Phase Transitions and Renormalization Group: (Oxford Graduate Texts)

This work tries to provide 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, we will emphasize the role of gaussian distributions and their relations with the mean field approximation and Landau's theory of critical phenomena. We will show that quasi-gaussian or mean-field approximations cannot describe correctly phase transitions in three space dimensions. We will assign 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. We thus discuss 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, we construct a general functional renormalization group, which can be used when perturbative methods are inadequate.

Table of Contents:
1: Quantum Field Theory and Renormalization Group 2: Gaussian Expectation Values. Steepest Descent Method . 3: Universality and Continuum Limit 4: Classical Statistical Physics: One Dimension 5: Continuum Limit and Path Integral 6: Ferromagnetic Systems. Correlations 7: Phase transitions: Generalities and Examples 8: Quasi-Gaussian Approximation: Universality, Critical Dimension 9: Renormalization Group: General Formulation 10: Perturbative Renormalization Group: Explicit Calculations 11: Renormalization group: N-component fields 12: Statistical Field Theory: Perturbative Expansion 13: The sigma4 Field Theory near Dimension 4 14: The O(N) Symmetric (phi2)2 Field Theory: Large N Limit 15: The Non-Linear sigma-Model 16: Functional Renormalization Group Appendix 1: Quantum Field Theory and Renormalization Group 2: Gaussian Expectation Values. Steepest Descent Method . 3: Universality and Continuum Limit 4: Classical Statistical Physics: One Dimension 5: Continuum Limit and Path Integral 6: Ferromagnetic Systems. Correlations 7: Phase transitions: Generalities and Examples 8: Quasi-Gaussian Approximation: Universality, Critical Dimension 9: Renormalization Group: General Formulation 10: Perturbative Renormalization Group: Explicit Calculations 11: Renormalization group: N-component fields 12: Statistical Field Theory: Perturbative Expansion 13: The sigma4 Field Theory near Dimension 4 14: The O(N) Symmetric (phi2)2 Field Theory: Large N Limit 15: The Non-Linear sigma-Model 16: Functional Renormalization Group Appendix

About the Author :
Professor Jean Zinn-Justin Head of Department, Dapnia, CEA/Saclay, France

Review :
`A subject of lasting importance, presented by one of the best qualified authors internationally.' John Chalker, University of Oxford `The topic is good, with renewed interest in the renormalization group by the new generation of string theorists and particle theorists.' Randall Kamien, University of Pennsylvania `The clear exposition of the main ideas and the simple and agile notation the author uses help facilitate the comprehension of the different concepts presented. Researchers familiar with statistical physics methods will find a self-contained framework to grasp the essence of quantum field theory and the renormalization group and to elucidate the prominent role they play at present in physics. For this reason, this book is highly recommendable due to the insight it gives into quantum field theories, providing sound basis for further research.' Journal of Statistical Physics