Hilbert Modules in Quantum Electro Dynamics and Quantum Probability

Abstract:

A physical system of the form \(\) with a distinguished state on \(\) may be described in a natural way on a Hilbert \(\)-module. Following the ideas of Accardi and Lu [1], we apply this possibility to a concrete system consisting of a boson field in the vacuum state coupled to a free electron.

We show that the physical system is described adequately on a new type of Fock module: the symmetric Fock module. It turns out that a module has to fulfill an algebraic condition in order to allow for the construction of a symmetric Fock module.

We prove in a central limit theorem that in the stochastic limit the moments of the collective operators (i.e. more or less the time-integrated interaction Hamiltonian) converge to the moments of free creators and annihilators on a full Fock module. In the sense of Voiculescu [22] and Speicher [20] these operators form a free white noise over the algebra \(\).