QUANTUM BUNDLES AND THEIR SYMMETRIES

Abstract

Wave functions in the domain of observables such as the Hamiltonian are not always smooth functions on the classical configuration space Q. Rather, they are often best regarded as functions on a G bundle EG over Q or as sections of an associated bundle. If H is a classical group which acts on Q, its quantum version HG, which acts on EG, is not always H, but an extension of H by G. A powerful and physically transparent construction of EG and HG, where G= U(1) and H1(Q, Z)=0, has been developed using the path space [Formula: see text]. ([Formula: see text] consists of paths on Q from a fixed point.) In this paper we show how to construct EG and HG when G is U(1) or U(1)×π1(Q) and there is no restriction on H1(Q, Z). The method is illustrated with concrete examples, such as a system of charges and monopoles. We argue also that [Formula: see text] is a sort of superbundle from which a large variety of bundles can be obtained by imposing suitable equivalence relations.