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Lining BR98B0289 ISSN 0029-3865 lining BR98B0289 CBPF - CENTRO BRAS1LE1RO DE PESQUtSAS FtSICAS Rio de Janeiro Notas de Fisica CBPF-NF-026/97 March 1997 A Course on: "Quantum Field Theory and Local Observables CBPF Rio de Janeiro February 1997 Bert Sdiroor Lit CNPq - Conselho Nacional de Desenvolvimento Cientifico e Tecnologico CBPF-NF-026/97 A Course on: "Quantum Field Theory and Local Observables" CBPF, Rio de Janeiro, February 1997 by Bert Schroer* Centro Brasileiro de Pesquisas Fisicas - CBPF Rua Dr. Xavier Sigaud, 150 22290-180 - Rio de Janeiro, RJ - Brazil Institut fur Theoretische Physik der FU-Berlin, Arnimallee 14, 14195 Berlin Germany. e-mail: [email protected] "Temporary Adress: CBPF, Rio de Janeiro Brazil, e-mail: [email protected] Contents 0.1 Introduction 2 The Basics of Quantum Theory 7 1.1 Multiparticle Wave Functions, Particle Statistics and Superselec- tion Sectors 7 1.2 The Superselection Sectors of CG 14 1.3 Illustration of Important Quantum Concepts 20 1.4 Measurement and Superselection Rules 25 The Construction of Fock-Space. 28 2.1 The Bosonic Fock-Space 28 2.2 The Fermion Fockspace 36 2.3 The CCR and CAR Algebras 38 2.4 Quasifree States 41 2.5 Temperature States and KMS condition 43 2.6 The CCR- and CAR-Functors 45 Poincare Symmetry and Quantum Theory 50 3.1 The Symmetry Concept of General Quantum Theory 50 3.2 One Particle Representations of the Poincare Group 54 3.3 Wigner Theory and Free Fields 65 3.4 The Equivalence Class of a Free Field 72 3.5 A First Look at Modular Localization 76 3.6 Rindler Wedges and Hawking Temperature 84 3.7 Fields associated with Free Fields 85 3.8 Special Features of m=0, d=l+l Fields 91 Elementary Approach to Perturbative Interactions 100 4.1 Kinematical Decompositions 100 4.2 Elementary Notion of Interaction and Perturbation 102 4.3 Second Order Perturbation and the Adiabatic Parametrization . 105 4.4 Invariant Parametrizations, Regularization 113 4.5 Specialities of Perturbative Gauge Theories 118 4.6 Perturbative Thermo-QFT 123 4.7 Functional techniques 123 4.8 Interactions with External Fields, CST-Problems 123 The General Framework of QFT 128 5.1 Model-independent Properties of pointlike Fields 128 5.2 Simple Structural Properties 133 5.3 Euclidean Fields 139 5.4 Constructive use of Euclidean Fields 145 5.5 Scattering Theory 145 Modular Localization and Bootstrap-FormfactorProgram 150 6.1 Introductory comments 150 6.2 Modular Localization and Interaction 154 6.3 The Bootstrap-Formfactor Program 155 6.3.1 Properties of Factorizing S-Matrices 156 6.3.2 Generalized Formfactors 159 6.3.3 Modular Theory and the Formfactor Program 162 6.4 Open Ends and Outlook 166 Introduction to Algebraic QFT 172 7.1 Some Useful Theorems 172 7.2 Abstracting Principles from Standard Setting 176 7.3 Starting the Reverse: the DHR Endomorphisms 178 7.4 Remarks on Broken Symmetries 187 7.5 Chiral Conformal Algebraic QFT 188 7.6 Constructive Aspects of Plektons 192 Tentative Resume and Outlook 193 0.1 Introduction A new monography or review on an old and established subject as QFT should justify and measure itself relative to the many existing review articles and text- books. The main motivation underlying these notes consists in the desire to unify two presently largely disconnected branches of QFT: (1) the standard (canonical, functional) approach which is mainly perturbative in the sense of an infinitesimal "deformation" of free fields1 and (2) nonperturbative constructions of low-dimensional models as the formfactor-bootstrap approach (which for the time being is limited to factorizable models in d=l+l spacetime dimensions) and the non-Lagrangian construction of conformal chiral QFT's.. The synthesis requires a significant step byond the concepts which were used in order to formulate the two mentioned separate branches. On the physical side the S-matrix regains some of its early prominence, however unlike in the pro- posals of Heisenberg (and later in the S-matrix bootstrap approach of Chew 1 Unfortunately only in the sense of an infinitesimal deformation around free fields, and not in that of an approximation of an existing QFT. CBPF-NF-026/97 -3- et al.), here it remains subservient to the locality and causality principles en- capsulated in the theory of local observables (the heartpiece of algebraic QFT). In the new context of the TCP- and the Tomita-reflection operators explained in these notes, the S-matrix takes the role of a powerful constructive tool in QFT. In particular local fields turn out to be just "coordinatizations" of local algebras, and different fields in the same local class describe the same physics. In addition to the charge superselection rules (including statistics and spin), the S-matrix reveals itself as the most valuable "net" invariant of the field net which incorporates all charge sectors. On the mathematical side one needs the Tomita-Takesaki modular theory which is closely related to the concept of KMS states in Quantum Statisti- cal Mechanics. In our present context they will be mainly used to implement localization and geometrical properties in Local Quantum Physics. It is the proximity of these concepts to TCP symmetry and other fundamental notions in QFT which gives confidence in our approach. The change of paradigm which accompanies the new approach is that those structures of QFT which were al- ways thought of as fundamental but "kinematicaP properties as TCP and Spin & Statistics, now move into the center of the "dynamical" stage. Of course such statements should be taken with the full awareness that the cut between "kine- matics" and what was hitherto considered as "dynamics" was never as rigid and a priori as the textbooks make us believe. A new approach is also expected to cast some new light on past successes. For this reason we will attempt a rather systematic presentation which includes a substantial part of the standard material of QFT (especially chapter 2), but occasionally somewhat different from the standard textbook treatments by em- phasis and interpretation. Since the new method (build on old physical princi- ples of local quantum theory) is still in its infancy, it is to early to expect an exhaustive treatment of the algebraic approach and the modular construction method. In fact QFT, despite its age, is still in its conceptual infancy, especially from the new viewpoint. This goes contrary to the widespread (but unfounded) belief that the basic equations of the world below the Planck scale are already known, and that if we would be more clever in our computations (bigger computers etc.) we would have a description of all phenomena below that scale. It is interesting to understand how this overly optimistic (and somewhat dull) picture about QFT developed in time. The greatest success of QFT and the raison d'etre for its continuing existence was certainly the renormalization of QED and its more recent extension via the standard model3. But it is helpful to remind oneself that in physics (and also outside) success and disaster often are close together. A big step forward naturally lends itself to an extension of the formalism through which it was obtained. In the case at hand, the perturbation theory was streamlined by the functional formulation without which one would 'However the remarkable aspect of the electro-weak theory beyond QED is not to much its precision, but rather its power to combine a overwhelming body of experimental facts into a reasonable quantum field theoretical shell which appears to be a good starting point for future conceptual advances. CBPF-NF-026/97 -4- be unable to understand most contemporary publications. To press ahead with a successful formalism and to see what it leads to, if extended beyond laboratory energies, is of course in the best tradition of theoretical physics, even if at the end it should turn out to be the wrong path. Since formal extensions are easy and take less time than conceptional in- vestigations, their rapid pursuit is very reasonable indeed. It only may lead to desaster if this develops into the main research topic for several generations and the if broad conceptual basis which led to the first success gets narrowed in its potential conceptional richness and at the end becomes lost with the younger generation. Whether we already reached this state of physical stagnation and crisis is left up to the judgement of the reader. In any case this report is not written for those who still (after a quarter cen- tury of post standard model stagnation) are convinced that the extension of the standard formalism is all one needs in order to make progress. It rather tries to attract those physicists who, on the one hand still have not lost their conceptual curiosity, and on the other hand feel uneasy about the present predominance of formalism over conceptional insight. The present author has tried to analyse the contemporary situation in terms of the lost balance between the conceptual "Bohr-Heisenberg"- and the esthetical mathematical "Dirac"-approach3. Hopefully the reader will be able to appreciate the use of the unifying phys- ical point of view which is encapsulated in the "local nets" and which gives e.g. all the low-dimensional richness without invoking special structures (as e.g. Virasoro- and Kac-Moody- algebras) which are restricted to low space-time dimensions only. After all, the main physical value of low dimensional mod- els is their use for a better conceptual and structural understanding of general QFT (and perhaps also for basing universality explanations in condensed matter physics on firm grounds) and only in second place the increase of our mathe- matical culture.
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