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
3 - The Dirac equation
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 this chapter, we investigate the Dirac equation, which is named after P. A. M. Dirac, who is one of the fathers of quantum field theory. The Dirac equation is a relativistic quantum mechanical wave equation for spin-1/2 particles (e.g. electrons), which was derived by Dirac in 1928. The difficulties in finding a consistent single-particle theory from the Klein–Gordon equation led Dirac to search for an equation that
had a positive-definite conserved probability density and
was first order both in time and space.
One can show that these two conditions imply that a matrix equation is required. The reason why the Klein–Gordon equation did not yield a positive-definite probability density is connected with the second-order time derivative in this equation, which arises because the Klein–Gordon equation is related to the relativistic energy–momentum relation E2 = m2 + p2 via the correspondence principle that includes a term E2. Thus, a ‘better’ Lorentz covariant wave equation with a positive-definite probability density should have a first-order time derivative only. However, the equivalence of time and space coordinates in Minkowski space requires that such an equation also have only first-order space derivatives.
- Type
- Chapter
- Information
- Relativistic Quantum PhysicsFrom Advanced Quantum Mechanics to Introductory Quantum Field Theory, pp. 40 - 93Publisher: Cambridge University PressPrint publication year: 2011