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

The Zeeman-Hamilton operators of free charged particles are identified with the Laplacians of certain Riemannian manifolds, called Zeeman manifolds. The quantum Hilbert space decomposes into subspaces (Zeeman zones) which are invariant under the actions 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.