# nLab Wigner theorem

Contents

not to be confused with Wigner classification

# Contents

## Statement

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

1. 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 about the foundations and interpretation of quantum mechanics include:

## References

For quaternionic Hilbert spaces

• C. S. Sharma and D. F. Almeida, Additive isometries on a quaternionic Hilbert space, Journal of Mathematical Physics 31, 1035 (1990) (doi:10.1063/1.528779)