Book contents
- Frontmatter
- Contents
- Introduction
- 1 Vector spaces
- 2 Operators in Hilbert spaces
- 3 Tensor algebras
- 4 Analysisin L2 (ℝd)
- 5 Measures
- 6 Algebras
- 7 Anti-symmetric calculus
- 8 Canonical commutation relations
- 9 CCR on Fock space
- 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
- References
- Symbols index
- Subject index
Introduction
- Frontmatter
- Contents
- Introduction
- 1 Vector spaces
- 2 Operators in Hilbert spaces
- 3 Tensor algebras
- 4 Analysisin L2 (ℝd)
- 5 Measures
- 6 Algebras
- 7 Anti-symmetric calculus
- 8 Canonical commutation relations
- 9 CCR on Fock space
- 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
- References
- Symbols index
- Subject index
Summary
Quantum fields and quantization are concepts that come from quantum physics, the most intriguing physical theory developed in the twentieth century. In our work we would like to describe in a coherent and comprehensive way basic aspects of their mathematical structure.
Most of our work is devoted to the simplest kinds of quantum fields and of quantization. We will mostly discuss mathematical aspects of free quantum fields. We will consider the quantization only on linear phase spaces. The reader will see that even within such a restricted scope the subject is rich, involves many concepts and has important applications, both to quantum theory and to pure mathematics.
A distinguished role in our work will be played by representations of the canonical commutation and anti-commutation relations. Let us briefly discuss the origin and the meaning of these concepts.
- Type
- Chapter
- Information
- Mathematics of Quantization and Quantum Fields , pp. 1 - 7Publisher: Cambridge University PressPrint publication year: 2013