Physics from the G-bundle viewpoint

It is generally agreed that in quantum mechanics, the observables of a physical system are represented by self-adjoint operators on a Hilbert space. Some natural differential operators (and their spectra) associated with systems that are classically modeled on principle fiber bundles (defined as G bundles) which are equipped with a metric tensor on the base as well as a connection (ie gauge potential) are studied. It is argued that the eigenvalues of these operators are possible values of the kinetic energy of particles that respond to the gauge potential when the base is three, and mass (squared) spectra of these particles when the base is four dimensional (eg space-time or a Euclidean analog). These essentially covariant Laplace operators obtained from minimum coupling are relevant as all interactions may be derived from those of the gauge type. (U.K.)