Search Results

Recent progress in the treatment of Dirac fields in lattice gauge theory has allowed the chiral symmetry and associated anomaly on the lattice to be discussed in a manner similar to that in continuum theory. In particular, the index theorem on the lattice can be discussed. The analysis of the index theorem on the discrete lattice itself has certain subtle aspects, but lattice theory deals with completely regularized quantities, and thus some of the subtle aspects in continuum theory are now given a more rigorous basis. It is explained that all the results of chiral anomalies in continuum theory are reproduced in a suitable continuum limit of lattice gauge theory, providing a uniform and consistent treatment of both continuum and lattice theories.

This chapter discusses random growth models describing the evolution of an interface in the plane. For specific models, three basic questions are discussed. First, under appropriate scaling, what is the limiting shape of the interface and what is the partial differential equation governing its evolution? Second, how can random fluctuations around the limit behaviour be described? Third, how can atypical behaviour be characterized? The power of probabilistic tools is demonstrated by employing laws of large numbers, central limit theorems, and large deviation techniques to answer these questions, respectively.

This chapter, within the framework of classical statistical mechanics, discusses a family of models defined on one-dimensional lattices. It studies the simplest local examples: models that involve only interactions between nearest neighbours on the lattice. For such models, correlation functions can be calculated by a transfer matrix formalism. The chapter first describes some general properties of transfer matrices in one-dimensional models. This formalism is used to establish various properties of correlation functions, like the thermodynamic or infinite volume limit, the large-distance behaviour of the two-point correlation function, and introduces the very important concept of correlation length. Connected correlation functions, cumulants of the distribution, play a particularly important role. Indeed, these functions satisfy the cluster property, which characterizes their decay at large distance. The transfer matrix formalism is applied to the example of a Gaussian Boltzmann weight, which is studied in detail. The chapter calculates the partition function and correlation functions explicitly, and observes that , the correlation length diverges, making it possible to define a continuum limit. It shows that results of the continuum limit can be reproduced directly by solving a partial differential equation in which all traces of the initial lattice structure have disappeared. Finally, it exhibits a slightly more general class of models which share the same properties: divergent correlation length and continuum limit. Exercises are provided at the end of the chapter.

This chapter discusses the related questions of universality and macroscopic continuum limit in random systems with a large number of degrees of freedom. It first explains the notion of universality using the classical example of the central limit theorem in probability theory. It then discusses the properties of the random walk on a lattice, where universality is directly related to the continuum limit. In both examples, the chapter is interested in the collective properties of an infinite number of random variables in a situation where the probability of large deviations with respect to the mean value decreases fast enough. They differ in the sense that a random walk is based on a spatial structure that does not necessarily exist in the case of the central limit theorem. From the study of these first examples emerges the importance of Gaussian distributions, and this justifies the technical considerations of Chapter 2. The chapter introduces some transformations, acting on distributions, which decrease the number of random variables. It shows that Gaussian distributions are attractive fixed points for these transformations. This will provides the first, extremely simple, applications of the renormalization group (RG) ideas and allows the establishment of corresponding terminology. Finally, in this context of the random walk, a path integral representation is associated with the existence of a continuum limit. Exercises are provided at the end of the chapter.

This chapter provides a simple physical interpretation to the formal continuum limit that has led, from an integral over position variables corresponding to discrete times, to a path integral. It shows that the integral corresponding to discrete times can be considered as the partition function of a classical statistical system in one space dimension. The continuum limit, then, corresponds to a limit where the correlation length, which characterizes the decay of correlations at large distance, diverges. This limit has some universality properties in the sense that different discretized forms lead to the same path integral. In this statistical framework, the correlation functions that have been introduced earlier appear as continuum limits of the correlation functions of classical statistical models on a one-dimensional lattice. Thus, the path integral can be used to exhibit a mathematical relation between classical statistical physics on a line and quantum statistical physics of a point-like particle at thermal equilibrium.

This chapter shows that, as in the case of the random walk, one can associate to the continuum limit a path integral, which generalizes the path integral of the Brownian motion. It first studies the Gaussian example (which is simpler) and then the general case. Exercises are provided at the end of the chapter.

This chapter goes back to the discrete lattice in order to transform the basis to momentum eigenstates. In addition, the chapter takes the continuum limit of space and we recover known relations and the delta function.

The first chapter discusses the asymptotic properties at large time and space of the familiar example of the random walk. The universality of a large scale behaviour and, correspondingly, the existence of a macroscopic continuum limit emerge as collective properties of systems involving a large number of random variables whose individual distribution is sufficiently localized. These properties, as well as the appearance of an asymptotic Gaussian distribution when the random variables are statistically independent, are illustrated with the simple example of the random walk with discrete time steps. The emphasis here is on locality, universality, continuum limit, path integral, Brownian motion, Gaussian distribution and scaling. These properties are first derived from an exact solution and then recovered by renormalization group (RG) methods. This makes it possible to introduce all the RG terminology.

This chapter describes how the arguments in the previous part, which worked on discrete systems, can be generalized to the continuum limit. After reviewing Hamilton’s formulation of classical mechanics and Poisson brackets, the attention is shifted from the Lagrangian to the Lagrangian density. The chapter uses the electromagnetic field as a first example of this approach.

There are many excellent reviews on random matrices both in mathematics and in physics literature. This chapter presents only a very quick overview of some selected topics in random matrix theory: the 1/N expansion, the continuum limit, the double scaling limit and the Schwinger–Dyson equations. The results listed in this chapter for random matrices will be recovered one by one for random tensors in subsequent chapters.

This chapter presents the perturbative expansion of invariant tensor measures in terms of Feynman graphs. It is shown that, assuming that the perturbation and the Gaussian part scale at the same rate with N, the moments of such measures admit (as formal power series in the coupling constants) a 1/N expansion indexed by the degree and that (still in the perturbative sense) all such measures are properly uniformly bounded. In the second part of the chapter the continuum limit of random tensor models and their Schwinger–Dyson equations are discussed.