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I have typed up some notes for myself on the earlier parts of the proof, which I believe that I understand, but it's possible that they would just be confusing, since my notation is different. I'll post all of my notes if someone requests it. The following is a very brief summary of the stuff before the section that's causing me problems. (It's probably very hard to follow if you're not already somewhat familiar with Wigner's theorem). Let me know if you want me to clarify some detail in this summary, or in the earlier parts of Bargmann's proof.

We consider equivalence classes of vectors in a Hilbert space H. Members of the same class differ only by a complex factor c such that |c|=1. Bargmann calls these classes

*rays**. We're given a permutation T of the set of unit rays, which is assumed to preserve probabilities. We begin by extending this map to the set of all rays. Our goal is to define Ux for all x in H in a way that ensures that the map U is either unitary or antiunitary, and also that for all x, Ux is a member of T[x], where [x] is the ray that x belongs to. Bargmann picks an arbitrary unit vector e, and first defines Ux for all x of the form x=e+y, such that y is orthogonal to e. Then he defines Ux for all x that are orthogonal to e. Actually, he calls the "U" defined on that subspace "V" instead of "U" for some reason. Later, he extends the definition to all of H, but before that, he proves a bunch of things about this "V". That's the part that's causing me trouble.

*) I think it's more common to call the 1-dimensional subspaces rays, but I'm using Bargmann's terminology in this post.