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 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.

Applications of Noether conservation theorem to Hamiltonian systems

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Reflections on the four facets of symmetry: how physics exemplifies rational thinking

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Wigner’s Theorem in -Type Spaces

“…Wigner's theorem plays a fundamental role in quantum mechanics and has several equivalent formulations and extensions (see, for example, [1,2,4,[6][7][8][9][10]12]). In [4], a real version of Wigner's theorem was given by using the functional equation It is easy to see that, when X and Y are real normed spaces, all mappings f : X → Y that are phase equivalent to real linear isometries are also the solutions of the functional equation (1.1).…”