The Wigner semi--circle law in quantum electro dynamics

In the present paper, the basic ideas of the {\it stochastic limit of quantum theory} are applied to quantum electro-dynamics. This naturally leads to the study of a a new type of quantum stochastic calculus on a {\it Hilbert module}. Our main result is that in the weak coupling limit of a system composed of a free particle (electron, atom,...) interacting, via the minimal coupling, with the quantum electromagnetic field, a new type of quantum noise arises, living on a Hilbert module rather than a Hilbert space. Moreover we prove that the vacuum distribution of the limiting field operator is not Gaussian, as usual, but a nonlinear deformation of the Wigner semi--circle law. A third new object arising from the present theory, is the so called {\it interacting Fock space}. A kind of Fock space in which the $n$ quanta, in the $n$-particle space, are not independent, but interact. The origin of all these new features is that we do not introduce the dipole approximation, but we keep the exponential response term, coupling the electron to the quantum electromagnetic field. This produces a nonlinear interaction among all the modes of the limit master field (quantum noise) whose explicit expression, that we find, can be considered as a nonlinear generalization of the {\it Fermi golden rule\/}.