MiS Preprint Repository

This paper evolves a new non-perturbative theory by which the problem of infinities appearing in quantum physics can be handled. Its most important application is an exact derivation of the Lamb shift formula by using no renormalization. The Lamb shift experiment (1947) gave rise to one of the greatest challenges whose explanation brought the modern renormalization technique into life. Since then this is the only tool for handling these infinities. The relation between this renormalization theory and our non-perturbative theory is also discussed in this paper.

Our key insight is the realization that the natural complex Heisenberg group representation splits the Hilbert space, $L{2}_{\mathbb{C}}({\mathbb{R}}^{2\kappa })$, of complex valued functions defined on an even dimensional Euclidean space into irreducible subspaces (alias zones) which are invariant also under the action of the Landau-Zeeman operator. After a natural modification, also the Coulomb operator can be involved into this zonal theory. Thus these zones can be separately investigated, both from geometrical and physical point of view. In the literature only the zone spanned by the holomorphic polynomials has been investigated so far. This zone is the well known Fock space.

This paper explicitly explores also the ignored (infinitely many) other zones. It turns out that quantities appearing as infinities on the total Hilbert space are finite in the zonal setting. Even the zonal Feynman integrals are well defined. In a sense, the desired finite quantities are provided here by an extended particle theory where these extended objects show up also on the rigorously developed mathematical level. Name de Broglie geometry was chosen to suggest this feature of the zonal theory.