Mathematics of Quantization and Quantum Fields

by Derezinski, Jan; Gerard, Christian

• ISBN13: 9781107011113
• ISBN10: 1107011116
• Format: Hardcover
• Copyright: 2013-04-15
• Publisher: Cambridge Univ Pr

Summary

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.

Table of Contents

Preface

Vector spaces

Operators in Hilbert spaces

Tensor algebras

Analysis in L2(Rd)

Measures

Algebras

Anti-symmetric calculus

Canonical commutation relations

CCR on Fock spaces

Symplectic invariance of CCR in finite dimensions

Symplectic invariance of the CCR on Fock spaces

Canonical anti-commutation relations

CAR on Fock spaces

Orthogonal invariance of CAR algebras

Clifford relations

Orthogonal invariance of the CAR on Fock spaces

Quasi-free states

Dynamics of quantum fields

Quantum fields on space-time

Diagrammatics

Euclidean approach for bosons

Interacting bosonic fields

Subject index

Symbols index

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