## Renormalization group and scaling

*Hidetoshi Nishimori and Gerardo Ortiz*

### in Elements of Phase Transitions and Critical Phenomena

- 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 ... More

## Elements of Phase Transitions and Critical Phenomena

*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

## Implementation of the renormalization group

*Hidetoshi Nishimori and Gerardo Ortiz*

### in Elements of Phase Transitions and Critical Phenomena

- 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 ... More

## Introduction to Renormalization Theory. Renormalization Group Equations

*JEAN ZINN-JUSTIN*

### in Quantum Field Theory and Critical Phenomena

- 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 ... More

## The Devil in the Details: Asymptotic Reasoning in Explanation, Reduction, and Emergence

*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. ... More

## The Theory Of QCD

*GÜNTHER DISSERTORI, IAN G. KNOWLES, and MICHAEL SCHMELLING*

### in Quantum Chromodynamics: High Energy Experiments and Theory

- 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 ... More

## THE WEYL ANOMALY AND RENORMALIZATION GROUP

*Kazuo Fujikawa and Hiroshi Suzuki*

### in Path Integrals and Quantum Anomalies

- 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 ... More

## Dimensional Regularization, Minimal Subtraction: RG Functions

*JEAN ZINN-JUSTIN*

### in Quantum Field Theory and Critical Phenomena

- 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 ... More

## Renormalization group and problem of radiation

*Israel Michael Sigal*

### in Quantum Theory from Small to Large Scales: Lecture Notes of the Les Houches Summer School: Volume 95, August 2010

- 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 ... More

## Quantum field theory and the renormalization group

*Jean Zinn-Justin*

### in Phase Transitions and Renormalization Group

- 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 ... More

## Functional renormalization group

*Jean Zinn-Justin*

### in Phase Transitions and Renormalization Group

- 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 ... More

## Scales III: Short-distance structure of quantum field theory

*Anthony Duncan*

### in The Conceptual Framework of Quantum Field Theory

- 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 ... More

## Phase Transitions, Critical Phenomena, and Finite-Size Scaling

*Hans-Peter Eckle*

### in Models of Quantum Matter: A First Course on Integrability and the Bethe Ansatz

- Published in print:
- 2019
- Published Online:
- September 2019
- ISBN:
- 9780199678839
- eISBN:
- 9780191878589
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780199678839.003.0005
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics, Condensed Matter Physics / Materials

Interacting many-particle systems may undergo phase transitions and exhibit critical phenomena in the limit of infinite system size, while the precursors of these phenomena are studied in the theory ... More

## Renormalization Theory and Effective Field Theories

*Matthias Neubert*

### in Effective Field Theory in Particle Physics and Cosmology: Lecture Notes of the Les Houches Summer School: Volume 108, July 2017

- Published in print:
- 2020
- Published Online:
- August 2020
- ISBN:
- 9780198855743
- eISBN:
- 9780191889370
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198855743.003.0001
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

Chapter 1 features lectures that review the formalism of renormalization in quantum field theories with special regard to effective quantum field theories. While renormalization theory is part of ... More

## FLUCTUATION THERMODYNAMICS

*Anatoly Larkin and Andrei Varlamov*

### in Theory of Fluctuations in Superconductors

- 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 ... More

## Phase transitions and critical phenomena

*Hidetoshi Nishimori and Gerardo Ortiz*

### in Elements of Phase Transitions and Critical Phenomena

- 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, ... More

## The s 4 field theory near dimension 4

*Jean Zinn-Justin*

### in Phase Transitions and Renormalization Group

- 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 ... More

## General Renormalization Group. The Critical Theory Near Dimension Four

*JEAN ZINN-JUSTIN*

### in Quantum Field Theory and Critical Phenomena

- 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 ... More

## Asymptotic Explanation

*Robert W. Batterman*

### in The Devil in the Details: Asymptotic Reasoning in Explanation, Reduction, and Emergence

- 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 ... More

## Hierarchies, Complex Fractal Dimensions, and Log-Periodicity

*Didier Sornette*

### in Why Stock Markets Crash: Critical Events in Complex Financial Systems

- 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 ... More