The Feynman integral in quantum field theory

Abstract

In a sense quantum field theory is just quantum mechanics in an infinite number of dimensions. Hence, the mathematical scheme of section 1.1 applies. However, in contradistinction to quantum mechanics the von Neumann theorem about the unitary equivalence of various representations of the canonical commutation relations does not apply in an infinite number of dimensions [116]. We must choose the representation (the choice is not arbitrary but is determined by the interaction). There would be many models of quantum field theory (QFT) if we did not demand relativistic invariance. This invariance implies a singular behaviour of the vacuum correlation functions of quantum fields. The singularities are at the origin of divergences of most of the terms of the perturbation series. The non-perturbative methods of the construction of the Hamiltonian started in the sixties. These methods were soon replaced by constructive methods at imaginary time. There are efficient non-perturbative methods of constructing quantum fields at imaginary time [161]. In principle, after the construction of Euclidean invariant fields one can obtain through an analytic continuation the relativistic ones [311]. However, this is not a realistic way of the investigation of quantum dynamics. Moreover, the Euclidean methods were successful in a construction of quantum models only in less than four dimensions.