nLab
Wigner theorem

not to be confused with Wigner classification

Context

AQFT

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory: classical, pre-quantum, quantum, perturbative quantum

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

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 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?).

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

References

See also

Last revised on December 28, 2017 at 17:36:57. See the history of this page for a list of all contributions to it.