nLab
Wigner theorem

Contents

Statement

Wigner’s theorem asserts that a function f:HHf : H \to H from a Hilbert space to itself (not assumed to be a linear functor)
is linear and in fact a (anti-)unitary operator (up to a phase) if only the function is

  1. surjective;

  2. norm-preserving.

Role in quantum mechanics

In quantum mechanics every symmetric operation needs to be a norm-preserving bijection from a Hilbert space of states to itself. Hence Wigner’s theorem asserts that in quantum mechanics symmetries are presented by unitary operators (or more rarely anti-unitary operator?s, as for example time reversal?).

Other theorems on the foundations and interpretation of quantum mechanics include:

References

Revised on January 11, 2014 13:12:37 by Urs Schreiber (89.204.139.199)