nLab
Wigner theorem

Statement

Wigner’s theorem asserts that a function $f:H\to H$ 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?).

Related concepts

• Gleason's theorem

• Kochen-Specker theorem

References

• Dan Freed, On Wigner’s theorem (arXiv:1112.2133)