Theory of zones on Zeeman manifolds A new approach to the infinities of QED

The Zeeman-Hamilton operator of free charged particles are identified with the Laplacians of certain Riemannian manifolds, called Zeeman manifolds. The quantum Hilbert space, H, decomposes into subspaces (Zeeman zones) which are invariant under the action both of the Zeeman operator and the natural 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 those divergent on the global setting appear to be finite on the zonal setting. Even the zonal Feynman integral is well defined. This zonal interpretation of particles has fundamental effect both on the physical and mathematical view of these objects. The points are non-existing on a zone, for instance. One should introduce the concept of zonal point-spread, defined by certain wave functions. As a result, the zonal particles are not pointbut point-spread-objects. The theory developed below includes explicit computation of objects such as the waves defining the point-spreads, the zonal Wiener-Kac and Dirac-Feynman flows (which then define the corresponding measures on the path-spaces), and the corresponding zonal Feynman-Kac formulas. It will be tested against several well known effect such as the Aharanov-Bohm effect and Lamb shift. There is also explained why these extended charged particles do not blow up. ∗Lehman College of CUNY, Bronx, NY, 10468, and Rényi Institute, Budapest, Hungary.