nLab
Wigner theorem

not to be confused with Wigner classification

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. 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 theorems

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

• order-theoretic structure in quantum mechanics

  • Kochen-Specker theorem

  • Alfsen-Shultz theorem

  • Harding-Döring-Hamhalter theorem

• Fell's theorem

• Gleason's theorem

• Bell's theorem

• Bub-Clifton theorem

• Kadison-Singer problem

References

• Valentine Bargman, Note on Wigner’s theorem on symmetry transformations, Journal of Mathematical Physics 5.7 (1964): 862-868 (doi:10.1063/1.1704188, pdf)

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

• Valter Moretti, Spectral Theory and Quantum Mechanics 2nd Ed. 2018 Springer, ch.12

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)

See also,

• Wikipedia, Wigner’s theorem