Quantum theory Foundations and measurement Theorem

Wigner's theorem on symmetries

Builds on Hilbert space Projective Hilbert space Unitary operator

Statement

Let $\mathcal{H}$ be a complex Hilbert space and let $\mathbb{P}\mathcal{H}$ be its projective Hilbert space, the set of rays $[\psi] = \set{c\psi : c \in \C}$ of unit vectors $\psi$ . For rays $[\psi]$ and $[\varphi]$ , the transition probability $\abs{\langle \psi, \varphi \rangle}^2$ depends only on the rays. Let $T \colon \mathbb{P}\mathcal{H} \to \mathbb{P}\mathcal{H}$ be a surjective map with $\abs{\langle \psi', \varphi' \rangle}^2 = \abs{\langle \psi, \varphi \rangle}^2 \quad \text{whenever } T[\psi] = [\psi'] \text{ and } T[\varphi] = [\varphi'].$ Then there is an operator $U \colon \mathcal{H} \to \mathcal{H}$ that is either unitary or antiunitary such that $T[\psi] = [U\psi]$ for every unit vector $\psi$ . Here $U$ is antiunitary if it is bijective, $U(a\psi + b\varphi) = \bar{a}U\psi + \bar{b}U\varphi$ and $\langle U\psi, U\varphi \rangle = \overline{\langle \psi, \varphi \rangle}$ for all $a, b \in \C$ and $\psi, \varphi \in \mathcal{H}$ . If $\dim \mathcal{H} \ge 2$ , then $U$ is unique up to a factor $e^{i\theta}$ with $\theta \in \R$ , and whether $U$ is unitary or antiunitary is determined by $T$ .

Remarks

A proof fixes an orthonormal basis $(e_k)$ , chooses representatives of the image rays $T[e_k]$ , and fixes their phases using the rays of the vectors $e_1 + e_k$ . The images of the remaining vectors are then forced, up to the choice between a linear and an antilinear extension, by the preserved transition probabilities. Example: complex conjugation of coordinates, $K(z_1, \dots, z_n) = (\bar{z}_1, \dots, \bar{z}_n)$ on $\C^n$ , preserves all transition probabilities and is antiunitary; it is not induced by any unitary operator when $n \ge 2$ . Time reversal in quantum mechanics is implemented by an antiunitary operator of this kind. In dimension $1$ there is only one ray, so uniqueness fails: the identity map of $\mathbb{P}\C$ is induced both by $\id$ and by $K$ .

Going deeper

For a student

The theorem is the starting point of the representation theory of symmetry groups in quantum mechanics. A symmetry group acting on rays gives operators $U(g)$ that multiply only up to phases, $U(g)U(h) = \omega(g, h)U(gh)$ , that is, a projective representation. For a connected Lie group, every element is a product of squares, and the square of a unitary or antiunitary operator is unitary, so all $U(g)$ are unitary. Bargmann's theorem then says when the phases can be removed, and this leads to spin and to Wigner's classification of particles.

Reading path

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Hilbert space concept
Projective Hilbert space concept
Unitary operator concept
Wigner's theorem on symmetries statement

Tag	`2KZCM5V7YN`
URI	`m2://2KZCM5V7YN`
Kind	statement, theorem
Writing	W0 stub
MSC	81R05, 81P16, 46N50
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About	Projective Hilbert space, Unitary operator
Uses	Hilbert space, Projective Hilbert space, Unitary operator
Same as	Wikidata Q1135224 Wikipedia Wigner's theorem
Changed	3 November 2026