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

[chriskindler]

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

[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!

[lexinwonderland]

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)

[Robinson]

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

[pronounjow]

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

[oscareriksson]

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

[Raspberry_aim]

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

[mr_rede_de_stone]

I like the idea of explicitly showing the 4D spaces you're talking about, it kinds of demystifies them in a nice, visual way.

[pacificll]

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

[BiswajitBhattacharjee-upvv]

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.

[jlowe]

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

[markawbolton]

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

[rift]

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.

[FirstLast-oejm]

"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!!!)

[DrDeuteron]

9:38 I like the phase animation. Note that for a particle at rest, the phase is constant in space and it rotates in time. Mass is the the thing that tells us how fast it rotates in time at zero wavenumber.

[williamharr]

Great video as always! Commenting to support the video

[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

[tune]

I am really excited for the next video :D

[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?

[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. 🤗