I'm a mathematician interested in abstract QFT. I'm trying to undersand why, under certain (all?) circumstances, we must have T2=−1 rather than T2=+1, where T is the time reversal operator. I understand from the Wikipedia article that requiring that energy stay positive forces T to be represented by an anti-unitary operator. But I don't see how this forces T2=−1. (Or maybe it doesn't force it, it merely allows it?)
Here's another version of my question. There are two distinct double covers of the Lie group O(n) which restrict to the familiar Spin(n)→SO(n) cover on SO(n); they are called Pin+(n) and Pin−(n). If R∈O(n) is a reflection and ˜R∈Pin±(n) covers R, then ˜R2=±1. So saying that T2=−1 means we are in Pin− rather than Pin+. (I'm assuming Euclidean signature here.) My question (version 2): Under what circumstances are we forced to use Pin− rather than Pin+ here?
(I posted a similar question on physics.stackexchange.com last week, but there were no replies.)
EDIT: Thanks to the half-integer spin hint in the comments below, I was able to do a more effective web search. If I understand correctly, Kramer's theorem says that for even-dimensional (half integer spin) representations of the Spin group, T must satisfy T2=−1, while for the odd-dimensional representations (integer spin), we have T2=1. I guess at this point it becomes a straightforward question in representation theory: Given an irreducible representation of Spin(n), we can ask whether it is possible to extend it to Pin−(n) (or Pin+(n)) so that the lifted reflections ˜R (e.g. T) act as an anti-unitary operator.
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