Laplace spectra on open and compact Zeeman manifolds

Abstract

By a recent observation, the Laplacians on the Riemannian manifolds the author used for isospectrality constructions are nothing but the Zeeman-Hamilton operators of free charged particles. These manifolds can be considered as prototypes of the so called Zeeman manifolds. This observation allows to develop a spectral theory both on open Z-manifolds and their compact submanifolds. The theory on open manifolds leads to a new nonperturbative approach to the infinities of QED. This idea exploits that the quantum Hilbert space decomposes into subspaces (Zeeman zones) which are invariant under the actions both of this Zeeman-Laplace operator and the natural complex Heisenberg group representation. Thus a well defined particle theory and zonal geometry can be developed on each zone separately. The most surprising result is that quantities divergent on the global setting are finite on the zonal setting. Even the zonal Feynman integral is well defined. The results include explicit computations of objects such as the zonal spectra, the waves defining the zonal point-spreads, the zonal Wiener-Kac resp. Dirac-Feynman flows, and the corresponding partition functions. The observation adds new view-point also to the problem of finding intertwining operators by which isospectral pairs of metrics with different local geometries on compact submanifolds can be constructed. Among the examples the author constructed the most surprising are the isospectrality families containing both homogeneous and locally inhomogeneous metrics. The observation provides even quantum physical interpretation to the isospectrality.