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  Time reversal symmetry and T^2 = -1

+ 7 like - 0 dislike

I'm a mathematician interested in abstract QFT. I'm trying to undersand why, under certain (all?) circumstances, we must have $T^2 = -1$ rather than $T^2 = +1$, 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 $T^2=-1$. (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 $Spin(n)\to SO(n)$ cover on $SO(n)$; they are called $Pin_+(n)$ and $Pin_-(n)$. If $R\in O(n)$ is a reflection and $\tilde{R}\in Pin_\pm(n)$ covers $R$, then $\tilde{R}^2 = \pm 1$. So saying that $T^2=-1$ 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 $T^2=-1$, while for the odd-dimensional representations (integer spin), we have $T^2=1$. 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 $\tilde R$ (e.g. $T$) act as an anti-unitary operator.

This post has been migrated from (A51.SE)
asked Jan 20, 2012 in Theoretical Physics by Kevin Walker (65 points) [ no revision ]
retagged Mar 7, 2014 by dimension10
$T^2=-1$ is only true for states with half integer spin.

This post has been migrated from (A51.SE)
Thanks, that's helpful. With that hint I was able to do a more fruitful web search and came across Kramer's theorem. I'll edit the question accordingly.

This post has been migrated from (A51.SE)
It is not unheard of that if the OP figures out the answer to their question, they post it and can even accept it if they think it is correct.

This post has been migrated from (A51.SE)
Note that signature is important. Space reflections correspond to unitary operators whereas time reflections correspond to anti unitary ones

This post has been migrated from (A51.SE)
I could better understand a question $Pin(3,1)$ vs $Pin(1,3)$. How we could answer what we are using in Euclidean case? Even in Lorentzian case we sometimes have to talk about experimental data (e.g. discovery of antiparticles, search for Majorana neutrino, etc.) to clarify such questions.

This post has been migrated from (A51.SE)
Kramer's Theorem and the issue of antiunitary time reversal symmetries are both dealt with in some detail by the first chapter of [Haake](http://www.amazon.ca/Quantum-Signatures-Chaos-Fritz-Haake/dp/3540677232), for those interested in reading more on the subject.

This post has been migrated from (A51.SE)

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