Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction and Overview
- 2 Classical Mechanics
- 3 Hilbert Space: The Arena of Quantum Mechanics
- 4 Quantum Mechanics
- 5 Scalar Quantum Field Theory
- 6 Expanding the Data Base
- 7 Rotationally Symmetric Models
- 8 Continuous and Discontinuous Perturbations
- 9 Independent-Value Models
- 10 Ultralocal Models
- 11 Summary and Outlook
- References
- Index
6 - Expanding the Data Base
Published online by Cambridge University Press: 15 September 2009
- Frontmatter
- Contents
- Preface
- 1 Introduction and Overview
- 2 Classical Mechanics
- 3 Hilbert Space: The Arena of Quantum Mechanics
- 4 Quantum Mechanics
- 5 Scalar Quantum Field Theory
- 6 Expanding the Data Base
- 7 Rotationally Symmetric Models
- 8 Continuous and Discontinuous Perturbations
- 9 Independent-Value Models
- 10 Ultralocal Models
- 11 Summary and Outlook
- References
- Index
Summary
WHAT TO LOOK FOR
In the previous chapters, we have developed a rather standard view toward classical and quantum theory illustrating the three canonical quantization procedures, namely those of Schrӧdinger, Heisenberg, and Feynman. In this chapter we want to return to Hilbert space and analyze several new issues as well as some old issues in greater depth than was previously the case. These issues will include operators and their domains, bilinear forms, infinite-product representations, irreducible representations of the canonical operators, as well as the important notion of “tags” (unitary invariants of operator representations). Some general remarks on measures, probability distributions, characteristic functions, and infinitely divisible distributions are also included here. Additionally, a few comments about Brownian motion are included. We close with some general remarks regarding canonical quantum scalar fields. The properties developed in this chapter will find application in our study of model quantum field theories in subsequent chapters.
Hilbert space and operators, revisited
In Chapter 3 we already introduced and discussed at some length properties of Hilbert spaces suitable for quantum mechanical applications. In all earlier applications we have used the elegant notation of Dirac, and this notation is generally appropriate for most applications.
- Type
- Chapter
- Information
- Beyond Conventional Quantization , pp. 117 - 144Publisher: Cambridge University PressPrint publication year: 1999