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.^[1] 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 ${\displaystyle \psi _{1}}$ and ${\displaystyle \psi _{2}}$ are separated by a selection rule if ${\displaystyle \langle \psi _{1}|H|\psi _{2}\rangle =0}$ for the given Hamiltonian ${\displaystyle H}$, while they are separated by a superselection rule if ${\displaystyle \langle \psi _{1}|A|\psi _{2}\rangle =0}$ for all physical observables ${\displaystyle A}$. Because no observable connects ${\displaystyle \langle \psi _{1}|}$ and ${\displaystyle |\psi _{2}\rangle }$ they cannot be put into a quantum superposition ${\displaystyle \alpha |\psi _{1}\rangle +\beta |\psi _{2}\rangle }$, 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.^[2]

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).