Principles of discrete time mechanics:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge
Cambridge Univ. Press
2014
|
| Edition: | 1. publ. |
| Series: | Cambridge monographs on mathematical physics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XIV, 365 S. graph. Darst. |
| ISBN: | 9781107034297 1107034299 |
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Record in the Search Index
| _version_ | 1804152199051214848 |
|---|---|
| adam_text | Contents
Й
Preface -page
xiii
Part I
Discrete
time concepts
1
Introduction
3
1.1
What is time?
3
1.2
The architecture of time
5
1.3
The chronon: historical perspectives
14
1.4
The chronon: some modern perspectives
17
1.5
Plan of this book
23
2
The physics of discreteness
24
2.1
The natural occurrence of discreteness
24
2.2
Fourier-transform scales
25
2.3
Atomic scales of time
27
2.4
De Broglie
scales
28
2.5
Hadronic scales
30
2.6
Grand unified scales
30
2.7
Planck scales
31
3
The road to calculus
32
3.1
The origins of calculus
32
3.2
The infinitesimal calculus and its variants
37
3.3
Non-standard analysis
40
3.4
q-Calculus
41
4
Temporal discretization
46
4.1
Why discretize time?
46
4.2
Notation
47
4.3
Some useful results
50
4.4
Discrete analogues of some generalized functions
52
4.5
Discrete first derivatives
53
4.6
Difference equations
55
4.7
Discrete Wronskians
58
viii Contents
5
Discrete
time dynamics architecture
61
5.1
Mappings, functions
61
5.2
Generalized sequences
65
5.3
Causality
66
5.4
Discrete time
67
5.5
Second-order architectures
69
6
Some models
6.1
Reverse engineering solutions
71
6.2
Reverse engineering constants of the motion
74
6.3
First-order discrete time causality
75
6.4
The Laplace-transform method
78
7
Classical cellular automata
80
7.1
Classical cellular automata
80
7.2
One-dimensional cellular automata
82
7.3
Spreadsheet mechanics
85
7.4
The Game of Life
88
7.5
Cellular time dilation
89
7.6
Classical register mechanics
97
Part II Classical discrete time mechanics
8
The action sum 111
8.1
Configuration-space manifolds 111
8.2
Continuous time action principles
112
8.3
The discrete time action principle
117
8.4
The discrete time equations of motion
119
8.5
The discrete time Noether theorem
119
8.6
Conserved quantities via the discrete time Weiss action principle
121
9
Worked examples
122
9.1
The complex harmonic oscillator
122
9.2
The anharmonic oscillator
124
9.3
Relativistic-particle models
126
10
Lee s approach to discrete time mechanics
129
10.1
Lee s discretization
129
10.2
The standard particle system
131
10.3
Discussion
133
10.4
Return to the relativistic point particle
134
Contents ix
11
Elliptic billiards
136
11.1
The general scenario
136
11.2
Elliptic billiards via the geometrical approach
137
11.3
Elliptic billiards via Lee mechanics
140
11.4
Complex-plane billiards
142
12
The construction of system functions
144
12.1
Phase space
144
12.2
Hamilton s principal function
145
12.3
Virtual-path construction of system functions
148
13
The classical discrete time oscillator
151
13.1
The discrete time oscillator
151
13.2
The Newtonian oscillator
152
13.3
Temporal discretization of the Newtonian oscillator
153
13.4
The generalized oscillator
154
13.5
Solutions
154
13.6
The three regimes
155
13.7
The Logan invariant
156
13.8
The oscillator in three dimensions
157
13.9
The anharmonic oscillator
158
14
Type-2 temporal discretization
160
14.1
Introduction
160
14.2
q-Mechanics
161
14.3
Phi-functions
164
14.4
The phi-
derivat
ive
165
14.5
Phi-integrals
166
14.6
The summation formula
166
14.7
Conserved currents
168
15
Intermission
1.70
15.1
The continuous time Lagrangian approach
171
15.2
The discrete time Lagrangian approach
173
15.3
Extended discrete time mechanics
175
Part III Discrete time quantum mechanics
16
Discrete time quantum mechanics
181
16.1
Quantization
16.2
Quantum dynamics
16.3
The
Schrödinger
picture
x
Contents
16.4 Position eigenstates 185
16.5 Normal-coordinate
systems
188
16.6
Compatible operators
190
17
The quantized discrete time oscillator
192
17.1
Introduction
192
17.2
Canonical quantization
193
17.3
The inhomogeneous oscillator
197
17.4
The elliptic regime
199
17.5
The hyperbolic regime
202
17.6
The time-dependent oscillator
203
18
Path integrals
209
18.1
Introduction
209
18.2
Feynman s path integrals
209
18.3
Lee s path integral
215
19
Quantum encoding
217
19.1
Introduction
217
19.2
First-order quantum encoding
218
19.3
Second-order quantum encoding
220
19.4
Invariants of the motion
221
Part IV Discrete time classical field theory
20
Discrete time classical field equations
227
20.1
Introduction
227
20.2
System functions for discrete time field theories
227
20.3
System functions for node variables
228
20.4
Equations of motion for node variables
230
20.5
Exact and near symmetry invariants
231
20.6
Linear momentum
233
20.7
Orbital angular momentum
234
20.8
Link variables
234
21
The discrete time
Schrödinger
equation
236
21.1
Introduction
236
21.2
Stationary states
239
21.3
Vibrancy relations
242
21.4
Linear independence and inner products
242
21.5
Conservation of charge
244
Contents xi
22
The discrete time Klein—Gordon equation
246
22.1
Introduction
246
22.2
Linear momentum
248
22.3
Orbital angular momentum
249
22.4
The free-charged Klein-Gordon equation
250
23
The discrete time Dirac equation
253
23.1
Introduction
253
23.2 Grassmann
variables in mechanics
254
23.3
The Grassmannian oscillator in continuous time
256
23.4
The Grassmannian oscillator in discrete time
258
23.5
The discrete time free Dirac equation
260
23.6
Charge and charge density
262
24
Discrete time Maxwell equations
265
24.1
Classical electrodynamical fields
265
24.2
Gauge
invariance
267
24.3
The inhomogeneous equations
269
24.4
The charge-free equations
270
24.5
Gauge transformations and virtual paths
271
24.6
Coupling to matter fields
272
25
The discrete time Skyrme model
275
25.1
The Skyrme model
275
25.2
The SU(2) particle
277
25.3
The
σ
model
281
25.4
Further considerations
282
Part V Discrete time quantum field theory
26
Discrete time quantum field theory
287
26.1
Introduction
287
26.2
The discrete time free quantized scalar field
289
26.3
The discrete time free quantized Dirac field
292
26.4
The discrete time free quantized Maxwell fields
297
27
Interacting discrete time scalar fields
306
27.1
Reduction formulae
307
27.2
Interacting fields: scalar field theory
308
27.3
Feynman rules for discrete time-ordered products
310
27.4
The two-two box scattering diagram
313
27.5
The vertex functions 316
xii Contents
27.6
The propagators
316
27.7
Rules for scattering amplitudes
318
Part VI Further developments
28
Space, time and gravitation
323
28.1
Snyder s quantized spacetime
323
28.2
Discrete time quantum fields on Robertson-Walker spacetimes
328
28.3
Regge
calculus
331
29
Causality and observation
333
29.1
Introduction
333
29.2
Causal sets
334
29-3
Quantum causal sets
336
29-4
Discrete time and the evolving observer
336
30
Concluding remarks
341
Appendix A Coherent states
343
Appendix
В
The time-dependent oscillator
345
Appendix
С
Quaternions
347
Appendix
D
Quantum registers
348
References
353
Index 36
!
PRINCIPLES OF DISCRETE TIME MECHANICS
Could time be discrete on some unimaginably small scale? Exploring the idea
in depth, this unique introduction to discrete time mechanics systematically
builds the theory up from scratch, beginning with the historical, physical and
mathematical background to the chronon hypothesis.
Covering classical and quantum discrete time mechanics, this book presents all
the tools needed to formulate and develop applications of discrete time mechanics
in a number of areas, including spreadsheet mechanics, classical and quantum
register mechanics, and classical and quantum mechanics and field theories. A
consistent emphasis on contextuality and the observer-system relationship is
maintained throughout.
George Jaroszkiewicz is an Associate Professor at the School of Math¬
ematical Sciences, University of Nottingham, having formerly held positions at
the University of Oxford and the University of Kent.
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| spelling | Jaroszkiewicz, George Verfasser aut Principles of discrete time mechanics George Jaroszkiewicz 1. publ. Cambridge Cambridge Univ. Press 2014 XIV, 365 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Cambridge monographs on mathematical physics Includes bibliographical references and index Quantum theory / Mathematics Mechanics Mathematik Quantentheorie Zeit (DE-588)4067461-7 gnd rswk-swf Quantisierung Physik (DE-588)4176603-9 gnd rswk-swf Zeit (DE-588)4067461-7 s Quantisierung Physik (DE-588)4176603-9 s DE-604 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027296500&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027296500&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Jaroszkiewicz, George Principles of discrete time mechanics Quantum theory / Mathematics Mechanics Mathematik Quantentheorie Zeit (DE-588)4067461-7 gnd Quantisierung Physik (DE-588)4176603-9 gnd |
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| title | Principles of discrete time mechanics |
| title_auth | Principles of discrete time mechanics |
| title_exact_search | Principles of discrete time mechanics |
| title_full | Principles of discrete time mechanics George Jaroszkiewicz |
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| title_full_unstemmed | Principles of discrete time mechanics George Jaroszkiewicz |
| title_short | Principles of discrete time mechanics |
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| topic | Quantum theory / Mathematics Mechanics Mathematik Quantentheorie Zeit (DE-588)4067461-7 gnd Quantisierung Physik (DE-588)4176603-9 gnd |
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