The Wigner semi-circle law in quantum electro dynamics

Abstract

In the present paper, the basic ideas of thestochastic limit of quantum theory are applied to quantum electro-dynamics. This naturally leads to the study of a new type of quantum stochastic calculus on aHilbert 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-calledinteracting Fock space. A kind of Fock space in which then quanta, in then-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 theFermi golden rule.