Mathematical Introduction

In this chapter, we discuss the mathematical concepts that are essential to an understanding of the subsequent material in the book. We delve into fundamental concepts such as the functional, functional derivatives, and functional integrals. Additionally, we explore Gaussian functional integrals and integrals over a Gaussian measure. We introduce the cumulant expansion method and prove the Bogoliubov’s inequality. Furthermore, we present methods for approximating functional integrals over Gaussian measures, including Gaussian approximation, variation method, and Gaussian equivalent representation.