nLab Wigner theorem

not to be confused with Wigner classification

Idea

What is known as Wigner’s theorem (in honor of the appendix of §20 in Wigner 1931/59) is one of the basic results on the foundations of quantum mechanics (generally of quantum physics). The theorem characterizes the intrinsic notion of symmetries in quantum physics (“quantum symmetries”) assuming only observable properties of (pure) quantum states (namely their projectivity subject to the Born rule) and deriving from this that quantum symmetries are necessarily represented by unitary operators or anti-unitary operators on the Hilbert space of states.

Notice that a prominent class of examples of anti-unitary operators appearing as quantum symmetries are time-reversal symmetries (Wigner 1959, §26) such as appearing in the CPT-theorem.

A key applications of Wigner’s theorem is within the K-theory classification of topological phases of matter, where the (anti-)unitary quantum symmetries of effectively free electrons in a crystal are identified with possible twistings of the equivariant K-theory of the crystal’s Brillouin torus (Freed & Moore 2013, §1, see also SS22, §2.2).

Preliminaries

Let $\mathcal{H}$ be a complex separable Hilbert space and write

(1)$$P\mathcal{H} \;\coloneqq\; \big( \mathcal{H} \setminus \{0\} \big)/ \mathbb{C}^\times$$

for its complex projective space (here $\mathbb{C}^\times = \mathbb{C} \setminus \{0\}$ denotes the group of units of the ring of complex numbers and $\mathrm{U}(1) \subset \mathbb{C}^\times$ will denote the unitary group U(1)).

In quantum physics one often says that a pure quantum state is an element (hence a vector, then often called a “wavefunction”) $\psi \,\in\, \mathcal{H}$, but by the Born rule it is really only the corresponding complex line

$$[\psi] \;\coloneqq\; \big\{ c \cdot \psi \,\big\vert\, c \in \mathrm{U}(1) \big\} \;\;\; \in \; P\mathcal{H}$$

which is physically observable. In fact, what is concretely observable (still according to the Born rule) are the transition probabilities, given by the following function to the closed interval of possible probability values:

(2)$$\array{ P\mathcal{H} \times P\mathcal{H} &\xrightarrow{\;\;\;p\;\;\;}& [0,1] \\ \big( [\psi_1] ,\, [\psi_2] \big) & \overset{\;\;\;\;}{\mapsto} & \frac{ \langle \psi_1 \vert \psi_2 \rangle \langle \psi_2 \vert \psi_1 \rangle }{ \langle \psi_1 \vert \psi_1 \rangle \langle \psi_2 \vert \psi_2 \rangle } \mathrlap{\,,} }$$

where $\langle - \vert - \rangle$ denotes the given inner product on $\mathcal{H}$.

Hence a “quantum symmetry” in the sense of a re-shuffling of the pure states of a quantum system which leaves the observable physics invariant must be a bijection on the projective space $P\mathcal{H}$ (1) which preserves the observable probabilities (2):

(3)$$S \;\colon\; P\mathcal{H} \xrightarrow{\;\sim\;} P\mathcal{H} \,, \;\;\;\;\;\; p \big( S(-) ,\, S(-) \big) \;=\; p(-,\,-) \,.$$

An example of a quantum symmetry (3) is provided by any complex-linear and unitary operator on the underlying Hilbert space,

$$U \,\colon\, \mathcal{H} \xrightarrow{\;\;} \mathcal{U} \,, \;\;\;\;\;\;\;\; \big\langle U(-) \big\vert U(-) \big\rangle \;=\; \big\langle - \big\vert - \big\rangle \,,$$

in that the induced projectivization

(4)$$\array{ P\mathcal{H} &\xrightarrow{\;\;[U]\;\;}& P\mathcal{H} \\ [\psi] &\mapsto& \big[ U(\psi) \big] }$$

evidently satisfies (3).

On the one hand, there are more quantum symmetries than arise from unitary operators this way. Namely, if $U$ is an anti-linear anti-unitary operator, then (4) still holds.

On the other hand, this does now already exhaust the most general situation: Wigner’s theorem (Prop. below) says that for every quantum symmetry $S \,\colon\, P\mathcal{H} \xrightarrow{\;} P\mathcal{H}$ (3) there exists a map $U \,\colon\, \mathcal{H} \xrightarrow{\;} \mathcal{H}$ which is either a unitary operator or an anti-unitary operator, such that $S \,=\, [U]$ (4).

Statement

Proposition

(Wigner’s theorem)
Every quantum symmetry $S \,\colon\, P\mathcal{H} \xrightarrow{\;} P\mathcal{H}$ (3) is the projectivization (4) of a map $U \,\colon\, \mathcal{H} \xrightarrow{\;} \mathcal{H}$ which is either a unitary operator or an anti-unitary operator.

The first full proof of this statement seems to be that due to Bargmann 1964, §1.3, following indications in the appendix of §20 in Wigner 1931/59. A geometric proof via the Fubini-Study metric is given in Freed 2012, Thm. 8. Statement and proof in the greater generality of possibly non-pure quantum states is given in Moretti 2017, Thm. 12.11.

Related theorems

Other theorems about the foundations and interpretation of quantum mechanics include:

• order-theoretic structure in quantum mechanics

  • Kochen-Specker theorem

  • Alfsen-Shultz theorem

  • Harding-Döring-Hamhalter theorem

• Fell's theorem

• Gleason's theorem

• Bell's theorem

• Bub-Clifton theorem

• Kadison-Singer problem

References

The original:

• Eugene P. Wigner: §20(app.) Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren, Springer (1931) [doi:10.1007/978-3-663-02555-9, pdf]

and its English translation:

• Eugene P. Wigner, §20(app.) and §26 in: Group theory: And its application to the quantum mechanics of atomic spectra, Academic Press (1959) [doi:978-0-12-750550-3]

The first full proof:

• Valentine Bargmann, Note on Wigner’s theorem on symmetry transformations, Journal of Mathematical Physics 5 7 (1964) 862-868 [doi:10.1063/1.1704188, pdf, pdf]

New proof using the Fubini-Study metric:

• Daniel Freed, On Wigner’s theorem, Geometry & Topology Monographs 18 (2012) 83–89 [arXiv:1112.2133, doi:10.2140/gtm.2012.18.83]

A proof in the greater generality of possibly non-pure quantum states:

• Valter Moretti, Thm. 12.11 of: Spectral Theory and Quantum Mechanics, Springer (2017) [doi:10.1007/978-3-319-70706-8]

See also:

• Wikipedia, Wigner’s theorem

Discussion for quaternionic Hilbert spaces:

• C. S. Sharma and D. F. Almeida, Additive isometries on a quaternionic Hilbert space, Journal of Mathematical Physics 31, 1035 (1990) (doi:10.1063/1.528779)

In the context of the K-theory classification of topological phases of matter:

• Daniel Freed, Gregory Moore, §1 of: Twisted equivariant matter, Ann. Henri Poincaré 14 (2013) 1927-2023 [arXiv:1208.5055, doi:10.1007/s00023-013-0236-x]