Quantum Scalar Field Theory in Ads and the Ads/CFT Correspondence
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Perturbative Versus Non-Perturbative Quantum Field Theory: Tao’S Method, the Casimir Effect, and Interacting Wightman Theories
universe Review Perturbative versus Non-Perturbative Quantum Field Theory: Tao’s Method, the Casimir Effect, and Interacting Wightman Theories Walter Felipe Wreszinski Instituto de Física, Universidade de São Paulo, São Paulo 05508-090, SP, Brazil; [email protected] Abstract: We dwell upon certain points concerning the meaning of quantum field theory: the problems with the perturbative approach, and the question raised by ’t Hooft of the existence of the theory in a well-defined (rigorous) mathematical sense, as well as some of the few existent mathematically precise results on fully quantized field theories. Emphasis is brought on how the mathematical contributions help to elucidate or illuminate certain conceptual aspects of the theory when applied to real physical phenomena, in particular, the singular nature of quantum fields. In a first part, we present a comprehensive review of divergent versus asymptotic series, with qed as background example, as well as a method due to Terence Tao which conveys mathematical sense to divergent series. In a second part, we apply Tao’s method to the Casimir effect in its simplest form, consisting of perfectly conducting parallel plates, arguing that the usual theory, which makes use of the Euler-MacLaurin formula, still contains a residual infinity, which is eliminated in our approach. In the third part, we revisit the general theory of nonperturbative quantum fields, in the form of newly proposed (with Christian Jaekel) Wightman axioms for interacting field theories, with applications to “dressed” electrons in a theory with massless particles (such as qed), as well as Citation: Wreszinski, W.F. unstable particles. Various problems (mostly open) are finally discussed in connection with concrete Perturbative versus Non-Perturbative models. -
Arxiv:1910.11245V1 [Hep-Ph] 24 Oct 2019 Inclusion of a Magnetic field Can Affect This Result Since the Magnetic field Contribute to the Charge Screening
Magnetic Field Effect in the Fine-Structure Constant and Electron Dynamical Mass E. J. Ferrer1 and A. Sanchez2 1Dept. of Physics and Astronomy, Univ. of Texas Rio Grande Valley, 1201 West University Dr., Edinburg, TX 78539 and CUNY-Graduate Center, New York 10314, USA 2Facultad de Ciencias, Universidad Nacional Aut´onomade M´exico, Apartado Postal 50-542, Ciudad de M´exico 04510, Mexico. We investigate the effect of an applied constant and uniform magnetic field in the fine-structure constant of massive and massless QED. In massive QED, it is shown that a strong magnetic field removes the so called Landau pole and that the fine-structure constant becomes anisotropic having different values along and transverse to the field direction. Contrary to other results in the literature, we find that the anisotropic fine-structure constant always decreases with the field. We also study the effect of the running of the coupling constant with the magnetic field on the electron mass. We find that in both cases of massive and massless QED, the electron dynamical mass always decreases with the magnetic field, what can be interpreted as an inverse magnetic catalysis effect. PACS numbers: 11.30.Rd, 12.20.-m, 03.75.Hh, 03.70.+k I. INTRODUCTION The effect of magnetic fields on the different properties of quantum particles has always attracted great interest [1]. At present, it has been reinforced by the fact that there is the capability to generate very strong magnetic fields in non-central heavy ion collisions, and because of the discovery of strongly magnetized compact stars, which have been named magnetars. -
Renormalisation Circumvents Haag's Theorem
Some no-go results & canonical quantum fields Axiomatics & Haag's theorem Renormalisation bypasses Haag's theorem Renormalisation circumvents Haag's theorem Lutz Klaczynski, HU Berlin Theory seminar DESY Zeuthen, 31.03.2016 Lutz Klaczynski, HU Berlin Renormalisation circumvents Haag's theorem Some no-go results & canonical quantum fields Axiomatics & Haag's theorem Renormalisation bypasses Haag's theorem Outline 1 Some no-go results & canonical quantum fields Sharp-spacetime quantum fields Interaction picture Theorems by Powers & Baumann (CCR/CAR) Schrader's result 2 Axiomatics & Haag's theorem Wightman axioms Haag vs. Gell-Mann & Low Haag's theorem for nontrivial spin 3 Renormalisation bypasses Haag's theorem Renormalisation Mass shift wrecks unitary equivalence Conclusion Lutz Klaczynski, HU Berlin Renormalisation circumvents Haag's theorem Sharp-spacetime quantum fields Some no-go results & canonical quantum fields Interaction picture Axiomatics & Haag's theorem Theorems by Powers & Baumann (CCR/CAR) Renormalisation bypasses Haag's theorem Schrader's result The quest for understanding canonical quantum fields 1 1 Constructive approaches by Glimm & Jaffe : d ≤ 3 2 2 Axiomatic quantum field theory by Wightman & G˚arding : axioms to construe quantum fields in terms of operator theory hard work, some results (eg PCT, spin-statistics theorem) interesting: triviality results (no-go theorems) 3 other axiomatic approaches, eg algebraic quantum field theory General form of no-go theorems φ quantum field with properties so-and-so ) φ is trivial. 3 forms -
275 – Quantum Field Theory
Quantum eld theory Lectures by Richard Borcherds, notes by Søren Fuglede Jørgensen Math 275, UC Berkeley, Fall 2010 Contents 1 Introduction 1 1.1 Dening a QFT . 1 1.2 Constructing a QFT . 2 1.3 Feynman diagrams . 4 1.4 Renormalization . 4 1.5 Constructing an interacting theory . 6 1.6 Anomalies . 6 1.7 Infrared divergences . 6 1.8 Summary of problems and solutions . 7 2 Review of Wightman axioms for a QFT 7 2.1 Free eld theories . 8 2.2 Fixing the Wightman axioms . 10 2.2.1 Reformulation in terms of distributions and states . 10 2.2.2 Extension of the Wightman axioms . 12 2.3 Free eld theory revisited . 12 2.4 Propagators . 13 2.5 Review of properties of distributions . 13 2.6 Construction of a massive propagator . 14 2.7 Wave front sets . 17 Disclaimer These are notes from the rst 4 weeks of a course given by Richard Borcherds in 2010.1 They are discontinued from the 7th lecture due to time constraints. They have been written and TeX'ed during the lecture and some parts have not been completely proofread, so there's bound to be a number of typos and mistakes that should be attributed to me rather than the lecturer. Also, I've made these notes primarily to be able to look back on what happened with more ease, and to get experience with TeX'ing live. That being said, feel very free to send any comments and or corrections to [email protected]. 1st lecture, August 26th 2010 1 Introduction 1.1 Dening a QFT The aim of the course is to dene a quantum eld theory, to nd out what a quantum eld theory is, and to dene Feynman measures, renormalization, anomalies, and regularization. -
The Renormalization Group in Quantum Field Theory
COARSE-GRAINING AS A ROUTE TO MICROSCOPIC PHYSICS: THE RENORMALIZATION GROUP IN QUANTUM FIELD THEORY BIHUI LI DEPARTMENT OF HISTORY AND PHILOSOPHY OF SCIENCE UNIVERSITY OF PITTSBURGH [email protected] ABSTRACT. The renormalization group (RG) has been characterized as merely a coarse-graining procedure that does not illuminate the microscopic content of quantum field theory (QFT), but merely gets us from that content, as given by axiomatic QFT, to macroscopic predictions. I argue that in the constructive field theory tradition, RG techniques do illuminate the microscopic dynamics of a QFT, which are not automatically given by axiomatic QFT. RG techniques in constructive field theory are also rigorous, so one cannot object to their foundational import on grounds of lack of rigor. Date: May 29, 2015. 1 Copyright Philosophy of Science 2015 Preprint (not copyedited or formatted) Please use DOI when citing or quoting 1. INTRODUCTION The renormalization group (RG) in quantum field theory (QFT) has received some attention from philosophers for how it relates physics at different scales and how it makes sense of perturbative renormalization (Huggett and Weingard 1995; Bain 2013). However, it has been relatively neglected by philosophers working in the axiomatic QFT tradition, who take axiomatic QFT to be the best vehicle for interpreting QFT. Doreen Fraser (2011) has argued that the RG is merely a way of getting from the microscopic principles of QFT, as provided by axiomatic QFT, to macroscopic experimental predictions. Thus, she argues, RG techniques do not illuminate the theoretical content of QFT, and we should stick to interpreting axiomatic QFT. David Wallace (2011), in contrast, has argued that the RG supports an effective field theory (EFT) interpretation of QFT, in which QFT does not apply to arbitrarily small length scales. -
Quantum Field Theory a Tourist Guide for Mathematicians
Mathematical Surveys and Monographs Volume 149 Quantum Field Theory A Tourist Guide for Mathematicians Gerald B. Folland American Mathematical Society http://dx.doi.org/10.1090/surv/149 Mathematical Surveys and Monographs Volume 149 Quantum Field Theory A Tourist Guide for Mathematicians Gerald B. Folland American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Jerry L. Bona Michael G. Eastwood Ralph L. Cohen Benjamin Sudakov J. T. Stafford, Chair 2010 Mathematics Subject Classification. Primary 81-01; Secondary 81T13, 81T15, 81U20, 81V10. For additional information and updates on this book, visit www.ams.org/bookpages/surv-149 Library of Congress Cataloging-in-Publication Data Folland, G. B. Quantum field theory : a tourist guide for mathematicians / Gerald B. Folland. p. cm. — (Mathematical surveys and monographs ; v. 149) Includes bibliographical references and index. ISBN 978-0-8218-4705-3 (alk. paper) 1. Quantum electrodynamics–Mathematics. 2. Quantum field theory–Mathematics. I. Title. QC680.F65 2008 530.1430151—dc22 2008021019 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. -
An Introduction to Rigorous Formulations of Quantum Field Theory
An introduction to rigorous formulations of quantum field theory Daniel Ranard Perimeter Institute for Theoretical Physics Advisors: Ryszard Kostecki and Lucien Hardy May 2015 Abstract No robust mathematical formalism exists for nonperturbative quantum field theory. However, the attempt to rigorously formulate field theory may help one understand its structure. Multiple approaches to axiomatization are discussed, with an emphasis on the different conceptual pictures that inspire each approach. 1 Contents 1 Introduction 3 1.1 Personal perspective . 3 1.2 Goals and outline . 3 2 Sketches of quantum field theory 4 2.1 Locality through tensor products . 4 2.2 Schr¨odinger(wavefunctional) representation . 6 2.3 Poincar´einvariance and microcausality . 9 2.4 Experimental predictions . 12 2.5 Path integral picture . 13 2.6 Particles and the Fock space . 17 2.7 Localized particles . 19 3 Seeking rigor 21 3.1 Hilbert spaces . 21 3.2 Fock space . 23 3.3 Smeared fields . 23 3.4 From Hilbert space to algebra . 25 3.5 Euclidean path integral . 30 4 Axioms for quantum field theory 31 4.1 Wightman axioms . 32 4.2 Haag-Kastler axioms . 32 4.3 Osterwalder-Schrader axioms . 33 4.4 Consequences of the axioms . 34 5 Outlook 35 6 Acknowledgments 35 2 1 Introduction 1.1 Personal perspective What is quantum field theory? Rather than ask how nature truly acts, simply ask: what is this theory? For a moment, strip the physical theory of its interpretation. What remains is the abstract mathematical arena in which one performs calculations. The theory of general relativity becomes geometry on a Lorentzian manifold; quantum theory becomes the analysis of Hilbert spaces and self-adjoint operators. -
Quantum Mechanics Quantum Field Theory(QFT)
Quantum Mechanics_quantum field theory(QFT) In theoretical physics, quantum field theory(QFT) is a theoretical framework for constructing quantum mechanical models of subatomic particles in particle physics and quasiparticles incondensed matter physics. A QFT treats particles as excited states of an underlying physical field, so these are called field quanta. For example, Quantum electrodynamics (QED) has one electron field and one photon field;Quantum chromodynamics (QCD) has one field for each type of quark; and, in condensed matter, there is an atomic displacement field that gives rise to phonon particles. Edward Witten describes QFT as "by far" the most difficult theory in modern physics.[1] In QFT, quantum mechanical interactions between particles are described by interaction terms between the corresponding underlying fields. QFT interaction terms are similar in spirit to those between charges with electric and magnetic fields in Maxwell's equations. However, unlike the classical fields of Maxwell's theory, fields in QFT generally exist in quantum superpositions of states and are subject to the laws of quantum mechanics. Quantum mechanical systems have a fixed number of particles, with each particle having a finite number of degrees of freedom. In contrast, the excited states of a QFT can represent any number of particles. This makes quantum field theories especially useful for describing systems where the particle count/number may change over time, a crucial feature of relativistic dynamics. Because the fields are continuous quantities over space, there exist excited states with arbitrarily large numbers of particles in them, providing QFT systems with an effectively infinite number of degrees of freedom. -
Ed Nelson's Work in Quantum Theory 1
ED NELSON’S WORK IN QUANTUM THEORY BARRY SIMON¤ Abstract. We review Edward Nelson’s contributions to nonrel- ativistic quantum theory and to quantum field theory. 1. Introduction It is a pleasure to contribute to this celebration of Ed Nelson’s scien- tific work, not only because of the importance of that work but because it allows me an opportunity to express my gratitude and acknowledge my enormous debt to Ed. He and Arthur Wightman were the key for- mulative influences on my education, not only as a graduate student but during my early postdoctoral years. Thanks, Ed! I was initially asked to talk about Ed’s work in quantum field theory (QFT), but I’m going to exceed my assignment by also discussing Ed’s impact on conventional nonrelativistic quantum mechanics (NRQM). There will be other talks on his work on unconventional quantum the- ory. After discussion of NRQM and the Nelson model, I’ll turn to the truly great contributions: the first control of a renormalization, albeit the Wick ordering that is now regarded as easy, and the seminal work on Euclidean QFT. Many important ideas I’ll discuss below involve crucial remarks of Ed that — in his typically generous fashion — he allowed others to publish. 2. NRQM Ed has very little published specifically on conventional NRQM but he had substantial impact through his lectures, students, and ideas from his papers that motivated work on NRQM. In particular, two of Date: August 12, 2004. ¤ Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125, USA. E-mail: [email protected]. -
Solutions in Constructive Field Theory
Solutions in Constructive Field Theory Bihui Li [email protected] April 26, 2016 Abstract Constructive field theory aims to rigorously construct concrete solutions to Lagrangians used in particle physics, where the solutions satisfy some relevant set of axioms. I examine the nature of solutions in constructive field theory and their relationship to both axiomatic and Lagrangian quantum field theory (QFT). I argue that Lagrangian QFT provides conditions for what counts as a successful constructive solution, and that it provides other important information that guides constructive field theorists to solutions. Solutions matter because they describe the behavior of systems in QFT, and thus what QFT says the world is like. Constructive field theory, in incorporating ingredients from both axiomatic and Lagrangian QFT, clarifies existing disputes about which parts of QFT are relevant to philosophers and the role of rigor in these disputes. 1 1 Introduction To date, philosophers of quantum field theory (QFT) have paid much attention to roughly two kinds of QFT: axiomatic approaches to algebraic QFT (Halvorson & M¨uger,2006), and Lagrangian-based QFT as used by particle physicists (Wallace, 2006). Comparatively less attention, however, has been paid to constructive QFT, an approach that aims to rigorously construct solutions of QFT for Lagrangians and Hamiltons that are important in particle physics, ensuring that such solutions satisfy certain axioms. In doing so, constructive QFT mediates between axiomatic approaches to QFT (of which algebraic QFT is one kind) and physicists' Lagrangian-based QFT. It ensures that the various axiom systems have a physically meaningful correspondence with the world. Since we usually take solutions in physical theories to describe the behavior of systems falling under the theory and constructive QFT aims to produce these solutions, constructive QFT deserves some philosophical attention. -
Footnotes Section 1 1) a Vivid Account of the of the Idea, Influenced
Footnotes Section 1 1) A vivid account of the of the idea, influenced particularly by remarks of Dirac [2J, has been given by FeYnman himself in [3J. 2) For the physical foundation see the original work of FeYnman and the book by FeYnman and Hibbs (Ref. [1J). Also e s g , [7J. 3) Actually, Holder continuous of index 1ess than 1/2 , see e s g , [8J. 4) For the definition by a "sequential limit", in more general situations, see e.g. [11J. 5) See also e.g. the references given in [41J and [42J. 6) The results are also briefly announced in [42J. 7) A theory of related pseudomeasures has inbetween been developed by P. Kree. See e.g. and references therein. 8) Besides the topics touched in this brief historical sketch of the mathematical study of FeYnman path integrals of non relativistic quantum mechanics there are others we did not mention, either because they concern problems other than those tackled later in this work or because no clear cut mathematical re sults are available. Let us mention however three more areas in which Feynman path integrals have been discussed and used, at least heuristically. a) Questions of the relation between FeYnman's quantization and the usual one: see e.g. [10J,6), [31J, [35J. b) FeYnman's path integrals on functions defined on manifolds other than Euclidean space, in particular for spin part icles. Attempts Using the sequential limit and analytic continuation approaches have been discussed to some extent, see e.g. [6J,7), [36J and references given therein. -
Classical and Quantum Integrability of 2D Dilaton Gravities in Euclidean Space
INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY Class. Quantum Grav. 22 (2005) 1361–1381 doi:10.1088/0264-9381/22/7/010 Classical and quantum integrability of 2D dilaton gravities in Euclidean space L Bergamin1, D Grumiller1,2, W Kummer1 and D V Vassilevich2,3 1 Institute for Theoretical Physics, TU Vienna, Wiedner Hauptstr. 8–10/136, A-1040 Vienna, Austria 2 Institute for Theoretical Physics, University of Leipzig, Augustusplatz 10–11, D-04103 Leipzig, Germany 3 V A Fock Institute of Physics, St Petersburg University, St Petersburg, Russia E-mail: [email protected], [email protected], [email protected] and [email protected] Received 2 December 2004, in final form 4 February 2005 Published 15 March 2005 Online at stacks.iop.org/CQG/22/1361 Abstract Euclidean dilaton gravity in two dimensions is studied exploiting its representation as a complexified first order gravity model. All local classical solutions are obtained. A global discussion reveals that for a given model only a restricted class of topologies is consistent with the metric and the dilaton. A particular case of string motivated Liouville gravity is studied in detail. Path integral quantization in generic Euclidean dilaton gravity is performed non-perturbatively by analogy to the Minkowskian case. PACS numbers: 04.20.Gz, 04.60.Kz 1. Introduction Two-dimensional gravities have attracted much attention over the last few decades since they retain many properties of their higher-dimensional counterparts but are considerably simpler. It has been demonstrated that all two-dimensional dilaton gravities are not only classically integrable, but this even holds at the (non-perturbative) quantum level (for a recent review cf [1]).