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.
Category: Updates
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
Notes Update 9/20/19; Fun SR Facts
I’ve posted an update to the notes with some reorganization of the section on quantum symmetry. Specifically, I included a subsection describing the momentum state eigenbasis and the Fourier relation between position and momentum bases.
I came across a fun article the other day that highlights something that is rather surprising about how length contraction would appear in the real world. Usually when I’ve described length contraction, we imagine it from the side in a rather two-dimensional sort of way. We draw a picture of a 2D car with the car’s image squeezed along the direction of its motion. This is fine for getting the idea across, but it doesn’t actually reflect what we’d really see. You see, objects are three dimensional and light from different parts of the object would reach us at different times. It turns out that as a result of this, length contraction would actually appear to be something like a rotated version of the object.
For more details and some very helpful graphics that explain what’s going on, check out this article from Physics World:
https://physicsworld.com/a/the-invisibility-of-length%E2%80%AFcontraction/