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E-grāmata: Perturbative Algebraic Quantum Field Theory: An Introduction for Mathematicians

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  • Formāts: PDF+DRM
  • Sērija : Mathematical Physics Studies
  • Izdošanas datums: 16-Mar-2016
  • Izdevniecība: Springer International Publishing AG
  • Valoda: eng
  • ISBN-13: 9783319259017
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Perturbative Algebraic Quantum Field Theory: An Introduction for MathematiciansPalielināt
  • Formāts: PDF+DRM
  • Sērija : Mathematical Physics Studies
  • Izdošanas datums: 16-Mar-2016
  • Izdevniecība: Springer International Publishing AG
  • Valoda: eng
  • ISBN-13: 9783319259017
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Perturbative Algebraic Quantum Field Theory (pAQFT), the subject of this book, is a complete and mathematically rigorous treatment of perturbative quantum field theory (pQFT) that doesn’t require the use of divergent quantities and works on a large class of Lorenzian manifolds.

We discuss in detail the examples of scalar fields, gauge theories and the effective quantum gravity.

pQFT models describe a wide range of physical phenomena and have remarkable agreement with experimental results. Despite this success, the theory suffers from many conceptual problems. pAQFT is a good candidate to solve many, if not all, of these conceptual problems.

Chapters 1-3 provide some background in mathematics and physics. Chapter 4 concerns classical theory of the scalar field, which is subsequently quantized in chapters 5 and 6. Chapter 7 covers gauge theory and chapter 8 discusses effective quantum gravity.

The book aims to be accessible to researchers and graduate students, who are interested in the mathematical foundations of pQFT.

Recenzijas

The author, who claims to be both a physicist and a mathematician, offers a useful and fascinating book which should be of interest and useful to professional mathematicians and students of both mathematics and physics. It is dedicated to prominent mathematical physicists who passed away recently: Rudolf Haag, Daniel Kastler, Uffe Haagerup, Raymond Stora, and John Roberts ... . The book intends to be pedagogical and self-contained. It establishes the mathematical foundation of perturbation theory in a convincing manner. (Gert Roepstorff, zbMATH 1347.81011, 2016)

1 Introduction
1(2)
2 Algebraic Approach to Quantum Theory
3(36)
2.1 Algebraic Quantum Mechanics
3(14)
2.1.1 Functional Analytic Preliminaries
3(3)
2.1.2 Observables and States
6(2)
2.1.3 Hilbert Space Representations
8(7)
2.1.4 Dynamics and the Interaction Picture
15(2)
2.2 Causality
17(4)
2.3 Haag--Kastler Axioms
21(2)
2.4 pAQFT Axioms
23(4)
2.4.1 More Functional Analysis
24(2)
2.4.2 Axioms
26(1)
2.5 Locally Covariant Quantum Field Theory
27(12)
References
34(5)
3 Kinematical Structure
39(20)
3.1 The Space of Field Configurations
39(1)
3.2 Functionals on the Configuration Space
40(6)
3.3 Fermionic Field Configurations
46(6)
3.4 Vector Fields
52(3)
3.5 Functorial Interpretation
55(4)
References
57(2)
4 Classical Theory
59(24)
4.1 Dynamics
59(4)
4.2 Natural Lagrangians
63(1)
4.3 Homological Characterization of the Solution Space
64(3)
4.4 The Net of Topological Poisson Algebras
67(7)
4.4.1 The Peierls Bracket and Microcausal Functionals
67(2)
4.4.2 Topologies on the Space of Microcausal Functionals
69(4)
4.4.3 The Classical Causal Net
73(1)
4.5 Analogy with Classical Mechanics
74(3)
4.6 Classical Møller Maps Off-Shell
77(6)
References
80(3)
5 Deformation Quantization
83(12)
5.1 Star Products
83(7)
5.2 The Star Product on the Space of Multivector Fields
90(2)
5.3 Kahler Structure
92(3)
References
93(2)
6 Interaction and Renormalization of the Scalar Field Theory
95(42)
6.1 Outline of the Approach
95(1)
6.2 Scattering Matrix and Time Ordered Products
96(17)
6.2.1 Wick Products
97(1)
6.2.2 Locally Covariant Wick Products
98(3)
6.2.3 Time-Ordered Products
101(2)
6.2.4 The Formal S-Matrix and Møller Operators
103(7)
6.2.5 Epstein--Glaser Axioms
110(3)
6.3 Renormalization Group
113(3)
6.4 Interacting Local Nets
116(4)
6.5 Construction of Time-Ordered Products
120(17)
6.5.1 Existence of Time-Ordered Products (Abstract Proof)
121(8)
6.5.2 Explicit Construction and Feynman Graphs
129(4)
6.5.3 Regularization of Distributions
133(2)
References
135(2)
7 Gauge Theories
137(20)
7.1 Classical Gauge Theory
137(8)
7.1.1 Dynamics and Symmetries
138(1)
7.1.2 The Koszul--Tate Complex
139(1)
7.1.3 The Chevalley--Eilenberg Complex
140(2)
7.1.4 The BV Complex
142(3)
7.2 Gauge-Fixing
145(6)
7.3 Quantization in the Batalin--Vilkoviski Formalism
151(6)
References
155(2)
8 Effective Quantum Gravity
157(16)
8.1 From LCQFT to Quantum Gravity
157(2)
8.2 Dynamics and Symmetries
159(3)
8.3 Linearized Theory
162(2)
8.4 Quantization
164(1)
8.5 Relational Observables
165(2)
8.6 Background Independence
167(6)
References
170(3)
Glossary 173(4)
Index 177
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