Statistical Field Theory

Abstract

Volume 1: From Brownian Motion to Renormalization and Lattice Gauge Theory. Volume 2: Strong Coupling, Monte Carlo Methods, Conformal Field Theory, and Random Systems. This two-volume work provides a comprehensive and timely survey of the application of the methods of quantum field theory to statistical physics, a very active and fruitful area of modern research. The first volume provides a pedagogical introduction to the subject, discussing Brownian motion, its anticommutative counterpart in the guise of Onsager's solution to the two-dimensional Ising model, the mean field or Landau approximation, scaling ideas exemplified by the Kosterlitz-Thouless theory for the XY transition, the continuous renormalization group applied to the standard phi-to the fourth theory (the simplest typical case) and lattice gauge theory as a pathway to the understanding of quark confinement in quantum chromodynamics. The second volume covers more diverse topics, including strong coupling expansions and their analysis, Monte Carlo simulations, two-dimensional conformal field theory, and simple disordered systems. The book concludes with a chapter on random geometry and the Polyakov model of random surfaces which illustrates the relations between string theory and statistical physics. The two volumes that make up this work will be useful to theoretical physicists and applied mathematicians who are interested in the exciting developments which have resulted from the synthesis of field theory and statistical physics.

Phase transition in tensionless surfaces

“…However, this value of m, along with our result that aV0; implies violation of the hyperscaling relation [11] a=2Àmd, for d=2. Hence, we try a fit motivated by the KT scenario [11]:…”

“…We can do further analysis by monitoring the dependence of the apparent critical temperature, T c (L) (defined as the temperature in which the specific heat shows the maximum for a given lattice size) on the lattice size. This apparent critical temperature scales, in an ordinary second order phase transition, as [11] …”

The universal equation of state near the critical point of QCD

“…The coupling g is expected to receive only logarithmic corrections (it corresponds to a marginal operator) cut off by the mass term. 3 Moreover, these corrections are expected to reduce g(k) and shift the potential closer to the Gaussian fixed point [29]. These conclusions are verified by the numerical integration of the full evolution equation (3).…”

What can be learnt from the nonperturbative renormalization group?

“…Since they all contribute algebraically, the resulting sums (integrals) diverge at very short (ultraviolet) or very long (infrared) wavelengths. Getting rid of these ultraviolet divergences is the subject of perturbative renormalization and is now well under control [1][2][3]. Second, even after renormalization, the perturbation series are in general non-convergent and, as such, difficult to use to obtain reliable quantitative results.…”

“…The low temperature expansion of the ferromagnetic XY (O(2)) model in two dimensions is identical at all orders to that of a free theory. This expansion is invalidated at finite temperature by the existence and liberation of vortices (Kosterlitz-Thouless transition) [3]. Quantum Chromodynamics (QCD) is incapable, at any finite order of perturbation, of describing quark confinement.…”