- 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)
Feb 4, 93
(auto. generated ps),
of related papers
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.