Superselection sector

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

Reference

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