Probabilistic Path Integrals in Quantum and Statistical Physics

Abstract

In this thesis, we explore the mathematical techniques for rigorously constructing path integral measures in quantum mechanics, statistical field theory, and relativistic quantum field theory. We begin with the quantum mechanical case, where the Feynman path integral is introduced via the Trotter product formula, and we discuss the challenges arising from the ill-defined nature of the so-called Feynman "measure." However, after performing a Wick rotation to imaginary time, the Feynman path integral becomes a well-defined Wiener integral over Wiener space. We demonstrate the connection between this Wiener integral and Brownian motion, and show how the associated Wiener measure corresponds to the Gibbs measure in statistical mechanics. In the late 1960s, Arthur Jaffe and James Glimm developed a rigorous framework known as constructive quantum field theory, in which the correlation functions of a relativistic quantum field theory are realized as integrals over probability measures defined on the space of tempered distributions. We show that these measures and integrals align with those used to model lattice systems near criticality through the scaling limit.Finally, we perform explicit calculations for the $\phi^4$-model and demonstrate that the resulting perturbation series is not convergent but asymptotic in nature. We compare these findings with results from the contemporary literature.