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 Gian Carlo Wick, Arthur Wightman, and Eugene Wigner to impose additional restrictions to quantum theory beyond those of selection rules. Mathematically speaking, two quantum states ψ 1 {\displaystyle \psi _{1}} and ψ 2 {\displaystyle \psi _{2}} are separated by a selection rule if ⟨ ψ 1 | H | ψ 2 ⟩ = 0 {\displaystyle \langle \psi _{1}|H|\psi _{2}\rangle =0} for the given Hamiltonian H {\displaystyle H} , while they are separated by a superselection rule if ⟨ ψ 1 | A | ψ 2 ⟩ = 0 {\displaystyle \langle \psi _{1}|A|\psi _{2}\rangle =0} for all physical observables A {\displaystyle A} . Because no observable connects ⟨ ψ 1 | {\displaystyle \langle \psi _{1}|} and | ψ 2 ⟩ {\displaystyle |\psi _{2}\rangle } they cannot be put into a quantum superposition α | ψ 1 ⟩ + β | ψ 2 ⟩ {\displaystyle \alpha |\psi _{1}\rangle +\beta |\psi _{2}\rangle } , 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).
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 ( a ⋅ ψ ) = ( g a ) ⋅ ( g ψ ) {\displaystyle g(a\cdot \psi )=(ga)\cdot (g\psi )}
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 an 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
ψ ( x + L ) = e i θ ψ ( x ) . {\displaystyle \psi (x+L)=e^{i\theta }\psi (x).}
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