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

Simon, R and Mukunda, N and Chaturvedi, S and Srinivasan, V (2008) Two elementary proofs of the Wigner theorem on symmetry in quantum mechanics. In: Physics Letters A, 372 (46). pp. 6847-6852.

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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 post-multiplication by a unitary symmetry, be taken into either the identity or complex conjugation. Our analysis often focuses on the behaviour of certain two-dimensional 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.

Item Type:	Journal Article
Publication:	Physics Letters A
Publisher:	Elsevier Science
Additional Information:	Copyright of this article belongs to Elsevier Science.
Keywords:	Wigner theorem;Symmetry groups;Unitary–antiunitary theorem;Quantum information.
Department/Centre:	Division of Physical & Mathematical Sciences > Centre for High Energy Physics
Date Deposited:	12 Apr 2010 08:46
Last Modified:	19 Sep 2010 05:59
URI:	http://eprints.iisc.ac.in/id/eprint/26868

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