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 transformations are represented on the Hilbert space of states. The physical states in a quantum theory are represented by unit vectors in Hilbert space up to a phase factor, i.e. by the complex line or ray the vector spans. In addition, by the Born rule the absolute value of the unit vector's inner product with a unit eigenvector, or equivalently the cosine squared of the angle between the lines the vectors span, corresponds to the transition probability. Ray space, in mathematics known as projective Hilbert space, is the space of all unit vectors in Hilbert space up to the equivalence relation of differing by a phase factor. By Wigner's theorem, any transformation of ray space that preserves the absolute value of the inner products can be represented by a unitary or antiunitary transformation of Hilbert space, which is unique up to a phase factor. As a consequence, the representation of a symmetry group on ray space can be lifted to a projective representation or sometimes even an ordinary representation on Hilbert space.

Rays and ray space It is a postulate of quantum mechanics that state vectors in complex separable Hilbert space H {\displaystyle H} that are scalar nonzero multiples of each other represent the same pure state, i.e., the vectors Ψ ∈ H ∖ { 0 } {\displaystyle \Psi \in H\setminus \{0\}} and λ Ψ {\displaystyle \lambda \Psi } , with λ ∈ C ∖ { 0 } {\displaystyle \lambda \in \mathbb {C} \setminus \{0\}} , represent the same state. By multiplying the state vectors with the phase factor, one obtains a set of vectors called the ray

Ψ _ = { e i α Ψ : α ∈ R } . {\displaystyle {\underline {\Psi }}=\left\{e^{i\alpha }\Psi :\alpha \in \mathbb {R} \right\}.}

Two nonzero vectors Ψ 1 , Ψ 2 {\displaystyle \Psi _{1},\Psi _{2}} define the same ray, if and only if they differ by some nonzero complex number: Ψ 1 = λ Ψ 2 {\displaystyle \Psi _{1}=\lambda \Psi _{2}} . Alternatively, we can consider a ray Ψ _ {\displaystyle {\underline {\Psi }}} as a set of vectors with norm 1, a unit ray, by intersecting the line Ψ _ {\displaystyle {\underline {\Psi }}} with the unit sphere

S H = { Φ ∈ H ∣ ‖ Φ ‖ 2 = 1 } {\displaystyle SH=\{\Phi \in H\mid \|\Phi \|^{2}=1\}} . Two unit vectors Ψ 1 , Ψ 2 {\displaystyle \Psi _{1},\Psi _{2}} then define the same unit ray Ψ 1 _ = Ψ 2 _ {\displaystyle {\underline {\Psi _{1}}}={\underline {\Psi _{2}}}} if they differ by a phase factor: Ψ 1 = e i α Ψ 2 {\displaystyle \Psi _{1}=e^{i\alpha }\Psi _{2}} . This is the more usual picture in physics. The set of rays is in one to one correspondence with the set of unit rays and we can identify them. There is also a one-to-one correspondence between physical pure states ρ {\displaystyle \rho } and (unit) rays Φ _ {\displaystyle {\underline {\Phi }}} given by

ρ = P Φ = | Φ ⟩ ⟨ Φ | ⟨ Φ | Φ ⟩ {\displaystyle \rho =P_{\Phi }={\frac {|\Phi \rangle \langle \Phi |}{\langle \Phi |\Phi \rangle }}}

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