In which field of mathematics do I learn path integrals?

Question

I don't mean line integrals, I am talking about path integrals or functional integrals like the ones that Feynman introduced to quantum mechanics. And what are the prerequisites to this field of study?

Answer 1

To study the precise mathematical formulation of path integrals, you actually need probabilistic tools. The path integral is a stochastic integral with suitable measures, such as the Wiener measure associated with brownian motion.

The ideas used by physicists are very useful, but not always mathematically accurate, and rely more or less on justification by approximation of these stochastic integrals.

So it really depends on the purpose for you to study them. If you plan to utilize them as a physicist do (or just to understand their physical meaning), not much mathematical background will be needed, apart from basic quantum mechanics knowledge (and Trotter formula) and the principle of least action of classical mechanics (a bit of calculus of variations, as already mentioned).

But if you are interested in a more rigorous study, perhaps oriented to mathematical physics, you really need to understand stochastic processes and probability.

Answer 2

I recommend two resources:

Feynman's original book called Quantum Mechanics and Path Integrals. This contains most of the prerequisites in the first two chapters, but you will need some maturity to get through them.

A. Zee's quantum field theory book Quantum Field Theory in a Nutshell for its friendly chapter on them.

Answer 3

1. Quantum Mechanics and Path Integrals: This is a book every physicist, or student of physics, should study. Here the author describes the principle of action in quantum physics. It is not a minimum action principle, like in classical mechanics: you can, however, derive the classical minimum principle from it, in the classical limit. Why is this important? Well, it so happens that the famous gauge field theories could only be quantized under this formalism. Feynman, of course, reformulates everything with his technique, so that the book is very enlightening: it is a rich experience to see well-known things under a different viewpoint. But there are many new things also. The applications are brilliant, covering just about everything: electrodynamics, statistical mechanics, you name it. A new mathematics is introduced by Feynman, a theory of integration in a space whose elements are curves (path integrals). As far as I know, the rigorous theory of this integration does not exist as of now. Undauntedly, Feynman is able to guide us to very important results by using intuitive methods, and checking the validity of a result by obtaining it by two different ways, for instance. Don't miss, by the way, his discussion on the role of rigor (in the mathematical sense) in physics. There is a section on that!
2. Path Integrals in Quantum Mechanics: The main goal of this book is to familiarize the reader with a tool, the path integral, that not only offers an alternative point of view on quantum mechanics, but more importantly, under a generalized form, has also become the key to a deeper understanding of quantum field theory and its applications, extending from particle physics to phase transitions or properties of quantum gases.
3. Mathematical Feynman Path Integrals and Their Applications: This volume provides a detailed, self-contained description of the mathematical difficulties as well as the possible techniques used to solve these difficulties. In particular, it gives a complete overview of the mathematical realization of Feynman path integrals in terms of well-defined functional integrals, that is, the infinite dimensional oscillatory integrals. It contains the traditional results on the topic as well as the more recent developments obtained by the author. Mathematical Feynman Path Integrals and Their Applications is devoted to both mathematicians and physicists, graduate students and researchers who are interested in the problem of mathematical foundations of Feynman path integrals
4. Path Integrals and Their Applications in Quantum Statistical and Solid State Physics: This book applies path integrals to statistical physics and solid state physics.
5. Path Integrals for Stochastic Processes - An Introduction: The aim of this book is to offer a brief but complete presentation of the path integral approach to stochastic processes. It could be used as an advanced textbook for graduate students and even ambitious undergraduates in physics. It describes how to apply these techniques for both Markov and non-Markov processes. The path expansion (or semiclassical approximation) is discussed and adapted to the stochastic context. Also, some examples of nonlinear transformations and some applications are discussed, as well as examples of rather unusual applications. An extensive bibliography is included. The book is detailed enough to capture the interest of the curious reader, and complete enough to provide a solid background to explore the research literature and start exploiting the learned material in real situations.
6. Path-Integral Methods and Their Applications: This book presents the major developments in this field with emphasis on application of path integration methods in diverse areas. After introducing the concept of path integrals, related topics like random walk, Brownian motion and Wiener integrals are discussed. Several techniques of path integration including global and local time transformations, numerical methods as well as approximation schemes have been presented. The book seeks to provide a proper perspective of some of the most recent exact results and approximation schemes for practical applications.
7. Mathematical Theory of Feynman Path Integrals: (From the Back Cover) Feynman path integrals, suggested heuristically by Feynman in the 40s, have become the basis of much of contemporary physics, from non-relativistic quantum mechanics to quantum fields, including gauge fields, gravitation, cosmology. Recently ideas based on Feynman path integrals have also played an important role in areas of mathematics like low-dimensional topology and differential geometry, algebraic geometry, infinite-dimensional analysis and geometry, and number theory.

Answer 4

Feynman's path integral formulation is closely related to the action principle of classical mechanics, which relies heavily on the calculus of variations. You need to learn, essentially, how to minimize a functional. Prerequisites are pretty much just calculus (multivariable, hopefully), as well some classical mechanics to understand the motivation behind the action principle.