Wigner's theorem

Wigner's Theorem states that any symmetry operation must be induced by a unitary or anti-unitary transformation. It was proved by Eugene Wigner in 1931 [E. P. Wigner, Gruppentheorie (Frederick Wieweg und Sohn, Braunschweig, Germany, 1931), pp. 251-254; Group Theory (Academic Press Inc., New York, 1959), pp. 233-236] .

More precisely, it states that a surjective map $T:H\; ightarrow\; H$ on a complex Hilbert space $H$ which satisfies::$|langle\; Tx,Ty\; angle|=|langle\; x,y\; angle|$

for all $x,y\; in\; H$, has the form $Tx=varphi(x)Ux$for all $xin\; H$ where $varphi:H\; ightarrow\; mathbb\{C\}$ is unimodular and $U:H\; ightarrow\; H$ is either unitary or antiunitary.

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

ee also

*Particle physics and representation theory