DOCSLIB.ORG

Bogoliubov-Parasiuk-Hepp-Zimmermann Renormalization in Configuration Space

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Bogoliubov-Parasiuk-Hepp-Zimmermann Renormalization in Configuration Space Bogoliubov-Parasiuk-Hepp-Zimmermann Renormalization in Configuration Space Von der Fakultät für Physik und Geowissenschaften der Universität Leipzig genehmigte DISSERTATION zur Erlangung des akademischen Grades DOCTOR RERUM NATURALIUM (Dr.rer.nat.) vorgelegt von Diplom-Physiker Steffen Pottel geboren am 09. Juli 1984 in Neubrandenburg (Deutschland) Gutachter: Professor Klaus Sibold Professor Raimar Wulkenhaar Tag der Verleihung: 18. Juni 2018 Bibliographische Daten Bogoliubov-Parasiuk-Hepp-Zimmermann Renormalization in Configuration Space (Bogoliubov-Parasiuk-Hepp-Zimmermann Renormierung im Konfigurationsraum) Pottel, Steffen Universität Leipzig, Dissertation, 2017 111 Seiten, 0 Abbildungen, 98 Referenzen Referat: In der vorliegenden Arbeit wird das Konzept der Renormierung im Impul- sraum nach Bogoliubov, Parasiuk, Hepp und Zimmermann in einen Ortsraumformalis- mus übertragen und auf analytische Raumzeiten im Rahmen von algebraischen Quan- tenfeldtheorien erweitert. Der Beweis des Schemas benutzt dabei keines der Argu- mente aus dem Impulsraum. Dennoch wird der Zusammenhang zwischen beiden For- mulierungen analysiert und Unterschiede sowie Grenzen unter Fouriertransformation aufgezeigt. Weiterhin werden Normalprodukte, die eine Verallgemeinerung der Wick- ordnung darstellen, Zimmermannidentitäten und die lineare Feldgleichung im Rahmen der neuen Renormierungsvorschrift hergeleitet. Acknowledgement I would like to thank Klaus Sibold for uncountable discussions and the opportunity to work on numerous topics in modern physics. His passion for renormalization theory and his veneration for Wolfhart Zimmermann served as inspiration for the present work. I am grateful for the support by my second supervisor Rainer Verch. Further, I would like to thank the members of the "Quantum Field Theory and Gravitation" group as well as the "Elementary Particle Theory" group at the Institute for Theoretical Physics for their support and creation of a productive working environment. In particular, the scientific coffee breaks organized by Gandalf Lechner and Ko Sanders are much appreciated. I would like to thank Jan Zschoche for many helpful discussion and proofreading parts of the present work. I am grateful for the kind hospitality at the Max Planck Institute for Mathematics in the Sciences and the financial support by the International Max Planck Research School "Mathematics in the Sciences". Above all, my family and Camilla deserve my sincerest thanks for their considerable amount of patience and unlimited support. Contents 1 Introduction 1 2 Preliminaries 9 2.1 The Wave Equation on Analytic Spacetimes . .9 2.2 Quantization of Fields . 15 3 BPHZ Renormalization in Configuration Space 28 3.1 Regularization . 28 3.2 Convergence of the R-operation . 32 3.2.1 Space Decomposition . 36 3.2.2 Reordering . 37 3.2.3 Recursion . 43 3.3 Recovering Time-Ordered Products . 49 4 Relation to the Momentum Space Method 56 4.1 Infrared Scaling . 56 4.2 Limit of Constant Coupling . 59 4.3 Additional Subtractions . 66 5 Normal Products 73 5.1 Zimmermann Identity . 74 5.2 Normal Products . 84 5.3 Field Equation . 92 6 Conclusion 96 Appendix 100 A Analytic Continuation . 100 B Taylor Operator . 102 Chapter 1 Introduction In laboratory experiments, scientists aim at extracting quantitative information about a physical system via observables. The system is prepared in a state and should produce the same result every time the experiment is repeated. This reproducibility does neither necessarily have to be exact for every measurement nor is it restricted to the same ex- act system. Specifically, a classical mechanical system is completely determined by its position and momentum at any given fixed time. In the Hamiltonian formulation, this corresponds to a point in phase space and the observables are represented by (smooth) functions. The system is said to be in a completely prepared state, since every mea- surement reproduces the same values any time it is repeated. For systems with a large particle number, an accurate preparation is often possible only with respect to certain macroscopic parameters like total energy, volume or total particle number. In theoretical approaches, one considers a large number of copies of the system which mathematically corresponds to the assignment of a distribution in phase space such that a measurement may be identified by the expectation value of a random variable, the observable. The latter experiences a sudden shift on microscopic scales where quantum effects set in. Namely, one does not want to change the notion of a system but let the observables account for the quantum character. For instance, in quantum mechanics, the simultane- ous measurement of conjugate observables, i.e. Fourier transform duals like position and momentum, is possible only up to a minimal uncertainty. Absorption and emission pro- cesses of photons on a microscopic level demand the incorporation of special relativity, thus the study of quantum fields. In the conventional approach to quantum field theory [IZ80] on Minkowski space, one greatly relies on its large symmetry group leading to a distinguished ground state, the particle interpretation of excitations and a preferred representation, the Fock space, based on asymptotic conditions [LSZ55, LSZ57]. However, physically most interesting models seem to be impenetrable since the involved quantum fields fulfill nonlinear partial dif- ferential equations, for which there is no general well-posedness theory available and, hence, generally no control over correlation functions. One usually evades the problem by using perturbation theory and approximates the interacting theory by a formal power 1 series in the nonlinearity about the free theory, i.e. the model with linear equation of motion. This formal expansion contains, apart from the lack of knowledge regarding convergence of the series in four and higher dimensions, some shortcomings related to physics. Undeniably, the self-interaction of the system is not fully reproduced, when the formation of bound states is neglected or long range interactions are not treatable satisfyingly. But even at finite order of the perturbative expansion, almost all physical quantities are ill-defined so that the approximation turns out to be too rough in order to cope with the properties of quantum fields. Due to Heisenberg’s uncertainty princi- ple, the latter have to be delocalized in spacetime and are promoted to distributions, by which physical quantities, which are nonlinear in the field, are afflicted with products of distributions defined at the same point. Nevertheless one may proceed and introduce a regularization, with which the physical quantities attain finite values, but this leads to the question whether the regularization is in concordance with all necessary physical principles. If it can be implemented for all physical quantities coherently and respects a set of physically reasonable axioms, the regularization prescription is called a renormal- ization scheme. In this regard, Feynman, Tomonaga and Schwinger were the pioneers with their works on loop corrections in quantum electrodynamics. Their results were studied more constructively by Dyson [Dys49a] leading to a broad and intense develop- ment in the field of renormalization theory [Vel76], which vests perturbative quantum field theory with high predictive power. Ultimately, the questionable recipe of simply summing the perturbation series up to some finite order and calling this a prediction is justified only by its overwhelming success: experiments can be described to be within accuracy parts per million. The existence of several renormalization prescriptions indicates that the problem of ren- dering perturbatively defined correlation functions among quantum fields well-defined does not have a unique solution. Indeed, it can be proved [Hep69] that any two renormal- ization schemes are related to each other by a finite change in the choice of ambiguities, the so-called counterterms. The first example of such equivalence was established between Bogoliubov-Parasiuk-Hepp, abbreviated BPH, and analytic renormalization. The idea of Bogoliubov and Parasiuk [BP57] is to employ a variation of the Hadamard regularization of singular integrals [RS81, Chapter 5] for the subtraction of divergent contributions, ab- breviated as R-operation. The variation of the Hadamard regularization is necessary due to the structure of the quantities in question. Specifically, the (time-ordered) correlation functions, evaluated in ground state, allow for an expansion in weights over so-called Feynman graphs, where the inner or interaction vertices may be treated as functions of spacetime rather than distributions. The aforementioned divergent contributions may occur at points of coinciding vertices in the integration over all configurations of such Feynman graphs in spacetime. For their analysis, it is convenient to consider the Fourier transform of correlation functions, since the complexity of the involved functions gets re- duced and the question of renormalizability changes over to the study of integrations over free (loop) momenta. The latter may be best understood in regard to the convolution theorem, i.e. the (generalized) convolutions of weights at incident edges of an integrated graph become standard pointwise products (including energy-momentum-conservation 2 at that vertex) and any pair of paths, which are disjoint with respect to their edge sets but share the same endpoints, transforms into a convolution. After choosing a basis of loops, the number of
Load more

Recommended publications
  • Wolfhart Zimmermann: Life and Work
    JID:NUPHB AID:14268 /FLA [m1+; v1.280; Prn:28/02/2018; 9:46] P.1(1-11) Available online at www.sciencedirect.com ScienceDirect Nuclear Physics B ••• (••••) •••–••• www.elsevier.com/locate/nuclphysb Wolfhart Zimmermann: Life and work Klaus Sibold Institut für Theoretische Physik, Universiät Leipzig, Postfach 100920, D-04009 Leipzig, Germany Received 19 December 2017; received in revised form 17 January 2018; accepted 22 January 2018 Editor: Hubert Saleur Abstract In this report, I briefly describe the life and work of Wolfhart Zimmermann. The highlights of his scien- tific achievements are sketched and some considerations are devoted to the man behind the scientist. The report is understood as being very personal: at various instances I shall illustrate facets of work and person by anecdotes. © 2018 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 1. Introduction The present report is based on a colloquium talk given at the end of a memorial symposium to honour Wolfhart Zimmermann: Max Planck Institute for Physics, Munich (May 22–23, 2017) I borrowed freely from the following obituaries: Physik-Journal 15 (2016) Nr. 12 S.50 W. Hollik, E. Seiler, K. Sibold Nucl. Phys. B193 (2016) 877–878 W. Hollik, E. Seiler, K. Sibold IAMP News Bulletin, Jan. 2017 p. 26–30 M. Salmhofer, E. Seiler, K. Sibold 2. The beginning Wolfhart Zimmermann was born on February 17, 1928 in Freiburg im Breisgau (Germany) as the son of a medical doctor. He had an older sister with whom he liked to play theater and, when E-mail address: [email protected].
    [Show full text]
  • Arxiv:1805.07240V4 [Gr-Qc]
    Living Rev. Relativ. manuscript No. (will be inserted by the editor) Hamiltonian formulation of general relativity and post-Newtonian dynamics of compact binaries Gerhard Sch¨afer · Piotr Jaranowski Received: / Accepted: Abstract Hamiltonian formalisms provide powerful tools for the computa- tion of approximate analytic solutions of the Einstein field equations. The post-Newtonian computations of the explicit analytic dynamics and motion of compact binaries are discussed within the most often applied Arnowitt–Deser– Misner formalism. The obtention of autonomous Hamiltonians is achieved by the transition to Routhians. Order reduction of higher derivative Hamiltonians results in standard Hamiltonians. Tetrad representation of general relativity is introduced for the tackling of compact binaries with spinning components. Configurations are treated where the absolute values of the spin vectors can be considered constant. Compact objects are modeled by use of Dirac delta functions and their derivatives. Consistency is achieved through transition to d- dimensional space and application of dimensional regularization. At the fourth post-Newtonian level, tail contributions to the binding energy show up. The conservative spin-dependent dynamics finds explicit presentation in Hamil- tonian form through next-to-next-to-leading-order spin-orbit and spin1-spin2 couplings and to leading-order in the cubic and quartic in spin interactions. The radiation reaction dynamics is presented explicitly through the third-and- half post-Newtonian order for spinless objects, and, for spinning bodies, to leading-order in the spin-orbit and spin1-spin2 couplings. The most important historical issues get pointed out. Keywords general relativity classical spin and gravity Hamiltonian formalism compact binary systems· canonical equations· of motion radiation reaction· and emission analytical· and dimensional regularization· · arXiv:1805.07240v4 [gr-qc] 7 Jan 2020 G.
    [Show full text]
  • The $\Phi^ 3 4 $ and $\Phi^ 3 6 $ Matricial QFT Models Have
    Φ3 Φ3 The 4 and 6 matricial QFT models have reflection positive two-point function Harald Grosse1, Akifumi Sako1,2 and Raimar Wulkenhaar3 1 Fakult¨at f¨ur Physik, Universit¨at Wien Boltzmanngasse 5, A-1090 Wien, Austria 2 Department of Mathematics, Faculty of Science Division II, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan 3 Mathematisches Institut der Westf¨alischen Wilhelms-Universit¨at Einsteinstraße 62, D-48149 M¨unster, Germany MSC 2010: 81T16, 81T08, 81R12 dedicated to the memory of Wolfhart Zimmermann (1928–2016) Abstract We extend our previous work (on D = 2) to give an exact solution of the Φ3 large- D N matrix model (or renormalised Kontsevich model) in D = 4 and D = 6 dimensions. Induction proofs and the difficult combinatorics are unchanged compared with D = 2, but the renormalisation – performed according to Zimmermann – is much more involved. As main result we prove that the Schwinger 2-point function resulting from 3 the ΦD-QFT model on Moyal space satisfies, for real coupling constant, reflection positivity in D = 4 and D = 6 dimensions. The K¨all´en-Lehmann mass spectrum of the associated Wightman 2-point function describes a scattering part p 2 2µ2 | | ≥ and an isolated fuzzy mass shell around p 2 = µ2. | | 1 Introduction arXiv:1612.07584v2 [math-ph] 2 Jun 2017 The Kontsevich model [1, 2] is of paramount importance because it elegantly proves Witten’s conjecture [3] about the equivalence of two approaches to quantum gravity in two dimensions: the Hermitean one-matrix model [4, 5, 6] versus the intersection theory on the moduli space of Riemann surfaces [7, 8, 9].
    [Show full text]
  • Dispersion Relations in Gauge Theories with Confinement
    EFI 95-54 MPI-Ph/95-82 Dispersion Relations in Gauge Theories with Confinement 1 Reinhard Oehme Enrico Fermi Institute and Department of Physics University of Chicago, Chicago, Illinois, 60637, USA 2 and Max-Planck-Institut f¨ur Physik - Werner-Heisenberg-Institut - 80805 Munich, Germany Abstract The analytic structure of physical amplitudes is considered for gauge the- ories with confinement of excitations corresponding to the elementary fields. Confinement is defined in terms of the BRST algebra. BRST-invariant, local, composite fields are introduced, which interpolate between physical asymp- totic states. It is shown that the singularities of physical amplitudes are arXiv:hep-th/9511007v1 1 Nov 1995 the same as in an effective theory with only physical fields. In particular, there are no structure singularities (anomalous thresholds) associated with confined constituents, like quarks and gluons. The old proofs of dispersion relations for hadronic amplitudes remain valid in QCD. 1Plenary talk presented at the XVIIIth International Workshop on High Energy Physics and Field Theory, Moscow-Protvino, June 1995. To be published in the Proceedings. 2Permanent Address It is the purpose of this talk, to give a survey of the problems involved in the derivation of analytic properties of physical amplitudes in gauge theories with confinement. This report is restricted to a brief resum´eof the essential points discussed in the talk. 3 Dispersion relations for amplitudes describing reactions between hadrons, and for form factors describing the structure of particles, have long played an important rˆole in particle physics [3, 4, 5, 6]. Analytic properties of Green’s functions are fundamental for proving many important results in quantum field theory.
    [Show full text]
  • Zimmermann's Forest Formula, Infrared Divergences and the QCD
    Zimmermann's Forest Formula, Infrared Divergences and the QCD Beta Function Franz Herzog Nikhef Theory Group, Science Park 105, 1098 XG Amsterdam, The Netherlands Abstract We review Zimmermann's forest formula, which solves Bogoliubov's recur- sive R-operation for the subtraction of ultraviolet divergences in perturba- tive Quantum Field Theory. We further discuss a generalisation of the R- operation which subtracts besides ultraviolet also Euclidean infrared diver- gences. This generalisation, which goes under the name of the R∗-operation, can be used efficiently to compute renormalisation constants. We will discuss several results obtained by this method with focus on the QCD beta function at five loops as well as the application to hadronic Higgs boson decay rates at N4LO. This article summarizes a talk given at the Wolfhart Zimmermann Memorial Symposium. Keywords: Renormalization Group, QCD Email address: [email protected] (Franz Herzog) arXiv:1711.06121v1 [hep-ph] 16 Nov 2017 Preprint submitted to Nuclear Physics B November 17, 2017 1. Introduction Despite the enormous success in describing the interactions of elementary particles the appearance of ultraviolet (UV) divergences make it difficult to establish Quantum Field Theory as a fundamental theory of nature. A solution which at least grants the interpretation of Quantum Field Theories as low energy effective theories is given by the procedure of renormalisation. This is the procedure to absorb the troublesome infinities, present at small distances, into the physical parameters of the theory. An important development in the establishment of renormalisation the- ory has been the Bogoliubov-Parasiuk-Hepp-Zimmermann (BPHZ) renor- malisation scheme. This scheme was originally developed by Bogoliubov and Parasiuk [1] in terms of a recursive subtraction operation, often called Bo- goliubov's R-operation.
    [Show full text]
  • Ebook Download Finite Quantum Electrodynamics
    FINITE QUANTUM ELECTRODYNAMICS : THE CAUSAL APPROACH PDF, EPUB, EBOOK G. Scharf | 410 pages | 13 Nov 2012 | Springer-Verlag Berlin and Heidelberg GmbH & Co. KG | 9783642633454 | English | Berlin, Germany Finite Quantum Electrodynamics : The Causal Approach PDF Book Scharf 1 1. Show all. This scrupulously researched and abundantly illustrated book includes drawings based on renderings by artists of the period to achieve utmost accuracy and authenticity. Views Read Edit View history. See all 6 brand new listings. For a better shopping experience, please upgrade now. Causal Perturbation Theory. After a historical introduction and mathematical preliminaries the book turns to electrostatics in vacuum, whereby Dispersion Forces I. These up-to-date terms cover twenty-first century digital technologies and Renormalization group On-shell scheme Minimal subtraction scheme. Relativistic Quantum Mechanics. Equally suitable as a reference for researchers specialized in science and engineering. About this book Introduction In this textbook for graduate students in physics the author carefully analyses the role of causality in Quantum Electrodynamics. Journal of Physics A: Mathematical and General. Adventure awaits with this comic-strip story of a space agent and his daring escape from an interstellar prison. Dirac equation Minkowski space Second quantization quantum mechanics scattering theory. In this textbook for graduate students in physics the author carefully analyses the role of causality in Quantum Electrodynamics. Enabling JavaScript in your browser will allow you to experience all the features of our site. In this book the author carefully analyses the role of the concept of causality in Quantum Electrodynamics. McMillan , Hardcover 5. Brand new: Lowest price The lowest-priced brand-new, unused, unopened, undamaged item in its original packaging where packaging is applicable.
    [Show full text]
  • The Operator Product Expansion
    The Operator Product Expansion Nolan Smyth 6/10/2020 QFT 3 - Final Presentation Outline 1. The Big Picture 2. Renormalization Group Refresher/Setup 3. Introducing the Operator Product Expansion 4. Callen-Symanzik Equation for Wilson Coefficients 5. An example: QCD corrections to weak decays 1 The Big Picture The Big Picture We've spent much of this quarter learning to deal with divergences, typically those of an ultraviolet nature. These divergences led to shifting in mass parameters and coupling constants through the process of renormalization, but were otherwise rather innocuous. Renormalization gave us a great deal of freedom in that we were free to define our theory at any arbitrary momentum scale. But in order to describe the the correct physical theory, the renormalized Green's functions must be identical to the bare Green functions, up to the field strength renormalization. (n) −n=2 (n) GR (µ; x1; :::; xn) = Z Gb (x1; :::; xn) 2 The Big Picture We then asked, how would our Green's functions change if we looked at a different renormalization scale? This led to the Callen-Symanzik (renormalization group) equations. Solving these, we calculated how coupling constants and fields evolve with momentum scale. Until now, we've utilized correlation functions to perform this analysis. But we will see that it is particularly useful to study the renormalization flow at the level of the operators themselves. But to do so, we will need to deal with the non-locality of operator products. This is where we will meet the Operator Product Expansion (OPE). 3 The Big Picture After motivating the OPE, we will see that all of the interesting behavior at high energy (short distances) depends only on Wilson coefficents.
    [Show full text]
  • Applications of the Reduction of Couplings
    Applications of the Reduction of Couplings∗ Dedicated to Professor Wolfhart Zimmermann on the occasion of his 70th birthday Jisuke Kubo Institute for Theoretical Physics, Kanazawa University Kanazawa 920-1192, Japan Abstract Applications of the principle of reduction of couplings to the standard model and supersym- metric grand unified theories are reviewed. Phenomenological applications of renormalization arXiv:hep-ph/9903482v1 25 Mar 1999 group invariant sum rules for soft supersymmetry-breaking parameters are also reviewed. ∗ To appear in the Proceedings of the Ringberg Symposium on Quantum Field Theory, Tegernsee, Germany, June 1998. 1 Application to the Standard Model High energy physicists have been using renormalizability as the predictive tool, and also to decide whether or not a quantity is calculable. As we have learned in the previous talk by Professor Oehme, it is possible, using the method of reduction of couplings [1, 2, 3], to renormalize a theory with fewer number of counter terms then usually counted, implying that the traditional notion of renormalizability should be generalized in a certain sense 1. Consequently, the notion of the predictability and the calculability [5] may also be generalized with the help of reduction of couplings. Of course, whether the generalizations of these notions have anything to do with nature is another question. The question can be answered if one applies the idea of reduction of couplings to realistic models, make predictions that are specific for reduction of couplings, and then wait till experimentalists find positive results 2. In 1984 Professor Zimmermann, Klaus Sibold and myself [8] began to apply the idea of reduction of couplings to the SU(3) SU(2) U(1) gauge model for the strong and C × L × Y electroweak interactions.
    [Show full text]
  • Reduction of Coupling Parameters and Duality 1
    EFI 99-01 MPI-Ph/99-04 Reduction of Coupling Parameters and Duality 1 Reinhard Oehme 2 Enrico Fermi Institute and Department of Physics University of Chicago Chicago, Illinois, 60637, USA 3 and Max-Planck-Institut f¨ur Physik - Werner-Heisenberg-Institut - 80805 Munich, Germany Abstract The general method of the reduction in the number of coupling parameters is dis- cussed. Using renormalization group invariance, theories with several independent couplings are related to a set of theories with a single coupling parameter. The reduced theories may have particular symmetries, or they may not be related to any known symmetry. The method is more general than the imposition of invari- ance properties. Usually, there are only a few reduced theories with an asymptotic arXiv:hep-th/9903092v1 10 Mar 1999 power series expansion corresponding to a renormalizable Lagrangian. There also exist ‘general’ solutions containing non-integer powers and sometimes logarithmic factors. As an example for the use of the reduction method, the dual magnetic theories associated with certain supersymmetric gauge theories are discussed. They have a superpotential with a Yukawa coupling parameter. This parameter is ex- pressed as a function of the gauge coupling. Given some standard conditions, a unique, isolated power series solution of the reduction equations is obtained. After reparametrization, the Yukawa coupling is proportional to the square of the gauge 1Dedicated to Wolfhart Zimmermann on the Occasion of his 70th Birthday. To appear in Recent Developments in Quantum Field Theory, Springer Verlag, Heidelberg, New York. (Editors: P. Breitenlohner, D. Maison and J. Wess). Ringberg Symposium, June 1998. 2E-mail: [email protected] 3Permanent Address coupling parameter.
    [Show full text]
  • JHEP07(2008)103 July 9, 2008 July 24, 2008 April 15, 2008 Ries
    Published by Institute of Physics Publishing for SISSA Received: April 15, 2008 Accepted: July 9, 2008 Published: July 24, 2008 Polyakov formulas for GJMS operators from AdS/CFT JHEP07(2008)103 Danilo E. D´ıaz Humboldt-Universit¨at zu Berlin, Institut f¨ur Physik, Newtonstr.15, D-12489 Berlin, Germany E-mail: [email protected] Abstract: We argue that the AdS/CFT calculational prescription for double-trace de- formations leads to a holographic derivation of the conformal anomaly, and its conformal primitive, associated to the whole family of conformally covariant powers of the Laplacian (GJMS operators) at the conformal boundary. The bulk side involves a quantum 1-loop correction to the SUGRA action and the boundary counterpart accounts for a sub-leading term in the large-N limit. The sequence of GJMS conformal Laplacians shows up in the two-point function of the CFT operator dual to a bulk scalar field at certain values of its scaling dimension. The restriction to conformally flat boundary metrics reduces the bulk computation to that of volume renormalization which renders the universal type A anomaly. In this way, we directly connect two chief roles of the Q-curvature: the main term in Polyakov formulas on one hand, and its relation to the Poincare metrics of the Fefferman-Graham construction, on the other hand. We find agreement with previously conjectured patterns including a generic and simple formula for the type A anomaly coeffi- cient that matches all reported values in the literature concerning GJMS operators, to our knowledge. Keywords: AdS-CFT Correspondence, Anomalies in Field and String Theories.
    [Show full text]
  • Applications of the Reduction of Couplings∗
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Applications of the Reduction of Couplings∗ Dedicated to Professor Wolfhart Zimmermann on the occasion of his 70th birthday Jisuke Kubo Institute for Theoretical Physics, Kanazawa University Kanazawa 920-1192, Japan Abstract Applications of the principle of reduction of couplings to the standard model and supersym- metric grand unified theories are reviewed. Phenomenological applications of renormalization group invariant sum rules for soft supersymmetry-breaking parameters are also reviewed. ∗ To appear in the Proceedings of the Ringberg Symposium on Quantum Field Theory, Tegernsee, Germany, June 1998. 1 Application to the Standard Model High energy physicists have been using renormalizability as the predictive tool, and also to decide whether or not a quantity is calculable. As we have learned in the previous talk by Professor Oehme, it is possible, using the method of reduction of couplings [1, 2, 3], to renormalize a theory with fewer number of counter terms then usually counted, implying that the traditional notion of renormalizability should be generalized in a certain sense 1. Consequently, the notion of the predictability and the calculability [5] may also be generalized with the help of reduction of couplings. Of course, whether the generalizations of these notions have anything to do with nature is another question. The question can be answered if one applies the idea of reduction of couplings to realistic models, make predictions that are specific for reduction of couplings, and then wait till experimentalists find positive results 2. In 1984 Professor Zimmermann, Klaus Sibold and myself [8] began to apply the idea of reduction of couplings to the SU(3) SU(2) U(1) gauge model for the strong and C × L × Y electroweak interactions.
    [Show full text]
  • 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.
    [Show full text]