(Ebook) Principles of Discrete Time Mechanics 1st Edition by George Jaroszkiewicz ISBN 9781107034297 1107034299

(Ebook) Principles of Discrete Time Mechanics 1st Edition by George Jaroszkiewicz ISBN 9781107034297 1107034299

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ISBN 10: 1107034299
ISBN 13: 9781107034297
Author: George Jaroszkiewicz

(Ebook) Principles of Discrete Time Mechanics 1st Edition Table of contents:

Part I: Discrete Time Concepts

1. Introduction
    1.1 What is time?
    1.2 The architecture of time
    1.3 The chronon: historical perspectives
    1.4 The chronon: some modern perspectives
    1.5 Plan of this book

2. The Physics of Discreteness
    2.1 The natural occurrence of discreteness
    2.2 Fourier-transform scales
    2.3 Atomic scales of time
    2.4 De Broglie scales
    2.5 Hadronic scales
    2.6 Grand unified scales
    2.7 Planck scales

3. The Road to Calculus
    3.1 The origins of calculus
    3.2 The infinitesimal calculus and its variants
    3.3 Non-standard analysis
    3.4 q-Calculus

4. Temporal Discretization
    4.1 Why discretize time?
    4.2 Notation
    4.3 Some useful results
    4.4 Discrete analogues of some generalized functions
    4.5 Discrete first derivatives
    4.6 Difference equations
    4.7 Discrete Wronskians

5. Discrete Time Dynamics Architecture
    5.1 Mappings, functions
    5.2 Generalized sequences
    5.3 Causality
    5.4 Discrete time
    5.5 Second-order architectures

6. Some Models
    6.1 Reverse engineering solutions
    6.2 Reverse engineering constants of the motion
    6.3 First-order discrete time causality
    6.4 The Laplace-transform method

7. Classical Cellular Automata
    7.1 Classical cellular automata
    7.2 One-dimensional cellular automata
    7.3 Spreadsheet mechanics
    7.4 The Game of Life
    7.5 Cellular time dilation
    7.6 Classical register mechanics

Part II: Classical Discrete Time Mechanics
8. The Action Sum
 8.1 Configuration-space manifolds
 8.2 Continuous time action principles
 8.3 The discrete time action principle
 8.4 The discrete time equations of motion
 8.5 The discrete time Noether theorem
 8.6 Conserved quantities via the discrete time Weiss action principle

9. Worked Examples
    9.1 The complex harmonic oscillator
    9.2 The anharmonic oscillator
    9.3 Relativistic-particle models

10. Lee's Approach to Discrete Time Mechanics
     10.1 Lee's discretization
     10.2 The standard particle system
     10.3 Discussion
     10.4 Return to the relativistic point particle

11. Elliptic Billiards
     11.1 The general scenario
     11.2 Elliptic billiards via the geometrical approach
     11.3 Elliptic billiards via Lee mechanics
     11.4 Complex-plane billiards

12. The Construction of System Functions
     12.1 Phase space
     12.2 Hamilton's principal function
     12.3 Virtual-path construction of system functions

13. The Classical Discrete Time Oscillator
     13.1 The discrete time oscillator
     13.2 The Newtonian oscillator
     13.3 Temporal discretization of the Newtonian oscillator
     13.4 The generalized oscillator
     13.5 Solutions
     13.6 The three regimes
     13.7 The Logan invariant
     13.8 The oscillator in three dimensions
     13.9 The anharmonic oscillator

14. Type-2 Temporal Discretization
     14.1 Introduction
     14.2 q-Mechanics
     14.3 Phi-functions
     14.4 The phi-derivative
     14.5 Phi-integrals
     14.6 The summation formula
     14.7 Conserved currents

15. Intermission
     15.1 The continuous time Lagrangian approach
     15.2 The discrete time Lagrangian approach
     15.3 Extended discrete time mechanics

Part III: Discrete Time Quantum Mechanics
16. Discrete Time Quantum Mechanics
 16.1 Quantization
 16.2 Quantum dynamics
 16.3 The Schrödinger picture
 16.4 Position eigenstates
 16.5 Normal-coordinate systems
 16.6 Compatible operators

17. The Quantized Discrete Time Oscillator
     17.1 Introduction
     17.2 Canonical quantization
     17.3 The inhomogeneous oscillator
     17.4 The elliptic regime
     17.5 The hyperbolic regime
     17.6 The time-dependent oscillator

18. Path Integrals
     18.1 Introduction
     18.2 Feynman's path integrals
     18.3 Lee's path integral

19. Quantum Encoding
     19.1 Introduction
     19.2 First-order quantum encoding
     19.3 Second-order quantum encoding
     19.4 Invariants of the motion

Part IV: Discrete Time Classical Field Theory
20. Discrete Time Classical Field Equations
 20.1 Introduction
 20.2 System functions for discrete time field theories
 20.3 System functions for node variables
 20.4 Equations of motion for node variables
 20.5 Exact and near symmetry invariants
 20.6 Linear momentum
 20.7 Orbital angular momentum
 20.8 Link variables

21. The Discrete Time Schrödinger Equation
     21.1 Introduction
     21.2 Stationary states
     21.3 Vibrancy relations
     21.4 Linear independence and inner products
     21.5 Conservation of charge

22. The Discrete Time Klein–Gordon Equation
     22.1 Introduction
     22.2 Linear momentum
     22.3 Orbital angular momentum
     22.4 The free-charged Klein–Gordon equation

23. The Discrete Time Dirac Equation
     23.1 Introduction
     23.2 Grassmann variables in mechanics
     23.3 The Grassmannian oscillator in continuous time
     23.4 The Grassmannian oscillator in discrete time
     23.5 The discrete time free Dirac equation
     23.6 Charge and charge density

24. Discrete Time Maxwell Equations
     24.1 Classical electrodynamical fields
     24.2 Gauge invariance
     24.3 The inhomogeneous equations
     24.4 The charge-free equations
     24.5 Gauge transformations and virtual paths
     24.6 Coupling to matter fields

25. The Discrete Time Skyrme Model
     25.1 The Skyrme model
     25.2 The SU(2) particle
     25.3 The σ model
     25.4 Further considerations

Part V: Discrete Time Quantum Field Theory
26. Discrete Time Quantum Field Theory
 26.1 Introduction
 26.2 The discrete time free quantized scalar field
 26.3 The discrete time free quantized Dirac field
 26.4 The discrete time free quantized Maxwell fields

27. Interacting Discrete Time Scalar Fields
     27.1 Reduction formulae
     27.2 Interacting fields: scalar field theory
     27.3 Feynman rules for discrete time-ordered products
     27.4 The two–two box scattering diagram
     27.5 The vertex functions
     27.6 The propagators
     27.7 Rules for scattering amplitudes

Part VI: Further Developments
28. Space, Time and Gravitation
 28.1 Snyder's quantized spacetime
 28.2 Discrete time quantum fields on Robertson–Walker spacetimes
 28.3 Regge calculus

29. Causality and Observation
     29.1 Introduction
     29.2 Causal sets
     29.3 Quantum causal sets
     29.4 Discrete time and the evolving observer

30. Concluding Remarks

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Tags: George Jaroszkiewicz, Principles, Time Mechanics