# Wigner's Theorem/Antiunitary Transformation

by thoughtgaze
Tags: transformation, wigner
 P: 57 So I'm reading Gottfried and Yan's Quantum Mechanics: Fundamentals. On page 284, They state Wigner's Theorem and explain the two cases. One transformation leads to no complex conjugation of the expansion coefficients (unitary) and the other leads to a complex conjugation of the expansion coefficients (antiunitary). Anyway, I'm confused when he states the following. Applying an antiunitary operator twice results in a unitary operation, since the expansion coefficients are conjugated twice. Therefore the antiunitary operators cannot be represented as a continuous group because for any such operation (call it A) there exists the square root of that operation (A_(1/2)), which when applied twice gives an A and thus any A in the continuous group must be unitary for self-consistency. The part I don't get: He then goes on to say "by the same argument, candidates for an antiunitary transformation must be such that A^2 reproduces the original description" I don't understand why it necessarily has to reproduce the original description. I only understand why it has to be a discrete transformation. Anyone care to shed some light?
 PF Patron P: 1,412 I think that Gottfried's reasoning is lacking precision, using hand-waving arguments, fuzzy. Therefore I would not take too seriously his conclusions. But, when taking square leads to the original description, life is certainly easier. That is probably the only reason.
P: 3,119
 Quote by thoughtgaze He then goes on to say "by the same argument, candidates for an antiunitary transformation must be such that A^2 reproduces the original description"
Whatever he means, it is not correct. Google "Kramers degeneracy" or read master Wigner himself:http://www.digizeitschriften.de/dms/...DZPPN002509032
Also non-trivial representations of the group operations C P and T have been discussed.

P: 57