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,
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.
Read more about this topic: Superselection
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