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The understanding of phase transitions has long been a fundamental problem of statistical mechanics. It has made spectac ular progress during the last few years, largely because of the ideas of K.G. Wilson, in applying to an apparently quite different domain the methods of the renormalization group, which had been developped in the framework of the quantum theory of fields. The ability of these theoretical methods to lead to very precise predictions has, ~n turn, stimulated in the last few years more refined experiments in different areas. We now have entered a period where the theoretical results yielded by the renormalization group approach are suffi ciently precise and can be compared with those of the traditional method of high temperature series expansion on lattices, and with the experimental data. Although very similar, the results coming from the renormalization group and high temperature analysis seemed to indicate systematic discrepancies between the continuous field theory and lattice models. It was therefore important to appreciate the reliability of the predictions coming from both theoretical schemes, and to compare them to the latest experimental results. We think that this Cargese Summer Institute has been very successful 1 in this respect. Indeed, leading experts in the field, both experimentalists and theoreticians, have gathered and presented detailed analysis of the present situation. In particular, B.G. Nickel has produced longer high temperature series which seem to indicate that the discrepancies between series and renormalization group results have been previously overestimated.
Vols. for 1980- issued in three parts: Series, Authors, and Titles.
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 was 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 lecturers? Although much remains to be done before answering "Yes," there seems to be no major obstacle along the road. The other courses taught during the first week presented: a) a solid introduction to functional numerical techniques (A. Sokal) and their applications to functional integrals encountered in chemistry (N. Makri). b) integrals based on Poisson processes and their applications to wave propagation (S. K. Foong), in particular a wave-restorer or wave-designer algorithm yielding the initial wave profile when one can only observe its distortion through a dissipative medium. c) the formulation of a quantum equivalence principle (H. Kleinert) which. given the flat space theory, yields a well-defined quantum theory in spaces with curvature and torsion.
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