Normal zones on Zeeman manifolds with trace class heat and Feynman kernels and well defined zonal Feynman integrals

Abstract

The problem of infinities (divergent integrals) appearing in quantum field theory or elementary particle physics are treated by renormalization in the current theories. By this perturbative tool the desired finite quantities are produced by differences of infinities. This paper gives a new non-perturbative approach to this problem. Namely, the Hilbert space on which the quantum Hamilton operator $H$ is acting, is decomposed into subspaces, called Zeeman zones, which are invariant under the actions both of $H$ and the natural complex Heisenberg group representation. Thus well defined particle theory and zonal geometry can be developed on each zone separately. One of the most surprising result is that the quantities which appear as infinities on the global setting become well defined finite ones on the zonal setting. For instance, both the Wiener-Kac and the Dirac-Feynman kernels become of the trace class on the zones, both defining the corresponding zonal measures on the path-spaces rigorously. The Hamilton operators, $H$, considered in this paper are those corresponding to electrons orbiting in a constant magnetic field. This is one of the most important Hamiltonians, introduced for explaining the Zeeman effect. This Hamiltonian is identified with the Laplacian of certain Riemannian,so called Zeeman manifolds. The spectral Zeeman zone decomposition is introduced, by one of the definitions, by the spectrum of the angular momentum operator. Thus a zone exhibits the magnetic state of the particle occupying the zone. All the important spectral theoretical objects are explicitly established. They include the spectrum, the zonal projection operators, the zonal Wiener-Kac and Dirac-Feynman kernels, and the zonal partition functions.