........ published in NEWSLETTER # 64
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........ published in NEWSLETTER # 64
FUNCTIONAL INTEGRATION: BASICS AND APPLICATIONS
By Professor C. DeWitt-Morette, University of Texas, Austin/TX (U.S.A.)
This volume (NATO ASI SERIES B361) is an outcome of the NATO Advanced Study Institute held at the Institut d'Etudes Scientifiques de Cargese in Corsica in September 1996. It is not simply a set of lecture notes but a combination of lecture notes and articles organized in a book for a larger public than the Institute participants. It includes texts which can serve as an introduction to more specialized topics, namely:
- A.Sokal, "Monte Carlo Methods in Statistical Mechanics" (updated version of his Lausanne course) by permission of the Cours de Troisieme Cycle de la Physique en Suisse Romande,
- P. Cartier, "Developpements recents sur les groupes de tresses" by permission of l'Association des Collaborateurs de Nicolas Bourbaki,
- M. Blau, "Localization and Digonalization" (updated version of his J. Math.Phys.article) by permission of the American Institute of Physics.
The program of the Institute covered several aspects of functional integration - from a robust mathematical foundation to many applications, heuristic and rigorous, in mathematics, physics, and chemistry. It included analytic and numerical computational techniques. One of the goals was to encourage cross-fertilization between these various aspects and disciplines.
The first week focused on quantum and classical systems with a finite number of degrees of freedom; the second week on field theories.
During the first week the basic course, given by P. Cartier, was a presentation of a recent rigorous approach to functional integration which does not resort to discretization, nor to analytic continuation. It provides a definition of functional integrals simpler and more powerful than the original ones. Could this approach accommodate the works presented by the other lectures? Although much remains to be done before answering "Yes", there seem to be no major obstacles.
During the second week the basic course was given by L. Kauffman on the theory of knots and its interfaces with polymers, quantum groups, and functional integration, thus increasing the scope of functional integration in many directions and opening up promising areas for future investigation. Functional integrals in polymer and membrane theory were presented heuristically by B. Duplantier and rigorously by J. Magnen.
New results in basic quantum field theory (B. DeWitt) and in topological quantum field theory (M. Blau) using functional integration included:
a) a formulation of gauge theory without ghosts.
b) a discussion of a variety of formal functional integral techniques for low-dimensional gauge theories.
Needless to say, many aspects of functional integration were not covered in the formal lectures; some of them were discussed in informal presentations and discussions by a number of participants. All of them have been invited to contribute a short paragraph to the proceedings stating either an open problem or a recently solved one in their fields of interest.
Reference books: B337, B361