Quantum mechanics is undoubtedly one of the most accurate and important scientific theories in the history of science. The theoretical foundations of quantum mechanics have been discussed in depth in Baaquie (2013e), where the main focus is on the interpretation of the mathematical symbols of quantum mechanics and on its enigmatic superstructure. In contrast, the main focus of this book is on the mathematics of path integral quantum mechanics.

The traditional approach to quantum mechanics has been to study the Schrödinger equation, one of the cornerstones of quantum mechanics, and which is a special case of partial differential equations. Needless to say, the study of the Schrödinger equation continues to be a central task of quantum mechanics, yielding a steady stream of new and valuable results.

Interestingly enough, there are two other formulations of quantum mechanics, namely the operator approach of Heisenberg and the path integral approach of Dirac–Feynman, that provide a mathematical framework which is independent of the Schrödinger equation. In this book, the Schrödinger equation is never directly solved; instead the Hamiltonian operator is analyzed and path integrals for different quantum and classical random systems are studied to gain an understanding of quantum mathematics.

I became aware of path integrals when I was a graduate student, and what intrigued me most was the novelty, flexibility and versatility of their theoretical and mathematical framework. I have spent most of my research years in exploring and employing this framework.