Mathematics of Quantization and Quantum Fields

This 2013 book, now OA, offers a definitive review of mathematical aspects of quantization and quantum field theory.

Jan Derezi?ski (Author), Christian Gérard (Author)

9781009290838, Cambridge University Press

Paperback / softback, published 9 February 2023

688 pages
24.4 x 17.1 x 3.5 cm, 1.18 kg

'… offers much highly valuable material.' Stig Stenholm, Contemporary Physics

Unifying topics that are scattered throughout the literature, this book offers a definitive review of mathematical aspects of quantization and quantum field theory. It presents both basic and advanced topics of quantum field theory in a mathematically consistent way, focusing on canonical commutation and anti-commutation relations. It begins with a discussion of the mathematical structures underlying free bosonic or fermionic fields: tensors, algebras, Fock spaces, and CCR and CAR representations. 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 mathematics and physics. This title, first published in 2013, has been reissued as an Open Access publication on Cambridge Core.

Preface
1. Vector spaces
2. Operators in Hilbert spaces
3. Tensor algebras
4. Analysis in L2(Rd)
5. Measures
6. Algebras
7. Anti-symmetric calculus
8. Canonical commutation relations
9. CCR on Fock spaces
10. Symplectic invariance of CCR in finite dimensions
11. Symplectic invariance of the CCR on Fock spaces
12. Canonical anti-commutation relations
13. CAR on Fock spaces
14. Orthogonal invariance of CAR algebras
15. Clifford relations
16. Orthogonal invariance of the CAR on Fock spaces
17. Quasi-free states
18. Dynamics of quantum fields
19. Quantum fields on space-time
20. Diagrammatics
21. Euclidean approach for bosons
22. Interacting bosonic fields
Subject index
Symbols index.

Subject Areas: Mathematical physics [PHU], Physics [PH], Mathematics & science [P]