not to be confused with Wigner classification
algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
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
norm-preserving.
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:
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
See also,
Last revised on July 2, 2019 at 16:14:14. See the history of this page for a list of all contributions to it.