Wigner's theorem

Wigner's theorem, proved by Eugene Wigner in 1931, is a cornerstone of the mathematical formulation of quantum mechanics. The theorem specifies how physical symmetries such as rotations, translations, and CPT are represented on the Hilbert space of states.

According to the theorem, any symmetry transformation of ray space is represented by a linear and unitary or antilinear and antiunitary transformation of Hilbert space. The representation of a symmetry group on Hilbert space is either an ordinary representation or a projective representation.

It is a postulate of quantum mechanics that vectors in Hilbert space that are scalar nonzero multiples of each other represent the same pure state. A ray is a set

and a ray whose vectors have unit norm is called a unit ray. If Φ ∈ Ψ, then Φ is a representative of Ψ. There is a one-to-one correspondence between physical pure states and unit rays. The space of all rays is called ray space.

Formally, if H is a complex Hilbert space, then let B be the subset

of vectors with unit norm. If H is finite-dimensional with complex dimension N, then B has real dimension 2N − 1. Define a relation ≅ on B by

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