Hidetoshi Nishimori and Gerardo Ortiz
- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780199577224
- eISBN:
- 9780191722943
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199577224.003.0003
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Mean-field theory is usually taken as a first step toward understanding critical phenomena, providing an overview that reveals qualitative behaviour of physical quantities. However, it is necessary ...
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Mean-field theory is usually taken as a first step toward understanding critical phenomena, providing an overview that reveals qualitative behaviour of physical quantities. However, it is necessary to proceed beyond the mean-field theory to better understand the situation, both qualitatively and quantitatively, when fluctuations play vital roles leading to exponents that cannot be explained by dimensional analysis, thus introducing anomalous dimensions. The present chapter explains the basic concepts of the renormalization group and scaling theory, which allow us to analyze critical phenomena with fluctuations systematically taken into account. The essential step in a renormalization group calculation consists of establishing recursion relations between the parameters defining the Hamiltonian of the system. These recursion or renormalization group equations define a flow with well-defined fixed points. Details other than the values of the relevant operators have no influence on the critical exponents and this represents universality.Less
Mean-field theory is usually taken as a first step toward understanding critical phenomena, providing an overview that reveals qualitative behaviour of physical quantities. However, it is necessary to proceed beyond the mean-field theory to better understand the situation, both qualitatively and quantitatively, when fluctuations play vital roles leading to exponents that cannot be explained by dimensional analysis, thus introducing anomalous dimensions. The present chapter explains the basic concepts of the renormalization group and scaling theory, which allow us to analyze critical phenomena with fluctuations systematically taken into account. The essential step in a renormalization group calculation consists of establishing recursion relations between the parameters defining the Hamiltonian of the system. These recursion or renormalization group equations define a flow with well-defined fixed points. Details other than the values of the relevant operators have no influence on the critical exponents and this represents universality.
Hidetoshi Nishimori and Gerardo Ortiz
- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780199577224
- eISBN:
- 9780191722943
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199577224.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Phase transitions and critical phenomena have consistently been among the principal subjects of active studies in statistical physics. The simple act of transforming one state of matter or phase into ...
More
Phase transitions and critical phenomena have consistently been among the principal subjects of active studies in statistical physics. The simple act of transforming one state of matter or phase into another, for instance by changing the temperature, has always captivated the curious mind. This book provides an introductory account on the theory of phase transitions and critical phenomena, a subject now recognized to be indispensable for students and researchers from many fields of physics and related disciplines. The first five chapters are very basic and quintessential, and cover standard topics such as mean-field theories, the renormalization group and scaling, universality, and statistical field theory methods. The remaining chapters develop more advanced concepts, including conformal field theory, the Kosterlitz-Thouless transition, the effects of randomness, percolation, exactly solvable models, series expansions, duality transformations, and numerical techniques. Moreover, a comprehensive series of appendices expand and clarify several issues not developed in the main text. The important role played by symmetry and topology in understanding the competition between phases and the resulting emergent collective behaviour, giving rise to rigidity and soft elementary excitations, is stressed throughout the book. Serious attempts have been directed toward a self-contained modular approach so that the reader does not have to refer to other sources for supplementary information. Accordingly, most of the concepts and calculations are described in detail, sometimes with additional/auxiliary descriptions given in appendices and exercises. The latter are presented as the topics develop with solutions found at the end of the book, thus giving the text a self-learning character.Less
Phase transitions and critical phenomena have consistently been among the principal subjects of active studies in statistical physics. The simple act of transforming one state of matter or phase into another, for instance by changing the temperature, has always captivated the curious mind. This book provides an introductory account on the theory of phase transitions and critical phenomena, a subject now recognized to be indispensable for students and researchers from many fields of physics and related disciplines. The first five chapters are very basic and quintessential, and cover standard topics such as mean-field theories, the renormalization group and scaling, universality, and statistical field theory methods. The remaining chapters develop more advanced concepts, including conformal field theory, the Kosterlitz-Thouless transition, the effects of randomness, percolation, exactly solvable models, series expansions, duality transformations, and numerical techniques. Moreover, a comprehensive series of appendices expand and clarify several issues not developed in the main text. The important role played by symmetry and topology in understanding the competition between phases and the resulting emergent collective behaviour, giving rise to rigidity and soft elementary excitations, is stressed throughout the book. Serious attempts have been directed toward a self-contained modular approach so that the reader does not have to refer to other sources for supplementary information. Accordingly, most of the concepts and calculations are described in detail, sometimes with additional/auxiliary descriptions given in appendices and exercises. The latter are presented as the topics develop with solutions found at the end of the book, thus giving the text a self-learning character.
Hidetoshi Nishimori and Gerardo Ortiz
- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780199577224
- eISBN:
- 9780191722943
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199577224.003.0004
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Actual computations of fixed points and eigenvalues usually involve approximations, often crude ones, except for a very limited number of simple cases such as the one-dimensional Ising model of the ...
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Actual computations of fixed points and eigenvalues usually involve approximations, often crude ones, except for a very limited number of simple cases such as the one-dimensional Ising model of the previous chapter. In real- and momentum-space renormalization group theory, there are no general prescriptions to systematically improve the degree of the approximation with a modest amount of effort. There are established methods to systematically improve precision, but they usually need a large amount of numerical calculations. The scope of the present chapter is modest as we limit ourselves to basic examples, including the epsilon expansion about the Gaussian fixed-point of the Landau-Ginzburg-Wilson model. Finally, the last section illustrates the extension of the renormalization group framework to study quantum phase transitions.Less
Actual computations of fixed points and eigenvalues usually involve approximations, often crude ones, except for a very limited number of simple cases such as the one-dimensional Ising model of the previous chapter. In real- and momentum-space renormalization group theory, there are no general prescriptions to systematically improve the degree of the approximation with a modest amount of effort. There are established methods to systematically improve precision, but they usually need a large amount of numerical calculations. The scope of the present chapter is modest as we limit ourselves to basic examples, including the epsilon expansion about the Gaussian fixed-point of the Landau-Ginzburg-Wilson model. Finally, the last section illustrates the extension of the renormalization group framework to study quantum phase transitions.
JEAN ZINN-JUSTIN
- Published in print:
- 2002
- Published Online:
- January 2010
- ISBN:
- 9780198509233
- eISBN:
- 9780191708732
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509233.003.0010
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter presents the essential steps of the proof of the renormalizability of a simple scalar field theory: the φ4 field theory in d = 4 dimensions. However, all the fundamental difficulties of ...
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This chapter presents the essential steps of the proof of the renormalizability of a simple scalar field theory: the φ4 field theory in d = 4 dimensions. However, all the fundamental difficulties of renormalization theory are already present in this particular example and it will eventually become clear that the extension to other theories is not difficult. The elegant presentation of Callan (Les Houches 1975) is followed, which allows renormalizability and renormalization group (Callan–Symanzik) equations to be proved at once. This presentation is specially suited to the chapter's general purpose since a large part of this work is devoted to applications of renormalization group (RG). Moreover, it emphasizes already at this technical level the equivalence between renormalizability and the existence of a renormalization.Less
This chapter presents the essential steps of the proof of the renormalizability of a simple scalar field theory: the φ4 field theory in d = 4 dimensions. However, all the fundamental difficulties of renormalization theory are already present in this particular example and it will eventually become clear that the extension to other theories is not difficult. The elegant presentation of Callan (Les Houches 1975) is followed, which allows renormalizability and renormalization group (Callan–Symanzik) equations to be proved at once. This presentation is specially suited to the chapter's general purpose since a large part of this work is devoted to applications of renormalization group (RG). Moreover, it emphasizes already at this technical level the equivalence between renormalizability and the existence of a renormalization.
Robert W. Batterman
- Published in print:
- 2001
- Published Online:
- November 2003
- ISBN:
- 9780195146479
- eISBN:
- 9780199833078
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0195146476.001.0001
- Subject:
- Philosophy, Philosophy of Science
This book focuses on a form of reasoning in science that I call “asymptotic reasoning.” At base, this type of reasoning involves methods that eliminate details and, in some sense, precision. ...
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This book focuses on a form of reasoning in science that I call “asymptotic reasoning.” At base, this type of reasoning involves methods that eliminate details and, in some sense, precision. Asymptotic reasoning has received systematic treatment in physics and applied mathematics, but virtually no attention has been paid to it by philosophers of science. I argue that once one understands the role played by asymptotic reasoning in explanatory arguments of scientists, our philosophical conceptions of explanation, reduction, and emergence require significant modification.Less
This book focuses on a form of reasoning in science that I call “asymptotic reasoning.” At base, this type of reasoning involves methods that eliminate details and, in some sense, precision. Asymptotic reasoning has received systematic treatment in physics and applied mathematics, but virtually no attention has been paid to it by philosophers of science. I argue that once one understands the role played by asymptotic reasoning in explanatory arguments of scientists, our philosophical conceptions of explanation, reduction, and emergence require significant modification.
GÜNTHER DISSERTORI, IAN G. KNOWLES, and MICHAEL SCHMELLING
- Published in print:
- 2009
- Published Online:
- January 2010
- ISBN:
- 9780199566419
- eISBN:
- 9780191708060
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199566419.003.0003
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter begins with the formal derivation of QCD as a gauge theory based on the symmetry group SU(3). It then describes the basic reactions of lepton-lepton, lepton-hadron, and hadron-hadron ...
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This chapter begins with the formal derivation of QCD as a gauge theory based on the symmetry group SU(3). It then describes the basic reactions of lepton-lepton, lepton-hadron, and hadron-hadron scattering in the language of QCD, and derives the born-level expressions for the basic QCD cross-sections. The chapter addresses the issue of ultraviolet divergences and renormalization. It discusses the renormalization group equations and the issue of infrared-safety in the context of the QCD improved parton model. The treatment of soft gluons then leads naturally to the various hadronization models, which form the basis of the Monte Carlo models discussed in the following chapter.Less
This chapter begins with the formal derivation of QCD as a gauge theory based on the symmetry group SU(3). It then describes the basic reactions of lepton-lepton, lepton-hadron, and hadron-hadron scattering in the language of QCD, and derives the born-level expressions for the basic QCD cross-sections. The chapter addresses the issue of ultraviolet divergences and renormalization. It discusses the renormalization group equations and the issue of infrared-safety in the context of the QCD improved parton model. The treatment of soft gluons then leads naturally to the various hadronization models, which form the basis of the Monte Carlo models discussed in the following chapter.
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/9780198529132.003.0007
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter discusses the quantum anomaly associated with the scale transformation of space-time coordinates, or the transformation generally called the Weyl transformation. In flat space-time, this ...
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This chapter discusses the quantum anomaly associated with the scale transformation of space-time coordinates, or the transformation generally called the Weyl transformation. In flat space-time, this anomaly is related to the renormalization group and the calculation of the $β$ function in the renormalization group equation is related to the calculation of the Weyl anomaly. In other words, the renormalization group equation is regarded as an expression of the Weyl anomaly in terms of Green’s functions. The calculation of the one-loop functions in QED and QCD by means of the Jacobians for the Weyl symmetry is illustrated. The Weyl anomalies in curved space-time are briefly explained. An improved finite energy-momentum tensor in renormalizable theory is also mentioned based on an analysis of the Weyl anomaly.Less
This chapter discusses the quantum anomaly associated with the scale transformation of space-time coordinates, or the transformation generally called the Weyl transformation. In flat space-time, this anomaly is related to the renormalization group and the calculation of the $β$ function in the renormalization group equation is related to the calculation of the Weyl anomaly. In other words, the renormalization group equation is regarded as an expression of the Weyl anomaly in terms of Green’s functions. The calculation of the one-loop functions in QED and QCD by means of the Jacobians for the Weyl symmetry is illustrated. The Weyl anomalies in curved space-time are briefly explained. An improved finite energy-momentum tensor in renormalizable theory is also mentioned based on an analysis of the Weyl anomaly.
JEAN ZINN-JUSTIN
- Published in print:
- 2002
- Published Online:
- January 2010
- ISBN:
- 9780198509233
- eISBN:
- 9780191708732
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509233.003.0011
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter introduces the concept of renormalization by minimal subtraction, within the framework of dimensional regularization. It first discusses the structure of renormalization constants and ...
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This chapter introduces the concept of renormalization by minimal subtraction, within the framework of dimensional regularization. It first discusses the structure of renormalization constants and renormalization group functions β(g), η(g), η2(g), and then shows that it is specially simple in the minimal subtraction scheme. It performs explicit calculations at two-loop order first in the simple one-component φ4 field and then in an N-component field theory with a general four-field interaction. These calculations will be useful for the theory of Critical Phenomena. Finally, to give an example of a theory involving fermions, the renormalization group (RG) functions are calculated at one-loop order in a theory containing fermions interacting through a Yukawa-like interaction with a scalar boson: the Gross–Neveu–Yukawa model.Less
This chapter introduces the concept of renormalization by minimal subtraction, within the framework of dimensional regularization. It first discusses the structure of renormalization constants and renormalization group functions β(g), η(g), η2(g), and then shows that it is specially simple in the minimal subtraction scheme. It performs explicit calculations at two-loop order first in the simple one-component φ4 field and then in an N-component field theory with a general four-field interaction. These calculations will be useful for the theory of Critical Phenomena. Finally, to give an example of a theory involving fermions, the renormalization group (RG) functions are calculated at one-loop order in a theory containing fermions interacting through a Yukawa-like interaction with a scalar boson: the Gross–Neveu–Yukawa model.
Israel Michael Sigal
- Published in print:
- 2012
- Published Online:
- September 2012
- ISBN:
- 9780199652495
- eISBN:
- 9780191741203
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199652495.003.0012
- Subject:
- Physics, Atomic, Laser, and Optical Physics
The standard model of non-relativistic quantum electrodynamics describes non-relativistic quantum matter, such as atoms and molecules, coupled to the quantized electromagnetic field. Within this ...
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The standard model of non-relativistic quantum electrodynamics describes non-relativistic quantum matter, such as atoms and molecules, coupled to the quantized electromagnetic field. Within this model, this chapter reviews basic notions, results, and techniques in theory radiation. It describes the key technique in this area — the spectral renormalization group. The review is based on joint works with Volker Bach and Jürg Fröhlich and with Walid Abou Salem, Thomas Chen, Jérémy Faupin, and Marcel Griesemer. A brief discussion of related contributions is given at the end of these lectures.Less
The standard model of non-relativistic quantum electrodynamics describes non-relativistic quantum matter, such as atoms and molecules, coupled to the quantized electromagnetic field. Within this model, this chapter reviews basic notions, results, and techniques in theory radiation. It describes the key technique in this area — the spectral renormalization group. The review is based on joint works with Volker Bach and Jürg Fröhlich and with Walid Abou Salem, Thomas Chen, Jérémy Faupin, and Marcel Griesemer. A brief discussion of related contributions is given at the end of these lectures.
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/9780199227198.003.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter presents a brief history of the origin and development of quantum field theory, and of the evolution of the interpretation of renormalization and the renormalization group which has led ...
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This chapter presents a brief history of the origin and development of quantum field theory, and of the evolution of the interpretation of renormalization and the renormalization group which has led to our present understanding. This history has two aspects: one directly related to the theory of fundamental interactions that describes physics at the microscopic scale; and another one related to the theory of phase transitions in macroscopic physics and their universal properties. That two so vastly different domains of physics have required the development of the same theoretical framework is extremely surprising. It is one of the attractions of theoretical physics that such relations can sometimes be found.Less
This chapter presents a brief history of the origin and development of quantum field theory, and of the evolution of the interpretation of renormalization and the renormalization group which has led to our present understanding. This history has two aspects: one directly related to the theory of fundamental interactions that describes physics at the microscopic scale; and another one related to the theory of phase transitions in macroscopic physics and their universal properties. That two so vastly different domains of physics have required the development of the same theoretical framework is extremely surprising. It is one of the attractions of theoretical physics that such relations can sometimes be found.
Anthony Duncan
- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199573264
- eISBN:
- 9780191743313
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199573264.003.0018
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter discusses one of the most fertile manifestations of scale separation in field theory: the Wilson operator product expansion (OPE), which provides a precise characterization of the ...
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This chapter discusses one of the most fertile manifestations of scale separation in field theory: the Wilson operator product expansion (OPE), which provides a precise characterization of the short-distance asymptotics of field theory amplitudes in terms of factorized products of ‘short-’ and ‘long-’distance terms. The useful application of the OPE in particular processes depends on the presence and structure of mass singularities in the relevant amplitudes. The role of the renormalization group in studying high-energy (or short-distance) behaviour is outlined.Less
This chapter discusses one of the most fertile manifestations of scale separation in field theory: the Wilson operator product expansion (OPE), which provides a precise characterization of the short-distance asymptotics of field theory amplitudes in terms of factorized products of ‘short-’ and ‘long-’distance terms. The useful application of the OPE in particular processes depends on the presence and structure of mass singularities in the relevant amplitudes. The role of the renormalization group in studying high-energy (or short-distance) behaviour is outlined.
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/9780199227198.003.0016
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter describes a general approach to the renormalization group (RG) close to ideas initially developed by Wegner and Wilson, and based on a partial integration over the large-momentum modes ...
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This chapter describes a general approach to the renormalization group (RG) close to ideas initially developed by Wegner and Wilson, and based on a partial integration over the large-momentum modes of fields. This RG takes the form of functional renormalization group (FRG) equations that express the equivalence between a change of a scale parameter related to microscopic physics and a change of the parameters of the Hamiltonian. Some forms of these renormalization group equations (RGE) are exact and one then also speaks of the exact renormalization group.Less
This chapter describes a general approach to the renormalization group (RG) close to ideas initially developed by Wegner and Wilson, and based on a partial integration over the large-momentum modes of fields. This RG takes the form of functional renormalization group (FRG) equations that express the equivalence between a change of a scale parameter related to microscopic physics and a change of the parameters of the Hamiltonian. Some forms of these renormalization group equations (RGE) are exact and one then also speaks of the exact renormalization group.
Anatoly Larkin and Andrei Varlamov
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198528159
- eISBN:
- 9780191713521
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528159.003.0002
- Subject:
- Physics, Condensed Matter Physics / Materials
This chapter introduces the Ginzburg-Landau functional and provides a general description of fluctuation thermodynamics within the framework of the functional integration over the fluctuation fields ...
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This chapter introduces the Ginzburg-Landau functional and provides a general description of fluctuation thermodynamics within the framework of the functional integration over the fluctuation fields approach. The method, in its harmonic approximation, is applied to the effect of fluctuations on heat capacity and magnetization of a superconductor at the critical temperature. The criterion of its validity (Ginzburg-Levanyuk criterion) is derived. An extension of the theory on close vicinity of transition is demonstrated within the framework of the renormalization group approach. The shift in critical temperature by fluctuations of the order parameter and electromagnetic fields is obtained.Less
This chapter introduces the Ginzburg-Landau functional and provides a general description of fluctuation thermodynamics within the framework of the functional integration over the fluctuation fields approach. The method, in its harmonic approximation, is applied to the effect of fluctuations on heat capacity and magnetization of a superconductor at the critical temperature. The criterion of its validity (Ginzburg-Levanyuk criterion) is derived. An extension of the theory on close vicinity of transition is demonstrated within the framework of the renormalization group approach. The shift in critical temperature by fluctuations of the order parameter and electromagnetic fields is obtained.
Hidetoshi Nishimori and Gerardo Ortiz
- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780199577224
- eISBN:
- 9780191722943
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199577224.003.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
As an introduction to the physics of phase transitions and critical phenomena, this chapter explains a number of basic and fundamental ideas such as phases, phase transitions, phase diagrams, ...
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As an introduction to the physics of phase transitions and critical phenomena, this chapter explains a number of basic and fundamental ideas such as phases, phase transitions, phase diagrams, universality, and critical phenomena. Especially important is the concept of order parameter, a quantity that measures the degree of asymmetry in the broken symmetry phase. Intuitive accounts are given to the concepts of coarse-graining, and scale and renormalization group transformations, which are powerful, systematic tools to analyze critical behaviour of macroscopic systems. Also explained are several spin and lattice gas model systems, on the basis of which phase transitions and critical phenomena will be studied.Less
As an introduction to the physics of phase transitions and critical phenomena, this chapter explains a number of basic and fundamental ideas such as phases, phase transitions, phase diagrams, universality, and critical phenomena. Especially important is the concept of order parameter, a quantity that measures the degree of asymmetry in the broken symmetry phase. Intuitive accounts are given to the concepts of coarse-graining, and scale and renormalization group transformations, which are powerful, systematic tools to analyze critical behaviour of macroscopic systems. Also explained are several spin and lattice gas model systems, on the basis of which phase transitions and critical phenomena will be studied.
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/9780199227198.003.0013
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter explains how the asymptotic renormalization group equations (RGE), introduced without too much justification in Section 10.6, can be proved within the framework of statistical (or ...
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This chapter explains how the asymptotic renormalization group equations (RGE), introduced without too much justification in Section 10.6, can be proved within the framework of statistical (or quantum) field theory. The proof is based on the methods of perturbative statistical field theory introduced in Chapter 12, and a few assumptions that it is thus possible to clarify. The discussion is restricted to theories with an Ising type symmetry and the field s has only one component. Generalization to models with N-component fields and O (N) symmetry is simple.Less
This chapter explains how the asymptotic renormalization group equations (RGE), introduced without too much justification in Section 10.6, can be proved within the framework of statistical (or quantum) field theory. The proof is based on the methods of perturbative statistical field theory introduced in Chapter 12, and a few assumptions that it is thus possible to clarify. The discussion is restricted to theories with an Ising type symmetry and the field s has only one component. Generalization to models with N-component fields and O (N) symmetry is simple.
JEAN ZINN-JUSTIN
- Published in print:
- 2002
- Published Online:
- January 2010
- ISBN:
- 9780198509233
- eISBN:
- 9780191708732
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509233.003.0025
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
The RG theory, as applied to Critical Phenomena, has been developed by Kadano, Wilson, Wegner, and many others. This chapter first describes the basic renormalization group ideas in a somewhat ...
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The RG theory, as applied to Critical Phenomena, has been developed by Kadano, Wilson, Wegner, and many others. This chapter first describes the basic renormalization group ideas in a somewhat abstract and intuitive framework. The formulation will lack precision and the arguments will be largely heuristic. The importance of fixed points in hamiltonian space will be stressed. The special role of gaussian models and their universal properties will be related to the existence of fixed point, the gaussian fixed point. It is shown that the RG equations which appear as a consequence of the necessity of renormalization of local field theories are directly connected with the abstract RG equations introduced in Section 25.1. Universality in the theory of critical phenomena is thus directly related to the property that local field theories are insensitive to the short distance structure, and physics can, therefore, be described by renormalized correlation functions. Conversely, in the statistical sense, QFTs are always close to criticality and their existence, beyond perturbation theory, relies, from the abstract RG point of view, on the presence of IR fixed points in hamiltonian space.Less
The RG theory, as applied to Critical Phenomena, has been developed by Kadano, Wilson, Wegner, and many others. This chapter first describes the basic renormalization group ideas in a somewhat abstract and intuitive framework. The formulation will lack precision and the arguments will be largely heuristic. The importance of fixed points in hamiltonian space will be stressed. The special role of gaussian models and their universal properties will be related to the existence of fixed point, the gaussian fixed point. It is shown that the RG equations which appear as a consequence of the necessity of renormalization of local field theories are directly connected with the abstract RG equations introduced in Section 25.1. Universality in the theory of critical phenomena is thus directly related to the property that local field theories are insensitive to the short distance structure, and physics can, therefore, be described by renormalized correlation functions. Conversely, in the statistical sense, QFTs are always close to criticality and their existence, beyond perturbation theory, relies, from the abstract RG point of view, on the presence of IR fixed points in hamiltonian space.
Robert W. Batterman
- Published in print:
- 2001
- Published Online:
- November 2003
- ISBN:
- 9780195146479
- eISBN:
- 9780199833078
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0195146476.003.0004
- Subject:
- Philosophy, Philosophy of Science
This chapter provides a fairly detailed discussion of the renormalization group account of the universality of critical phenomena. This discussion allows one to determine the distinctive features of ...
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This chapter provides a fairly detailed discussion of the renormalization group account of the universality of critical phenomena. This discussion allows one to determine the distinctive features of asymptotic explanation in general. Two other, superficially quite different, explanatory accounts involving “intermediate asymptotics” are then discussed. It is argued that these different examples exhibit the same general asymptotic explanatory strategy – one that is ubiquitous in physics and applied mathematics. The chapter concludes with a discussion of the importance of stability considerations in the asymptotic explanations.Less
This chapter provides a fairly detailed discussion of the renormalization group account of the universality of critical phenomena. This discussion allows one to determine the distinctive features of asymptotic explanation in general. Two other, superficially quite different, explanatory accounts involving “intermediate asymptotics” are then discussed. It is argued that these different examples exhibit the same general asymptotic explanatory strategy – one that is ubiquitous in physics and applied mathematics. The chapter concludes with a discussion of the importance of stability considerations in the asymptotic explanations.
Didier Sornette
- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691175959
- eISBN:
- 9781400885091
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175959.003.0006
- Subject:
- Business and Management, Finance, Accounting, and Banking
This chapter describes the concept of fractals and their self-similarity, including fractals with complex dimensions. It shows how these geometric and mathematical objects enable one to codify the ...
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This chapter describes the concept of fractals and their self-similarity, including fractals with complex dimensions. It shows how these geometric and mathematical objects enable one to codify the information contained in the precursory patterns before large stock market crashes. The chapter first considers how models of cooperative behaviors resulting from imitation between agents organized within a hierarchical structure exhibit the announced critical phenomena decorated with “log-periodicity.” It then examines the underlying hierarchical structure of social networks, critical behavior in hierarchical networks, a hierarchical model of financial bubbles, and discrete scale invariance. It also discusses a technique, called the “renormalization group,” and a simple model exhibiting a finite-time singularity due to a positive feedback induced by trend following investment strategies. Finally, it looks at scenarios leading to discrete scale invariance and log-periodicity.Less
This chapter describes the concept of fractals and their self-similarity, including fractals with complex dimensions. It shows how these geometric and mathematical objects enable one to codify the information contained in the precursory patterns before large stock market crashes. The chapter first considers how models of cooperative behaviors resulting from imitation between agents organized within a hierarchical structure exhibit the announced critical phenomena decorated with “log-periodicity.” It then examines the underlying hierarchical structure of social networks, critical behavior in hierarchical networks, a hierarchical model of financial bubbles, and discrete scale invariance. It also discusses a technique, called the “renormalization group,” and a simple model exhibiting a finite-time singularity due to a positive feedback induced by trend following investment strategies. Finally, it looks at scenarios leading to discrete scale invariance and log-periodicity.
JAGDISH MEHRA and KIMBALL A. MILTON
- Published in print:
- 2003
- Published Online:
- February 2010
- ISBN:
- 9780198527459
- eISBN:
- 9780191709593
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198527459.003.0014
- Subject:
- Physics, History of Physics
Julian Schwinger left Harvard University in 1971 to move to the University of California at Los Angeles (UCLA), perhaps because he perceived that his source theory had received a chilly reception at ...
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Julian Schwinger left Harvard University in 1971 to move to the University of California at Los Angeles (UCLA), perhaps because he perceived that his source theory had received a chilly reception at Harvard, and thought (rather erroneously as it turned out) that UCLA, where he intended to go, would be more hospitable. Schwinger wished his audiences to rid themselves of all the machinery of operator quantum field theory that they had so laboriously acquired. Less comprehensible was the outright hostility expressed in many quarters. But probably at least as important for his relocation was the fact he had been at Harvard for 25 years, and felt the need for a change. This chapter looks at Schwinger's renewed interest in strong-field electrodynamics while at UCLA, the discovery of an extremely narrow spin-1 resonance coupling to the photon by researchers at the Stanford Linear Accelerator Center and at Brookhaven National Laboratory, Schwinger's two papers on renormalisation group without renormalisation group, deep inelastic scattering and Schwinger's reaction to partons and quarks, source theory and general relativity, magnetic charge and dyons, and supersymmetry.Less
Julian Schwinger left Harvard University in 1971 to move to the University of California at Los Angeles (UCLA), perhaps because he perceived that his source theory had received a chilly reception at Harvard, and thought (rather erroneously as it turned out) that UCLA, where he intended to go, would be more hospitable. Schwinger wished his audiences to rid themselves of all the machinery of operator quantum field theory that they had so laboriously acquired. Less comprehensible was the outright hostility expressed in many quarters. But probably at least as important for his relocation was the fact he had been at Harvard for 25 years, and felt the need for a change. This chapter looks at Schwinger's renewed interest in strong-field electrodynamics while at UCLA, the discovery of an extremely narrow spin-1 resonance coupling to the photon by researchers at the Stanford Linear Accelerator Center and at Brookhaven National Laboratory, Schwinger's two papers on renormalisation group without renormalisation group, deep inelastic scattering and Schwinger's reaction to partons and quarks, source theory and general relativity, magnetic charge and dyons, and supersymmetry.
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/9780199227198.003.0009
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter introduces the general concept of renormalization group in the spirit of the work. It studies the role of fixed points and their stability properties. It exhibits a particular fixed ...
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This chapter introduces the general concept of renormalization group in the spirit of the work. It studies the role of fixed points and their stability properties. It exhibits a particular fixed point, the Gaussian fixed point, which is stable in dimension larger than four. It identifies the leading perturbation to the Gaussian fixed point in dimension = four. It discusses the possible existence of a non-Gaussian fixed point near dimension four.Less
This chapter introduces the general concept of renormalization group in the spirit of the work. It studies the role of fixed points and their stability properties. It exhibits a particular fixed point, the Gaussian fixed point, which is stable in dimension larger than four. It identifies the leading perturbation to the Gaussian fixed point in dimension = four. It discusses the possible existence of a non-Gaussian fixed point near dimension four.
