# Superselection

## Rule forbidding the coherence of certain states / From Wikipedia, the free encyclopedia

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