nLab PCT quantum symmetries -- section

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:

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

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

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

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

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

which we may call the group of PCT symmetries.

Definition

A PCT quantum symmetry is a lift of a subgroup of PCT symmetries (3) to a quantum symmetry (2), hence a dashed group homomorphism making the following diagram commute:

where for $g \in G$ we denote by

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

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 $\widehat{P}$ (4) such that

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

2. if $T \in G$ then $\big[\widehat{T}\big]$ has a representative $\widehat{T}$ (4) such that

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

3. if $C \in G$ then $\big[\widehat{C}\big]$ has a representative $\widehat{C}$ (4) 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 (1) 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).