93-24 David C.Brydges Notes with the collaboration of R.Fernandez

Functional Integrals and their Applications (Notes for a course for the ``Troisieme Cycle de la Physique en Suisse Romande'' given in Lausanne, Switzerland, during the summer of 1992) (683K, Postscript) Feb 4, 93

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Abstract. These lectures are concerned with the analysis and applications of functional integrals defined by small perturbations of Gaussian measures. The central topic is the renormalization-group. Following Wilson and Polchinski, an effective potential is studied as a function of an ultra-violet cutoff. By changing the cutoff in a continuous manner, one obtains a differential equation for the effective potential. It is shown that by converting this equation to an integral equation and generating an iterative solution one obtains the Mayer expansion of classical statistical mechanics. Some results on the convergence of such an expansion are deduced, with applications to Coulomb and Yukawa gases. However, this method of solving for the effective potential turns out to be of limited value due to ``large-field problems'', which we explain. To achieve better results we abandon the effective action and represent the partition function as a polymer gas. The polymer gas representation has enough in common with the effective action that we are able to exactly repeat the previous method of iterating an integral equation, concluding with the flow of the activities of the polymers given by ``cluster expansions''. This is expounded in considerable detail, using as illustration the example of dipole gases. In addition, there are two introductory sections---independent of the other lectures---in which, as a motivation, problems in the theory of Coulomb gases and polymer physics are shown to be related to functional integrals of the type studied here. In particular, a heuristic discussion is presented on how quantum effects can destroy Debye screening.

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