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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.
Table of Contents
Operators in Hilbert spaces
Analysis in L2(Rd)
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
Orthogonal invariance of the CAR on Fock spaces
Dynamics of quantum fields
Quantum fields on space-time
Euclidean approach for bosons
Interacting bosonic fields
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