The geometry of Heisenberg groups: with applications in signal theory, optics, quantization, and field quantization
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
American Mathematical Soc.
2008
|
| Series: | Mathematical surveys and monographs
151 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references (p. 289-294) and index |
| Physical Description: | XVI, 299 S. |
| ISBN: | 9780821844953 0821844954 |
Staff View
MARC
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| 245 | 1 | 0 | |a The geometry of Heisenberg groups |b with applications in signal theory, optics, quantization, and field quantization |c Ernst Binz, Sonja Pods ; with an appendix by Serge Preston |
| 264 | 1 | |a Providence, RI |b American Mathematical Soc. |c 2008 | |
| 300 | |a XVI, 299 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mathematical surveys and monographs |v 151 | |
| 500 | |a Includes bibliographical references (p. 289-294) and index | ||
| 650 | 4 | |a Mathematik | |
| 650 | 4 | |a Quantentheorie | |
| 650 | 4 | |a Heisenberg uncertainty principle |x Mathematics | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
Introduction
ix
Chapter
1.
The Skew Field of Quaternions
1
1.1.
Definition of the Field of Quaternions
И
and Elementary Formulae
1
1.2.
Embeddings of
С
into the Quaternions and Natural Unitary Groups
7
1.3.
С
-linear Structures, Symplectic Structures and Orientation,
Pauli
Elements
9
1.4.
Inner Automorphisms of
H
12
1.5.
The Oriented Rotation Angle of the Inner Automorphism with
Respect to the Natural Minkowski Metric
18
1.6.
Link between Space-Time Geometry and Euclidean Geometry on the
Quaternions
20
Chapter
2.
Elements of the Geometry of S3,
Hopf
Bundles and Spin
Representations
25
2.1.
One-Parameter Groups of SU(2) and SO(E)
25
2.2.
Parallels of Latitude and Meridians on S3
27
2.3.
One-Parameter Subgroups of SU(2) and
Hopf
Bundles
28
2.4.
Spin Representations
40
2.5.
The Infinitesimal Spin^-Representation
43
Chapter
3.
Internal Variables of Singularity Free Vector Fields in a Euclidean
Space
47
3.1.
The Complex Line Bundle Fa
47
3.2.
Symplectic and Hermitian Structures on Fa
52
3.3.
Gradient Fields
54
3.4.
Curvature Forms on Level Surfaces
56
3.5.
Vector Fields Defined by Two-Forms
58
3.6.
The Principal bundle Fa and its Natural Connection Form
60
3.7.
The Characteristic Principal Bundle
65
3.8.
Horizontal and Periodic Lifts of Integral Curves
69
Chapter
4.
Isomorphism Classes,
Chem
Classes and Homotopy Classes of
Singularity Free Vector Fields in 3-Space
73
4.1.
Isomorphism Classes of Characteristic Principal Bundles of Vector
Fields
73
4.2.
The Structure of Isomorphism Classes
77
4.3.
Chern Classes
80
4.4.
Mapping Degree and First Chern-de Rham Classes
94
4.5.
Hodge-Morrey Decomposition
100
vi
CONTENTS
Chapter
5. Heisenberg
Algebras,
Heisenberg
Groups, Minkowski Metrics,
Jordan Algebras and SL(2, C)
107
5.1.
Natural Symplectic Structure on a Plane in 3-Space
107
5.2.
The Notion of
a
Heisenberg
Algebra
112
5.3. Heisenberg
Group and its Lie Algebra
115
5.4.
9ęea
as a Semi-direct Product
119
5.5.
A
Heisenberg
Algebra Structure on sp(F)
121
5.6.
The Spin Group and the Skew Field of Quaternions are Determined
by Only One
Heisenberg
Group
124
5.7.
Scalar Products and Minkowski Metrics on the
Heisenberg
Algebra
126
5.8.
Symplectic Group, Special Linear Groups and
Lorentz
group
128
Chapter
6.
The
Heisenberg
Group and Natural C*-Algebras of a Vector Field
in 3-Space
131
6.1.
The
Heisenberg
Group Bundle of a Vector Field
132
6.2.
Infinite Dimensional
Heisenberg
Algebras and Infinite Dimensional
Heisenberg
Groups of Vector Fields
135
6.3.
Maps Determined by Homomorphisms
140
6.4.
Group Algebras of Infinite Dimensional
Heisenberg
Groups
143
6.5.
The C*-Group Algebra and the Twisted Convolution, the Weyl
Algebra and the
Poisson
Algebra
153
Chapter
7.
The
Schrödinger
Representation and the Metaplectic
Representation
161
7.1.
Definition of the
Schrödinger
Representation and Phase Space
161
7.2.
Characteristic Ingredients of the
Schrödinger
Representation
166
7.3.
The Infinitesimal
Schrödinger
Representation and Phase Space
175
7.4.
Projective
Representations of the Symplectic Group Constructed via
the
Schrödinger
Representation
176
7.5.
A Realization of the Metaplectic Group and the Metaplectic
Representation
180
Chapter
8.
The
Heisenberg
Group: A Basic Geometric Background of Signal
Analysis and Geometric Optics
191
8.1.
The Notion of a Signal
192
8.2.
Time-Frequency Analysis and the Uncertainty Principle
193
8.3.
Further Tools of Time-Frequency Analysis
196
8.4.
Reconstruction Formulae
201
8.5.
The Geometry Underlying Time-Frequency Analysis
202
8.6.
The Radar Ambiguity Function
204
8.7.
The Stone-von Neumann Theorem in Time-Frequency Analysis
205
8.8.
Geometric Optics
206
8.9.
Holography
210
Chapter
9.
Quantization of Quadratic Polynomials
215
9.1.
Elementary Observations on Information and its Transmission
215
9.2.
Preservation of Information
217
9.3.
The
Poisson
Algebra of all Homogeneous Quadratic Polynomials in
Two Variables
221
9.4.
The Quantization of Inhomogeneous Quadratic Polynomials
230
CONTENTS
vii
9.5.
The Schrödinger
Equation
237
9.6.
State Spaces and
Observables,
Elements of Stochastic Interpretation
238
Chapter
10.
Field Theoretic Weyl Quantization of a Vector Field in 3-Space
247
10.1.
The Mathematical Setting
247
10.2.
The Idea of Weyl Quantization of X
248
10.3.
Weyl Quantization of Singularity Free Vector Fields in 3-Space
251
10.4.
The Relation to the GNS Representation
258
10.5.
The Influence of the Topology on the Weyl Quantization
263
Appendix A. Thermodynamics, Geometry and the
Heisenberg
Group by
Serge Preston
269
A.I. Introduction
269
A.2. The Contact Structure of Homogeneous Thermodynamics
270
A.3. Gibbs Space. Legendre Surfaces of Equilibrium
270
A.4. Thermodynamical Metrics of
Weinhold
and Ruppeiner
271
A.
5.
Indefinite Thermodynamical Metric
G
of R.
Mrugała
272
A.6.
Levi-Civitá
Connection of the Metric
G
273
A.7. Curvature Properties of
G
274
A.8. The
Heisenberg
Group as the Thermodynamical Phase Space
275
A.9. Geodesies of the Metric
G
279
АЛО.
Symplectization of the Manifold
(Ρ, Θ,
G)
281
A.ll. Properties of the Metric
G
282
A.
12.
Constitutive Hypersurface and its Lift to
Ρ
283
A.
13.
Hyperbolic Rotations and the Projectivization of
Ρ
284
A.
14.
Group Action of
9Í„
and the Partial Orbit Structure of
Ρ
285
Appendix. Bibliography
289
Bibliography
291
Index
295
|
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| dewey-ones | 512 - Algebra |
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| dewey-sort | 3512 3482 |
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| discipline | Mathematik |
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| indexdate | 2024-07-09T22:04:04Z |
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| language | English |
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| spelling | Binz, Ernst 1939- Verfasser (DE-588)138916306 aut The geometry of Heisenberg groups with applications in signal theory, optics, quantization, and field quantization Ernst Binz, Sonja Pods ; with an appendix by Serge Preston Providence, RI American Mathematical Soc. 2008 XVI, 299 S. txt rdacontent n rdamedia nc rdacarrier Mathematical surveys and monographs 151 Includes bibliographical references (p. 289-294) and index Mathematik Quantentheorie Heisenberg uncertainty principle Mathematics Quantum theory Mathematics Heisenberg-Gruppe (DE-588)4314104-3 gnd rswk-swf Heisenberg-Gruppe (DE-588)4314104-3 s DE-604 Pods, Sonja 1974- Sonstige (DE-588)124926134 oth Erscheint auch als Online-Ausgabe 978-1-4704-1378-1 Mathematical surveys and monographs 151 (DE-604)BV000018014 151 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018627583&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Binz, Ernst 1939- The geometry of Heisenberg groups with applications in signal theory, optics, quantization, and field quantization Mathematical surveys and monographs Mathematik Quantentheorie Heisenberg uncertainty principle Mathematics Quantum theory Mathematics Heisenberg-Gruppe (DE-588)4314104-3 gnd |
| subject_GND | (DE-588)4314104-3 |
| title | The geometry of Heisenberg groups with applications in signal theory, optics, quantization, and field quantization |
| title_auth | The geometry of Heisenberg groups with applications in signal theory, optics, quantization, and field quantization |
| title_exact_search | The geometry of Heisenberg groups with applications in signal theory, optics, quantization, and field quantization |
| title_full | The geometry of Heisenberg groups with applications in signal theory, optics, quantization, and field quantization Ernst Binz, Sonja Pods ; with an appendix by Serge Preston |
| title_fullStr | The geometry of Heisenberg groups with applications in signal theory, optics, quantization, and field quantization Ernst Binz, Sonja Pods ; with an appendix by Serge Preston |
| title_full_unstemmed | The geometry of Heisenberg groups with applications in signal theory, optics, quantization, and field quantization Ernst Binz, Sonja Pods ; with an appendix by Serge Preston |
| title_short | The geometry of Heisenberg groups |
| title_sort | the geometry of heisenberg groups with applications in signal theory optics quantization and field quantization |
| title_sub | with applications in signal theory, optics, quantization, and field quantization |
| topic | Mathematik Quantentheorie Heisenberg uncertainty principle Mathematics Quantum theory Mathematics Heisenberg-Gruppe (DE-588)4314104-3 gnd |
| topic_facet | Mathematik Quantentheorie Heisenberg uncertainty principle Mathematics Quantum theory Mathematics Heisenberg-Gruppe |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018627583&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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