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Superselection theory is about identifying superselection sectors in quantum field theory. In quantum field theory there are conservation laws that specify that in an isolated system the total value of certain charges (e.g. electric charges, magnetic charges) cannot change due to any interactions taking place in the system. A superselection sector can be specified by prescribing a number to each of these charges, it is the space of all states of the system with these values for the charges.
There are two complementary viewpoints about superselection sectors:
In the picture of (Wick-Wigner-Wightman 52) all possible states of a system form one big Hilbert space and superselection sectors are subspaces of this Hilbert space, and no measurement, that is no observable, can map a state in one superselection sector to another. Further, superpositions of states in different sectors do not exist in physical reality.
Hence, mathematically: superselection sectors are irreducible representations of a given algebra of observables inside the space of quantum states.
An example would be a superposition of a state containing one electron with a state containing two electrons.
In the AQFT approach to quantum field theory the observables of the theory are selfadjoint elements of a C-star algebra. A concrete physical system is a state of this algebra, which is accompanied via the GNS construction with a representation of the algebra. A superselection sector from this viewpoint is an equivalence class of unitarily equivalent representations, see representation of a C-star algebra.
Usually, in AQFT not all representations of the algebra of observables are considered to be physically relevant, so that superselection theory starts with the statement of conditions that representations have to fulfill in order to be admissible, and only those are considered. Example:
For the Wick–Wightman–Wigner picture of superselection sectors see
A brief survey of that is in section 2.3 of
See also AQFT, QFT and Haag-Kastler axioms.