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.

Further properties

PCT Quantum Symmetries

We discuss how PCT quantum symmetries (Def. below) are classified (Prop. below) by a “10-fold way” (Cor. below).

The definition and the argument is quite straightforward. Just for completeness we offer some Preliminaries, but the reader may want to skip right ahead to the Definition and Classification.

The 10-fold classification of these PCT quantum symmetries immediately induces other incarnations of the 10-fold way, notably:

• the classification of flavors of topological K-theory (KU-theory, KO-theory, hence KR-theory in any degree)

  (by the Atiyah-Jänich theorem in terms of graded Fredholm operators, see there)

and with that

• the 10-fold way in the K-theory classification of topological phases of matter.

Preliminaries

Let:

• $\mathscr{H}$ be a countably infinite-dimensional complex Hilbert space,

• $\mathrm{U}(\mathscr{H})$ denote its unitary group,

• $T$ be a real structure on $\mathscr{H}$, hence a complex antilinear involution,

• $C_2^{(t)} \coloneqq \{id, T\}$ denote the group generated by $T$.

Observe that the semidirect product of the unitary group with that generated by $T$ is isomorphic the group consisting of unitary operators and anti-unitary operators:

(5)$$\begin{array}{ccc} \mathrm{U}(\mathscr{H}) \rtimes C_2^{(t)} &\overset{\sim}{\longrightarrow}& \mathrm{U}(\mathscr{H}) \sqcup \mathrm{U}_{anti}(\mathscr{H}) \\ (U, id) &\mapsto& U \\ (U, T) &\mapsto& U \circ T \mathrlap{\,.} \end{array}$$

This relation passes to projective unitary groups:

$$\begin{aligned} P \mathrm{U}(\mathscr{H}) &\coloneqq\; \mathrm{U}(\mathscr{H})/\mathrm{U}(1) \\ P \mathrm{U}_{anti}(\mathscr{H}) &\coloneqq\; \mathrm{U}_{anti}(\mathscr{H})/\mathrm{U}(1) \end{aligned}$$

in that we have an analogous isomorphism

$$\begin{array}{ccc} P \mathrm{U}(\mathscr{H}) \rtimes C_2^{(t)} &\overset{\sim}{\longrightarrow}& P\mathrm{U}(\mathscr{H}) \sqcup P\mathrm{U}_{anti}(\mathscr{H}) \\ ([U], id) &\mapsto& [U] \\ ([U], T) &\mapsto& [U \circ T] \mathrlap{\,.} \end{array}$$

This group of projective unitary/antiunitary operators is the group of quantum symmetries as usually considered (Wigner's theorem).

We next enlarge this a little more to include what may be thought of as particle/antiparticle symmetry, often referred to as charge conjugation symmetry.

To that end, let

• $\mathscr{H}_{gr} \coloneqq \mathscr{H} \ominus \mathscr{H}$ denote the $\mathbb{Z}_2$-graded Hilbert space,

• $P$ the grading involution on $\mathscr{H}$:

  $$\begin{array}{ccc} \mathscr{H} &\overset{P}{\longrightarrow}& \mathscr{H} \\ \left( \begin{matrix} \psi_+ \\ \psi_- \end{matrix} \right) &\mapsto& \left( \begin{matrix} \psi_- \\ \psi_+ \end{matrix} \right) \end{array}$$

• $C_2^{(p)} \coloneqq \{id, P\}$ denote the group generated by $P$,

• $C_2^{(c)} \coloneqq \{id, C \coloneqq P T\}$ denote the group generated by $P \circ T$,

• $\mathrm{U}_{gr}(\mathscr{H}_{gr}) \coloneqq \mathrm{U}(\mathscr{H})^2 \rtimes C_2^{(p)}$ denote the semidirect product which is isomorphic to the subgroup of $\mathrm{U}(\mathscr{H}_{gr})$ on the operators which are of homogeneous degree (either even or odd),

• $P\mathrm{U}_{gr}(\mathscr{H}_{gr}) \coloneqq \frac{\mathrm{U}(\mathscr{H})^2}{\mathrm{U}(1)} \rtimes C_2^{(p)}$ denote the corresponding projective group

Finally, combine all this to consider the following:

Definition and Classification

Definition

The group of graded quantum symmetries is the semidirect product

(6)$$QS \;\coloneqq\; \frac{ \mathrm{U}(\mathscr{H})^2 }{ \mathrm{U}(1) } \rtimes \big( C_2^{(t)} \times C_2^{(c)} \big) \mathrlap{\,,}$$

of the even graded projective unitary group with the operations of degree involution $P$ and complex involution $T$.

This is a group extension

$$\tfrac{ \mathrm{U}(\mathscr{H})^2 }{ \mathrm{U}(1) } \hookrightarrow QS \twoheadrightarrow C_2^{(t)} \times C_2^{(c)}$$

of

(7)$$C_2^{(t)} \times C_2^{(c)} \;=\; \big\{ id, T, C, P \coloneqq C T \big\} \,,$$

which we may call the group of PCT symmetries.

Definition

A PCT quantum symmetry is a lift of a subgroup of PCT symmetries (7) to a quantum symmetry (6), hence a dashed group homomorphism making the following diagram commute:

where for $g \in G$ we denote by

(8)$$\widehat{g} \;\in\; \mathrm{U}(\mathscr{H})^2 \rtimes \big( C_2^{(t)} \times C_2^{(c)} \big)$$

a representative of the $\mathrm{U}(1)$-coset equivalence class $\big[\widehat{g}\big]$.

Proposition

Given a PCT quantum symmetry $\widehat{G}$ (Def. ), we have

1. if $G = C_2^{(p)} = \big\{id, P = C T\big\}$, then $\big[\widehat{P}\big]$ has a representative $\widehat{P}$ (8) such that

   $$\widehat{P}{}^2 = id \mathrlap{\,,}$$

2. if $T \in G$ then $\big[\widehat{T}\big]$ has a representative $\widehat{T}$ (8) such that

   $$\widehat{T}{}^2 \in \{\pm id\} \mathrlap{\,,}$$

3. if $C \in G$ then $\big[\widehat{C}\big]$ has a representative $\widehat{C}$ (8) such that

   $$\widehat{C}{}^2 \in \{\pm id\} \mathrlap{\,,}$$

and all these cases occur.

Proof

1. That $\big[\widehat{(-)}\big]$ is a group homomorphism means equivalently that

   $$\begin{aligned} & \big[\widehat{P}\big]^2 \;=\; id \\ \Leftrightarrow \;\; & \widehat{P}^2 \;=\; \omega\, id \;\;\; \text{for some} \; \omega \in \mathrm{U}(1) \,. \end{aligned}$$

   But since for $G = C_2^{(p)}$ the operator $\widehat{P}$ must be unitary according to (5) and hence in particular complex-linear, its rescaling by any square root $\sqrt{\omega} \in \mathrm{U}(1)$ yields an alternative representative

   $$\big[ \tfrac{1}{\sqrt{\omega}} \widehat{P} \big] \;=\; \big[ \widehat{P} \big]$$

   with the desired property:

   $$\begin{aligned} \big( \tfrac{1}{\sqrt{\omega}} \widehat{P} \big)^2 & = \tfrac{1}{\sqrt{\omega}} \widehat{P} \tfrac{1}{\sqrt{\omega}} \widehat{P} \\ & = \tfrac{1}{\sqrt{\omega}} \tfrac{1}{\sqrt{\omega}} \widehat{P} \widehat{P} \\ & = id \mathrlap{\,.} \end{aligned}$$

2. Group homomorphy again requires that

   $$\begin{aligned} & \big[\widehat{T}\big]^2 \;=\; id \\ \Leftrightarrow \;\; & \widehat{T}^2 \;=\; \omega\, id \;\;\; \text{for some} \; \omega \in \mathrm{U}(1) \,, \end{aligned}$$

   but now that $\widehat{T}$ is anti-unitary and hence in particular complex-antilinear, there is first all a further constraint, namely

   $$\begin{aligned} & \widehat{T} \widehat{T}^2 = \widehat{T}^2 \widehatT \\ \Rightarrow\;\; & \omega^\ast \widehat{T} = \omega \widehat{T} \\ \Leftrightarrow\;\; & \omega^\ast = \omega \\ \Leftrightarrow\;\; & \omega \,\in\, \mathrm{U}(1) \cap \mathbb{R} \\ \Leftrightarrow\;\; & \omega \,\in\, \{\pm 1\} \mathrlap{\,.} \end{aligned}$$

   On the other hand, for the same reason the single non-trivial value for $\omega$ may no longer be scaled away as before, since now

   $$\begin{aligned} \big( \pm \mathrm{i} \widehat{T} \big)^2 & = (\pm\mathrm{i})\widehat{T} (\pm\mathrm{i})\widehat{T} \\ & = (\pm\mathrm{i})(\mp\mathrm{i})\widehat{T}\widehat{T} \\ & = \widehat{T}^2 \mathrlap{\,.} \end{aligned}$$

3. Same argument as in (2.).

Corollary

(10-fold way of PCT quantum symmetries)
The set of PCT quantum symmetries (Def. ) falls, by Prop. , into ten classes, according to the following table:

Remark

The labels in the last row of the table in Cor. are traditional — early appearance in Schnyder, Ryu, Furusaki & Ludwig 2008 table 1 (where, beware, the labels “DII” and “CII” are swapped with regards to the modern convention), reviewed in Chiu, Teo, Schnyder & Ryu 2016), with historical origin in Cartan’s 1926 classification of symmetric spaces in terms of simple Lie algebras (see there).

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

Related entries

• particle statistics

• K-theory classification of topological phases of matter

• fermionic group

References

The original:

• Eugene P. Wigner §20 (appndx) of: 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

• Armin Uhlmann: Anti- (Conjugate) Linearity, Sci. China Phys. Mech. Astron. 59 (2016) 630301 [arXiv:1507.06545, doi:10.1007/s11433-015-5777-1]

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]