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]

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/img/?PPN=GDZPPN002509032
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_group#Magnetic_groups_and_time_reversal