Quantum Fields

Abstract

In this chapter we fix some notations which shall be used throughout the book. In particular some connections between probabilistic and quantum field theoretical notions are established. Since, in the stochastic limit, strongly nonlinear interactions break the validity of the standard commutation relations, leading to some deformations (even operator deformations) of them, in the following we shall need a notion of a free quantum field more general than the usual one [BoLoOkTo87], in the sense that it does not postulate a priori any commutation relation. For this reason we will introduce the notion of a Gaussian quantum field independently of the choice of specific commutation relations. On the other hand, it is known that the standard (Bose or Fermi) commutation relations follow from the Gaussian statistics, while the converse is true only in the Fock case. This suggests the general idea that statistics (i.e. the correlations) is a more basic physical notion than algebra (i.e. the commutation relations); in fact the former can be, at least in principle, compared with experiments, while the latter corresponds to an assumption directly on the mathematical model. Since the notion of a quantum field is a particular case of (generalized) quantum stochastic process, one can apply the reconstruction theorem of [AFL82] to conclude that the limit, in the sense of correlators, of a family of quantum fields is still a quantum field.