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arXiv · 0808.0779

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

R. Simon·N. Mukunda·S. Chaturvedi·V. Srinivasan
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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 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.

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R. Simon, N. Mukunda, S. Chaturvedi, V. Srinivasan. 2008-08-21. Two elementary proofs of the Wigner theorem on symmetry in quantum mechanics. https://doi.org/10.1016/j.physleta.2008.09.052

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