The Geometry of Heisenberg Groups: With Applications in Signal Theory, Optics, Quantization, and Field QuantizationAmerican Mathematical Soc., 2008 - 299 páginas The three-dimensional Heisenberg group, being the simplest 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 well 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. |
Contenido
| 1 | |
Chapter 2 Elements of the Geometry of S3 Hopf Bundles and Spin Representations | 25 |
Chapter 3 Internal Variables of Singularity Free Vector Fields in a Euclidean Space | 47 |
Chapter 4 Isomorphism Classes Chern Classes and Homotopy Classes | 73 |
Chapter 5 Heisenberg Algebras Heisenberg Groups Minkowski Metrics | 107 |
Chapter 6 The Heisenberg Group and Natural CAlgebras of a Vector Field | 131 |
Chapter 7 The Schrodinger Representation and the Metaplectic Representation | 161 |
A Basic Geometric Background of Signal Analysis | 191 |
Chapter 9 Quantization of Quadratic Polynomials | 215 |
Chapter 10 Field Theoretic Weyl Quantization of a Vector Field in 3Space | 247 |
Appendix A Thermodynamics Geometry and the Heisenberg Group | 269 |
| 289 | |
| 291 | |
| 295 | |
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Términos y frases comunes
automorphism bracket C*-algebra called chapter Chern class cohomology commutator complex line bundle coordinate curvature defined denoted determined element equation Fae G Fae TFª fibre Fourier transform free vector field geometry given grad h₁ h₂ Heisenberg algebra Heisenberg group Hence Hermitian Hilbert space holds true homomorphism infinite dimensional isomorphism LEMMA Lie algebra linear map linear space matrix metaplectic group metaplectic representation metric G Minkowski metric Moreover Mp(F multiplication natural Obviously operator optics oriented orthogonal phase space plane Poisson algebra prHopf principal bundle PROPOSITION quaternions R-linear respectively rotation scalar product Schrödinger representation signal singularity free vector skew field smooth SO(E Sp(F Sp(Fa symplectic form symplectic structure THª theorem topology trivial TS² unit vector unitary vector bundle Weyl algebra Weyl quantization yields