Mathematics of Quantization and Quantum Fields

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Cambridge University Press, Mar 7, 2013 - Science - 674 pages
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Unifying a range of topics that are currently scattered throughout the literature, this book offers a unique and definitive review of mathematical aspects of quantization and quantum field theory. The authors present both basic and more advanced topics of quantum field theory in a mathematically consistent way, focusing on canonical commutation and anti-commutation relations. They begin with a discussion of the mathematical structures underlying free bosonic or fermionic fields, like tensors, algebras, Fock spaces, and CCR and CAR representations (including their symplectic and orthogonal invariance). Applications of these topics to physical problems are discussed in later chapters. Although most of the book is devoted to free quantum fields, it also contains an exposition of two important aspects of interacting fields: diagrammatics and the Euclidean approach to constructive quantum field theory. With its in-depth coverage, this text is essential reading for graduate students and researchers in departments of mathematics and physics.
 

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Contents

Introduction
1
Operators in Hilbert spaces
36
Tensor algebras
57
Analysis in L2 Rd
92
Measures
111
Algebras
142
Antisymmetric calculus
159
Canonical commutation relations
173
CAR on Fock spaces
337
Orthogonal invariance of CAR algebras
351
Clifiord relations
368
Orthogonal invariance of the CAR on Fock spaces
386
Quasifree states
423
Dynamics of quantum fields
475
Quantum fields on spacetime
512
Diagrammatics
555

CCR on Fock space
212
Symplectic invariance of CCR
239
Symplectic invariance of the
266
Canonical anticommutation relations
313
Euclidean approach for bosons
605
Interacting bosonic fields
641
References
661
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About the author (2013)

Jan Dereziński is a Professor in the Faculty of Physics at the University of Warsaw. His research interests cover various aspects of quantum physics and quantum field theory, especially from the rigorous point of view.

Christian Gérard is a Professor in the Département de Mathématiques at the Université Paris-Sud. He was previously Directeur de Recherche at the CNRS (Centre National de la Recherche Scientifique). His research interests are the spectral and scattering theory in non-relativistic quantum mechanics and in quantum field theory.

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