Superselection

In quantum mechanics, superselection extends the concept of selection rules. Superselection rules are postulated rules forbidding the preparation of quantum states that exhibit coherence between eigenstates of certain observables. It was originally introduced by Wick, Wightman, and Wigner to impose additional restrictions to quantum theory beyond those of selection rules. Mathematically speaking, two quantum states and are separated by a selection rule if for the given Hamiltonian , while they are separated by a superselection rule if for all physical observables . Because no observable connects and they cannot be put into a quantum superposition , and/or a quantum superposition cannot be distinguished from a classical mixture of the two states. It also implies that there is a classically conserved quantity that differs between the two states. 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). 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. 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, 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).