take for example the plots at 14:50. As you state, there's activity even at rest in the relativistic wave equation: this makes sense, since we know from SR that mass is a form of "bundled" energy. In protons, the mass comes from the internal dynamics of quarks and gluons. What if the mass of ALL particles is indicative of internal dynamics? We know electrons DO have internal dynamics, because they have spin. The difference between an electron and a positron would then be a time reversal of this internal dynamic, corresponding to opposite time evolution of the internal phases. When i think of moebius strips (which are geometrical representations of spin 1/2) i also see that their geometry is CHIRAL. This adds an additional two valued degree of freedom, because a current on the surface of a single strip can go in two directions. If the topology of the strip is responsible for charge, then the phase circulation would be responsible for the magnetic moment. This stuff gets wild when you reinterpret the Dirac matrices for spin 1/2 as unit quaternions... Which in turn bring you back to the hopf fibration. Rotations on a 4D sphere. I'm picturing pretty vortices in spacetime.

[@Sol-En]

Thanks, great explanation !

[@RichBehiel]

Thanks for watching! :)

[@TriangularCosmos]

This is great even though I have little understanding 👍

[@user-jv5ch6mb7r]

Thanks.❤

[@surendra7856]

GREAT 'GURUJI' 🙏

[@truthbetold444]

Wow, great job!

[@RichBehiel]

Thanks! :)

[@felixgent4084]

Gorgeous!

[@RichBehiel]

Thanks! :)

[@WDDG]

I really enjoy your videos. I was wondering what software and/or programming languages you are using to do the visualizations. Thanks!

[@RichBehiel]

Thanks, I’m glad to hear that! :) I use python, usually matplotlib but sometimes plotly for 3D stuff.

[@tune490]

Fantastic! :D

[@kmg3658]

Thank you!

[@RichBehiel]

You’re welcome! :)

[@kaustubhpandey3259]

I love your videos❤ Eager for the hydrogen p3 video!

[@RichBehiel]

Thanks! :) I’m working on a couple other videos first (Spinors, Electromagnetism as a Gauge Theory), then will return to hydrogen using those concepts.

[@kaustubhpandey3259]

I just spent a day watching your videos I trust your process. I insist on it Just keep doing what you do❤

[@gaslaireuh6119]

Hello, your channel is awesome! I have a question: why do we have to work with second partial derivatives in this equation, and more generally in quantum mechanics? What is the physical implication of this mathematical consideration? Thank you for your help.

[@johnchase2148]

When you mentioned fluid reminded me of looking at the Sun with a telescope when it looked like a golden fluid like waves in motion.

[@gaHuJIa_Macmep]

20:22 - good discussion of the group velocity (pretty standard, BTW). But had you mentioned here that phase velocity, which is omega/k, is always GREATER than c (which can be seen immediately, derivation is even simpler) - you presentation would have gained alot.

[@RichBehiel]

True, I should have included that. I kinda hinted at it, talking about how the phase velocity in the example was way faster than the speed of light, but I should have been more articulate on that point.

[@beamshooter]

17:00 beautiful illustration

[@beamshooter]

20:00 already see where this is going .... my relativity bone is tingling

[@RichBehiel]

Thanks! :)

[@Doozy_Titter]

Thank you. The fact that this is free is unreal😂

[@RichBehiel]

Thanks for watching! :)

[@jensphiliphohmann1876]

16:20 f Yes, the phase state is well over c but note that in the previous image about 15:00, it's infinite.

[@pelegsap]

Great! btw, spinors are actually not that complicated when you use the right tools. I highly recommend watching Eigenchris' videos (his still WIP "spinors for beginners" series).

[@RichBehiel]

I love that series! There are some aspects of spinors that are easy enough to understand, or at least easy enough to work with. But then there are aspects of them that honestly I find very mysterious. I don’t really know how to draw one, for example, in the same way that I can draw a vector.

[@____uncompetative]

"A spinor is a Rank ½ Tensor" - Sir Michael Atiyah

[@gaHuJIa_Macmep]

Could you give a precise link for convenience? Thanks.

[@greenguo1424]

23:15 most important thing is passion and don't give up

[@erebology]

❤❤❤ Thank you so much! ❤❤❤

[@RichBehiel]

Thanks for watching! :)

[@raymondbrowning762]

very nice

[@ehfik]

beautiful!

[@RichBehiel]

Thanks! :)

[@YossiSirote]

Excellent!!!!!❤❤❤❤❤

[@RichBehiel]

Thanks! :)

[@johnwickfromfortnite5744]

A main question still remains for me: In what way is Klein-Gordon a quantum equation, i.e. how are the commutation relations of quantum mechanics built into it? The construction we have here is just some kind of relativistic field equation that admits plane wave solutions as a complete space of basis vectors, i fail to see the quantum nature of it.

[@RichBehiel]

Great question! :) The answer is much the same as for Schrödinger. Consider momentum and position, for example. A momentum eigenstate is a plane wave that fills all space, so its position is smeared out everywhere. Conversely, imagine localizing a KG wavefunction to a single point, now the particle’s momentum is smeared out everywhere in momentum space. Between these two extremes, there’s always a trade off between how much you can localize the position and how much you can “localize” the momentum, and that’s just baked into the Fourier analysis of the situation. The Heisenberg uncertainty principle follows from that.

[@jyotirmayakhatua9365]

I m just searching Dirac equations and here it is....... please make a video on propagator spinor and vector field

[@jostpuur]

If you use the Hamiltonian of Klein-Gordon field, it assigns positive energies to both the positive and the negative frequency solutions. I don't think that those two type of solutions should be called the positive and the negative energy solutions.

[@JAYMOAP]

Spin zero particle is a zero frequency gap amplitude in reality it's not a particle btw. It's a collective excitation mode . Nice channel. Subscribed.

[@MattHudsonAtx]

Ok, the manim work in this video is en pointe. And of course, the math is excellent too.

[@RichBehiel]

Thanks! :)

[@KipIngram]

I would very much like to see the Noether's theorem method of getting to probability densities.

[@RichBehiel]

Funny you should say that! I’m working on Electromagnetism as a Gauge Theory now, and am debating whether or not to include that derivation. I will for sure include the derivation of local charge conservation (this pops out while deriving the inhomogeneous Maxwell’s equations). The derivation of probability density/current from Noether’s theorem is a bit simpler, as it just relies on the inherent global U(1) symmetry of the Dirac equation, so I should probably include that too. Although it might be a bit tangential. Idk.

[@KipIngram]

@@RichBehiel Oh, please do, and I look very forward to that video. Good luck with it!

[@RichBehiel]

@KipIngram thanks! :)

[@manipulativer]

question about group and phase velocity; Is there a similar thing with sound? Cause on the sound level even though the speed of sound is like 300m/s but individual air molecules speed is 500m/s?

[@RichBehiel]

Sound is an interesting thing. If you look up “monoatomic chain speed of sound”, you can find a derivation of the speed of sound. In general, the phase velocity and group velocity of sound are almost identical, although depending on the medium there can be some subtle differences, since waves of different frequency may travel at slightly different speeds.

[@stoneyyy19]

37:04 are the space and time derivatives of psi and psi comp. conj. commutative? Wouldn't the product rule of differentiation be broken by factoring d/dt and del like this?

[@RichBehiel]

Same as my other comment, psi is complex-valued in KG :)

[@hexane360]

23:40 When considering a distribution on the mass shell like this, does this imply the particle is localized in time, in analogy to how uncertain momentum localizes it in space?

[@RichBehiel]

Good question! Momentum is to space as energy is to time. A particle that’s perfectly localized in space will have totally uncertain momentum, and vice versa, so likewise a particle that’s perfectly localized in time will have totally uncertain energy, and vice versa. Unifying those concepts, we have a picture where the space of all four-momentum, and spacetime, are reciprocals of each other. A point in one is a total spread in the other. A tight distribution in one is a loose distribution in the other. So when a particle is spread out as a Gaussian on the mass shell, it will also have some spread in spacetime, and these two spreads will be inversely related.

[@hexane360]

@@RichBehielI guess my question is: It's easy for me to interpret a particle being localized in space once I have a sum of different momentums (i.e. a wavepacket). But I'm not sure how to interpret a particle being localized in time. Does it just decay in the future? In classical Schrodinger, I understand that time-independent solutions have a well defined energy but are spread over all time. Time-bounded wavefunctions have a spread of energy instead. But I'm having trouble generalizing this to the relativistic case.

[@RichBehiel]

@hexane360 oh, I see what you mean! I think the essence of it is the same in relativistic and classical QM. As you say, energy eigenstates are spread over all time, and that’s as true for Schrödinger as for KG. Likewise, you’re right that a state which comes into being for a finite amount of time must have a spread of energies. I think those ideas directly transfer over to the relativistic context. One difference would be that time and space are unified in KG, as part of the same four-vector, and so the parallels between momentum/space and energy/time are more manifest in the theory, whereas in Schrödinger we find that space and time are sort of fractured into two different things. So KG gives us more confidence that ideas about momentum and space can be directly applied to energy and time.

[@stoneyyy19]

35:16 sry if this question sounds stupid, but isn't psi anticommutative? In the first equation you multiply -psi comp. conj. on the left side, which is reflected by it being on the left of both psis, in the second equation psi is on the left side of the d'Alembertian times first psi comp. conj. as well as on the right side of the second psi comp. conjugate. So is psi multiplied on the left or right side of the second equation and shouldn't there be a minus sign either before the first psi or before mu squared? Or am I just dumb and missing something?

[@RichBehiel]

Great question! I should have emphasized this in the video, but for KG, psi is still a complex-valued function, same as Schrödinger, and complex numbers commute. It’s only once we get into Dirac that psi becomes a bispinor.

[@officiallyaninja]

30:20 wait a minute! Is the time reversal of antimatter because reflecting a line about the horizontal axis is the same as reflecting it about the vertical axis??? That's wild

[@RichBehiel]

Essentially, yes. Although notice how the sequence of colors flips around when you reflect horizontally but not vertically, so time reversal and reversing the direction in space are slightly different things. But still, we have that degree of freedom in the mass shell. The two halves of the mass shell represent matter and antimatter, respectively.

[@petergramma1873]

Can someone explain what units each coordinate represents at around the 25 minute mark for the complex coefficients of the mass shell?

[@KipIngram]

I was astounded when I learned that in his later years Dirac regarded his life's work as a failure. My God - if Dirac was a failure, then we have no successes. He's literally at the top of the heap as far as I'm concerned.

[@RichBehiel]

I agree, Dirac is a legendary genius. In his defense though, he would probably consider all of theoretical physics since the ‘70s to be a failure. He was really trying hard to calculate the fine structure constant from deeper principles. It seems that he was obsessed with trying to understand the nature of the electron. He knew the power, and the limits, of his own equation.

[@KipIngram]

@@RichBehiel I certainly didn't mean for that comment to be any sort of negative judgment of Dirac - I thought it just reflected a man of true humility, which is a rare find in a person of such prominence.

[@kaustubhpandey3259]

Tenet: "A positron is an electron moving back in time"

[@davidwilkie9551]

As an almost completely academically unqualified amateur, promoting reorientation to parallel observation practices in Logarithmic Time, a pure-math sequence of omnidirectional-dimensional holography-quantization stages for each prime-cofactor resonance bonding stage is probably also the job of qualified Geometers who has internalised the principles of Singularity-point positioning resonance of Bose-Einsteinian coherence-cohesion objective location in the picture-plane containment landscape and "solids of rotational substantiation" POV in Euler's Unit Circle derivivation of instantaneous inside-outside differentiates in temporal superposition Calculus. Otherwise, the video looks like some very satisfying mathematics, a job that should be done.

[@jensphiliphohmann1876]

15:05 This lifelessness can easily be fixed by "artificially" adding the particle's rest energy to the non-relativistic solition. The original SCHROEDINGER solution uses an ansatz which still hadn't recognized mass as a form of energy, thus vastly underestimating the frequency at which the wave function ought oscillate, leading to zero frequency for a particle whos sum of potential and kinetic energy is zero. However, E₀=mc² was already known since 1905, so one could easily have taken this to state mc²/ħ as the minimum frequency of any particle's wave function.

[@gaHuJIa_Macmep]

25:16 there is some confusion here. When you do a fair Fourier transform, you take a psi(x) where x is a 4-vector and you get a psi(p) where p is also a 4-vector. And the integration in the Fourier formula goes over all the R^4 which you denoted by lower and upper limits (a little slovenly but let's keep it like that). But what you've done here is something else. As I understand you, you are talking about decomposing a wave packet's wave function into eigenfunctions of the KG equation (i.e. planar waves) but not all of them which exist in nature but only those which correspond to the given mass shell. Right? That means for a given particle's mass (because there are many mass shells corresponding to various masses; you choose a particular one). So you should integrate not over the whole R^4 but over this complex manifold (the mass shell) which is nontrivial (there is some intricate Jacobian inside) and it's a sin to denote the integral just by lower and upper limits (those infinities). It misleads. Also all sorts of questions arise here: which functions can be represented this way, what is the space of them and whether we really need the second component corresponding to "negative energies". You mention this but never justify this. This is a bunch of work here! Math work, I mean. And to call it just a "Fourier transform"... well, it resembles one but is in no way equivalent to it in a functional analysis sense. We know that Fourier acts from L_2 to L_2 but this is irrelevant here as you have a more contrived situation...

[@RichBehiel]

I often say “Fourier transform” colloquially and imprecisely whenever talking about a change of basis between position and momentum, or vice versa. Reason being, as long as I include the formula on the screen, that can take care of all the details. But I could be more careful about this in the future. In the example at 25:16, it’s a bit ambiguous and I should have clarified, but I’m showing a 1D example there, so the momentum axis just goes from -inf to inf. The E-p-m relation (mass shell) is implied, although if we want to be formal, we can slap on a Dirac delta function to enforce that constraint. But let’s talk about R4. In the case that we’re integrating over the entire hypervolume of R4, then we definitely want to equip our integrand with a Dirac delta function that’s only nonzero for four-momenta on the mass shell. That’ll carve up the 4D interval into the two separate curved 3D halves of the mass shell. I’ve seen it written that way before, usually toward the beginning of a derivation. But somewhere along the lines, people usually split up the integral over R4 into two integrals over R3, one for each half of the mass shell, in which E is simply a function of p. In that case the 3D interval is pure and straightforward. But conceptually both approaches are the same.

[@gaHuJIa_Macmep]

@@RichBehiel Yes, I see my mistake: you are right, we don't integrate over R^4, just R^3 for momenta. In this case yes, it's just ordinary Fourier. You see: confusion here stems from the fact that we have both 3-momentum (ordinary one) and a 4-momentum (which is a relativistic 4-vector of Energy-momentum), and they are both denoted by p! So you tend to mix things up sometimes, especially when you adopt a relativistic point of view at some point and tend to abandon all those "dirty" 3-vectors and deal only with relativistic, covariant entities. I'm sorry. But the question remained: what if we restricted ourselves with integrating over a single component of the mass shell? What kind of problems would have we stumbled upon? In what respect would our decomposition be inferior? Does it have to do anything with "negative frequencies" in a regular Fourier decomposition (which you are forced to take in order to get a real function, not just an analytic signal), or is it something different in nature? It's unclear from your video, you just mention it...

[@HebaruSan]

34:09 - "swap the real and imaginary parts" - I thought the complex conjugate was multiplying the imaginary component by -1?

[@RichBehiel]

Yup, you’re totally right, I misspoke 😔 Meant to say “swap the sign on the imaginary part”.

[@michaelparis6039]

μ has units [1/length], so what are the units of \psi - what does this operator measure?

[@-_Nuke_-]

I have no way of understanding any of this! But I liked the visuals :D

[@roundchaos]

God I wish I understood even half of this. Dirac seems like a genius among geniuses.

[@RichBehiel]

Dirac is right up there with Einstein and Newton. Truly a visionary, and I think his equation contains surprises that still haven’t been fully explored yet.

[@roundchaos]

@@RichBehiel Can't wait for your next video, your channel is incredible.

[@RichBehiel]

Thanks! :)

[@l.h.308]

Yes, Dirac was incredible and yet not quite satisfied with himself. His equation is for spin 1/2 particles. Has anyone tried to find similar equations for particles with spin 3/2, 5/2,... ? (They exist, don't they? Hyperons?)

[@yousefmohsen6287]

Thx

[@RichBehiel]

Thanks for watching! :)

[@gaHuJIa_Macmep]

7:16 you have mu in both roles: as a summand index and as "mass" as well. A little bit confusing. Better use nu for the former to avoid it...

[@RichBehiel]

Very true! It’s strange, I didn’t even notice I had used mu for both, until after posting the video 😔 Will be sure to avoid this ambiguity in the future.

[@Mouse-qm8wn]

As a non- physicist can you please explain to me if Klein Gordon and Dirac equations have been tested using practical experiments? If yes, Can you please give me hint where to look on the internet? Cheers from Denmark 😊🇩🇰

[@whoijacket]

“The math is about to get a little bit fancy…”. Has already used d’Alembertian Operators….

[@RichBehiel]

😂

[@beamshooter]

5:20 why didn't they make the "square" squared for consistency..... also saying square squared is fun!

[@RichBehiel]

I agree! Sometimes the laplacian is written without the 2, but then the triangle is flipped upside-down instead. So instead of nabla squared, it’s written as a delta. That notation aligns more with the d’Alembertian, but it’s confusing because delta is commonly used in other contexts.

[@beamshooter]

@@RichBehielthey really just want to confuse us.

[@ronaldjorgensen6839]

THANK YOU, 6,"30 AM DENVER

[@johnchase2148]

I started looking at the Sun 20+ years ago. I had instant connection with the Sun that was observed by witnesses and appearance changed the longer I observed. Since then I have believed that my thoughts was faster than the speed of light. Would like to have your thoughts for ,what energy and why me? I feel entangled.

[@KipIngram]

13:28 - Couldn't we just note that the KG equation is linear and move on? That implies the whole superposition thing.

[@RichBehiel]

Yes, you’re correct. With these videos, I have to think about what the audience will know, and what will be helpful to cover, in order to make the ideas as accessible as possible. It’s a balance between boring the people who already know the prerequisites, and leaving behind those who don’t. I try to find the right balance, but because there’s a distribution of knowledge in the audience, any choices I make will necessarily bore some people and confuse others. As I keep making these videos, I hope to get better at finding that balance. The goal is to make the ideas as accessible as possible, while also actually showing the ideas in detail (as opposed to just talking about them in a very digestible but vague way, as for example PBS Spacetime does - I’m a huge fan of their vids, but I strive to be more technical, even if that means appealing to a narrower niche). Based on your comments, I can tell you’re someone who really has a passionate interest in this subject matter, so if I were just talking to someone with your perspective, I’d just say the equation is linear and would move on. But I figured it might be helpful to show the superposition concept for those who haven’t seen it before.

[@tariq3erwa]

Can you take a detour to Pauli's theory?

[@lukeobrient2444]

31:00

[@b.clarenc9517]

41:50 Yeah, it would have been the wrong... Diraction.

[@Pradowpradow]

41:54 it would have been the wrong ... Diraction

[@RichBehiel]

😂

[@gaHuJIa_Macmep]

32:20 - well, yeah, this thing about KG being the 2nd order (and needing derivative as an initial condition) versus Schrödinger's being the 1st order also appeared to me. I even foresaw you will be talking about it in advance... he-he. But we are comfortable with this in classical mechanics where all our equations (being it Newton's 2nd law or Euler-Lagrange equations coming from Hamilton principle) are 2nd order, and what's the problem with this here?? Why could we tolerate it there and refuse to tolerate it now??? And the remedy for curing this "problem" is well-known (from ODE theory): phase space. You just invite the derivative to the team: you unite both the function and its derivative into a phase vector, and - voilà! - you have a 1st order equation (but a vector one) again! Why not do it here?..

[@RichBehiel]

That’s a really great question. And it’s tricky to answer, because many of Dirac’s original concerns with KG have since been somewhat obviated by the development of quantum field theory. Viewing it from a modern perspective, Klein-Gordon isn’t actually problematic, if you already know about bosons and fermions, spin-statistics theorem and all that, then yeah, Klein-Gordon works great if you happen to have a spin-0 particle (these are weird particles, pi mesons and Higgs bosons and such). Anyway, physics is much more mature now, so everything has its place, including things like Klein-Gordon when viewed in its proper context. But it was a different world back in 1929. For example, nobody knew that spinors, which were first explored I think by Cartan back in 1913, had such a fundamental constitutive role to play in the wavefunction of an electron. And as we saw in the Dirac equation video, going from 2nd order to 1st order is precisely what makes spinors pop out of the mass shell. So, we can say that Dirac was correct in taking issue with KG being 2nd-order, because making it 1st-order is precisely what needed to be done in order to get the correct character of the wavefunction. But that’s a retrospective explanation that doesn’t really address what Dirac was thinking at the time. One of his concerns with the 2nd-order nature of KG is that it led to probability densities which were not positive-definite, as shown in this video; if KG were first-order instead, then the four-current would not have the problematic time derivative term that makes the probability sometimes go negative (even this is not a problem from a modern QFT perspective, when accounting for creation and annihilation). So anyway, the derivation of the probability density and current provides one mathematically precise explanation of what at the time seemed like a fundamental issue with KG. There are also some related concerns regarding unitary time evolution. I hope this answer shed some light on the situation. I’m sure there’s a lot I’ve left out here.

[@gaHuJIa_Macmep]

@@RichBehiel yes, thanks. Just a couple of comments: 1. You are right in that we are talking in hindsight now, and it's always more instructive to find out what was the original thinking back then. Although oftentimes the original intentions turn out to be false... 2. This conserved quantity, being not necessarily positive (non-negative) may not be a probability density but, say, some charge density (say, baryonic charge) and as such it can be either sign. So it shouldn't constitute the problem in itself... In one word, I think, Dirac was led by his own motto: "Laws of nature should possess mathematical beauty", and it is what was his guiding principle...

[@RichBehiel]

@gaHuJIa_Macmep yup. That’s the view that a lot of Dirac’s contemporaries took at the time, that it’s charge density. The problem they couldn’t solve is how to treat spin. Equipping KG with a classical angular momentum term wouldn’t have worked, for example. So Dirac’s intuition turned out to be productive, even if misguided. In modern times it has become fashionable to say that Dirac was right, but for the wrong reasons, or something to that effect. You see this attitude in Weinberg’s Quantum Theory of Fields volume 1, for example, and that book has set the tone of a generation. But I think it’s a bit unfair. Given the state of physics at the time, Dirac’s adamant refusal to go with the herd, and instead to follow his intuition and the pursuit of beauty and logical elegance, turned out to be exactly what was needed to bring clarity to the relativistic electron. And his equation is still the basis of so much physics to this day. So it’s not wrong to examine the validity of Dirac’s assumptions in a modern context, but it’s also important to remember and appreciate the historical context.

[@gaHuJIa_Macmep]

@@RichBehiel Yeah... In one word - great man. Such people like Dirac, Feynman, Einstein, Arnold - let us reconcile with Mankind's existence and give it a meaning and justification...

[@gaHuJIa_Macmep]

BTW, thank you for the term: "mass shell", I learned it from your video. Back then, in 1990-ties when I was learning subatomic physics our lecturer used the term "mass surface" for it (in another language) and I thought it's a literal translation from English; it turned out you call it "shell", not just "surface" (for obvious reasons, of course), but this is a bit of terminology which you can't deduce, you just need to know it.

[@erawanpencil]

@27:15 have you become Daoist yet? ;-) "Opposites are not contradictory, but complementary" -Bohr

[@RichBehiel]

Personally, I’m not particularly religious. In my opinion, physics contains the last credible residue of religion, and even that is probably a mirage. But I really enjoyed the Tao Te Ching, and definitely resonated with its overall vibe.

[@angelamusiemangela]

Quindi questo teorema cosa dice:? Che....?????? (Suggerimento) Quali sono i numeri reali? e fino a quale numero sono primi,partendo da 0 che , può essere compreso o nullo? Quale la teoria che porta alla centralità della figura geometrica riscontrata,in piano, cioè la circonferenza ,di Dirac? Qual'e' il punto massimo in percentuale di risultato trovato? Quindi... Il teorema è:.........

[@tuk7raz]

Hello from Kazakhstan! You are familiar with Michelson's experiments. (1881) and its improvements for the discovery of gravitational waves (2015) - And that's only 50%. It is possible to continue this experience; Use "from the gyroscopes two non-circular spools of optical fiber." This device, which is located inside the car, measures: inertial speed in a straight line. (the device does not record the speed of the satellite or planet in orbit). I can share ideas for joint invention - Light is an ordered vibration of gravitational quanta. There is a company in China that makes (fiber optic angular velocity meter) they will be able to create a hybrid device. Please, can you come to an agreement with them? I guarantee payment at cost on my part.

[@KipIngram]

Hold it - when you "factored out" the derivative operation, you carried that operator across psi and psi*. That seems illegitimate to me - psi varies in space and time. In your final for the outer derivative is going to act on those. Ok, wait - it seems that the extra term you pick up might wind up canceling out, since you're doing this to both terms in the difference. But... that probably should have gotten some attention. It does happen to work out given your full equation, but it was a "suspect step" just the same - you should show WHY it's ok.

[@RichBehiel]

Thanks for pointing that out. I’m in a bit of a rush this evening to get some things done, so I don’t really have time to review it in depth at the moment. But if I recall correctly, I did gloss over a few details at that step. The math checks out, but it’s not as simple as I might have implied.

[@gaHuJIa_Macmep]

38:59 - a little incorrectness here. I understand what you mean here but the numbers you compare are purely imaginary (because in the expression for rho it is multiplied by i with something), so, strictly speaking, you can't compare them, you can't use the "

[@gaHuJIa_Macmep]

And one more little thing (or, maybe, not that little). You derived the continuity equation - well done. Meaning that there is some "conserved quantity", you call it rho. In the sense that derivative of it with respect to time is equal to divergence of some vector field. So, we call this quantity a density of something, and the said vector field - a flux density of this something. Ok. But where do you get the fact that this something is a probability density of the particle from? It doesn't correspond to the standard quantum-mechanical modulus of psi squared, so a leap of faith is needed here... Or, can you justify it somehow? Or is it a new, independent postulate?..

[@KartikPatel-nt4ff]

😅😮😅😅😮😅😮😅😅😅😅😅😅well ingormeti0n.Good show more content 😅😅

[@Valentino-wb4hf]

Wha the fue

[@RichConnerGMN]

what

[@RichBehiel]

huh