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Quantum Mechanics and Path Integrals: Richard P. Feynman, A. R. Hibbs

The fundamental physical and mathematical concepts which underlie the path integral approach to quantum mechanics were first developed by R. P. Feynman in the course of his graduate studies at Princeton, although more fully developed ideas, such as those described in this volume, were not worked out until a few years later. These early inquiries were involved with the problem of the infinite self energy of the electron. In working on that problem, a ``least action'' principle using half advanced and half retarded potentials were discovered. The principle could deal successfullly with the infinity arising in the application of classical electrodynamics.

Amazon Review:

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!

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