- #1
Petr Mugver
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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?
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.
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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