Frederic Paugam: “Towards the Mathematics of Quantum Field Theory”

Abstract

Quantum field theory can be considered as one of the cornerstones of modern physics. Historically, it originates from the problem of quantizing the electromagnetic field, dating back to shortly after the advent of quantum mechanics. This amounts to understanding the quantization of a system with infinitely many degrees of freedom, namely, the values of the electromagnetic field at all spacetime points. It turned out that precisely this property would form the main obstacle in the search for a mathematical formalism for quantum field theories. In the 1950s this led to a separation of the field. On the one hand perturbation theory led to a computationally yet undefeated approach, where in particular Feynman’s graphical method streamlined the analysis of physical probability amplitudes. On the other hand, an elegant axiomatic approach to quantum fields emerged through the work of Haag, Kastler, Wightman and others, leading to the subject known as algebraic quantum field theory. For a historical account we refer to [7]. Even though the two approaches ought to be complementary, it turned out that the separation only became larger and larger towards the end of the last century. This was mainly due to the fact that physically relevant interacting quantum fields were found difficult to be included in the strict algebraic formulation, while at the same time the computational approach of Feynman amplitudes mistified the mathematical structure behind this successful branch of physics.