Perturbative QFT: off-shell fields, deformation quantization and causal perturbation theory

Abstract: Perturbative QFT is developed in terms of off-shell fields (that is, functionals on the configuration space not restricted by any field equation), and by quantizing the (underlying) free theory by an $\hbar$-dependent deformation of the classical product (i.e., the pointwise product of functionals). The time-ordered product of local fields is defined axiomatically, and constructed by induction on the number of factors using Stora's version of the Epstein--Glaser construction; in particular, the interaction is adiabatically switched off. The set of solutions of these axioms can be understood as the orbit of the St\"uckelberg--Petermann renormalization group when acting on a particular solution. Interacting fields are defined in terms of the time-ordered product by Bogoliubov's formula; they satisfy the following, physically desired properties: causality, spacelike commutativity, (off-shell) field equation and existence of the classical limit. Local, algebraic properties of the observables can be obtained without performing the adiabatic limit (i.e., the limit removing the adiabatic switching of the interaction).