Introduction to Algebraic and Constructive Quantum Field Theory

by Baez, John C.; Segal, Irving Ezra; Zhou, Zhengfang

• ISBN13: 9780691085463
• ISBN10: 0691085463
• Format: Hardcover
• Copyright: 1992-08-01
• Publisher: Princeton Univ Pr

Summary

The authors present a rigorous treatment of the first principles of the algebraic and analytic core of quantum field theory. Their aim is to correlate modern mathematical theory with the explanation of the observed process of particle production and of particle-wave duality that heuristic quantum field theory provides. Many topics are treated here in book form for the first time, from the origins of complex structures to the quantization of tachyons and domains of dependence for quantized wave equations.
This work begins with a comprehensive analysis, in a universal format, of the structure and characterization of free fields, which is illustrated by applications to specific fields. Nonlinear local functions of both free fields (or Wick products) and interacting fields are established mathematically in a way that is consistent with the basic physical constraints and practice. Among other topics discussed are functional integration, Fourier transforms in Hilbert space, and implementability of canonical transformations.
The authors address readers interested in fundamental mathematical physics and who have at least the training of an entering graduate student. A series of lexicons connects the mathematical development with the underlying physical motivation or interpretation. The examples and problems illustrate the theory and relate it to the scientific literature.

Table of Contents

Introduction
The Free Boson Field	p. 3
Weyl and Heisenberg systems	p. 4
Functional integration	p. 15
Quasi-invariant distributions	p. 25
Absolute continuity	p. 30
Irreducibility and ergodicity	p. 37
The Fourier-Wiener transform	p. 41
The structure of [Gamma] and wave-particle duality	p. 47
Implications of wave-particle duality	p. 57
Characterization of the free boson field	p. 62
The complex wave representation	p. 64
Analytic features of the complex wave representation	p. 70
The Free Fermion Field	p. 75
Clifford systems	p. 75
Existence of the free fermion field	p. 80
The real wave representation	p. 82
The complex wave representation	p. 89
Properties of the Free Fields	p. 96
The exponential laws	p. 97
Irreducibility	p. 99
Representation of the orthogonal group by measure-preserving transformations	p. 100
Bosonic quantization of symplectic dynamics	p. 105
Fermionic quantization of orthogonal dynamics	p. 113
Absolute Continuity and Unitary Implementability	p. 118
Equivalence of distributions	p. 119
Quasi-invariant distributions and Weyl systems	p. 121
Ergodicity and irreducibility of Weyl pairs	p. 124
Infinite products of Hilbert spaces	p. 125
Affine transforms of the isonormal distribution	p. 130
Implementability of orthogonal transformations on the fermion field	p. 135
C*-Algebraic Quantization	p. 142
Weyl algebras over a linear symplectic space	p. 143
Regular states of the general boson field	p. 147
The representation-independent Clifford algebra	p. 149
Lexicon: The distribution of occupation numbers	p. 150
Quantization of Linear Differential Equations	p. 153
The Schrodinger equation	p. 154
Quantization of second-order equations	p. 158
Finite propagation velocity	p. 160
Quantization of the Dirac equation	p. 162
Quantization of global spaces of wave functions	p. 168
Renormalized Products of Quantum Fields	p. 174
The algebra of additive renormalization	p. 174
Renormalized products of the free boson field	p. 184
Regularity properties of boson field operators	p. 190
Renormalized local products of field operators	p. 200
Construction of Nonlinear Quantized Fields	p. 208
The L[subscript p] scale	p. 210
Renormalized products at fixed time	p. 214
Properties of fixed-time renormalization	p. 223
The semigroup generated by the interaction Hamiltonian	p. 227
The pseudo-interacting field	p. 229
Dynamic causality	p. 233
The local quantized equation of motion	p. 237
Appendix A Principal Notations	p. 251
Appendix B Universal Fields and the Quantization of Wave Equations	p. 254
Glossary	p. 258
Bibliography	p. 281
Index	p. 289
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