Superselection sector

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A superselection sector is a concept used in quantum mechanics.

1 Formulation
2 Physical meaning
3 Relationship to symmetry
4 Greater generality
5 Examples

[edit] Formulation

One of the insights of quantum mechanics is that not all self-adjoint operators are observables.

Suppose given an operator unital *-algebra A and an observable unital *-subalgebra O. The observables would then correspond to the self-adjoint elements of O. A reducible unitary representation of O is decomposable into the direct sum of inequivalent irreducible unitary representations of O. Each irreducible representation is called a superselection sector.

[edit] Physical meaning

Observables map a state in each irrep into another state in the same irrep. The relative phase of a superposition of nonzero states from different irreps is not observable (the expectation values of the observables can't distinguish between them). In the density state formulation where states are positive linear functionals of O where the unit of O is mapped to 1 (the unit, 1_O is "intuitively" an observable, a trivial one, no doubt), this would correspond to a mixed state; and in fact, all possible values for the relative phase would give rise to the same state.

In the density state formulation, the direct sum of two or more equivalent irreps would give rise to exactly the same states as a single rep alone, so we can cut off all but one irrep.

[edit] Relationship to symmetry

Symmetries often give rise to superselections sectors; but this is not the only reason for the occurrence of superselection sectors. In the study of decoherence, for example, if we restrict ourselves to observations in a local region, we can have approximate superselection sectors.

Say a group G acts upon A, and H is a unitary rep of A, and also a unitary rep of G such that for all g in G, a in A and $|\psi\rangle$ in H,

$g\left[a|\psi \rangle \right] = g \left[a\right]g\left[|\psi\rangle\right]$

(i.e. the representation of H, as a unitary rep of A is a G-intertwiner).

O is an invariant subalgebra of A under G (all observables are invariant under G, but not every self-adjoint operator invariant under G is necessarily an observable). H decomposes into superselection sectors, each of which is the direct product of an irrep of G and a rep of O. The irrep of G acts trivially under O. Using the same argument as above, we can reduce it to a rep of O alone. However, we can still keep the irrep of G as a label for the superselection sector, if we wish.

[edit] Greater generality

Insisting that H is a rep of G is unnecessarily restrictive. To take an example, let G be the Lorentz group and A be an operator algebra rep of G. H could be a Hilbert space of A which contains, among other things states with an odd number of fermions. This would mean H is not a rep of the Lorentz group, although it is a rep of its double cover.

Or let's say G is a group with an extension K. A is again an algebra rep of G. (Any rep of G can be turned into a rep of K.) Then, it is possible to have a unitary rep of A, H which is a unitary rep of K but not G.

Replacing G with a Lie algebra, Lie superalgebra or a Hopf algebra would still work. See algebra representation of a Lie superalgebra, unitary representation of a star Lie superalgebra, algebra representation of a Hopf algebra and representation of a Hopf algebra.

[edit] Examples

A simple example would be a quantum mechanical particle confined to a closed loop (i.e. a periodic line of period L). The superselection sectors are labeled by an angle θ between 0 and 2π. All the wave functions within a single superselection sector satisfy

$\psi\left(x+L\right)=e^{i \theta}\psi\left(x\right)$