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 two proofs of this theorem which are both elementary and economical. Central to our proofs is the recognition that a given Wigner symmetry can, by postmultiplication by a unitary symmetry, be taken into either the identity or complex conjugation. Our analysis often focuses on the behaviour of certain twodimensional subspaces of the Hilbert space under the action of a given Wigner symmetry, but the relevance of this behaviour to the larger picture of the whole Hilbert space is made transparent at every stage.
 Publication:

Physics Letters A
 Pub Date:
 November 2008
 DOI:
 10.1016/j.physleta.2008.09.052
 arXiv:
 arXiv:0808.0779
 Bibcode:
 2008PhLA..372.6847S
 Keywords:

 11.30.j;
 03.56.Ta;
 03.65.Fd;
 Symmetry and conservation laws;
 Algebraic methods;
 Quantum Physics
 EPrint:
 A second proof of Wigner's theorem added and, consequently, the title slightly changed. 8 pages, Latex, no figures