Boutet de Monvel / Buchholz / Iagolnitzer Rigorous Quantum Field Theory

A Festschrift for Jacques Bros
1. Auflage 2006
ISBN: 978-3-7643-7434-1
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 251, 326 Seiten

Reihe: Progress in Mathematics

Jacques Bros has greatly advanced our present understanding of rigorous quantum field theory through numerous contributions; this book arose from an international symposium held in honour of Bros on the occasion of his 70th birthday. Key topics in this volume include: Analytic structures of Quantum Field Theory (QFT), renormalization group methods, gauge QFT, stability properties and extension of the axiomatic framework, QFT on models of curved spacetimes, QFT on noncommutative Minkowski spacetime.

Autoren/Hrsg.

Buchholz, Anne

Iagolnitzer, Detlev

Moschella, Daniel

Boutet de Monvel, Anne

Buchholz, Detlev

Iagolnitzer, Daniel

Moschella, Ugo

Inhaltsverzeichnis

1;Contents;6
2;1 Introduction;8
2.1;References;16
3;2 Local Counterterms on the Noncommutative Minkowski Space;18
3.1;2.1 Free fields and perturbation theory;19
3.2;2.2 Local counterterms;26
3.3;2.3 Quasiplanar Wick products;29
3.4;2.4 Outlook;32
3.5;References;32
4;3 Massless Scalar Field in a Two-dimensional de Sitter Universe;34
4.1;3.1 de Sitter geometry and the massive scalar field;34
4.2;3.2 Massless limit;36
4.3;3.3 Invariant Krein space;37
4.4;3.4 Equation of motion and gauge invariance;41
4.5;3.5 Conclusions;44
4.6;References;45
5;4 Locally Covariant Quantum Field Theories;46
5.1;4.1 Locality and general covariance;46
5.2;4.2 Quantum Field Theory as a Functor;47
5.3;4.3 Beyond simple functoriality: Equivalence, Dynamics, Fields, Scattering, and more.;49
5.4;4.4 Conclusions and Outlook;53
5.5;References;53
6;5 Asymptotic Abelianness and Braided Tensor C* - Categories;56
6.1;5.1 Introduction;56
6.2;5.2 Asymptotically Abelian Intertwiners;58
6.3;5.3 The Emergence of Braiding;64
6.4;5.4 Algebraic Quantum Field Theory;67
6.5;Acknowledgements;70
6.6;References;70
7;6 Yang–Mills and Some Related Algebras;72
7.1;6.1 Introduction;72
7.2;6.2 Homogeneous algebras;73
7.3;6.3 The Yang–Mills algebra;76
7.4;6.4 The super Yang–Mills algebra;78
7.5;6.5 The super self-duality algebra;80
7.6;6.6 Deformations;83
7.7;References;85
8;7 Remarks on the Anti-de Sitter Space-Time;86
8.1;7.1 Introduction;86
8.2;7.2 Some Lorentzian geometry in AdS;89
8.3;7.3 Real Scalar Field on;91
8.4;7.4 The n - point Tuboids;93
8.5;7.5 Spectral Condition for AdS;94
8.6;7.6 Two-Point Functions, Generalized Free Fields;95
8.7;7.7 n- Point Permuted Tuboids;96
8.8;7.8 Bisognano-Wichmann-KMS Property, CTP;97
8.9;7.9 Wick Rotations for Xd;98
8.10;Appendix. q-Sheeted AdS Covers and Exotic Locality;98
8.11;References;100
9;8 Quantum Energy Inequalities and Stability Conditions in Quantum Field Theory;102
9.1;8.1 Introduction;102
9.2;8.2 Quantum Energy Inequalities;103
9.3;8.3 Stability at Three Scales;105
9.4;8.4 Connections with Nuclearity;112
9.5;8.5 Conclusion;115
9.6;References;116
10;9 Action Ward Identity and the Stückelberg–Petermann Renormalization Group;120
10.1;9.1 Introduction;120
10.2;9.2 Basic properties required for interacting fields;121
10.3;9.3 Construction of solutions;123
10.4;9.4 Renormalization group;125
10.5;9.5 Local nets and local fields;126
10.6;References;129
11;10 On the Relativistic KMS Condition for the Model;132
11.1;10.1 Introduction;132
11.2;10.2 The spatially-cutoff model at positive temperature;133
11.3;10.3 Euclidean approach;137
11.4;10.4 The relativistic KMS condition;140
11.5;10.5 Outlook;146
11.6;References;146
12;11 The Analyticity Program in Axiomatic Quantum Field Theory;148
12.1;11.1 Introduction;148
12.2;11.2 The axiomatic framework;149
12.3;11.3 Causality and Analyticity: mathematical results;153
12.4;11.4 The linear axiomatic program;156
12.5;11.5 Causality and local analyticity of chronological functions: physical discussion;160
12.6;11.6 The nonlinear program: some results;162
12.7;11.7 The Analyticity Program in Constructive QFT;165
13;12 Renormalization Theory Based on Flow Equations;168
13.1;12.1 Introduction;168
13.2;12.2 Renormalization of .4 Theory;169
13.3;12.3 Relativistic Theory;175
13.4;12.4 A short look at further results;178
13.5;References;179
14;13 Towards the Construction of Quantum Field Theories from a Factorizing S- Matrix;182
14.1;13.1 Introduction;182
14.2;13.2 Wedge-local fields;184
14.3;13.3 Existence of local observables;187
14.4;13.4 Modular Compactness for Wedge Algebras;190
14.5;13.5 Summary and Outlook;200
14.6;Appendix;200
14.7;References;202
15;14 String-Localized Covariant Quantum Fields;206
15.1;14.1 Introduction;206
15.2;14.2 Wigner Particles;208
15.3;14.3 Modular Localization Structure for;209
15.4;14.4 String-Localized Covariant Wave Functions;211
15.5;14.5 Summary and Outlook;214
15.6;A Extension of the Representations to P+;215
15.7;B Intertwiners and Localization Structure for the Principal Series Representations;216
15.8;References;219
16;15 Quantum Anosov Systems;220
16.1;15.1 Introduction;220
16.2;15.2 The classical system;221
16.3;15.3 Spectral properties and clustering properties;222
16.4;15.4 Verification of the Anosov structure in crossed product constructions;223
16.5;15.5 The Type III Case;225
16.6;15.6 Application to Quantum field Theory;228
16.7;References;229
17;16 DFR Perturbative Quantum Field Theory on Quantum Space Time, and Wick Reduction;232
17.1;16.1 Introduction;232
17.2;16.2 DFR Quantum Space-time, and All That;233
17.3;16.3 Nonlocal Dyson Diagrams;238
17.4;16.4 Conclusions;241
17.5;Appendix. Twisted Products;241
17.6;References;243
18;17 On Local Boundary CFT and Non- Local CFT on the Boundary;246
18.1;17.1 Introduction;246
18.2;17.2 Algebraic boundary conformal QFT;247
18.3;17.3 The charge structure of BCFT fields;252
18.4;17.4 Nimreps and non-vacuum BCFT;253
18.5;17.5 Appendix: Modular Theory in QFT and in BCFT;256
18.6;References;258
19;18 Algebraic Holography in Asymptotically Simple, Asymptotically AdS Space- times;260
19.1;18.1 Introduction;260
19.2;18.2 Doing away with coordinates in Rehren duality;262
19.3;18.3 Properties of the Rehren bijection in AAdS space-times;267
19.4;18.4 Perspectives and open problems;275
19.5;Acknowledgements;276
19.6;References;276
20;19 Non-Commutative Renormalization;278
20.1;19.1 Introduction;278
20.2;19.2 The UV-IR Problem ;280
20.3;19.3 The Covariant Theory;281
20.4;19.4 Smooth slices;285
20.5;19.5 Towards Non-commutative Constructive Field Theory;286
20.6;References;288
21;20 New Constructions in Local Quantum Physics;290
21.1;20.1 How modular theory entered particle physics;290
21.2;20.2 Modular localization and the bootstrap-formfactor program;295
21.3;20.3 Constructive aspects of lightfront holography;300
21.4;References;306
22;21 Physical Fields in QED;308
22.1;21.1 Introduction;308
22.2;21.2 Formal Considerations;309
22.3;21.3 Problems, and a Solution;310
22.4;Appendix: Proving Equation (21.17);314
22.5;References;315
23;22 Complex Angular Momentum Analysis and Diagonalization of the Bethe – Salpeter Structure in Axiomatic Quantum Field Theory;318
23.1;22.1 Introduction: Phenomenological Motivation;318
23.2;22.2 Harmonic Analysis on Complex Hyperboloids;320
23.3;22.3 Complex Angular Momentum in General Quantum Field Theory;326
23.4;References;332

Leseproben

1 Introduction (P. 1)

After the great scienti.c revolutions of Special Relativity and of Quantum Mechanics in the first half of the twentieth century, Relativistic Quantum Field Theory (QFT) was introduced to provide a synthesis of these two new paradigms (in Kuhn’s words). As such, QFT is one of the major advances for theoretical physics in the second half of the last century.

The main reason for emphasizing the importance of QFT lies in the impressive effort that it represents towards a unified understanding of the structure of matter at subatomic scales, as it emerges in the collision phenomena of particle physics in the range of energies presently explored in particle accelerators. But another reason is the extraordinary mathematical wealth of this theoretical framework, which makes it a fascinating domain of research for mathematical physics and may also stimulate the interest of mathematicians.

Foreseen by Dirac in 1933 and finaly established discovered by Feynman around 1949 for the treatment of quantum electrodynamics, the so-called "path-integral formalism" of quantum field theory is also now considered as a powerful tool for providing perturbative methods of computation in many problems of theoretical physics, namely in statistical mechanics and in string theory. However, the use of relativistic quantum fields as a basic concept of mathematical physics underlying all the phenomena of particle physics in a very large range of energies represents a much more ambitious program.

This program was indeed stimulated by the success of the quantum electrodynamics (QED) formalism for computing the electron-photon, electronelectron and electron-positron scattering amplitudes. Even today, it is by no means understood why the perturbative expansion of QED in powers of the coupling parameter, that is the electric charge of the electron, provides such a spectacular agreement with experimental data, although in practice it is reduced to the computation of the very first terms of the series (the only ones which are computable).

This is even more surprising since we now know that the series cannot be convergent, so that, paraphrasing Wigner’s words, one may wonder about the "unreasonable effectiveness" of perturbation theory in four-dimensional QED. In the 1950s, however, the success of these computations was credited to the smallness of the coupling parameter of QED, and this situation stimulated research for other methods of investigation of QFT, which could apply to quantum field models with large coupling: it was in fact the very concept of a quantum field which appeared as the most powerful and promising one for explaining the phenomena of strong nuclear interaction of particle physics.

During all that period the study of phenomena of weak nuclear interaction also benefitted, like QED, from the success of a perturbative QFT formalism reduced to very few terms. The demand for a more general nonperturbative treatment of QFT motivated an important community of mathematical physicists to work out a model-independent axiomatic approach to relativistic quantum field theory [2,8,9,11,14].

Their first task was to provide a mathematically meaningful concept of a relativistic quantum field in terms of "operator-valued distribution in the Hilbert space of quantum physical states", in such a way that the notion of ingoing and outgoing particle states could be introduced in terms of appropriate asymptotic forms of the field operators.