Functional Integration: Action and Symmetries (Cambridge Monographs on Mathematical Physics) by Pierre Cartier (PDF)

Description

Functional integration successfully entered physics as path integrals in the 1942 PhD dissertation of Richard P. Feynman, but it made no sense at all as a mathematical definition. Cartier and DeWitt-Morette have created, in this book, a fresh approach to functional integration. The book is self-contained: mathematical ideas are introduced, developed, generalised and applied. In the authors’ hands, functional integration is shown to be a robust, user-friendly and multi-purpose tool that can be applied to a great variety of situations, for example: systems of indistinguishable particles; Aharonov–Bohm systems; supersymmetry; non-gaussian integrals. Problems in quantum field theory are also considered. In the final part the authors outline topics that can be profitably pursued using material already presented.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I got this book because when I was taking Steve Carlip’s course on quantum gravity, someone asked a few questions about functional integration and Carlip replied, “Well, there is a good book on the subject called ‘Functional Integration’…”A functional integral is something of the form (using TeX) int g(x) exp(I(x [dx], just to make explanations clear [dx] is the “functional measure”. Usually I(x) is the action (obtained from the Lagrangian’s integral), and g(x) is usually 1.This unique book fits a niche that other books on the subject gloss over (namely: what really is a functional integral?!).Although this book has a mathematical bent (the aim is rigorously defining the functional integral, after all), it is fairly readable. The pre-requisites, I’d say, is take a course on measure theory (failing that, perhaps read Folland’s

⭐) because the problem lies in figuring out what exactly is the functional measure [dx].But that supposes you know what a measure is! Hence you need to know measure theory, to some degree.The book is quite accurate in its citations, which may be scary (e.g. citing Bourbaki for the history of a “Polish space”). But never fear, the skilled authors explain it beautifully so even physicists can understand it.This book is mostly expository with only 6 exercises (3 in section 9.3; the other 3 in appendix G), and around 16 projects (in chapter 19) some are open problems. Others “do not cover new ground; but new approaches to the subject matter may shed light on issues whose familiarity has become a substitute for understanding.”The book is a gem, anyone who has any questions regarding how to well-define functional integration should buy a copy.

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