Spinor Lorentz Transformations | How to Boost a Spinor

In this video, we will show you how a Dirac spinor transforms under a Lorentz transformation.

Contents:
00:00 Our Goal
00:38 Determining S
01:23 Determining T
02:30 Finite Transformation

References:
[1] Peskin, Schroeder, "An Introduction to Quantum Field Theory".

If you want to help us get rid of ads on YouTube, you can support us on Patreon!

[aminamouhamed]

Do you think you can make a video or more than one about Groups and representations and how to use them in different situations? Like SO(3), SU(2), Clifford Algebras, Dirac reps . I find them very confusing. That would be amazing. Thank you for your work!

[redandblue]

I really can't quite understand what's going on, but I'm so glad I found this channel! I hope to study physics so there's lots to learn here.

[NikolajKuntner]

Thanks for making such videos. I'm interested in how you choose which topics you cover or what you think your intended audience is? I made some youtube videos on geometric topics, but I feel I'd always want to take a step back and discuss a topic that's more on the base and required to understand the topic at hand. If you're tackling physics concepts, like here, and if the videos are also short, then I suppose the videos do either spark interest in the watchers, or otherwise are tailored to a rather small group of people that are knowledgable enough to get it but had problems to understand those issues from books.

[wernerheisenberg]

Again you helped me so much! Thank you! <3

[SpotterVideo]

What do the Twistors of Roger Penrose and the Hopf Fibrations of Eric Weinstein and the "Belt Trick" of Paul Dirac have in common?
In Spinors it takes two complete turns to get down the "rabbit hole" (Alpha Funnel 3D--->4D) to produce one twist cycle (1 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, with each twist cycle of the 4D Hypertube proportional to Planck’s Constant.

In this model Alpha equals the compactification ratio within the twistor cone, which is approximately 1/137.

1= Hypertubule diameter at 4D interface

137= Cone’s larger end diameter at 3D interface where the photons are absorbed or emitted.

The 4D twisted Hypertubule gets longer or shorter as twisting or untwisting occurs. (720 degrees per twist cycle.)

If quarks have not been isolated and gluons have not been isolated, how do we know they are not parts of the same thing? The tentacles of an octopus and the body of an octopus are parts of the same creature.

Is there an alternative interpretation of "Asymptotic Freedom"? What if Quarks are actually made up of twisted tubes which become physically entangled with two other twisted tubes to produce a proton? Instead of the Strong Force being mediated by the constant exchange of gluons, it would be mediated by the physical entanglement of these twisted tubes. When only two twisted tubules are entangled, a meson is produced which is unstable and rapidly unwinds (decays) into something else. A proton would be analogous to three twisted rubber bands becoming entangled and the "Quarks" would be the places where the tubes are tangled together. The behavior would be the same as rubber balls (representing the Quarks) connected with twisted rubber bands being separated from each other or placed closer together producing the exact same phenomenon as "Asymptotic Freedom" in protons and neutrons. The force would become greater as the balls are separated, but the force would become less if the balls were placed closer together. Therefore, the gluon is a synthetic particle (zero mass, zero charge) invented to explain the Strong Force. The "Color Force" is a consequence of the XYZ orientation entanglement of the twisted tubules. The two twisted tubule entanglement of Mesons is not stable and unwinds. It takes the entanglement of three twisted tubules to produce the stable proton.

[Cosmalano]

1:07 why did you multiply by the inverse on only one side?

[richardthomas]

Excellent! Right around 1:15 you left multiply each side of the equation by lambda sigma-nu (a Lorentz transformation matrix). How do you know that it can slide past the S^(-1) matrix -- i.e., commute with it? After all, the S matrix is a function of the lambda matrices, which do not commute with each other. Thanks!

[erbgzuwcv]

You used the exponential map for the Lorentz group O(1, 3). But how can you be sure that this map is surjective? For example I cannot generate O(3) by taking the exponentials of skew-symmetric matrices, because the determinant of these matrix exponentials is always +1. Therefore I only generated SO(3).