The Mass Shell (Relativistic Energy-Momentum-Mass Relation)

In this video, we look at the Mass Shell, a way of visualizing the relativistic energy-momentum-mass relation, which is a central concept in special relativity. A good understanding of the mass shell will set us up for our upcoming explorations into relativistic wave equations.
Stay tuned for the next videos, in which we will derive relativistic wave equations, explore the four-potential and gauge symmetry, and eventually will return to hydrogen with a relativistic treatment of the electron! :)
For further reading, please check out Introduction to Elementary Particles, by David Griffiths. That book provides a ton of insightful context around the ideas in this quantum physics playlist.
Chapters:
0:00 Intro
0:59 Four-Momentum
3:57 Mass Shell in 1+1 Dimensions
5:50 Mass Shell in Higher Dimensions
9:27 Example: Klein-Gordon Free Particle
#physics #quantum #math

[@DrBrianKeating]

so good!!! thanks

[@RichBehiel]

Thanks Brian! :)

[@chriskindler10]

your videos are simply excellent. I have been so invested in QFT for the past years that I was just doing calculations and taking the core ideas for granted

[@RichBehiel]

Thanks, I’m glad you’ve enjoyed the videos! :) With QFT there are so many equations that it’s easy to get caught up in the technical details, but beneath it all are some very elegant ideas.

[@BiswajitBhattacharjee-up8vv]

You have made the field open to all other field learners. With all our blessings your all too good videos will be good resource and learning materials for AI. It's YI FOR US.

[@lexinwonderland5741]

I'm the most excited about the Klein-Gordon equation, particularly because I recently learned about the *sine*-Gordon eqn (u_tt-u_xx+sin(u)=0 instead of u_tt-u_xx+u=0) and I'm excited to hear your illustration of the concept. The sine-Gordon eqn is a way to generate pseudospherical surfaces (i'm a mathematician interested in differential geometry, not a physicist haha, I just love watching higher physics videos bc they're very in-depth examples of what i'm studying elsewhere)

[@Valerii_Pivovarov]

@lexinwonderland5741, ФИЗИКА изучает законы Природы. Это не придуманные человеком законы, ибо их создала сама Природа, а человек их только открывает. Именно поэтому ФИЗИКА является наукой о Природе. А вот, математика - это плод человеческого творчества. Она для физика является всего лишь великолепным инструментом для проверки собственных идей. Но, если в результате проверки получен достойный физический смысл, то объяснить ВНЯТНО этот физический смысл можно и без высшей математики. Заметьте, получив проверенный результат, физик может объяснить его смысл на базе обычной алгебры и его поймёт даже школьник. Повторяю - ПОЙМЁТ даже ШКОЛЬНИК. Поэтому ФИЗИК должен в совершенстве знать высшую математику, все её достоинства и, главное, недостатки. Но и математики не «лыком шиты», ибо их заслуга в том, что они обеспечивают развитие математики и тем самым облегчают деятельность ФИЗИКОВ. Но, беда, если математик (и уповающий только на математику физико-математик) сам вторгается в ФИЗИКУ, создавая собственные математические модели. Во-первых, математик считает, чем сложнее его математическая модель, тем она правильнее, а он сам - гениальнее. Он даже не понимает Святое Правило: Истина в простоте. Не зря над входом в физическую аудиторию Гёттингенского университета большими золотыми буквами по-латыни начертан девиз: «Simplex sigillum veri» (Простота - печать Истины). Во-вторых, математик даже не догадывается, что основой в ФИЗИКЕ является ФИЗИЧЕСКИЙ СМЫСЛ. Чтобы не быть голословным, привожу конкретный пример: Птолемей в своё время создал сложнейшие эпициклы, доказывая недоказуемое, что Солнце вращается вокруг Земли. Физического смысла в его теории не было, но ею пользовались почти полторы тысячи лет, пока не появились открытые великим Кеплером ПРОСТЫЕ и внятные законы Природы (особенно третий Закон Кеплера), объясняющие движение планет (в том числе и Земли) вокруг Солнца. С трудом и не сразу, но затем даже математики признали эту фальсификацию Птолемея.

[@daniellewilson8527]

I didn’t know pseudospheres were a thing, that’s cool

[@ArnavBarbaad]

Omg thank you for this! I have recently finished learning about Noether's theorem and how it's a "on shell" theorem, but couldn't understand what on and off shelf really meant. Gonna watch this one with popcorn!

[@Robinson8491]

Always heard Nima Arkani Hamed talk about being on shell and off shell. Thanks for finally giving an image to those words, and even an eqaution

[@Raspberry_aim]

Thank you so much for making these videos! The depth of information, explanations, graphics, and presentation are all excellent!

[@RichBehiel]

Thanks, I’m glad you’re enjoying the videos! :)

[@oscareriksson9414]

I have no idea what you were saying but this diagram was one of the coolest things I've ever seen

[@pronounjow]

That's the trippiest tesseract animation I've ever seen! 🤩

[@rift1067]

Question at 2:50 regarding the "third thing we have to know": how did we conclude that the magnitude of the four-momentum will simply equal mc? Did we deduce this from Einstein's initial derivation in his paper where he used energy-momentum relations of electromagnetic fields in the framework of SR, or is there a more organic way of integrating mass into the relation without invoking a specific scenario? I am so looking forward to the rest of the series. I love QFT, but I've struggled to an embarrassing degree in my nuclear physics undergrad course, so I have a LOT of gaps in my knowledge. I hope this series will allow me to jog my memory and clear any misunderstandings, misinterpretations and complete obliviousness towards the more abstract details of this subject as I dive into more difficult topics in high energy physics for my masters.

[@RichBehiel]

Great question! In this video I just took that as a given, but it can be derived from deeper principles. There’s the photon in a box thought experiment, for example. But I don’t actually know how this was originally derived. That could be an interesting video.

[@rift1067]

@@RichBehiel thank you for the insight. I would love to watch that video if you do decide to do one.

[@angelmendez-rivera351]

The Minkowski norm can be derived by noting that the four-momentum is equal to the Lorentz mass (i.e., the rest mass) scaling the derivative of the four-position with respect to the proper time. The four position is the four-vector (ct, x, y, z). When differentiating with respect to proper time, one gets γ(v)(c, vx, vy, vz). You can verify that the Minkowski norm of this four-vector is c, and when multiplying by m to get the four-momentum, the Minkowski norm is mc.

[@ChasSimpson]

I saw some comment earlier about you being the physics version 3blue 1brown. I think you're even better! Thank you for the vidoes

[@RichBehiel]

Wow, that’s the highest compliment! :) Grant is a role model, and an inspiration for the channel.

[@pacificll8762]

Great work! Can’t wait for the next videos!

[@RichBehiel]

Thanks! :)

[@PawlTV]

This is beyond fantastic.

[@RichBehiel]

Thanks, I’m glad you enjoyed it! :)

[@markawbolton]

Great timbre on your Voce, good crisp well presented Narrration... a real pleasure to listen too...

[@millamulisha]

Commenting on the connections between curvature and mass… (You may already know this but…) There is an interesting similarity between how ‘effective mass’ in the theory of material mobility (electrical conduction) is calculated and this picture you’ve given us. Long story: If you are given the lattice structure of a material you can derive a potential for it, using this potential you solve the Schrödinger equation for an electron in this potential, from the solution you can derive an Energy vs Wave number relation (along different directions of the lattice), this relation allows you to look at the various wells and peaks of the conduction band (as well as a valance band which I’ll ignore), from this diagram evaluated at the various minima you determine the curvature, in the theory this curvature is inversely related to the ‘effective mass’ (in the theory) where higher curvature implies lower effective mass, given there are various minima along different directions you take the harmonic mean of these to obtain the average effective mass of conductivity. This is not too dissimilar to the way in which mass is related in the mass-shell picture (higher curvature at the minima of the ‘Energy vs Momentum’ diagram implies smaller mass). Anyway, I just thought the correspondence was neat - a connection to material science. 🤗

[@mistertheguy3073]

Love this so much

[@jlowe8059]

I rarely like and subscribe, but this channel earned it.

[@RichBehiel]

Thank you! :)

[@FirstLast-oe2jm]

"there's three things you gotta know,,," *tesseract* all jokes aside I've used your videos to help get a conceptual level paper and pen hasn't, thank you for your work (also please!; It was an offhand in your part 2 video but, a view on the real value equations would be cool and valuable!!!)

[@williamharr7338]

Great video as always! Commenting to support the video

[@RichBehiel]

Thanks! :)

[@corsaircaruso471]

My 🍃 brain has been exploring the relation of the mass, energy, and momentum.

[@tune490]

I am really excited for the next video :D

[@richardjowsey]

Those ±√ energy-momentum shells appear to express the Poincaré group's reflection (mirror) symmetry

[@RichBehiel]

I’ve usually heard it explained in the context of CPT symmetry, which is closely related to mirror symmetry. In QFT, the discrete charge conjugation symmetry takes, for example, a spin-up electron and converts it to a spin-down positron, a kind of CP switch, which by CPT is equivalent to time reversal. So positrons can be thought of like electrons that move backwards in time. But then looking at the relationship between energy and time in quantum mechanics, for example in an energy eigenstate, we see that the energy rotates the complex phase everywhere with a certain angular frequency given by E/hbar. In that context, negative energy is the same as positive energy, but the phase just goes the other way around in the complex plane. So if positrons are imagined as electrons moving backwards in time, their “negative” energy is actually positive, or rather, the distinction between negative and positive energy is undermined by the symmetries involved with spin and charge.

[@davebradshaw3188]

Keen for your dirac equation/notation video :)

[@DeclanMBrennan]

That was beautifully explained and both revealed and clarified a lot for me. Thanks. For m=0, the shells appear to end up being a double cone. This cone seems to be almost the dual of the light (or null) cone where position is the dual of momentum and time is the dual of energy. I realize E has been divided by c to give it momentum units so it's not an exact dual, but this is just a constant scaling factor on the vertical axis.

[@RichBehiel]

Thanks, I’m glad you enjoyed the video! :) And great observation! As we’ll see in the next video, the idea that the mass shell for a massless particle is the dual of the light cone is very deep. Since the momentum operator is -i*hbar*gradient, and the energy operator is i*hbar*d/dt, we can combine those operators into the four-gradient, then apply it twice to get the d’Alembert operator, and the E/c goes along nicely with the ct in the position four-vector since the d’Alembertian divides by c^2.

[@DeclanMBrennan]

​@@RichBehiel I can't claim to fully understand that yet 🙂, but it is tantalizing. I'm very much looking forward to the rest of the series.

[@dullyvampir83]

Great video! Just a question if I want to study a particle I would set 4 Numbers to describe it: Energy and the 3 momenta, doesnt't matter if it is a photon or electron or other point mass. With this the minkowsky distance is set and all paths that have the same distance are physical valid?

[@ColeCoug]

Great videos! I am wondering what a Fourier transform of the mass shell would look like and could we see it plotted in something like spacetime diagrams

[@RichBehiel]

Thanks! :) Imagine some complex-valued function defined on the mass shell. Its Fourier transform, up to some factors of hbar and c, is a wavefunction in spacetime that obeys the Klein-Gordon equation for a free particle. That’s hinted at in the equation near the end of the video, but we’ll dive into it much more deeply in upcoming vids. The space of all complex-valued functions on the mass shell can be mapped to the space of all free-particle Klein-Gordon solutions, assuming the negative half of the shell is included too. That’s actually probably the best argument flor why the negative energy shell is needed - without it, we wouldn’t have a complete set of states.

[@mosquitobight]

I like the cycling hypercube, even though it didn't really help me visualize four-momentum.

[@RichBehiel]

I probably should have lingered on that topic for a bit longer. Basically imagine the space of all possible momentum vectors, it’s just a 3D space, since momentum can point in any direction and can have any magnitude. So we can use a cube to represent that space, although the space extends beyond the boundaries of the cube. Now add another dimension for the energy. So for each momentum vector, we can assign an energy scalar as well. This is like extruding our cube along an energy axis, which is shown with the color gradient in the hypercube. In classical physics, energy and momentum are treated as two separate but related quantities. In relativity, we can unify them into the four-momentum, so that rather than a vector momentum and a scalar energy, we have a four-vector for this energy/momentum thing. And that lives in a 4D space. But the energy and momentum are related. More momentum, more energy. So actually each momentum vector has only two possible energy values to go with it (positive and negative). This constrains the 4D space into a 3D subspace that’s shaped like a couple of shells - the mass shell. The four-momentum is still a four-dimensional quantity, but its range of values is restricted to the shell. In that way, the mass shell provides a core constraint on relativistic physics. And we’ll see later that making relativistic waves is basically just a matter of doing Fourier transforms from the mass shell into spacetime. Although this becomes much more nontrivial when using the Dirac equation. I hope that helped clear things up :)

[@mosquitobight]

@@RichBehiel Thanks for the detailed explanation. I'll do my best to work my way through it. :) So as I understand it, if the speed of light is the scaling factor between a time dimension and a space dimension, then an object's energy is just its "momentum" in the time direction?

[@RichBehiel]

@mosquitobight that’s correct! Energy is like the component of momentum that points in time, that is, the component that endures without spatial preference. Personally I like to imagine the time component of any four-vector as being a kind of microscopic swirling. Motion without net propagation. But that’s only a loose metaphor.

[@HebaruSan]

For a function with a scalar return value and vector3 input, my go-to visualization is fog of varying density over the space of a room. Is that valid here? I guess it would need a way to represent positive and negative, maybe blue and red fog?

[@5ty717]

Legend

[@AstroPatel]

Your visualizations are amazing. Do you have a publicly available package? I would've loved something like this during grad school

[@Twas-RightHere]

He has a video specifically showing how he makes his visualisations.

[@AstroPatel]

@@Twas-RightHere thanks!

[@Twas-RightHere]

@@AstroPatelWait sorry, just checked. I’m thinking of a different channel. However, from the look of the video my guess would be he’s using 3Blue1Brown’s python library called “Manim”.

[@RichBehiel]

Thanks! :) I don’t, unfortunately. I always make these in such a hurry that it’s all just spaghetti code. One of these days I’d like to focus more on the animation codes, but for now I want to put most of my time into making the vids, at least up until part 3 of the hydrogen atom series.

[@AstroPatel]

@@RichBehiel makes sense, I made my fair share of smooth animations but nothing quite on this level. I await your future videos!

[@carly09et]

Interesting projection of a phaZe space. Is E a 'complex' argument? The factor E/c is complex with respect to three space geometry. The nature of c^2 == e_0*u_0 and mass being a function of density in 'space' .

[@feynstein1004]

3:58 You've got mail 😋

[@RichBehiel]

😂

[@renscience]

Can you sum up the Minkowski metric in your usual easy to understand ways please?

[@purplenanite]

This is a bit off topic, but... If I added an additional time dimension, the equation would be (i think) (mc)^2 = (E_0/c)^2 + (E_1/c)^2 -p^2 . So that forms a different conic section - If i am interpreting that correctly, would that mean there would be a way to continuously transform matter into antimatter (in a universe with 2 time dimensions), or am i off base?

[@RichBehiel]

Very interesting question! Honestly I’m not sure. I’m used to thinking about antimatter in the context of the Clifford algebra of Minkowski spacetime, so adding another time dimensions kind of blows up that picture. One way to answer that question would be to plot in 3D the (p,E1,E2) surface and see how it looks. You’d have axial symmetry along the p dimension. I suspect you’d have the same kind of hyperboloids for fixed mass, but the way the mass parametrizes the family of hyperboloids might be inverted. Idk though. Very interesting question!

[@purplenanite]

@@RichBehiel I mean, I do suppose you could do clifford algebra in that space - you would just have two timelike basis vectors. (+,+,-,-,-) The mass shell does seem to be simply connected in this case. There are two shells only in instances where there is one dimension of 1 type and N of the other But yeah, it does tend to blow stuff up - sorry for that. I love your videos!

[@RichBehiel]

@purplenanite no need to apologize! This idea is going to be on my mind for days 😅 I haven’t though about this at all, but a simply connected mass shell would lead to so many possibilities. Although it’s hard to imagine a 3+2D space. I’m glad to hear you love the videos! Thanks for watching, and for the interesting comments! :)

[@purplenanite]

@@RichBehiel Thanks! if you figure anything out about this, I would love to hear more about it!

[@dantebroggi3734]

This is interesting, and I too would find it interesting to hear more about it. A bit of raw speculation I found recently on other weird things one might have with 2+ time dimensions: kzbin.info/www/bejne/n5inn6SQfJVgrNU

[@magicponyrides]

If I understand this video will I finally understand what the "on-shell" condition or assumption is?

[@RichBehiel]

Yes! :) “on-shell” means the energy and momentum obey the relativistic energy-momentum-mass relation, in other words it’s a real particle. “Off-shell” means it’s a virtual particle. In the 3+1D space of four-momenta, the mass shell is the 3D subspace of four-momenta that a real particle with some mass can have.

[@magicponyrides]

@@RichBehiel Oh man, thank you so much.

[@ChildishBerbino]

Ooh yeah tell me about the mass shell ;)

[@forheuristiclifeksh7836]

1:36

[@_a.no.n_]

why do you rapresent the 4 dimensional shells as a hypercube? Can't you add time to the 3D, making it change in time? (move/change shape)?

[@RichBehiel]

Sure, that’s another way to do it! :)

[@_a.no.n_]

@@RichBehiel But it would be visualizable! :ppp 😀

[@FunkyDexter]

Great video. Isn't treating negative energy solutions as antimatter inconsistent though? Negative energy would produce an opposite sign curvature, unless I'm misunderstanding. Wouldn't that mean antimatter has antigravitational effects? Yet it falls down just like normal matter.

[@RichBehiel]

Very good question! This question bothered Dirac enough to come up with his Dirac sea concept, which although far-fetched, was the only remotely viable way to make the problem go away at the time. Problem is, the negative energy solutions are required in order to have a complete set of states. They can’t just be disregarded as nonphysical, or else for example the positive energy plane waves on their own can’t span the space of solutions to the relativistic wave equations. It’ll take a couple videos to unpack this in more depth. We’ll have to explore the symmetries of the Clifford algebra of Minkowski spacetime, and think about the relationship between CPT symmetry and the connection between energy and the direction of time.

[@FunkyDexter]

@@RichBehiel I've looked it up, and apparently the different way QFT and GR treat negative energy (the former as antimatter with positive energy, the latter as some sort of exotic solution which violates energy conditions) is one of the central issues in reconciling the two theories, is that correct?

[@RichBehiel]

@FunkyDexter honestly I don’t know. But I suspect you’re right. Definitely one of the major issues with combining QFT and GR is how to handle the vacuum energy. QFT predicts that there should be way more energy density in the vacuum than our universe’s tiny cosmological constant can account for. That problem is known as the vacuum catastrophe. I’m not sure how negative energy plays into it, but that definitely seems like a related question.

[@DestroManiak]

She sells mass shells by the hyperbola shore

[@KingShinyRotom]

Whereeas this when I was writing my thesis ToT

[@phrygianphreak4428]

Only two shells? And i thought figuring out how to use three shells was hard enough

[@richardspectacular5327]

So you're telling me that empty space is a semiconductor? I wonder if one could make the vacuum tube equivalent of a BJT using antimatter... 🤯

[@angelmendez-rivera351]

What? No. He said nothing remotely close to this.

[@ShinobiNatrix]

hum hum first

[@RichBehiel]

Nice!

[@JH-le4sd]

Is the Minkowski metric motivated by anything else? Doesn't this mean either energy/time or momentum is imaginary?

[@RichBehiel]

Great question. It’s usually introduced in the context of spacetime separation of events. Say two events are separated by some four-vector [ct,x,y,z] as measured in some inertial frame. Another observer in another frame will measure a different separation, due to the Lorentz transform. But, both of those four-vectors will have a magnitude which both observers agree on, if the Minkowski metric is used. So, both observers agree that s^2 = (ct)^2 - x^2 - y^2 -z^2 units of spacetime are separating the two events, where s is the spacetime interval. When s = 0, the two events are light-like separated, as in a beam of light could shine from one event to the other. Or, whether s is positive or negative determines if the events are more separated in space or more separated in time. To your second question, check out the “imaginary time” page on Wikipedia. It’s an interesting framework. As far as I’m aware, it’s a different but essentially isomorphic way of looking at things. One of my upcoming videos is going to be on how we don’t actually need imaginary numbers when doing complex algebra.

[@evgeniibarannik8996]

But why there is strange asymmetry between py, pz, E/c on the one side and px on the other side? First 3 values share one vertex on your 4d graph, but px is separated compared to them.

[@RichBehiel]

Great question. It has to do with the way the plot is projected from 4D to 3D, while the 4D objects are being rotated. To see that the asymmetry isn’t inherent in the actual datapoints, imagine rotating the plot around the vertical axis in 3D. Then the same question arises about py instead of px.

[@Robinson8491]

Mass of the particle x the speed of light = the identity of the particle

[@angelmendez-rivera351]

??

[@Robinson8491]

@@angelmendez-rivera351 yeah I know I was in a phase. But basically it's the observable in quantum mechanics, a.k.a. a 'particle' I guess. And that's when something has an identity. As a wave function it doesn't. It's defined as a particle, and thus an identity. Or something like that

[@angelmendez-rivera351]

@@Robinson8491 Your response does not help much, and it is still largely incomprehensible. •A particle does not represent an observable, nor does an observable represent a particle. A particle is represented by a Hilbert space, on which the observables operate on. •mc is just a scalar, not an observable. •Your usage of the word "identity" has no meaning in physics.

[@Robinson8491]

@@angelmendez-rivera351 so according to you the wave function is also part of the particle? Doesn't the wave function move with light speed as a maximum according to the Schrodinger picture? And could you please tell me about the quantum flux and its relation to the special relativistic light speed limitation? Thank you

[@thomasolson7447]

(w^2-x^2-y^2-z^2)^2+(2*w*x)^2+(2*w*y)^2+(2*w*z)^2=(w^2+x^2+y^2+z^2)^2 This is kind of like a double angle I guess. It would be the angle formed from [1,0,0,0] and whatever 2d plane it makes. If this means something here, the double angle in 2d also means something. I always dismissed it, to be honest. I suppose I should look into it. But we're using the wrong variables. It can all be done with polynomial zeros, or algebra. Not sure how to address that. Energy and mass are in the square root, not scaled on the outside of the expression. Also, your graph is wrong. It should touch the corners and form a grid. And you don't have that potential energy tent between -1 and 1 on z.

[@angelamusiemangela]

Shellyne! Over-size..gravity leopardian...stack of shery i pole..... The end.Baoliiiii .....mi corazon!😘😘😘😘😘😘 Love youuuu! Baci baci.

[@Valerii_Pivovarov]

Уравнение Клейна - Гордона используется для описания быстро движущихся частиц, имеющих массу покоя, будто кто-либо видел элементарные частицы в покое. В Природе буквально всё находится в движении. Это уравнение для вышеупомянутой свободной частицы имеет простое решение в виде синусоидальных плоских волн. Так всё-таки, ЧАСТИЦА или ВОЛНА? Ребята всё запутали до предела. И главное. Чтобы понять бездарность СТО, достаточно обратиться к общеизвестным фактам. Первое. Знал ли Майкельсон, а вслед за ним и наш «гений», что свет распространяется в электромагнитном поле в виде КОЛЕБАНИЙ этой материальной среды? И этот факт уже давно общепризнан. Майкельсон этого не знал, ибо думал, что свет распространяется в «неподвижном эфире». Второе. Электромагнитное поле Земли (в котором и распространяется свет) не стоит на месте, а движется вместе с Землёй. Это означает, что в опыте Майкельсона источник и приёмник света, а так же среда, в которой этот свет распространяется, имели одинаковую скорость движения. То есть, в системе отсчёта, связанной с центром потенциального поля Земли, сам Майкельсон, его интерферометр и электромагнитное поле Земли находились друг относительно друга в «состоянии покоя». Этот факт тоже никому не удалось опровергнуть. Какой «ветер», при этом, надеялся «поймать» Майкельсон? Понятно, что данный опыт является совершенно бессмысленным. Следовательно, такой же бессмысленной является и основанная на этом опыте бездарная теория нашего «гения». Поэтому меньше обращайте внимания на математические изыски, лишённые элементарного физического смысла.

[@angelamusiemangela]

Questo di due punti di folata di vento, continuo e costante derivato dalle folate delle fiandre.E ci vogliono bene per sempre, quindi un' infinità di baci e un abbraccio 😘💯