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Quantum Field Theory and Beyond QUANTUM FIELD THEORY AND BEYOND Essays in Honor of Wolfhart Zimmermann This page intentionally left blank QUANTUM FIELD THEORY AND BEYOND Essays in Honor of Wolfhart Zimmermann Proceedings of the Symposium in Honor of Wolfhart Zimmermann’s 80th Birthday Ringberg Castle, Tegernsee, Germany 3 – 6 February 2008 editors Erhard Seiler Max-Planck-Institut für Physik, Germany Klaus Sibold Universität Leipzig, Germany World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. QUANTUM FIELD THEORY AND BEYOND Essays in Honor of Wolfhart Zimmermann Proceedings of the Symposium in Honor of Wolfhart Zimmermann’s 80th Birthday Copyright © 2008 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN-13 978-981-283-354-9 ISBN-10 981-283-354-4 Printed in Singapore. LaiFun - QaunFieldTheory.pmd 1 9/5/2008, 8:35 AM September 12, 2008 14:27 WSPC - Proceedings Trim Size: 9in x 6in festschrift v PREFACE This volume collects twelve essays written in honor of Wolfhart Zimmer- mann on the occasion of his 80th birthday. Ten of them are based on talks given at a Symposium in his honor, held at the Ringberg castle of the Max Planck Society from February 3rd to 6th, 2008. Wolfhart Zimmermann has been in the forefront of research in Quantum Field Theory since the 1950s, when the famous work of ‘LSZ’ (Lehmann, Symanzik, Zimmermann) was created, which is at the basis of all modern applications of Quantum Field Theory to Accelerator Physics, and without which results expected from the LHC at CERN – scheduled to start oper- ation this year – would not be possible. But he was also the first person to construct composite operators in Quantum Field Theory, thus laying the groundwork for the mathematical description of symmetries and their breaking in perturbation theory with its undisputable success in the stan- dard model of particle physics. On the more abstract level this led to the understanding of anomalies: all anomalies known up to date have their origin in an identity which he proved for normal products of different sub- traction degrees. In particular anomaly coefficients are given in terms of the coefficients of this identity. The range of this observation has not yet been fully exhausted until the present day, but remains a source of new structural relations. The perturbatively formulated operator product expansion which he established gave safe ground to the corresponding non-perturbative conjec- ture of Kenneth Wilson which lead to quantitatively successful results in Quantum Chromodynamics. His study of asymptotic freedom in the case of several coupling constants introduced the concept of reduction of couplings which enlarges the notion of symmetry and under mild assumptions builds a bridge from perturbation theory to the non-perturbative regime of a theory. The proof that theories exist which have vanishing dilatation and conformal anomalies to all orders of perturbation theory is based on this concept. Amongst these models is the famous supersymmetric Yang-Mills theory with four supersymmetries. Some of the contributors of this volume were his students, some were September 12, 2008 14:27 WSPC - Proceedings Trim Size: 9in x 6in festschrift vi Preface his collaborators, some just shared his interest in the subject, but all of them are inspired by him and his work. Analogously one might classify the contributions by content. Some continue work originally initiated by him and employ his tools directly. Some are devoted to lay new groundwork in quantum physics or to broaden and to deepen applications. But all are close in spirit and originality. Two of the contributions are particularly remarkable by the fact that they employ ideas and concepts of Quantum Field Theory in different areas of physics, such as the mathematical theory of electrons in disordered media and the theory of dynamical systems. They thus prove how fruitfully renormalization theory has transcended its origin in particle physics and leads to new insight in otherwise seemingly disjoint parts of physics. A glance at the subjects covered also reveals the enormous richness and diversity of a theory that originally aimed at a mathematically sound the- ory of elementary particles. New concepts of spacetime are being checked, fundamentals of quantum mechanics are formulated; Quantum Field The- ory is embedded in new structures. The essays in this volume attest both to Wolfhart Zimmermann’s inspiring influence and the power and continuing vigor of Quantum Field Theory in our days. We have tried to arrange the contributions roughly according to in- creasing distance from Wolfhart Zimmermanns own work in Quantum Field Theory; of course the linear order required by the presentation cannot do justice at all to the various interconnections between the different articles. We would like to thank all contributors for their carefully written es- says which provide an entertaining tour through part of todays theoretical physics. For funding the symposium thanks are due to the Max Planck Institute for Physics in Munich and to the Max Planck Institute for Mathematics in the Sciences in Leipzig. We are most grateful to A. H¨ormann and his team for perfect organization and warm hospitality at the Ringberg castle. Fi- nally we would like to thank T. Hahn for providing the picture on which the cover is based and P. Breitenlohner for technical help with the preparation of this volume. Erhard Seiler Munich, Germany Klaus Sibold Leipzig, Germany 31 July 2008 September 12, 2008 14:27 WSPC - Proceedings Trim Size: 9in x 6in festschrift vii CONTENTS Preface v Zimmermann’s Subtraction Scheme and the Perturbative Solution to the Renormalization Group Evolution Equations 1 C. Becchi A New Look at the Higgs-Kibble Model 16 O. Steinmann LargeRegularQCDCouplingatLowEnergy? 34 D. V. Shirkov The Dihedral Group as a Family Group 46 J. Kubo On the Consequences of Twisted Poincar´eSymmetry Upon QFT on Moyal Noncommutative Spaces 64 G. Fiore Taming the Landau Ghost in Noncommutative Quantum Field Theory 85 H. Grosse Warped Convolutions: A Novel Tool in the Construction of Quantum Field Theories 107 D. Buchholz and S. J. Summers Quantum (or Averaged) Energy Inequalities in Quantum Field Theory 122 R. Verch September 12, 2008 14:27 WSPC - Proceedings Trim Size: 9in x 6in festschrift viii Contents Field Theory and Brane Dynamics 141 T. E. Clark Knots as Possible Excitations of the Quantum Yang-Mills Fields 156 L. D. Faddeev Feynman Graphs and Renormalization in Quantum Diffusion 167 L. Erd˝os, M. Salmhofer and H.-T. Yau Renormalization in Chaotic and Pseudochaotic Dynamical Systems 183 J. H. Lowenstein Author Index 207 September 12, 2008 14:27 WSPC - Proceedings Trim Size: 9in x 6in festschrift 1 ZIMMERMANN’S SUBTRACTION SCHEME AND THE PERTURBATIVE SOLUTION TO THE RENORMALIZATION GROUP EVOLUTION EQUATIONS CARLO BECCHI∗ Universit`adi Genova, Dipartimento di Fisica and I.N.F.N. Sezione di Genova via Dodecaneso 33, Genova I-16146, Italy ∗E-mail: [email protected] In the framework of Euclidean field theory we show that an infrared safe slightly modified version of Zimmermann’s subtraction scheme generates the perturba- tive solutions to the Wilson-Polchinski renormalization group equations. Keywords: Wilson-Polchinski Renormalization Group; BPHZ renormalization 1. Introduction On the occasion of Wolfhart Zimmermann’s 80th birthday I think that a short look at the present status of Quantum Field Theory is certainly timely. I would like in particular to give an example of the persisting fundamental role of many Zimmermann’s contributions in the development of Quantum Field Theory. No doubt quantum field theory is one of the major achievements of twenty’s century physics.1 Even if no interacting four dimensional model has yet been solved, an axiomatic framework leading to a well defined scat- tering theory is now clearly defined and different constructive approaches have been set up for a class of models. Lehmann-Symanzik-Zimmermann construction of scattering amplitudes has been and remains a basic step in the construction of a complete theory. Among the constructive methods the most important are loop ordered perturbative renormalization2 and Wil- son’s renormalization group (R.G.).3 I think that a short comparison of the use of these methods in the framework of perturbation theory is timely. Loop ordered perturbative renormalization is the natural development of QED and has produced exceptionally successful phenomenological anal- September 12, 2008 14:27 WSPC - Proceedings Trim Size: 9in x 6in festschrift 2 C. Becchi yses in the framework of the Standard Model of Electro-Weak and Strong Interactions. Forgetting the problems related to infra-red divergences the construction
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