1 - Quantum mechanics, functional integrals

Summary

Quantum field theories are quantum mechanical systems with an infinite number of degrees of freedom. We start this book with a review of quantum mechanics in a form that is particularly suited for the field theory generalization. This form utilizes the Heisenberg picture and the functional integral representation of probability amplitudes — the transformation functions of Dirac. Some time will be spent developing and discussing the functional or path integral representation of quantum mechanics since this will enable us to learn a new formalism in a simple and familiar physical context — we won't have to learn too many new things all at once. The functional formulation will be illustrated by applying it at some length to the harmonic oscillator which is not only the simplest (and exactly soluble) non-trivial quantum system but which is also the dynamics of a single mode of a free field theory. The purpose of this review of quantum mechanics is to introduce in the simplest form much of the notation and methods used later in the book. Throughout this chapter we will explicitly work only with systems that have a single degree of freedom (one coordinate variable). The extension to systems with several degrees of freedom is straightforward.