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Functional Integrals and their Applications
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 comm...
An inductive realization of loop vertex expansion is proposed and is applied to the construction of the ϕ14 theory. It appears simpler and more natural than the standard one at least for some situations.
We study the invariant unstable manifold of the trivial renormalization group fixed point tangent to the φ4 vertex in three dimensions. We parametrize it by a running φ4 coupling with linear step β-function. It is shown to have a renormalized double expansion in the running coupling and its logarithm.
We use Ginibre's general formulation of Griffiths' inequalities to derive new correlation inequalities for two-component classical and quantum mechanical systems of distinguishable particles interacting via two body potentials of positive type. As a consequence we obtain existence of the thermodynamic limit of the thermodynamic and correlation functions in the grand canonical ensemble at arbitrary temperatures and chemical potentials. For a large class of systems we show that the limiting correlation functions are clustering. (In a subsequent article these results are extended to the correlation functions of two-component quantum mechanical gases with Bose-Einstein statistics). Finally, a general construction of the thermodynamic limit of the pressure for gases which are not H-stable, above collapse temperature, is presented.
There are many problems in statistical mechanics and field theory where systems of interacting particles (or particle-like defects, contours, molecules, clusters, etc.) arise. Understanding these systems would be simpler if the particles were noninteracting. If the particles are dilute and weakly interacting, then the corrections to the independent particle approximation are small. These can be isolated and regarded as new kinds of (nonlocal) particles by a procedure known as the Mayer expansion. The nonlocal particles (or clusters) carry all the information about the long-distance behavior of the system.
The standard approach to Coulomb systems in equilibrium classical statistical mechanics (at low density or high temperature) was invented by Debye and Hückel. It may be found under their names in most text books on physical chemistry. Typically one considers a charge fixed in a background sea of other charges which are approxmated by a continuous charge distribution density. The Debye Hückel theory shows that this background charge distribution arranges itself in a spherically symmetric way about the fixed charge so that by Newton’s theorem the view from distance r is equivalent to a single charge at r = 0 whose magnitude equals the total charge of the fixed charge plus that of the “cloud” of radius r about it. Furthermore this total charge behaves as exp(-const × r) so that the charge is “shielded” and its “effective potential” is a Yukawa, exp(−cr)/r.
Ferromagnetic lattice spin systems can be expressed as gases of random walks interacting via a soft core repulsion. By using a mixed spinrandom walk representation we present a unified approach to many recently established correlation inequalities. As an application of these inequalities we obtain a simple proof of the mass gap for the λ(φ 4) 2 quantum field model. We also establish new upper bounds on critical temperatures.
Renormalization group ideas have been extremely important to progress in our understanding of gauge field theory. Particularly the idea of asymptotic freedom leads us to hope that nonabelian gauge theories exist in four dimensions and yet are capable of producing the physics we observe — quarks confined in meson and baryon states. For a thorough understanding of the ultraviolet behavior of gauge theories, we need to go beyond the approximation of the theory at some momentum scale by theories with one or a small number of coupling constants. In other words, we need a method of performing exact renormalization group transformations, keeping control of higher order effects, nonlocal effects, and large field effects that are usually ignored.
The subject of this book is equilibrium statistical mechanics, in particular the theory of critical phenomena, and quantum field theory. The central theme is the use of random-walk representations as a tool to derive correlation inequalities. The consequences of these inequalities for critical-exponent theory are expounded in detail. The book contains some previously unpublished results. It addresses both the researcher and the graduate student in modern statistical mechanics and quantum field theory.
A method is presented for calculating the grand‐partition function of a one‐dimensional gas with the potential V(x) = + ∞ for 0 ≤ x < δ and V(x) = −α e−γx for x ≥ δ.