Solutions in Constructive Field Theory
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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. -
Defining Physics at Imaginary Time: Reflection Positivity for Certain
Defining physics at imaginary time: reflection positivity for certain Riemannian manifolds A thesis presented by Christian Coolidge Anderson [email protected] (978) 204-7656 to the Department of Mathematics in partial fulfillment of the requirements for an honors degree. Advised by Professor Arthur Jaffe. Harvard University Cambridge, Massachusetts March 2013 Contents 1 Introduction 1 2 Axiomatic quantum field theory 2 3 Definition of reflection positivity 4 4 Reflection positivity on a Riemannian manifold M 7 4.1 Function space E over M ..................... 7 4.2 Reflection on M .......................... 10 4.3 Reflection positive inner product on E+ ⊂ E . 11 5 The Osterwalder-Schrader construction 12 5.1 Quantization of operators . 13 5.2 Examples of quantizable operators . 14 5.3 Quantization domains . 16 5.4 The Hamiltonian . 17 6 Reflection positivity on the level of group representations 17 6.1 Weakened quantization condition . 18 6.2 Symmetric local semigroups . 19 6.3 A unitary representation for Glor . 20 7 Construction of reflection positive measures 22 7.1 Nuclear spaces . 23 7.2 Construction of nuclear space over M . 24 7.3 Gaussian measures . 27 7.4 Construction of Gaussian measure . 28 7.5 OS axioms for the Gaussian measure . 30 8 Reflection positivity for the Laplacian covariance 31 9 Reflection positivity for the Dirac covariance 34 9.1 Introduction to the Dirac operator . 35 9.2 Proof of reflection positivity . 38 10 Conclusion 40 11 Appendix A: Cited theorems 40 12 Acknowledgments 41 1 Introduction Two concepts dominate contemporary physics: relativity and quantum me- chanics. They unite to describe the physics of interacting particles, which live in relativistic spacetime while exhibiting quantum behavior. -
Mathematisches Forschungsinstitut Oberwolfach Reflection Positivity
Mathematisches Forschungsinstitut Oberwolfach Report No. 55/2017 DOI: 10.4171/OWR/2017/55 Reflection Positivity Organised by Arthur Jaffe, Harvard Karl-Hermann Neeb, Erlangen Gestur Olafsson, Baton Rouge Benjamin Schlein, Z¨urich 26 November – 2 December 2017 Abstract. The main theme of the workshop was reflection positivity and its occurences in various areas of mathematics and physics, such as Representa- tion Theory, Quantum Field Theory, Noncommutative Geometry, Dynamical Systems, Analysis and Statistical Mechanics. Accordingly, the program was intrinsically interdisciplinary and included talks covering different aspects of reflection positivity. Mathematics Subject Classification (2010): 17B10, 22E65, 22E70, 81T08. Introduction by the Organisers The workshop on Reflection Positivity was organized by Arthur Jaffe (Cambridge, MA), Karl-Hermann Neeb (Erlangen), Gestur Olafsson´ (Baton Rouge) and Ben- jamin Schlein (Z¨urich) during the week November 27 to December 1, 2017. The meeting was attended by 53 participants from all over the world. It was organized around 24 lectures each of 50 minutes duration representing major recent advances or introducing to a specific aspect or application of reflection positivity. The meeting was exciting and highly successful. The quality of the lectures was outstanding. The exceptional atmosphere of the Oberwolfach Institute provided the optimal environment for bringing people from different areas together and to create an atmosphere of scientific interaction and cross-fertilization. In particular, people from different subcommunities exchanged ideas and this lead to new col- laborations that will probably stimulate progress in unexpected directions. 3264 Oberwolfach Report 55/2017 Reflection positivity (RP) emerged in the early 1970s in the work of Osterwalder and Schrader as one of their axioms for constructive quantum field theory ensuring the equivalence of their euclidean setup with Wightman fields. -
Partial Differential Equations
CALENDAR OF AMS MEETINGS THIS CALENDAR lists all meetings which have been approved by the Council pnor to the date this issue of the Nouces was sent to press. The summer and annual meetings are joint meetings of the Mathematical Association of America and the Ameri· can Mathematical Society. The meeting dates which fall rather far in the future are subject to change; this is particularly true of meetings to which no numbers have yet been assigned. Programs of the meetings will appear in the issues indicated below. First and second announcements of the meetings will have appeared in earlier issues. ABSTRACTS OF PAPERS presented at a meeting of the Society are published in the journal Abstracts of papers presented to the American Mathematical Society in the issue corresponding to that of the Notices which contains the program of the meet ing. Abstracts should be submitted on special forms which are available in many departments of mathematics and from the office of the Society in Providence. Abstracts of papers to be presented at the meeting must be received at the headquarters of the Society in Providence, Rhode Island, on or before the deadline given below for the meeting. Note that the deadline for ab stracts submitted for consideration for presentation at special sessions is usually three weeks earlier than that specified below. For additional information consult the meeting announcement and the list of organizers of special sessions. MEETING ABSTRACT NUMBER DATE PLACE DEADLINE ISSUE 778 June 20-21, 1980 Ellensburg, Washington APRIL 21 June 1980 779 August 18-22, 1980 Ann Arbor, Michigan JUNE 3 August 1980 (84th Summer Meeting) October 17-18, 1980 Storrs, Connecticut October 31-November 1, 1980 Kenosha, Wisconsin January 7-11, 1981 San Francisco, California (87th Annual Meeting) January 13-17, 1982 Cincinnati, Ohio (88th Annual Meeting) Notices DEADLINES ISSUE NEWS ADVERTISING June 1980 April 18 April 29 August 1980 June 3 June 18 Deadlines for announcements intended for the Special Meetings section are the same as for News. -
6. L CPT Reversal
6. CPT reversal l Précis. On the representation view, there may be weak and strong arrows of time. A weak arrow exists. A strong arrow might too, but not in current physics due to CPT symmetry. There are many ways to reverse time besides the time reversal operator T. Writing P for spatial reflection or ‘parity’, and C for matter-antimatter exchange or ‘charge conjugation’, we find that PT, CT, and CPT are all time reversing transformations, when they exist. More generally, for any non-time-reversing (unitary) transformation U, we find that UT is time reversing. Are any of these transformations relevant for the arrow of time? Feynman (1949) captured many of our imaginations with the proposal that to understand the direction of time, we must actually consider the exchange of matter and antimatter as well.1 Writing about it years later, he said: “A backwards-moving electron when viewed with time moving forwards appears the same as an ordinary electron, except it’s attracted to normal electrons — we say it has positive charge. For this reason it’s called a ‘positron’. The positron is a sister to the electron, and it is an example of an ‘anti-particle’. This phenomenon is quite general. Every particle in Nature has an amplitude to move backwards in time, and therefore has an anti-particle.”(Feynman 1985, p.98) Philosophers Arntzenius and Greaves (2009, p.584) have defended Feynman’s proposal, 1In his Nobel prize speech, Feynman (1972, pg.163) attributes this idea to Wheeler in the development of their absorber theory (Wheeler and Feynman 1945). -
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. -
Arthur Strong Wightman (1922–2013)
Obituary Arthur Strong Wightman (1922–2013) Arthur Wightman, a founding father of modern mathematical physics, passed away on January 13, 2013 at the age of 90. His own scientific work had an enormous impact in clar- ifying the compatibility of relativity with quantum theory in the framework of quantum field theory. But his stature and influence was linked with an enormous cadre of students, scientific collaborators, and friends whose careers shaped fields both in mathematics and theoretical physics. Princeton has a long tradition in mathematical physics, with university faculty from Sir James Jeans through H.P. Robertson, Hermann Weyl, John von Neumann, Eugene Wigner, and Valentine Bargmann, as well as a long history of close collaborations with colleagues at the Institute for Advanced Study. Princeton became a mecca for quantum field theorists as well as other mathematical physicists during the Wightman era. Ever since the advent of “axiomatic quantum field theory”, many researchers flocked to cross the threshold of his open office door—both in Palmer and later in Jadwin—for Arthur was renowned for his generosity in sharing ideas and research directions. In fact, some students wondered whether Arthur might be too generous with his time helping others, to the extent that it took time away from his own research. Arthur had voracious intellectual appetites and breadth of interests. Through his interactions with others and his guidance of students and postdocs, he had profound impact not only on axiomatic and constructive quantum field theory but on the de- velopment of the mathematical approaches to statistical mechanics, classical mechanics, dynamical systems, transport theory, non-relativistic quantum mechanics, scattering the- ory, perturbation of eigenvalues, perturbative renormalization theory, algebraic quantum field theory, representations of C⇤-algebras, classification of von Neumann algebras, and higher spin equations. -
1995 Steele Prizes
steele.qxp 4/27/98 3:29 PM Page 1288 1995 Steele Prizes Three Leroy P. Steele Prizes were presented at The text that follows contains, for each award, the awards banquet during the Summer Math- the committee’s citation, the recipient’s response fest in Burlington, Vermont, in early August. upon receiving the award, and a brief bio- These prizes were established in 1970 in honor graphical sketch of the recipient. of George David Birkhoff, William Fogg Osgood, and William Caspar Graustein Edward Nelson: 1995 Steele Prize for and are endowed under the Seminal Contribution to Research terms of a bequest from The 1995 Leroy P. Steele award for research of Leroy P. Steele. seminal importance goes to Professor Edward The Steele Prizes are Nelson of Princeton University for the following ...for a research awarded in three categories: two papers in mathematical physics character- for a research paper of fun- ized by leaders of the field as extremely innov- paper of damental and lasting impor- ative: tance, for expository writing, 1. “A quartic interaction in two dimensions” in fundamental and for cumulative influence Mathematical Theory of Elementary Particles, and lasting extending over a career, in- MIT Press, 1966, pages 69–73; cluding the education of doc- 2. “Construction of quantum fields from Markoff importance, toral students. The current fields” in Journal of Functional Analysis 12 award is $4,000 in each cate- (1973), 97–112. for expository gory. In these papers he showed for the first time The recipients of the 1995 how to use the powerful tools of probability writing, and for Steele Prizes are Edward Nel- theory to attack the hard analytic questions of son for seminal contribution constructive quantum field theory, controlling cumulative to research, Jean-Pierre Serre renormalizations with Lp estimates in the first influence for mathematical exposition, paper and, in the second, turning Euclidean and John T. -
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.