Two elementary proofs of the Wigner theorem on symmetry in quantum mechanics
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.
Keywords
Cite
@article{arxiv.0808.0779,
title = {Two elementary proofs of the Wigner theorem on symmetry in quantum mechanics},
author = {R. Simon and N. Mukunda and S. Chaturvedi and V. Srinivasan},
journal= {arXiv preprint arXiv:0808.0779},
year = {2009}
}
Comments
A second proof of Wigner's theorem added and, consequently, the title slightly changed. 8 pages, Latex, no figures