Wigner's theorem

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Wigner's Theorem states that any symmetry operation must be induced by a unitary or anti-unitary transformation.

More precisely it states that a surjective map $T:H\rightarrow H$ on a complex Hilbert space $H$ which satisfies:

$|\langle Tx,Ty\rangle|=|\langle x,y\rangle|$

for all $x,y \in H$ , has the form $T x = φ(x) U x$ for all $x\in H$ where $\phi:H\rightarrow \mathbb{C}$ is unimodular and $U:H\rightarrow H$ is either unitary or antiunitary.

[edit] References

Bargmann, V. "Note on Wigner's Theorem on Symmetry Operations". Journal of Mathematical Physics Vol 5, no. 7, Jul 1964.
Molnar, Lajos. "An Algebraic Approach to Wigner's Unitary-Antiunitary Theorem". http://arxiv.org/abs/math/9808033