On the role of space–time topology in quantum phenomena: Superselection of charge and emergence of nontrivial vacua

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Abstract
Schwarzschild–Kruskal space–time admits a two-parameter family of everywhere regular, static, source-free Maxwell fields. It is shown that there exists a corresponding two-parameter family of unitarily inequivalent representations of the canonical commutation relations. Elements of the underlying Hilbert space may be interpreted as ’’quantum fluctuations of the Maxwell field off nontrivial classical vacua.’’ The representation corresponding to the ’’trivial’’ sector—i.e., the zero classical solution— is the usual Fock representation. All others are ’’non-Fock.’’ In particular, in all other sectors, the Maxwell field develops a nonzero vacuum expectation value. The parameters labelling the family can be interpreted as electric and magnetic charges. Therefore, unitary inequivalence naturally leads to superselection rules for these charges. These features arise in spite of the linearity of field equations only because the space–time topology is ’’nontrivial.’’ Also, because of linearity, an exact analysis is possible at the quantum level; recourse to perturbation theory is unnecessary.