Wigner's Theorem/Antiunitary Transformation

thoughtgaze

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?

arkajad

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.

DrDu

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.

thoughtgaze

Thank you so much guys, I've been very confused about this.

DrDu

I think a non-trivial but interesting example are magnetic symmetry groups on a lattice.
Consider a regular lattice of magnetic moments pointing up and down alternantly. The inversion of the magnetic moment corresponds to time inversion (and obviously is anti-unitary) but is not a symmetry of the lattice. However a combination of a translation by the nearest moment distance (one unit) and time inversion is (and is anti-unitary). Repeating this operation is equal to a unitary transformation, namely the shift by two units which is certainly different from the identity.
See
http://en.wikipedia.org/wiki/Space_g..._time_reversal