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!

Image result for picard facepalm meme

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!

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