A superselection sector is a concept used in quantum mechanics when a representation of a *-algebra is decomposed into irreducible components. It formalizes the idea that not all self-adjoint operators are observables because the relative phase of a superposition of nonzero states from different irreducible components is not observable (the expectation values of the observables can't distinguish between them).

Formulation

Suppose A is a unital *-algebra and O is a unital *-subalgebra whose self-adjoint elements correspond to observables. A unitary representation of O may be decomposed as the direct sum of irreducible unitary representations of O. Each isotypic component in this decomposition is called a superselection sector. Observables preserve the superselection sectors.

Relationship to symmetry

Symmetries often give rise to superselection sectors (although this is not the only way they occur). Suppose a group G acts upon A, and that H is a unitary representation of both A and G which is equivariant in the sense that for all g in G, a in A and ψ in H,

$g\; (acdotpsi)\; =\; (ga)cdot\; (gpsi)$

Suppose that 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 tensor product of in irreducible representation of G with a representation of O.

This can be generalized by assuming that H is only a representation of an extension or cover K of G. (For instance G could be the Lorentz group, and K the corresponding spin double cover.) Alternatively, one can replace G by a Lie algebra, Lie superalgebra or a Hopf algebra.

Examples

Consider 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

$psi(x+L)=e^\{i\; theta\}psi(x).$

Reference

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