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Wigner's Theorem/Antiunitary Transformation 
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#1
Feb2112, 10:16 PM

P: 63

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 selfconsistency. 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? 


#2
Feb2212, 02:23 AM

P: 1,411

I think that Gottfried's reasoning is lacking precision, using handwaving 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.



#3
Feb2212, 03:05 AM

Sci Advisor
P: 3,593

Also nontrivial representations of the group operations C P and T have been discussed. 


#4
Feb2212, 09:13 AM

P: 63

Wigner's Theorem/Antiunitary Transformation
Thank you so much guys, I've been very confused about this.



#5
Feb2212, 09:24 AM

Sci Advisor
P: 3,593

I think a nontrivial 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 antiunitary) 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 antiunitary). 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 


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