I’ve added new material to the non-trivial spin part of the notes (in section 4). This material covers Lorentz transformations of angular momentum and taking the massless limit. I’ve also added a couple of new exercises.
Month: October 2019
Mistakes and Muddles and Notes Update (10/11/19)
Last class (on 10/10/19) I got a bit confused during what I expected would be a simple justification for the Lorentz transformation of the spacetime momentum operator. As it turns out, the calculation was telling me that there was an inconsistency in the formulas that I wrote down!
The issue I ran into is connected to the difference between a passive and an active transformation. You can think of an active transformation as taking our particle and boosting it so that it has a different momentum in our reference frame. By contrast, you can think of a passive transformation as boosting an observer who initially is in our frame to a new inertial frame. Now, given a particle in a state of definite momentum p, if you boost the particle then you with your original momentum operator will measure the particle’s momentum to be Λp. If you also consider a boosted observer who moves along with the boosted particle then they should observe the original momentum p. Therein lies the error I made!
I’ve decided to punish my students for the error of my ways and I’ve added an exercise (4.7) to my notes!
More On Spinors
I came across a very nice paper that goes into good detail about spinors in 3D space and 3+1 dimensional spacetime. I should add however that in other spatial dimensions certain features of this explanation don’t work (I don’t think)—basically for the same reason that you can only define the cross-product of two vectors in three-dimensions.
https://arxiv.org/pdf/1312.3824.pdf
Big Notes Update: 10/1/19
I’ve updated the notes with a whole new section on relativistic quantum mechanics. I’ve also added a document QFT.SR.Solutions.pdf that contains solutions to the special relativity section problems that I assigned.