Book contents
- Frontmatter
- Contents
- Preface
- Part I Discrete time concepts
- Part II Classical discrete time mechanics
- 8 The action sum
- 9 Worked examples
- 10 Lee's approach to discrete time mechanics
- 11 Elliptic billiards
- 12 The construction of system functions
- 13 The classical discrete time oscillator
- 14 Type-2 temporal discretization
- 15 Intermission
- Part III Discrete time quantum mechanics
- Part IV Discrete time classical field theory
- 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
10 - Lee's approach to discrete time mechanics
from Part II - Classical discrete time mechanics
Published online by Cambridge University Press: 05 May 2014
- Frontmatter
- Contents
- Preface
- Part I Discrete time concepts
- Part II Classical discrete time mechanics
- 8 The action sum
- 9 Worked examples
- 10 Lee's approach to discrete time mechanics
- 11 Elliptic billiards
- 12 The construction of system functions
- 13 The classical discrete time oscillator
- 14 Type-2 temporal discretization
- 15 Intermission
- Part III Discrete time quantum mechanics
- Part IV Discrete time classical field theory
- 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
Lee's discretization
Time is a continuous parameter in standard Schrödinger wave mechanics, and quantum wavefunctions are differentiable functions of that parameter. We shall refer to this as temporal differentiability. Temporal differentiability is assumed in Dirac's more abstract formulation of quantum mechanics, where state vectors in some abstract Hilbert space and Hermitian operators over that space depend on continuous time (CT) (Dirac, 1958). The same is assumed in relativistic quantum field theories, where quantum field operators in the Heisenberg picture are differentiable over time and space.
Temporal differentiability is a necessary condition for the existence of the quantum wave equations and operator field equations found in such theories. Unfortunately, these differentiable equations are usually impossible to solve exactly, so various techniques such as perturbation theory and computer simulation are employed to make suitable approximations.
Numerical techniques on their own, however, are insufficient to answer questions of principle, such as the meaning of quantized spacetime. A more principled approach to the solution of quantum differential equations, namely the path integral (PI), was developed, exploited and popularized principally by Feynman (Feynman and Hibbs, 1965). In some situations, such as quantum gravity, the PI was found to be virtually the only technology available to discuss the quantum physics. We discuss this approach in greater detail in Chapter 18.
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- Principles of Discrete Time Mechanics , pp. 129 - 135Publisher: Cambridge University PressPrint publication year: 2014