Wigner's representation theorem says that any invertible transformation between rays of a Hilbert space that preserves transition probabilities can be implemented by a transformation on the Hilbert space itself, which is either unitary or antiunitary, depending on the particular transformation considered, right?(adsbygoogle = window.adsbygoogle || []).push({});

For example, you can find in any QM book that almost all symmetries are represented by linear operators, the only significant exception being time inversion, right?

I present here the most trivial example I can imagine: a one-dimensional Hilbert space. The two complex function of complex variable

[tex]f(z)=z\qquad\textrm{and}\qquad g(z)=z^*[/tex]

are, respectively, unitary and antiunitary, and they both induce the same transformation between rays of the Hilbert space [tex]H=\mathbb{C}[/tex], the identity transformation.

I know the solution of this apparent paradox should be easy, but I really can't see it! I know the proof of the theorem, and it doesn't help!

Any hint woukd be appreciated.

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# Very stupid question about Wigner's Theorem

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