Second quantization, projective modules, and local gauge invariance

Bundles and bundle structures have gained wide currency in modern approaches to certain topics in quantum physics, significant applications appearing in connection with gauge theories, theories of geometric quantization, and in numerous other contexts. It is argued that such structures can already be discerned in the most elementary notions of second quantization. An examination of the methods traditionally used by physicists in dealing quantum mechanically with systems exhibiting an infinite number of degrees of freedom reveals the implicit use of module structures over algebras of functions. By making these structures explicit and adapting some results of perturbation theory an association between bare particles and finitely generated projective modules is arrived at. In particular, rank one modules emerge naturally, for algebraic reasons, as the appropriate descriptors of bosons in this association. As a first application of the formalism the existence of phononlike excitations in general many-fermion systems is shown. When these ideas are further specialized (local) gauge theoretical notions arise in a natural way out of a consideration of the bundles. (author)