  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 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.
