Superselection in Algebraic Language

Algebraic superselection compares representations and states of one specified observable algebra. Central projections separate classical sector labels, while multiplicities and nontrivial commutants need not define distinct sectors. For infinite systems, quasi-equivalence and disjointness make these comparisons precise without requiring every representation to be a discrete sum of irreducibles.

Required background. Representations distinguishes equivalence and multiplicity. Superselection Rules explains the physical restriction of observables. Von Neumann Algebras supplies centers and normal states.

Central projections label observable sectors

Let $\pi$ represent a unital C*-algebra $\mathcal A$ and set $\mathcal M=\pi(\mathcal A)''$. Its center is

$$Z(\mathcal M)=\mathcal M\cap\mathcal M'.$$

A central projection $z$ splits both the Hilbert space and the represented algebra into mutually reducing pieces. Every observable in $\mathcal M$ is block diagonal with respect to $z$ and $I-z$.

For a normal state $\omega$, the numbers $\omega(z)$ are probabilities of central events. If $0<p=\omega(z)<1$, the conditional functionals

$$\omega_z(A)=\frac{\omega(zA)}p,\qquad \omega_{1-z}(A)=\frac{\omega((I-z)A)}{1-p}$$

are positive normalized states, because centrality makes $zA$ positive whenever $A$ is positive. They reconstruct $\omega$ as a convex combination.

The center is therefore a commutative algebra of classical decomposition data. A factor has center $\mathbb CI$, so no nontrivial central event splits it. It can nevertheless act reducibly and support mixed states.

A finite-dimensional model separates weights from internal states

Consider the observable algebra

$$\mathcal A=M_2(\mathbb C)\oplus M_3(\mathbb C).$$

Every state has the form

$$\omega(A\oplus B) =p\,\operatorname{Tr}(\rho_2A) +(1-p)\operatorname{Tr}(\rho_3B),$$

where $0\leq p\leq1$ and each density matrix is normalized when its coefficient is nonzero. The central projections $I_2\oplus0$ and $0\oplus I_3$ determine the weight $p$.

This formula distinguishes classical uncertainty about the summand from mixedness inside a summand. For $0<p<1$, the state is not pure even if both conditional density matrices are rank one. A pure state occupies one summand and is pure there.

Calling these summands physical superselection sectors assumes that this is the observable algebra of the system. Embedding it into $M_5$ introduces additional off-diagonal observables and changes which preparations are distinguishable. Algebra alone does not justify why those larger-algebra observables are physically excluded.

Equivalent copies are multiplicity, not central labels

Represent $M_2$ on $\mathbb C^2\otimes\mathbb C^r$ by $A\mapsto A\otimes I_r$. The commutant is $I_2\otimes M_r$, but the represented algebra’s center remains scalar. A projection selecting one multiplicity vector lies in the commutant and generally not in the represented algebra.

Consequently different copies are not central observable sectors. This is the same distinction between irreducible type and copy that appears in compact-group decompositions, now applied to the observable representation itself.

The example also shows why a nontrivial commutant alone does not establish physically distinct superselection sectors. One must identify central projections that distinguish observable alternatives or a physically justified class of inequivalent representations.

Normal state families define quasi-equivalence

The folium of a representation $\pi$ is the set of state functionals on $\mathcal A$ obtained by restricting normal states of $\pi(\mathcal A)''$. Concretely, they can be written using density operators on the representation Hilbert space:

$$a\longmapsto\operatorname{Tr}(\rho\,\pi(a)).$$

Different densities can give the same restricted functional.

Two representations are quasi-equivalent when their von Neumann closures are normally star-isomorphic by a map carrying $\pi_1(a)$ to $\pi_2(a)$ for every $a$. The quasi-equivalence theorem (Bratteli and Robinson, Section 2.4.4) makes this equivalent to equality of their folia. This theorem is imported, not inferred merely from abstract algebra isomorphism.

One copy and several identical copies of the defining representation of $M_n$ are quasi-equivalent: they admit exactly the same algebraic state functionals. They are not unitarily equivalent when their multiplicities differ. Quasi-equivalence removes this spectator multiplicity from the comparison.

Disjointness is stronger than inequivalence

Representations are disjoint when they have no unitarily equivalent nonzero subrepresentations. Equivalently, there is no nonzero bounded intertwiner between them. Distinct irreducible representations are disjoint if they are inequivalent, but this implication fails for reducible representations that share a common summand.

A direct sum of disjoint representations has separate central support projections in its von Neumann closure. By contrast, merely choosing two nonequivalent reducible representations does not guarantee such a split.

In general infinite systems, decomposition into factors may use direct integrals, and arbitrary representations need not have a manageable unique decomposition into irreducibles. Stronger sector classifications require extra assumptions, such as type-I structure or locality and a chosen physical selection criterion. Those conditions are additional theory, not consequences of the word “sector.”

Exercise: inequivalent but overlapping representations

Let $\pi_x$ be point evaluation of $C([0,1])$ at $x$. Are $\pi_0\oplus\pi_1$ and $\pi_0\oplus\pi_{1/2}$ disjoint? Are they quasi-equivalent?

Solution

They are not disjoint because both contain $\pi_0$. They are not quasi-equivalent either: the first folium contains the state $f\mapsto f(1)$, whereas every state in the second folium is supported on $\{0,1/2\}$. Thus neither unitary inequivalence nor failure of quasi-equivalence by itself implies disjointness.

References

• Bratteli, Ola, and Derek W. Robinson, Operator Algebras and Quantum Statistical Mechanics 1, 2nd ed., Springer, 1987, doi:10.1007/978-3-662-02520-8, Section 2.4.4 — quasi-equivalence and disjointness; Section 4.2.2 — central and subcentral decompositions.
• Naaijkens, Pieter, Quantum Spin Systems on Infinite Lattices, Lecture Notes in Physics 933, Springer, 2017, arXiv:1311.2717 — representations and sector questions for infinite quantum systems.