Mathematical Developments in the Rise of Yang-Mills Gauge Theories
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Renormalization Group Flows from Holography; Supersymmetry and a C-Theorem
© 1999 International Press Adv. Theor. Math. Phys. 3 (1999) 363-417 Renormalization Group Flows from Holography; Supersymmetry and a c-Theorem D. Z. Freedmana, S. S. Gubser6, K. Pilchc and N. P. Warnerd aDepartment of Mathematics and Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 b Department of Physics, Harvard University, Cambridge, MA 02138, USA c Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-0484, USA d Theory Division, CERN, CH-1211 Geneva 23, Switzerland Abstract We obtain first order equations that determine a super symmetric kink solution in five-dimensional Af = 8 gauged supergravity. The kink interpo- lates between an exterior anti-de Sitter region with maximal supersymme- try and an interior anti-de Sitter region with one quarter of the maximal supersymmetry. One eighth of supersymmetry is preserved by the kink as a whole. We interpret it as describing the renormalization group flow in J\f = 4 super-Yang-Mills theory broken to an Af = 1 theory by the addition of a mass term for one of the three adjoint chiral superfields. A detailed cor- respondence is obtained between fields of bulk supergravity in the interior anti-de Sitter region and composite operators of the infrared field theory. We also point out that the truncation used to find the reduced symmetry critical point can be extended to obtain a new Af = 4 gauged supergravity theory holographically dual to a sector of Af = 2 gauge theories based on quiver diagrams. e-print archive: http.y/xxx.lanl.gov/abs/hep-th/9904017 *On leave from Department of Physics and Astronomy, USC, Los Angeles, CA 90089 364 RENORMALIZATION GROUP FLOWS We consider more general kink geometries and construct a c-function that is positive and monotonic if a weak energy condition holds in the bulk gravity theory. -
Conformal Symmetry in Field Theory and in Quantum Gravity
universe Review Conformal Symmetry in Field Theory and in Quantum Gravity Lesław Rachwał Instituto de Física, Universidade de Brasília, Brasília DF 70910-900, Brazil; [email protected] Received: 29 August 2018; Accepted: 9 November 2018; Published: 15 November 2018 Abstract: Conformal symmetry always played an important role in field theory (both quantum and classical) and in gravity. We present construction of quantum conformal gravity and discuss its features regarding scattering amplitudes and quantum effective action. First, the long and complicated story of UV-divergences is recalled. With the development of UV-finite higher derivative (or non-local) gravitational theory, all problems with infinities and spacetime singularities might be completely solved. Moreover, the non-local quantum conformal theory reveals itself to be ghost-free, so the unitarity of the theory should be safe. After the construction of UV-finite theory, we focused on making it manifestly conformally invariant using the dilaton trick. We also argue that in this class of theories conformal anomaly can be taken to vanish by fine-tuning the couplings. As applications of this theory, the constraints of the conformal symmetry on the form of the effective action and on the scattering amplitudes are shown. We also remark about the preservation of the unitarity bound for scattering. Finally, the old model of conformal supergravity by Fradkin and Tseytlin is briefly presented. Keywords: quantum gravity; conformal gravity; quantum field theory; non-local gravity; super- renormalizable gravity; UV-finite gravity; conformal anomaly; scattering amplitudes; conformal symmetry; conformal supergravity 1. Introduction From the beginning of research on theories enjoying invariance under local spacetime-dependent transformations, conformal symmetry played a pivotal role—first introduced by Weyl related changes of meters to measure distances (and also due to relativity changes of periods of clocks to measure time intervals). -
Effective Quantum Field Theories Thomas Mannel Theoretical Physics I (Particle Physics) University of Siegen, Siegen, Germany
Generating Functionals Functional Integration Renormalization Introduction to Effective Quantum Field Theories Thomas Mannel Theoretical Physics I (Particle Physics) University of Siegen, Siegen, Germany 2nd Autumn School on High Energy Physics and Quantum Field Theory Yerevan, Armenia, 6-10 October, 2014 T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization Overview Lecture 1: Basics of Quantum Field Theory Generating Functionals Functional Integration Perturbation Theory Renormalization Lecture 2: Effective Field Thoeries Effective Actions Effective Lagrangians Identifying relevant degrees of freedom Renormalization and Renormalization Group T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization Lecture 3: Examples @ work From Standard Model to Fermi Theory From QCD to Heavy Quark Effective Theory From QCD to Chiral Perturbation Theory From New Physics to the Standard Model Lecture 4: Limitations: When Effective Field Theories become ineffective Dispersion theory and effective field theory Bound Systems of Quarks and anomalous thresholds When quarks are needed in QCD É. T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization Lecture 1: Basics of Quantum Field Theory Thomas Mannel Theoretische Physik I, Universität Siegen f q f et Yerevan, October 2014 T. Mannel, Siegen University Effective Quantum -
Renormalization Group Flows and Continual Lie Algebras
hep–th/0307154 July 2003 Renormalization group flows and continual Lie algebras Ioannis Bakas Department of Physics, University of Patras GR-26500 Patras, Greece [email protected] Abstract We study the renormalization group flows of two-dimensional metrics in sigma models us- ing the one-loop beta functions, and demonstrate that they provide a continual analogue of the Toda field equations in conformally flat coordinates. In this algebraic setting, the logarithm of the world-sheet length scale, t, is interpreted as Dynkin parameter on the root system of a novel continual Lie algebra, denoted by (d/dt; 1), with anti-symmetric G Cartan kernel K(t, t′) = δ′(t t′); as such, it coincides with the Cartan matrix of the − superalgebra sl(N N + 1) in the large N limit. The resulting Toda field equation is a | non-linear generalization of the heat equation, which is integrable in target space and arXiv:hep-th/0307154v1 17 Jul 2003 shares the same dissipative properties in time, t. We provide the general solution of the renormalization group flows in terms of free fields, via B¨acklund transformations, and present some simple examples that illustrate the validity of their formal power series expansion in terms of algebraic data. We study in detail the sausage model that arises as geometric deformation of the O(3) sigma model, and give a new interpretation to its ultra-violet limit by gluing together two copies of Witten’s two-dimensional black hole in the asymptotic region. We also provide some new solutions that describe the renormal- ization group flow of negatively curved spaces in different patches, which look like a cane in the infra-red region. -
POLCHINSKI* L.Vman L,Ahoratorv ~!/"Phvs,W
Nuclear Physics B231 (lq84) 269-295 ' Norlh-tlolland Publishing Company RENORMALIZATION AND EFFECTIVE LAGRANGIANS Joseph POLCHINSKI* l.vman l,ahoratorv ~!/"Phvs,w. Ilarcard Unn'er.~itv. ('amhrtdge. .'Was,~adntsett,s 0213b¢. USA Received 27 April 1983 There is a strong intuitive understanding of renormalization, due to Wilson, in terms of the scaling of effective lagrangians. We show that this can be made the basis for a proof of perturbative renormalization. We first study rcnormalizabilit,.: in the language of renormalization group flows for a toy renormalization group equation. We then derive an exact renormalization group equation for a four-dimensional X4,4 theorv with a momentum cutoff. Wc organize the cutoff dependence of the effective lagrangian into relevant and irrelevant parts, and derive a linear equation for the irrelevant part. A length} but straightforward argument establishes that the piece identified as irrelevant actually is so in perturbation theory. This implies renormalizabilitv The method extends immediately to an}' system in which a momentum-space cutoff can bc used. but the principle is more general and should apply for any physical cutoff. Neither Weinberg's theorem nor arguments based on the topology of graphs are needed. I. Introduction The understanding of renormalization has advanced greatly in the past two decades. Originally it was just a means of removing infinities from perturbative calculations. The question of why nature should be described by a renormalizable theory was not addressed. These were simply the only theories in which calculations could be done. A great improvement comes when one takes seriously the idea of a physical cutoff at a very large energy scale A. -
Notes on Statistical Field Theory
Lecture Notes on Statistical Field Theory Kevin Zhou [email protected] These notes cover statistical field theory and the renormalization group. The primary sources were: • Kardar, Statistical Physics of Fields. A concise and logically tight presentation of the subject, with good problems. Possibly a bit too terse unless paired with the 8.334 video lectures. • David Tong's Statistical Field Theory lecture notes. A readable, easygoing introduction covering the core material of Kardar's book, written to seamlessly pair with a standard course in quantum field theory. • Goldenfeld, Lectures on Phase Transitions and the Renormalization Group. Covers similar material to Kardar's book with a conversational tone, focusing on the conceptual basis for phase transitions and motivation for the renormalization group. The notes are structured around the MIT course based on Kardar's textbook, and were revised to include material from Part III Statistical Field Theory as lectured in 2017. Sections containing this additional material are marked with stars. The most recent version is here; please report any errors found to [email protected]. 2 Contents Contents 1 Introduction 3 1.1 Phonons...........................................3 1.2 Phase Transitions......................................6 1.3 Critical Behavior......................................8 2 Landau Theory 12 2.1 Landau{Ginzburg Hamiltonian.............................. 12 2.2 Mean Field Theory..................................... 13 2.3 Symmetry Breaking.................................... 16 3 Fluctuations 19 3.1 Scattering and Fluctuations................................ 19 3.2 Position Space Fluctuations................................ 20 3.3 Saddle Point Fluctuations................................. 23 3.4 ∗ Path Integral Methods.................................. 24 4 The Scaling Hypothesis 29 4.1 The Homogeneity Assumption............................... 29 4.2 Correlation Lengths.................................... 30 4.3 Renormalization Group (Conceptual).......................... -
Perturbative Algebraic Quantum Field Theory at Finite Temperature
Perturbative Algebraic Quantum Field Theory at Finite Temperature Dissertation zur Erlangung des Doktorgrades des Fachbereichs Physik der Universität Hamburg vorgelegt von Falk Lindner aus Zittau Hamburg 2013 Gutachter der Dissertation: Prof. Dr. K. Fredenhagen Prof. Dr. D. Bahns Gutachter der Disputation: Prof. Dr. K. Fredenhagen Prof. Dr. J. Louis Datum der Disputation: 01. 07. 2013 Vorsitzende des Prüfungsausschusses: Prof. Dr. C. Hagner Vorsitzender des Promotionsausschusses: Prof. Dr. P. Hauschildt Dekan der Fakultät für Mathematik, Informatik und Naturwissenschaften: Prof. Dr. H. Graener Zusammenfassung Der algebraische Zugang zur perturbativen Quantenfeldtheorie in der Minkowskiraum- zeit wird vorgestellt, wobei ein Schwerpunkt auf die inhärente Zustandsunabhängig- keit des Formalismus gelegt wird. Des Weiteren wird der Zustandsraum der pertur- bativen QFT eingehend untersucht. Die Dynamik wechselwirkender Theorien wird durch ein neues Verfahren konstruiert, das die Gültigkeit des Zeitschichtaxioms in der kausalen Störungstheorie systematisch ausnutzt. Dies beleuchtet einen bisher un- bekannten Zusammenhang zwischen dem statistischen Zugang der Quantenmechanik und der perturbativen Quantenfeldtheorie. Die entwickelten Methoden werden zur ex- pliziten Konstruktion von KMS- und Vakuumzuständen des wechselwirkenden, mas- siven Klein-Gordon Feldes benutzt und damit mögliche Infrarotdivergenzen der Theo- rie, also insbesondere der wechselwirkenden Wightman- und zeitgeordneten Funktio- nen des wechselwirkenden Feldes ausgeschlossen. Abstract We present the algebraic approach to perturbative quantum field theory for the real scalar field in Minkowski spacetime. In this work we put a special emphasis on the in- herent state-independence of the framework and provide a detailed analysis of the state space. The dynamics of the interacting system is constructed in a novel way by virtue of the time-slice axiom in causal perturbation theory. -
Regularization and Renormalization of Non-Perturbative Quantum Electrodynamics Via the Dyson-Schwinger Equations
University of Adelaide School of Chemistry and Physics Doctor of Philosophy Regularization and Renormalization of Non-Perturbative Quantum Electrodynamics via the Dyson-Schwinger Equations by Tom Sizer Supervisors: Professor A. G. Williams and Dr A. Kızılers¨u March 2014 Contents 1 Introduction 1 1.1 Introduction................................... 1 1.2 Dyson-SchwingerEquations . .. .. 2 1.3 Renormalization................................. 4 1.4 Dynamical Chiral Symmetry Breaking . 5 1.5 ChapterOutline................................. 5 1.6 Notation..................................... 7 2 Canonical QED 9 2.1 Canonically Quantized QED . 9 2.2 FeynmanRules ................................. 12 2.3 Analysis of Divergences & Weinberg’s Theorem . 14 2.4 ElectronPropagatorandSelf-Energy . 17 2.5 PhotonPropagatorandPolarizationTensor . 18 2.6 ProperVertex.................................. 20 2.7 Ward-TakahashiIdentity . 21 2.8 Skeleton Expansion and Dyson-Schwinger Equations . 22 2.9 Renormalization................................. 25 2.10 RenormalizedPerturbationTheory . 27 2.11 Outline Proof of Renormalizability of QED . 28 3 Functional QED 31 3.1 FullGreen’sFunctions ............................. 31 3.2 GeneratingFunctionals............................. 33 3.3 AbstractDyson-SchwingerEquations . 34 3.4 Connected and One-Particle Irreducible Green’s Functions . 35 3.5 Euclidean Field Theory . 39 3.6 QEDviaFunctionalIntegrals . 40 3.7 Regularization.................................. 42 3.7.1 Cutoff Regularization . 42 3.7.2 Pauli-Villars Regularization . 42 i 3.7.3 Lattice Regularization . 43 3.7.4 Dimensional Regularization . 44 3.8 RenormalizationoftheDSEs ......................... 45 3.9 RenormalizationGroup............................. 49 3.10BrokenScaleInvariance ............................ 53 4 The Choice of Vertex 55 4.1 Unrenormalized Quenched Formalism . 55 4.2 RainbowQED.................................. 57 4.2.1 Self-Energy Derivations . 58 4.2.2 Analytic Approximations . 60 4.2.3 Numerical Solutions . 62 4.3 Rainbow QED with a 4-Fermion Interaction . -
Arxiv:1809.08749V3 [Quant-Ph] 5 Dec 2018 a General Guiding Principle in Building the Theory of Fun- Damental Interactions (See, E.G., [1,3])
Resolution of Gauge Ambiguities in Ultrastrong-Coupling Cavity QED Omar Di Stefano,1 Alessio Settineri,2 Vincenzo Macr`ı,1 Luigi Garziano,1 Roberto Stassi,1 Salvatore Savasta,1, 2, ∗ and Franco Nori1, 3 1Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan 2Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Universit`adi Messina, I-98166 Messina, Italy 3Physics Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA Gauge invariance is the cornerstone of modern quantum field theory [1{4]. Recently, it has been shown that the quantum Rabi model, describing the dipolar coupling between a two-level atom and a quantized electromagnetic field, violates this principle [5{7]. This widely used model describes a plethora of quantum systems and physical processes under different interaction regimes [8,9]. In the ultrastrong coupling regime, it provides predictions which drastically depend on the chosen gauge. This failure is attributed to the finite-level truncation of the matter system. We show that a careful application of the gauge principle is able to restore gauge invariance even for extreme light- matter interaction regimes. The resulting quantum Rabi Hamiltonian in the Coulomb gauge differs significantly from the standard model and provides the same physical results obtained by using the dipole gauge. It contains field operators to all orders that cannot be neglected when the coupling strength is high. These results shed light on subtleties of gauge invariance in the nonperturbative and extreme interaction regimes, which are now experimentally accessible, and solve all the long-lasting controversies arising from gauge ambiguities in the quantum Rabi and Dicke models [5, 10{18]. -
Simulating Quantum Field Theory with A
Simulating quantum field theory with a quantum computer PoS(LATTICE2018)024 John Preskill∗ Institute for Quantum Information and Matter Walter Burke Institute for Theoretical Physics California Institute of Technology, Pasadena CA 91125, USA E-mail: [email protected] Forthcoming exascale digital computers will further advance our knowledge of quantum chromo- dynamics, but formidable challenges will remain. In particular, Euclidean Monte Carlo methods are not well suited for studying real-time evolution in hadronic collisions, or the properties of hadronic matter at nonzero temperature and chemical potential. Digital computers may never be able to achieve accurate simulations of such phenomena in QCD and other strongly-coupled field theories; quantum computers will do so eventually, though I’m not sure when. Progress toward quantum simulation of quantum field theory will require the collaborative efforts of quantumists and field theorists, and though the physics payoff may still be far away, it’s worthwhile to get started now. Today’s research can hasten the arrival of a new era in which quantum simulation fuels rapid progress in fundamental physics. The 36th Annual International Symposium on Lattice Field Theory - LATTICE2018 22-28 July, 2018 Michigan State University, East Lansing, Michigan, USA. ∗Speaker. c Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ Simulating quantum field theory with a quantum computer John Preskill 1. Introduction My talk at Lattice 2018 had two main parts. In the first part I commented on the near-term prospects for useful applications of quantum computing. -
Gauge Principle and QED∗
Gauge Principle and QED¤ Norbert Straumann Institute for Theoretical Physics University of Zurich,¨ Switzerland August, 2005 Abstract One of the major developments of twentieth century physics has been the gradual recognition that a common feature of the known fundamental inter- actions is their gauge structure. In this talk the early history of gauge theory is reviewed, emphasizing especially Weyl’s seminal contributions of 1918 and 1929. 1 Introduction The organizers of this conference asked me to review the early history of gauge theories. Because of space and time limitations I shall concentrate on Weyl’s seminal papers of 1918 and 1929. Important contributions by Fock, Klein and others, based on Kaluza’s five-dimensional unification attempt, will not be dis- cussed. (For this I refer to [30] and [31].) The history of gauge theories begins with GR, which can be regarded as a non- Abelian gauge theory of a special type. To a large extent the other gauge theories emerged in a slow and complicated process gradually from GR. Their common geometrical structure – best expressed in terms of connections of fiber bundles – is now widely recognized. It all began with H. Weyl [2] who made in 1918 the first attempt to extend GR in order to describe gravitation and electromagnetism within a unifying geomet- rical framework. This brilliant proposal contains the germs of all mathematical ¤Invited talk at PHOTON2005, The Photon: Its First Hundred Years and the Future, 31.8- 04.09, 2005, Warsaw. 1 aspects of non-Abelian gauge theory. The word ‘gauge’ (german: ‘Eich-’) trans- formation appeared for the first time in this paper, but in the everyday meaning of change of length or change of calibration. -
Quantum Mechanics of Topological Solitons
Imperial College London Department of Physics Quantum mechanics of topological solitons David J. Weir September 2011 Supervised by Arttu Rajantie Submitted in part fulfilment of the requirements for the degree of Doctor of Philosophy in Physics of Imperial College London and the Diploma of Imperial College London 1 Declaration I herewith certify that all material in this dissertation which is not my own work has been properly acknowledged. David J. Weir 3 Abstract Topological solitons { are of broad interest in physics. They are objects with localised energy and stability ensured by their topological properties. It is possible to create them during phase transitions which break some sym- metry in a frustrated system. They are ubiquitous in condensed matter, ranging from monopole excitations in spin ices to vortices in superconduc- tors. In such situations, their behaviour has been extensively studied. Less well understood and yet equally interesting are the symmetry-breaking phase transitions that could produce topological defects is the early universe. Grand unified theories generically admit the creation of cosmic strings and monopoles, amongst other objects. There is no reason to expect that the behaviour of such objects should be classical or, indeed, supersymmetric, so to fully understand the behaviour of these theories it is necessary to study the quantum properties of the associated topological defects. Unfortunately, the standard analytical tools for studying quantum field theory { including perturbation theory { do not work so well when applied to topological defects. Motivated by this realisation, this thesis presents numerical techniques for the study of topological solitons in quantum field theory. Calculations are carried out nonperturbatively within the framework of lattice Monte Carlo simulations.