DOCSLIB.ORG

From Algebraic Quantum Field Theory to Effective Quantum Gravity

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
From Algebraic Quantum Field Theory to Effective Quantum Gravity Algebraic approach to QFT Quantum gravity Locality and beyond: from algebraic quantum field theory to effective quantum gravity Kasia Rejzner University of York Local Quantum Physics and beyond in memoriam Rudolf Haag, 26.09.2016 Kasia Rejzner Locality and beyond 1 / 26 Algebraic approach to QFT Quantum gravity Outline of the talk 1 Algebraic approach to QFT AQFT LCQFT pAQFT 2 Quantum gravity Effective quantum gravity Observables The author of a beautiful book Local Quantum Physics. One of the fathers of LQP. We will all miss him. AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT The father of Local Quantum Physics Rudolf Haag (1922 – 2016). Kasia Rejzner Locality and beyond 2 / 26 One of the fathers of LQP. We will all miss him. AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT The father of Local Quantum Physics Rudolf Haag (1922 – 2016). The author of a beautiful book Local Quantum Physics. Kasia Rejzner Locality and beyond 2 / 26 We will all miss him. AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT The father of Local Quantum Physics Rudolf Haag (1922 – 2016). The author of a beautiful book Local Quantum Physics. One of the fathers of LQP. Kasia Rejzner Locality and beyond 2 / 26 AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT The father of Local Quantum Physics Rudolf Haag (1922 – 2016). The author of a beautiful book Local Quantum Physics. One of the fathers of LQP. We will all miss him. Kasia Rejzner Locality and beyond 2 / 26 It started as the axiomatic framework of Haag-Kastler[ Haag & Kastler 64]: a model is defined by associating to each region O of Minkowski spacetime the algebra A(O) of observables that can be measured in O. The physical notion of subsystems is realized by the condition of isotony, i.e.: O2 ⊃ O1 ) A(O2) ⊃ A(O1). We obtain a net of algebras. AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Local Quantum Physics Algebraic Quantum Field Theory (Local Quantum Physics) is a mathematically rigorous framework, which allows to investigate conceptual problems in QFT. Kasia Rejzner Locality and beyond 3 / 26 The physical notion of subsystems is realized by the condition of isotony, i.e.: O2 ⊃ O1 ) A(O2) ⊃ A(O1). We obtain a net of algebras. AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Local Quantum Physics Algebraic Quantum Field Theory (Local Quantum Physics) is a mathematically rigorous framework, which allows to investigate conceptual problems in QFT. It started as the axiomatic framework of Haag-Kastler[ Haag & Kastler 64]: a model is defined by associating to each region O of Minkowski spacetime the algebra A(O) of observables that can be measured in O. A(O1) O1 Kasia Rejzner Locality and beyond 3 / 26 AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Local Quantum Physics Algebraic Quantum Field Theory (Local Quantum Physics) is a mathematically rigorous framework, which allows to investigate conceptual problems in QFT. It started as the axiomatic framework of Haag-Kastler[ Haag & Kastler 64]: a model is defined by associating to each region O of Minkowski spacetime the algebra A(O) of observables that can be measured in O. The physical notion of subsystems A(O2) ⊃ A(O1) is realized by the condition of isotony, i.e.: O2 ⊃ O1 ) A(O2) ⊃ A(O1). We obtain a net of algebras. O2 ⊃ O1 Kasia Rejzner Locality and beyond 3 / 26 Different aspects of AQFT and relations to physics Renor- malization QFT, PARTICLE COSMOLOGY PHYSICS Quantum gravity KMS states Defor- mations PAQFT MODEL QUANTUM LCQFT STATISTICAL BUILDING QUANTUM MECHANICS CFT in 2D INFORMATION THEORY QFT ON NONCO- MUTATIVE SPACETIMES AQFT NONCOM- ALGEBRAIC MUTATIVE QIT GEOMETRY CONSTRUC- Foundations Entan- TIVE QFT of quantum glement theory Local algebras of observables A(O) are defined abstractly, the Hilbert space representation comes later (this deals with the non-uniqueness of the vacuum). Algebras A(O) are constructed using only the local data. Local features of the theory (observables) are separated from the global features (states). Main advantages The corresponding generalization of AQFT is called locally covariant quantum field theory[ Hollands-Wald 01, Brunetti-Fredenhagen-Verch 01, Fewster-Verch 12,. ]. AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Algebraic QFT on curved spacetimes Conceptual problems of QFT on curved spacetimes can be easily solved in the algebraic approach, because of the powerful principle of locality. Kasia Rejzner Locality and beyond 5 / 26 Local algebras of observables A(O) are defined abstractly, the Hilbert space representation comes later (this deals with the non-uniqueness of the vacuum). Algebras A(O) are constructed using only the local data. Local features of the theory (observables) are separated from the global features (states). Main advantages AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Algebraic QFT on curved spacetimes Conceptual problems of QFT on curved spacetimes can be easily solved in the algebraic approach, because of the powerful principle of locality. The corresponding generalization of AQFT is called locally covariant quantum field theory[ Hollands-Wald 01, Brunetti-Fredenhagen-Verch 01, Fewster-Verch 12,. ]. Kasia Rejzner Locality and beyond 5 / 26 Local algebras of observables A(O) are defined abstractly, the Hilbert space representation comes later (this deals with the non-uniqueness of the vacuum). Algebras A(O) are constructed using only the local data. Local features of the theory (observables) are separated from the global features (states). AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Algebraic QFT on curved spacetimes Conceptual problems of QFT on curved spacetimes can be easily solved in the algebraic approach, because of the powerful principle of locality. The corresponding generalization of AQFT is called locally covariant quantum field theory[ Hollands-Wald 01, Brunetti-Fredenhagen-Verch 01, Fewster-Verch 12,. ]. Main advantages Kasia Rejzner Locality and beyond 5 / 26 Algebras A(O) are constructed using only the local data. Local features of the theory (observables) are separated from the global features (states). AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Algebraic QFT on curved spacetimes Conceptual problems of QFT on curved spacetimes can be easily solved in the algebraic approach, because of the powerful principle of locality. The corresponding generalization of AQFT is called locally covariant quantum field theory[ Hollands-Wald 01, Brunetti-Fredenhagen-Verch 01, Fewster-Verch 12,. ]. Main advantages Local algebras of observables A(O) are defined abstractly, the Hilbert space representation comes later (this deals with the non-uniqueness of the vacuum). Kasia Rejzner Locality and beyond 5 / 26 Local features of the theory (observables) are separated from the global features (states). AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Algebraic QFT on curved spacetimes Conceptual problems of QFT on curved spacetimes can be easily solved in the algebraic approach, because of the powerful principle of locality. The corresponding generalization of AQFT is called locally covariant quantum field theory[ Hollands-Wald 01, Brunetti-Fredenhagen-Verch 01, Fewster-Verch 12,. ]. Main advantages Local algebras of observables A(O) are defined abstractly, the Hilbert space representation comes later (this deals with the non-uniqueness of the vacuum). Algebras A(O) are constructed using only the local data. Kasia Rejzner Locality and beyond 5 / 26 AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Algebraic QFT on curved spacetimes Conceptual problems of QFT on curved spacetimes can be easily solved in the algebraic approach, because of the powerful principle of locality. The corresponding generalization of AQFT is called locally covariant quantum field theory[ Hollands-Wald 01, Brunetti-Fredenhagen-Verch 01, Fewster-Verch 12,. ]. Main advantages Local algebras of observables A(O) are defined abstractly, the Hilbert space representation comes later (this deals with the non-uniqueness of the vacuum). Algebras A(O) are constructed using only the local data. Local features of the theory (observables) are separated from the global features (states). Kasia Rejzner Locality and beyond 5 / 26 Replace O1 and O2 with arbitrary spacetimes M = (M; g), N = (N; g0) and require the embedding : M ! N to be an isometry. Require that preserves orientations and the causal structure (no new causal links are created by the embedding). Assign to each spacetime M an algebra A(M) and to each admissible embedding a homomorphism of algebras A (notion of subsystems). This has to be done covariantly. AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Locally covariant quantum field theory (LCQFT) In the original AQFT axioms we associate algebras to regions of a fixed spacetimes. Now we go a step further. Kasia Rejzner Locality and beyond 6 / 26 Require that preserves orientations and the causal structure (no new causal links are created by the embedding). Assign to each spacetime M an algebra A(M) and to each admissible embedding a homomorphism of algebras A (notion of subsystems). This has to be done covariantly. AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Locally covariant quantum field theory (LCQFT) In the original AQFT axioms we associate algebras to regions of a fixed spacetimes. Now we go a step further. Replace O1 and O2 with arbitrary spacetimes M = (M; g), N = (N; g0) and require the embedding : M ! N to be an isometry. N M Kasia Rejzner Locality and beyond 6 / 26 Assign to each spacetime M an algebra A(M) and to each admissible embedding a homomorphism of algebras A (notion of subsystems). This has to be done covariantly. AQFT Algebraic approach to QFT LCQFT Quantum gravity pAQFT Locally covariant quantum field theory (LCQFT) In the original AQFT axioms we associate algebras to regions of a fixed spacetimes. Now we go a step further. Replace O1 and O2 with arbitrary spacetimes M = (M; g), N = (N; g0) and require the embedding : M ! N to be an isometry.
Load more

Recommended publications
  • Questions in Quantum Physics: a Personal View
    Questions in quantum physics: a personal view Rudolf Haag Waldschmidtstraße 4b, D–83727 Schliersee-Neuhaus, Germany Abstract An assessment of the present status of the theory, some immediate tasks which are suggested thereby and some questions whose answers may require a longer breath since they relate to significant changes in the conceptual and mathematical structure of the theory. 1 Introduction Personal views are shaped by past experiences and so it may be worth while pondering a little about accidental circumstances which channelled the course of one’s own thinking. Meeting the right person at the right time, stumbling across a book or article which suddenly opens a new window. Fifty years ago, as eager students at Munich University just entering the phase of our own scientific research, we were studying the enormous papers by Julian Schwinger on Quantum Electrodynamics, following the arguments line by line but not really grasping the message. I remember the feelings of frustration, realizing that we were far away from the centers of action. But, mixed with this, also some dismay which did not only refer to the enormous arsenal of formalism in the new arXiv:hep-th/0001006v1 3 Jan 2000 developments of QED but began with the standard presentation of the interpretation of Quantum Theory. I remember long discussions with my thesis advisor Fritz Bopp, often while circling some blocks of streets ten times late in the evening, where we looked in vain for some reality behind the enigma of wave-particle dualism. Why should physical quantities correspond to operators in Hilbert space? Why should probabilities be described as absolute squares of amplitudes?, etc., etc.
    [Show full text]
  • Quantum Mechanical Laws - Bogdan Mielnik and Oscar Rosas-Ruiz
    FUNDAMENTALS OF PHYSICS – Vol. I - Quantum Mechanical Laws - Bogdan Mielnik and Oscar Rosas-Ruiz QUANTUM MECHANICAL LAWS Bogdan Mielnik and Oscar Rosas-Ortiz Departamento de Física, Centro de Investigación y de Estudios Avanzados, México Keywords: Indeterminism, Quantum Observables, Probabilistic Interpretation, Schrödinger’s cat, Quantum Control, Entangled States, Einstein-Podolski-Rosen Paradox, Canonical Quantization, Bell Inequalities, Path integral, Quantum Cryptography, Quantum Teleportation, Quantum Computing. Contents: 1. Introduction 2. Black body radiation: the lateral problem becomes fundamental. 3. The discovery of photons 4. Compton’s effect: collisions confirm the existence of photons 5. Atoms: the contradictions of the planetary model 6. The mystery of the allowed energy levels 7. Luis de Broglie: particles or waves? 8. Schrödinger’s wave mechanics: wave vibrations explain the energy levels 9. The statistical interpretation 10. The Schrödinger’s picture of quantum theory 11. The uncertainty principle: instrumental and mathematical aspects. 12. Typical states and spectra 13. Unitary evolution 14. Canonical quantization: scientific or magic algorithm? 15. The mixed states 16. Quantum control: how to manipulate the particle? 17. Measurement theory and its conceptual consequences 18. Interpretational polemics and paradoxes 19. Entangled states 20. Dirac’s theory of the electron as the square root of the Klein-Gordon law 21. Feynman: the interference of virtual histories 22. Locality problems 23. The idea UNESCOof quantum computing and future – perspectives EOLSS 24. Open questions Glossary Bibliography Biographical SketchesSAMPLE CHAPTERS Summary The present day quantum laws unify simple empirical facts and fundamental principles describing the behavior of micro-systems. A parallel but different component is the symbolic language adequate to express the ‘logic’ of quantum phenomena.
    [Show full text]
  • Numdam:AIHPA 1996__64 4 385 0
    ANNALES DE L’I. H. P., SECTION A RUDOLF HAAG IZUMI OJIMA On the problem of defining a specific theory within the frame of local quantum physics Annales de l’I. H. P., section A, tome 64, no 4 (1996), p. 385-393 <http://www.numdam.org/item?id=AIHPA_1996__64_4_385_0> © Gauthier-Villars, 1996, tous droits réservés. L’accès aux archives de la revue « Annales de l’I. H. P., section A » implique l’accord avec les conditions générales d’utilisation (http://www.numdam. org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. Inst. Henri Poincare, Vol. 64, n° 4, 1996, 385 Physique theorique On the problem of defining a specific theory within the frame of local quantum physics Rudolf HAAG Waldschmidtstraße 4B D-83727 Schliersee, Germany. Izumi OJIMA Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-01, Japan. ABSTRACT. - The notion and use of germs of states are discussed. La notion de germe d’état et son utilisation sont 1’ objet de cette discussion. 1. INTRODUCTION The customary procedure of defining a specific theory or model in Quantum Field Theory starts by writing down a Lagrangean in terms of point-like fields. This defines first of all a classical field theory which is then subjected to a process called "quantization", either in the old fashioned form of replacing the numerical-valued fields by field operators with commutation relations or by a Feynman-Schwinger functional integral which should directly give the correlation functions of fields in the vacuum state.
    [Show full text]
  • V.Y. Glaser SOME RETROSPECTIVE REMARKS by Ph. Blanchard
    V.Y. Glaser SOME RETROSPECTIVE REMARKS by Ph. Blanchard OPENINGS MATHEMATICS IN PHYSICS GREAT ENCOUNTERS ALONG THE WAY Zagreb, Göttingen, Copenhagen, Geneva, Strasbourg, Bures sur Yvette USING MATHEMATICS WITH CLARITY AND ELEGANCE Quantum Mechanics, Quantum Field Theory O P E N I N G S I am happy to have been asked to speak about Yurko Glaser, his thinking and its actions. It is an honor for me to pay tribute to the brilliant achievements of this leading mathematical physicist, gifted teacher and exceptional friend. It was in Strasbourg at the spring meeting of the RCP 25, where we first met 1967. At this time I was in Zürich at the ETH, working on the Paul-Fierz model of the infrared catastrophe under the direction of Res Jost. Yurko was born on April 21, 1924 just before the discovery by Schrödinger, Heisenberg, Dirac, Born … of modern Quantum Theory in the mid 1920’s. Carlo Rubbia was also born in Gorizia, Görz, Friaul – Julisch Venetien. Quantum Theory before 1925 – the Old Quantum Theory (Planck, Einstein, Bohr, Sommerfeld …) – was part craft part art. Old principles had been founded wanting, new ones had not yet been discovered. Modern Quantum Theory was a real revolution of our understanding of physical process. Compared with this change, Einstein’s relativity, born in 1905, seem not much more than very interesting variations on nevertheless classical themes. Yurko studied at the University of Zagreb, where he received his Diploma in 1950 and his Ph.D in 1953 under the supervision of W. Heisenberg. He moved to Göttingen in 1951-1952 and made his first important contributions to physics, a book of QED published 1955 in Zagreb and outstanding results on QFT, the attempt to clarify the compatibility of special relativity theory with Quantum Theory.
    [Show full text]
  • 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.
    [Show full text]
  • 31 Aug 2021 Self-Normalizing Path Integrals
    Self-normalizing Path Integrals Ivan M. Burbano∗1 and Francisco Calder´on†2 1Department of Physics, University of California, 366 Physics North MC 7300, Berkeley, CA 94720-7300, USA 2Department of Philosophy, University of Michigan, 435 South State Street 2215 Angell Hall, Ann Arbor, MI 48109-1003, USA September 3, 2021 Abstract We address the problem of computing the overall normalization constant of path integrals using zeta-function regularization techniques. In particular, we study a phe- nomenon we called “self-normalization,” in which the ambiguity of the integral mea- sure, which would typically need to be renormalized, resolves itself. Hawking had already detected this phenomenon in the context of Gaussian integrals. However, our approach extends Hawking’s work for the cases in which the space of fields is not a vector space but instead has another structure which we call a “linear foliation.” After describing the general framework, we work out examples in one (the transition ampli- tudes and partition functions for the harmonic oscillator and the particle on a circle in the presence of a magnetic field) and two (the partition functions for the massive and compact bosons on the torus and the cylinder) spacetime dimensions in a detailed fashion. One of the applications of our results, explicitly shown in the examples, is the computation of the overall normalization of path integrals that do not self-normalize. That is usually done in the literature using different comparison methods involving arXiv:2109.00517v1 [hep-th] 31 Aug 2021 additional assumptions on the nature of this constant. Our method recovers the nor- malization without the need for those extra assumptions.
    [Show full text]
  • Rudolf Haag's Legacy of Local Quantum Physics And
    Rudolf Haag’s legacy of Local Quantum Physics and reminiscences about a cherished teacher and friend In memory of Rudolf Haag (1922-2016) submitted to the Eur. Phys. J. H Bert Schroer permanent address: Institut f¨ur Theoretische Physik FU-Berlin, Arnimallee 14, 14195 Berlin, Germany November 2016 Abstract After some personal recollectioms about Rudolf Haag and his thoughts which led him to ”Local Quantum Physics”, the present work recalls his ideas about scattering theory, the relation between local observables and localized fields and his contributions to the physical aspects of modu- lar operator theory which paved the way for an intrisic understanding of quantum causal localization in which fields ”coordinatize” the local algebras. The paper ends with the presentation of string-local fields whose con- struction and use in a new renormalization theory for higher spin fields is part of an ongoing reformulation of gauge theory in the conceptual setting of Haag’s LQP. 1 First encounter with Rudolf Haag arXiv:1612.00003v1 [math-ph] 30 Nov 2016 On his return from the Niels Bohr Institute in Copenhagen to the University of Munich Rudolf Haag passed through Hamburg to meet his colleague Harry Lehmann, at that time the newly appointed successor of Wilhelm Lenz who held the chair of theoretical physics since the 1920 foundation of the University of Hamburg. It was the year 1958 shortly after the decision to construct the DESY particle accelerator in Hamburg which created a lot of excitement. I had nearly completed my diploma thesis under Lehmann and begun to worry about my career.
    [Show full text]
  • Why the Quantum
    Deep Beauty: Mathematical Innovation and the Search for an Underlying Intelligibility of the Quantum World A Symposium and Book Publication Program Sponsored by Princeton University’s Department of Philosophy Supported by a Grant from the John Templeton Foundation Celebrating the Life and Legacy of John von Neumann and the 75th Anniversary of the Publication of His Classic Text: The Mathematical Foundations of Quantum Mechanics* Symposium: October 3 – October 4, 2007 Princeton, New Jersey _______________________ Revised 09-11-07, PContractor ________________ * von Neumann, Johann. Mathematische grundlagen der quantenmechanik. Berlin: J. Springer, 1932. Deep Beauty:Mathematical Innovation and the Search for an Underlying Intelligibility of the Quantum World John von Neumann,1903-1957 Courtesy of the Archives of the Institute for Advanced Study (Princeton)* The following photos are copyrighted by the Institute for Advanced Study, and they were photographed by Alan Richards unless otherwise specified. For copyright information, visit http://admin.ias.edu/hslib/archives.htm. *[ED. NOTE: ELLIPSIS WILL WRITE FOR PERMISSION IF PHOTO IS USED; SEE http://www.physics.umd.edu/robot/neumann.html] Page 2 of 14 Deep Beauty:Mathematical Innovation and the Search for an Underlying Intelligibility of the Quantum World Project Overview and Background DEEP BEAUTY: Mathematical Innovation and the Search for an Underlying Intelligibility of the Quantum World celebrates the life and legacy of the scientific and mathematical polymath John Von Neumann 50 years after his death and 75 years following the publication of his renowned, path- breaking classic text, The Mathematical Foundations of Quantum Mechanics.* The program, supported by a grant from the John Templeton Foundation, consists of (1) a two-day symposium sponsored by the Department of Philosophy at Princeton University to be held in Princeton October 3–4, 2007 and (2) a subsequent research volume to be published by a major academic press.
    [Show full text]
  • IAMP News Bulletin July 2016
    IAMP News Bulletin July 2016 International Association of Mathematical Physics Contents International Association of Mathematical Physics News Bulletin, July 2016 Contents CMP in the Twenty-First Century3 Obituary: Roland S´en´eor 10 Spectral Geometry Conference and the Legacy of Mark Kac 17 News from the IAMP Executive Committee 19 Contact Coordinates for this Issue 22 Bulletin Editor Editorial Board Evans Harrell Rafael Benguria, Virginie Bonnaillie-No¨el, Yasuyuki Kawahigashi, Manfred Salmhofer, Robert Sims Contacts. http://www.iamp.org and e-mail: [email protected] Cover picture: Mathematical physics conference at Rome, 1977. See the article by Michael Aizenman in this issue. The views expressed in this IAMP News Bulletin are those of the authors and do not necessarily represent those of the MIAMP Executive Committee, Editor or Editorial Board. Any complete or partialΦ performance or reproduction made without the consent of the author or of his successors in title or assigns shall be unlawful. All reproduction rights are henceforth reserved, and mention of the IAMP News Bulletin is obligatory in the reference. (Art.L.122-4 of the Code of Intellectual Property). ISSN 2304-7348 News Bulletin (International Association of Mathematical Physics) 2 ∩IAMP News Bulletin, July 2016 CMP in the Twenty-First Century CMP in the Twenty-First Century by Michael Aizenman (Princeton University) Michael Aizenman is Thomas D. Jones Professor of Mathematical Physics at Princeton University. He served on the IAMP Executive Committee in 1997{1999, as Editor of Communications in Mathematical Physics from 1986 to 1993, and as Editor-in-Chief from 2001 to 2012. To every age its art.
    [Show full text]
  • Introduction to N = 2 Gauge Theory
    Introduction to N = 2 Gauge Theory Chris Elliott January 13th, 2015 1 Introduction and Emphasis The goal of this seminar is to understand the Nekrasov partition function introduced by Nekrasov in 2003 [Nek03]. This is a generating function for correlation functions in an N = 2 supersymmetric gauge theory, so my goal today is to explain what N = 2 gauge theory is and why we might find it interesting. We begin with the latter question, why is this interesting? There are two intertwined answers. 1. Firstly, N = 2 gauge theory has been a rich source of powerful 4-manifold invariants, as well as an organising principle that has led to advances in actually computing these invariants in examples. The first invariants of this sort { Donaldson's polynomial invariants [Don83] { were originally inspired by ordinary SU(n) Yang- Mills theory. However, in a ground-breaking 1988 paper, Witten [Wit88] gave a construction of Donaldson invariants as expectation values of certain observables in a twist of N = 2 gauge theory (we'll talk more about this shortly). This opened the door to later work of Seiberg and Witten [SW94], who gave an alternative description of these invariants also coming from N = 2 gauge theory, using the fact that these observables in the twisted theory could be computed in an effective theory describing the far IR limit of the full gauge theory. The Seiberg-Witten invariants are more readibly computable than the Donaldson invariants, and the moduli space from which one computes the invariants has nicer properties (compactness, for instance). 2. Secondly, N = 2 gauge theory is a source of interesting dualities between quantum field theories, a particularly notable example of which is the Alday-Gaiotto-Tachikawa (AGT) correspondence [AGT10].
    [Show full text]
  • Haag's Theorem - Wikipedia, the Free Encyclopedia
    Haag's theorem - Wikipedia, the free encyclopedia Create account Log in Article Talk Read Edit View history Haag's theorem From Wikipedia, the free encyclopedia Main page Contents Rudolf Haag postulated [1] that the interaction picture does not exist in an interacting, relativistic Featured content quantum field theory (QFT), something now commonly known as Haag's Theorem. Haag's Current events original proof was subsequently generalized by a number of authors, notably Hall and Random article Wightman,[2] who reached the conclusion that a single, universal Hilbert space representation Donate to Wikipedia does not suffice for describing both free and interacting fields. In 1975, Reed and Simon proved Wikimedia Shop [3] that a Haag-like theorem also applies to free neutral scalar fields of different masses, which Interaction implies that the interaction picture cannot exist even under the absence of interactions. Help About Wikipedia Contents [hide] Community portal 1 Formal description of Haag's theorem Recent changes 2 Physical (heuristic) point of view Contact page 3 Workarounds Tools 4 Conflicting reactions of the practitioners of QFT What links here 5 References Related changes 6 Further reading Upload file Special pages Permanent link [edit] Page information Formal description of Haag's theorem Wikidata item In its modern form, the Haag theorem may be stated as follows:[4] Cite this page Consider two representations of the canonical commutation relations (CCR), and Print/export (where denote the respective Hilbert spaces and the collection of Create a book operators in the CCR). Both representations are called unitarily equivalent if and only if there Download as PDF Printable version exists some unitary mapping from Hilbert space to Hilbert space such that for each operator there exists an operator .
    [Show full text]
  • ESSAY REVIEW Local Quantum Physics
    Stud. Hist. Phil. Mod. Phyr, Vol. 27, No. 4, pp. 511-524, 1996 Published by Elsevier Science Ltd. Printed in Great Britain. 1355-2198/96 $15.00 + 0.00 ESSAYREVIEW Local Quantum Physics N. ?? Landsman * Rudolf Haag, Local Quantum Physics (Berlin: Springer-Verlag, 1992; corrected 2nd edn, 1993 1 ), xiv + 356 pp. ISBN 3-540-53610-S hardback f35.00; xv + 390 pp. ISBN 3-540-61049-9 softcover (revised and enlarged edition, 1996), f32.00 Our civilization is characterized by the word ‘progress’. Progress is its form rather than making progress one of its features. Typically it constructs. It is occupied with building an ever more complicated structure. And even clarity is only sought as a means to this end, not as an end in itself. For me on the contrary clarity, perspicuity are valuable in themselves. I am not interested in constructing a building, so much as in having a perspicuous view of the foundations of typical buildings Ludwig Wittgenstein 1. Introduction Local quantum physics is understood as the synthesis of special relativity and quantum physics. According to A. S. Wightman, himself no mean contributor to this field, ‘Rudolf Haag has thought as long and as deeply about the foundations of relativistic quantum mechanics as anyone alive’, and the present book is in many ways his scientific autobiography. Its principal subject matter, algebraic quantum field theory, has a reputation of being rather esoteric, axiomatic and mathematical (and, some pagans would say, irrelevant to physics). To the extent that this is true, at least for Haag himself the mathematics is merely there to serve the objective of providing a conceptual analysis of quantum field theory and related matters, which is as precise and deductive as possible.
    [Show full text]