Because the origin of the renormalization group is in quantum field theory and more importantly because applications of the theory in phase transitions have reflected back into quantum field theory it was decided that the conference should cover the renormalization group in critical phenomena and quantum field theory. The conference took place at the Temple University Conference Center at Sugarloaf, Chestnut Hill, Pa on May 29, 30 and 31, 1973. The present publication is the Proceedings of the Conference. (Author)
Because the origin of the renormalization group is in quantum field theory and more importantly because applications of the theory in phase transitions have reflected back into quantum field theory it was decided that the conference should cover the renormalization group in critical phenomena and quantum field theory. The conference took place at the Temple University Conference Center at Sugarloaf, Chestnut Hill, Pa on May 29, 30 and 31, 1973. The present publication is the Proceedings of the Conference. (Author)
In this chapter the basic concepts of the modern approach to equilibrium critical behaviour, conventionally grouped under the title ‘renormalization group’, are introduced. This terminology is rather unfortunate. The mathematical structure of the procedure, in the sense that it may be said to have any rigorous underpinnings, is certainly not that of a group. Neither is renormalization in quantum field theory an essential element, although it has an intimate connection with some formulations of the renormalization group. In fact, the renormalization group framework may be applied to problems quite unrelated to field theory. The origins of the name may be traced to the particle physics of the 1960s, when it was optimistically hoped that everything in fundamental physics might be explained in terms of symmetries and group theory, rather than dynamics. One of the earliest applications of renormalization group ideas, in fact, was to the rather esoteric subject of the high energy behaviour of renormalized quantum electrodynamics. It took the vision of K. Wilson to realise that these methods had a far wider field of application in the scaling theory of critical phenomena that was being formulated by Fisher, Kadanoff and others in the latter part of the decade. By then, however, the name had become firmly attached to the subject. Not only are the words ‘renormalization’ and ‘group’ examples of unfortunate terminology, the use of the definite article ‘the’ which usually precedes them is even more confusing.
Pertinent Concepts and Ideas in the Theory of Critical Phenomena Formulation of the Problem of Phase Transitions in Terms of Functional Integrals Functional Integrals in Quantum Field Theory Perturbation Theory and Feynman Graphs Vertex Functions and Symmetry Breaking Expansions in the Number of Loops and Components Renormalization The Renormalization Group and Scaling the Critical Region The Computation of the Critical Exponents Beyond Leading Scaling Universality Revisited Critical Behavior with Several Couplings Crossover Phenomena Critical Phenomena near Two Dimensions Real Space Methods Finite Size Scaling Monte Carlo Methods. Numerical Field Theory.
The nature and origins of renormalization group ideas in statistical physics and condensed matter theory are recounted informally, emphasizing those features of prime importance in these areas of science in contradistinction to quantum field theory, in particular: critical exponents and scaling, relevance, irrelevance and marginality, universality, and Wilson's crucial concept of flows and fixed points in a large space of Hamiltonians. Contents Foreword 1 Introduction 2 Whence came renormalization group theory? 3 Where stands the renormalization group? 4 Exponents, anomalous dimensions, scale invariance and scale dependence 5 The challenges posed by critical phenomena 6 Exponent relations, scaling, and irrelevance 7 Relevance, crossover, and marginality 8 The task for renormalization group theory 9 Kadanoff's scaling picture 10 Wilson's quest 11 The construction of renormalization group transformations: the epsilon expansion 12 Flows, fixed points, universality and scaling 13 Conclusions Acknowledgments Selected bibliography Appendix A Asymptotic behavior Appendix B Unitarity of the renormalization group Appendix C Nature of a semigroup Foreword It was a pleasure to participate in the Colloquium cosponsored by the Departments of Philosophy and of Physics at Boston University “On the Foundations of Quantum Field Theory.” In the full title, this was preceded by the phrase: “A Historical Examination and Philosophical Reflections,” which set the aims of the meeting. Naturally, the participants were mainly high-energy physicists, experts in field theories, and interested philosophers of science.
This special issue of Journal of Physics A: Mathematical and General is a collection of articles dealing with the application of renormalization group methods to a wide range of problems in physics. Most articles are based on contributions presented at the 6th International Conference, Renormalization Group 2005 (RG 2005), held at the University of Helsinki from 30 August to 3--September 2005. This was the sixth of a series of conferences on this topic, the first three of which took place in 1986, 1991 and 1996 at the Joint Institute for Nuclear Research in Dubna, Russia. The fourth (RG99) was held in 1999 in Taxco, Mexico, and the fifth (RG02) in 2002 in Tatranska Strba, Slovakia. During the second half of the 20th century a number of important and very successful new paradigms such as scale-dependent (`running') coupling constants, scaling, and universality became firmly established in physics. To a large extent, these advances were due to the development of renormalization group ideas and their successful implementation as powerful calculational schemes to tackle problems with many length scales. While the concept of the renormalization group had originally been introduced in elementary particle physics, its modern form eventually emerged from the study of problems in condensed matter physics, such as critical phenomena and the Kondo effect. This in turn has significantly reshaped our understanding of field theories in high energy physics, and produced powerful computational tools for dealing with difficult many-length-scale problems in distinct fields of physics, ranging from elementary particle and condensed matter physics to hydrodynamic turbulence and growth phenomena in nonequilibrium physics. The history of the renormalization group, characterized by frequent exchanges of ideas between elementary particle and condensed matter physics, bears testimony to the enormous benefits to which such exchanges can lead. The main aim of the series of renormalization group conferences has been to bring together scientists with backgrounds and expertise in the application of the renormalization group to problems in broadly different areas of physics, offering regular events for continuing such fruiful exchanges of ideas. We hope that the present collection of contributions to quantum field theory, mathematical and methodological aspects of the renormalization group, statistical mechanics, condensed matter physics, and stochastic dynamics conveys this spirit. We feel it amply demonstrates the breadth of activity of research based on the renormalization group and its potential for future work. On behalf of the organizers of the conference, we would like to thank the sixty participants of the conference and the members of the Advisory Board for their contributions. Financial and organizational support of the University of Helsinki, the Academy of Finland, the Finnish Society of Sciences and Letters and the Naval Academy is gratefully acknowledged. We are very grateful to all authors who contributed to this special issue. Finally, we owe special thanks to all who assisted in the preparation of it, in particular to the staff of Journal of Physics A: Mathematical and General .
Abstract Chapter 6 describes how the perturbative renormalization group (RG) of quantum field theory has made it possible to derive universal properties of continuous macroscopic phase transitions. The RG initially emerged as a consequence of the necessity to cancel infinities that appear in the perturbative expansion (the renormalization procedure) and the possibility of defining the parameters of the renormalized theory at different momentum scales. Although the field theory RG is now understood to be an asymptotic form, it has made it possible to confirm the Wilson–Fisher fixed point and led to an understanding of universality for a large class of critical phenomena. In the framework of dimensional continuation, zeros of RG beta functions, which correspond to Wilson–Fisher’s fixed points, have made it possible to recover Wilson–Fisher’s epsilon expansion, proving scaling relations and calculating critical exponents. Series summation methods have then been used to generate precise values of exponents.