Gaussian Integrals

Abstract

In this chapter, we lay the mathematical foundations for the functional-integral formalism that we develop in later chapters. We start with introducing the Gaussian probability distribution together with the corresponding integrals over this distribution, called Gaussian integrals. These concepts are then generalized to higher dimensions, to the complex plane, and to what are called Grassmann variables. The multidimensional Gaussian integral is of great importance for the rest of this book. In Chap. 7, we show that it leads to an exact solution of noninteracting quantum gases, which then also forms the basis for a perturbative description of interacting quantum gases. The goal of this chapter is to highlight the practical use of several important mathematical results that are needed to understand the rest of the book. The chapter is not intended to be a full mathematical account of all the above-mentioned topics, meaning that proofs will often be omitted or replaced by illustrative examples. The more experienced reader who is already familiar with Gaussian integrals, complex analysis, and Grassmann algebras, can use this chapter for reference.