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[资源] Functional Integrals and their Applications

Introduction

These lectures are concerned with the analysis and applications of functional
integrals de ned by small perturbations of Gaussian measures The central topic
is the renormalization group Following Wilson and Polchinski an e ective po
tential is studied as a function of an ultra violet cuto By changing the cuto in a
continuous manner one obtains a di erential equation for the e ective potential
It is shown that by converting this equation to an integral equation and generat
ing 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 solv
ing for the e ective potential turns out to be of limited value due to large eld
problems which we explain To achieve better results we abandon the e ective
action and represent the partition function as a polymer gas The polymer gas
representation has enough in common with the e ective action that we are able to
exactly repeat the previous method of iterating an integral equation concluding
with the ow 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
e ects can destroy Debye screening

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[ Last edited by zhq025 on 2008-1-16 at 10:38 ]

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