The Path Integral Formulation of Quantum Mechanics

Abstract

We turn now to a topic that is important already for ordinary quantum mechanics and essential in quantum field theory: the so-called path integral. In the setting of ordinary quantum mechanics (of the sort we have been considering in this book), the integrals in question are over spaces of “paths,” that is, maps of some interval [a, b] into ${\mathbb{R}}^{n}.$ In the setting of quantum field theory, the integrals are integrals over spaces of “fields,” that is, maps of some region inside ${\mathbb{R}}^{d}$ into ${\mathbb{R}}^{n}.$ Formal integrals of this sort abound in the physics literature, and it is typically difficult to make rigorous mathematical sense of them—although much effort has been expended in the attempt! In this chapter, we will develop a rigorous integral over spaces of paths by using the Wiener measure, resulting in the Feynman–Kac formula.