Second quantization, projective modules, and local gauge invariance

Abstract

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 (e.g., Atiyah and Jones, 1978), theories of geometric quantization (e.g., Kostant, 1970; Sniatycki, 1974), and in numerous other contexts. In this paper we argue that such structures can already be discerned in the most elementary notions of second quantization, albeit in disguised form. An examination of the methods traditionally used by physicists in dealing quantum mechanically with systems exhibiting an infinite number of degrees of freedom reveals, almost from the outset, the implicit use of module structures over algebras of functions (Section 2). By making these structures explicit and adapting some results of perturbation theory we arrive at an association between bare particles and finitely generated projective modules (Sections 3 and 4). In particular, rank one modules emerge naturally, for algebraic reasons, as the appropriate descriptors of bosons in this association. (This provides a possible setting for the development of standard geometric quantization theory.) As a first application of the formalism we show the existence of phononlike excitations in general many-fermion systems. When these ideas are further specialized (local) gauge theoretical notions arise in a natural way out of a consideration of the bundles obtained via Swan's theorem. These theories emerge moreover equipped with an interpretation linked directly to the geometrical entities associated with the underlying bundles. Thus for example in the line bundle (or rank one) case,