Quantum Field Theory

Abstract

This chapter provides a quick—and therefore incomplete—introduction to quantum field theory. Those among our readers who know little about it will find here the basic material allowing them to appreciate and understand the remaining chapters of this book. Section 2.1 explains the canonical quantization of free fields, bosons and fermions, starting from a discrete formulation. It is appropriate for readers without any previous knowledge of quantum field theory; some experience with quantum mechanics remains an essential condition, however. Section 2.2 reviews the path-integral formalism of quantum mechanics for a single degree of freedom, and then for quantum fields, especially fermions. Section 2.3 introduces the central notion of a correlation function, both in the canonical and path-integral formalisms. The Wick rotation to imaginary time is performed, with the example of the free massive boson illustrating the exponential decay of correlations with distance. Section 2.4 explains the meaning of a symmetry transformation and the consequences of symmetries in classical and quantum field theories. This section deserves special attention—even from experienced readers—because the notion of a symmetry transformation and how it is implemented is fundamental to this work. Section 2.5 is devoted to the energy-momentum tensor, the conserved current associated with translation invariance, which plays a central role as the generator of conformal transformations when suitably modified.