**Mathematical Surveys and Monographs**

Volume: 151;
2008;
299 pp;
Hardcover

MSC: Primary 22; 43; 46; 53; 57; 78; 80;

Print ISBN: 978-0-8218-4495-3

Product Code: SURV/151

List Price: $93.00

AMS Member Price: $74.40

MAA member Price: $83.70

**Electronic ISBN: 978-1-4704-1378-1
Product Code: SURV/151.E**

List Price: $93.00

AMS Member Price: $74.40

MAA member Price: $83.70

#### Supplemental Materials

# The Geometry of Heisenberg Groups: With Applications in Signal Theory, Optics, Quantization, and Field Quantization

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*Ernst Binz; Sonja Pods*

The three-dimensional Heisenberg group, being a quite simple non-commutative
Lie group, appears prominently in various applications of mathematics. The
goal of this book is to present basic geometric and algebraic properties
of the Heisenberg group and its relation to other important mathematical
structures (the skew field of quaternions, symplectic structures, and
representations) and to describe some of its applications. In particular,
the authors address such subjects as signal analysis and processing,
geometric optics, and quantization. In each case, the authors present
necessary details of the applied topic being considered.

With no prerequisites beyond the standard mathematical curriculum, this
book manages to encompass a large variety of topics being easily
accessible in its fundamentals. It can be useful to students and
researchers working in mathematics and in applied mathematics.

#### Readership

Graduate students and research mathematicians interested in the use of analysis on Heisenberg groups to various problems in pure and applied mathematics.

#### Table of Contents

# Table of Contents

## The Geometry of Heisenberg Groups: With Applications in Signal Theory, Optics, Quantization, and Field Quantization

- Contents v7 free
- Introduction ix11 free
- Chapter 1. The Skew Field of Quaternions 119 free
- Chapter 2. Elements of the Geometry of $S^3$, Hopf Bundles and Spin Representations 2543
- Chapter 3. Internal Variables of Singularity Free Vector Fields in a Euclidean Space 4765
- Chapter 4. Isomorphism Classes, Chern Classes and Homotopy Classes... 7391
- Chapter 5. Heisenberg Algebras, Heisenberg Groups, Minkowski Metrics... 107125
- Chapter 6. The Heisenberg Group and Natural $C^*$-Algebras of a Vector Field ... 131149
- Chapter 7. The Schrodinger Representation and the Metaplectic Representation 161179
- Chapter 8. The Heisenberg Group: A Basic Geometric Background of Signal Analysis... 191209
- Chapter 9. Quantization of Quadratic Polynomials 215233
- Chapter 10. Field Theoretic Weyl Quantization of a Vector Field in 3-Space 247265
- Appendix A. Thermodynamics, Geometry and the Heisenberg Group... 269287
- Appendix Bibliography 289307
- Bibliography 291309
- Index 295313 free