Kazuo Fujikawa and Hiroshi Suzuki
- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198529132
- eISBN:
- 9780191712821
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198529132.003.0009
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
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 ...
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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.Less
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.
Jean Zinn-Justin
- Published in print:
- 2007
- Published Online:
- January 2010
- ISBN:
- 9780199227198
- eISBN:
- 9780191711107
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199227198.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This book provides an elementary introduction to the notions of continuum limit and universality in statistical systems with a large number of degrees of freedom. The existence of a continuum limit ...
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This book provides an elementary introduction to the notions of continuum limit and universality in statistical systems with a large number of degrees of freedom. The existence of a continuum limit requires the appearance of correlations at large distance, a situation that is encountered in second order phase transitions, near the critical temperature. In this context, the book emphasizes the role of gaussian distributions and their relations with the mean field approximation and Landau′s theory of critical phenomena. The book shows that quasi-gaussian or mean-field approximations cannot describe correctly phase transitions in three space dimensions. The book assigns this difficulty to the coupling of very different physical length scales, even though the systems we will consider have only local, that is, short range, interactions. To analyze the unusual situation, a new concept is required: the renormalization group, whose fixed points allow understanding the universality of physical properties at large distance, beyond mean-field theory. In the continuum limit, critical phenomena can be described by quantum field theories. In this framework, the renormalization group is directly related to the renormalization process; that is, the necessity to cancel the infinities that arise in straightforward formulations of the theory. The book discusses the renormalization group in the context of various relevant field theories. This leads to proofs of universality and to efficient tools for calculating universal quantities in a perturbative framework. Finally, the book constructs a general functional renormalization group, which can be used when perturbative methods are inadequate.Less
This book provides an elementary introduction to the notions of continuum limit and universality in statistical systems with a large number of degrees of freedom. The existence of a continuum limit requires the appearance of correlations at large distance, a situation that is encountered in second order phase transitions, near the critical temperature. In this context, the book emphasizes the role of gaussian distributions and their relations with the mean field approximation and Landau′s theory of critical phenomena. The book shows that quasi-gaussian or mean-field approximations cannot describe correctly phase transitions in three space dimensions. The book assigns this difficulty to the coupling of very different physical length scales, even though the systems we will consider have only local, that is, short range, interactions. To analyze the unusual situation, a new concept is required: the renormalization group, whose fixed points allow understanding the universality of physical properties at large distance, beyond mean-field theory. In the continuum limit, critical phenomena can be described by quantum field theories. In this framework, the renormalization group is directly related to the renormalization process; that is, the necessity to cancel the infinities that arise in straightforward formulations of the theory. The book discusses the renormalization group in the context of various relevant field theories. This leads to proofs of universality and to efficient tools for calculating universal quantities in a perturbative framework. Finally, the book constructs a general functional renormalization group, which can be used when perturbative methods are inadequate.
Timo Seppäläinen
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199239252
- eISBN:
- 9780191716911
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199239252.003.0001
- Subject:
- Mathematics, Probability / Statistics, Analysis
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 ...
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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.Less
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.
Jean Zinn-Justin
- Published in print:
- 2007
- Published Online:
- January 2010
- ISBN:
- 9780199227198
- eISBN:
- 9780191711107
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199227198.003.0004
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
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 ...
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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.Less
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.
Jean Zinn-Justin
- Published in print:
- 2007
- Published Online:
- January 2010
- ISBN:
- 9780199227198
- eISBN:
- 9780191711107
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199227198.003.0003
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
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 ...
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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.Less
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.
Jean Zinn-Justin
- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198566748
- eISBN:
- 9780191717994
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566748.003.0004
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics
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 ...
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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.Less
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.
Jean Zinn-Justin
- Published in print:
- 2007
- Published Online:
- January 2010
- ISBN:
- 9780199227198
- eISBN:
- 9780191711107
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199227198.003.0005
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
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 ...
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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.Less
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.
Efstratios Manousakis
- Published in print:
- 2015
- Published Online:
- December 2015
- ISBN:
- 9780198749349
- eISBN:
- 9780191813474
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198749349.003.0003
- Subject:
- Physics, Atomic, Laser, and Optical Physics
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 ...
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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.Less
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.
Jean Zinn-Justin
- Published in print:
- 2019
- Published Online:
- August 2019
- ISBN:
- 9780198787754
- eISBN:
- 9780191829840
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198787754.003.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
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 ...
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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.Less
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.
Jean Zinn-Justin
- Published in print:
- 2019
- Published Online:
- August 2019
- ISBN:
- 9780198787754
- eISBN:
- 9780191829840
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198787754.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Theoretical physics is a cornerstone of modern physics and provides a foundation for all modern quantitative science. It aims to describe all natural phenomena using mathematical theories and models ...
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Theoretical physics is a cornerstone of modern physics and provides a foundation for all modern quantitative science. It aims to describe all natural phenomena using mathematical theories and models and, in consequence, develops our understanding of the fundamental nature of the universe. This book offers an overview of major areas covering the recent developments in modern theoretical physics. Each chapter introduces a new key topic, and develops the discussion in a self-contained manner. At the same time, the selected topics have common themes running throughout the book, which connect the independent discussions. The main themes—renormalization group, fixed points, universality and continuum limit—open and conclude the work. Other important and related themes are path integrals and field integrals, effective field theories, gauge theories, the mathematical structure at the basis of the interactions in fundamental particle physics, including quantization problems and anomalies, stochastic dynamical equations and summation of perturbative series.Less
Theoretical physics is a cornerstone of modern physics and provides a foundation for all modern quantitative science. It aims to describe all natural phenomena using mathematical theories and models and, in consequence, develops our understanding of the fundamental nature of the universe. This book offers an overview of major areas covering the recent developments in modern theoretical physics. Each chapter introduces a new key topic, and develops the discussion in a self-contained manner. At the same time, the selected topics have common themes running throughout the book, which connect the independent discussions. The main themes—renormalization group, fixed points, universality and continuum limit—open and conclude the work. Other important and related themes are path integrals and field integrals, effective field theories, gauge theories, the mathematical structure at the basis of the interactions in fundamental particle physics, including quantization problems and anomalies, stochastic dynamical equations and summation of perturbative series.
Razvan Gurau
- Published in print:
- 2016
- Published Online:
- January 2017
- ISBN:
- 9780198787938
- eISBN:
- 9780191829918
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198787938.003.0007
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics, Particle Physics / Astrophysics / Cosmology
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: ...
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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.Less
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.
Tom Lancaster and Stephen J. Blundell
- Published in print:
- 2014
- Published Online:
- June 2014
- ISBN:
- 9780199699322
- eISBN:
- 9780191779435
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199699322.003.0006
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
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 ...
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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.Less
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.
Razvan Gurau
- Published in print:
- 2016
- Published Online:
- January 2017
- ISBN:
- 9780198787938
- eISBN:
- 9780191829918
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198787938.003.0008
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics, Particle Physics / Astrophysics / Cosmology
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 ...
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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.Less
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.