Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction to relativistic quantum mechanics
- 2 The Klein–Gordon equation
- 3 The Dirac equation
- 4 Quantization of the non-relativistic string
- 5 Introduction to relativistic quantum field theory: propagators, interactions, and all that
- 6 Quantization of the Klein–Gordon field
- 7 Quantization of the Dirac field
- 8 Maxwell's equations and quantization of the electromagnetic field
- 9 The electromagnetic Lagrangian and introduction to Yang–Mills theory
- 10 Asymptotic fields and the LSZ formalism
- 11 Perturbation theory
- 12 Elementary processes of quantum electrodynamics
- 13 Introduction to regularization, renormalization, and radiative corrections
- Appendix A A brief survey of group theory and its notation
- Bibliography
- Index
6 - Quantization of the Klein–Gordon field
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- 1 Introduction to relativistic quantum mechanics
- 2 The Klein–Gordon equation
- 3 The Dirac equation
- 4 Quantization of the non-relativistic string
- 5 Introduction to relativistic quantum field theory: propagators, interactions, and all that
- 6 Quantization of the Klein–Gordon field
- 7 Quantization of the Dirac field
- 8 Maxwell's equations and quantization of the electromagnetic field
- 9 The electromagnetic Lagrangian and introduction to Yang–Mills theory
- 10 Asymptotic fields and the LSZ formalism
- 11 Perturbation theory
- 12 Elementary processes of quantum electrodynamics
- 13 Introduction to regularization, renormalization, and radiative corrections
- Appendix A A brief survey of group theory and its notation
- Bibliography
- Index
Summary
In ordinary non-relativistic quantum mechanics, quantizing a point particle, we interpret the generalized coordinates qn (normally the three Cartesian spatial coordinates x, y, and z) and their corresponding 3-momenta pn as operators, which act on the wave function Ψ that is a representation of the vector of states, and they fulfil the commutation relations [qn, pm] = iħδnm. However, in field theory, the position coordinates xi have a different meaning and are used to label the infinitely many ‘coordinates’ φ(x) and their corresponding conjugate momenta π(x). In order to quantize a field theory, we impose on the fields φ(x) and π(x) the commutation relations [φ(t, x), π(t, y)] = iħδ(x - y), which are the natural generalizations of the canonical commutation relations for ordinary quantum mechanics. Note that, in what follows, we will again set ħ = 1. In addition, the discussion in this chapter will be performed for a free neutral Klein–Gordon field. However, in the last two sections, we will present two natural extensions. First, in Section 6.5, we will extend the discussion with an example of interactions in the form of a classical external source, and second, in Section 6.6, we will describe how a free charged Klein–Gordon field can be treated.
Canonical quantization
Next, we want to develop the canonical quantum field theory for the neutral Klein–Gordon field. This theory was invented in the 1930s and it is indeed a very successful theory.
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- Relativistic Quantum PhysicsFrom Advanced Quantum Mechanics to Introductory Quantum Field Theory, pp. 122 - 137Publisher: Cambridge University PressPrint publication year: 2011