Two elementary proofs of the Wigner theorem on symmetry in quantum mechanics

TL;DR

This paper provides two simple and efficient proofs of Wigner's theorem, demonstrating that quantum symmetries can be represented by either unitary or antiunitary operators, using a novel approach involving two-dimensional subspaces.

Contribution

It introduces two elementary proofs of Wigner's theorem, simplifying the understanding of quantum symmetries and their operator representations.

Findings

Wigner symmetry can be reduced to identity or complex conjugation.

The proofs involve analyzing the symmetry's effect on two-dimensional subspaces.

The approach is both elementary and economical.

Abstract

In quantum theory, symmetry has to be defined necessarily in terms of the family of unit rays, the state space. The theorem of Wigner asserts that a symmetry so defined at the level of rays can always be lifted into a linear unitary or an antilinear antiunitary operator acting on the underlying Hilbert space. We present a proof of this theorem which is both elementary and economical. Central to our proof is the recognition that a given Wigner symmetry can, by post-multiplication by a unitary symmetry, be taken into either the identity or complex conjugation. Our analysis involves a judicious interplay between the effect a given Wigner symmetry has on certain two-dimensional subspaces and the effect it has on the entire Hilbert space.