Book contents
- Frontmatter
- Contents
- Preface
- Part I Discrete time concepts
- Part II Classical discrete time mechanics
- Part III Discrete time quantum mechanics
- Part IV Discrete time classical field theory
- 20 Discrete time classical field equations
- 21 The discrete time Schrödinger equation
- 22 The discrete time Klein–Gordon equation
- 23 The discrete time Dirac equation
- 24 Discrete time Maxwell equations
- 25 The discrete time Skyrme model
- Part V Discrete time quantum field theory
- Part VI Further developments
- Appendix A Coherent states
- Appendix B The time-dependent oscillator
- Appendix C Quaternions
- Appendix D Quantum registers
- References
- Index
23 - The discrete time Dirac equation
from Part IV - Discrete time classical field theory
Published online by Cambridge University Press: 05 May 2014
- Frontmatter
- Contents
- Preface
- Part I Discrete time concepts
- Part II Classical discrete time mechanics
- Part III Discrete time quantum mechanics
- Part IV Discrete time classical field theory
- 20 Discrete time classical field equations
- 21 The discrete time Schrödinger equation
- 22 The discrete time Klein–Gordon equation
- 23 The discrete time Dirac equation
- 24 Discrete time Maxwell equations
- 25 The discrete time Skyrme model
- Part V Discrete time quantum field theory
- Part VI Further developments
- Appendix A Coherent states
- Appendix B The time-dependent oscillator
- Appendix C Quaternions
- Appendix D Quantum registers
- References
- Index
Summary
Introduction
In this chapter we apply temporal discretization to mechanical systems described by anticommuting variables rather than the commuting variables normally used in classical mechanics (CM). The anticommuting numbers representing such variables are called Grassmann(ian) numbers by mathematicians, after Hermann G. Grassmann (1809–1877), who did some pioneering work in linear algebra. On that account we shall refer to anticommuting numbers as g-numbers, in contrast to c-numbers (c for commuting1) when we refer to ordinary real or complex variables.
The significant difference between g-numbers and c-numbers is that, where as any two c numbers x and y satisfy the commutative multiplication rule xy = +yx, any two g-numbers θ and ϕ satisfy the anticommutation rule θϕ = –ϕθ. G-numbers and c-numbers can be multiplied together in any order, i.e., g-numbers and c-numbers commute. The product zθ of a c-number z and a g-number θ is a g-number.
The values of classical variables in ordinary CM are c-numbers, so we may refer to such variables as c-variables. It is possible to construct forms of CM where the classical variables are g-numbers, in which case we shall refer to such variables as g-variables.
G-variables should not be thought of as quantized versions of c-variables. They are just as classical as c-variables but much less familiar in terms of their applications to the real world. G-variables should not be confused with Dirac's q-numbers either.
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- Principles of Discrete Time Mechanics , pp. 253 - 264Publisher: Cambridge University PressPrint publication year: 2014