Sonja Pods

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

The skew field of quaternions Elements of the geometry of

Heisenberg groups: a unifying structure of signal theory, holography and quantum information theory

S^3

Heisenberg groups—the fundamental ingredient to describe information, its transmission and quantization

, Hopf bundles and spin representations Internal variables of singularity free vector fields in a Euclidean space Isomorphism classes, Chern classes and homotopy classes of singularity free vector fields in three-space Heisenberg algebras, Heisenberg groups, Minkowski metrics, Jordan algebras and SL

Spinor Geometry and Signal Transmission in Three‐Space

(2,\mathbb {C})

6 The Heisenberg group as a fundamental structure in nature

The Heisenberg group and natural

A Heisenberg Algebra Bundle of a Vector Field in Three‐Space and its Weyl Quantization

C*

The Heisenberg Group in Classical and Quantum Information Transmission

-algebras of a vector field in 3-space The Schrodinger representation and the metaplectic representation The Heisenberg group-A basic geometric background of signal analysis and geometric optics Quantization of quadratic polynomials Field theoretic Weyl quantization of a vector field in 3-space Thermodynamics, geometry and the Heisenberg group by Serge Preston Bibliography Index.

The skew field of quaternions

Vector fields in three-space admit bundles of internal variables such as a Heisenberg algebra bundle. Information transmission along field lines of vector fields is described by a wave linked to the Schrödinger representation in the realm of time-frequency analysis. The preservation of local information causes geometric optics and a quantization scheme. A natural circle bundle models quantum information visualized by holographic methods. Features of this setting are applied to magnetic resonance imaging.

Thermodynamics, geometry and the Heisenberg group by Serge Preston

For a singularity free gradient field in an open set of an oriented Euclidean space of dimension three we define a natural principal bundle out of an immanent complex line bundle. The fibres of this bundle encode information. The elements of both bundles are called internal variables. Several other natural bundles are associated with the principal bundle and, in turn, determine the vector field. Two examples are given and it is shown that for a constant vector field circular polarized waves with values in the principal bundle are associated with the vector field. These waves transmit information encoded in internal variables and, moreover, determine a Schrodinger representation. On U(2) a relation between spin representations and Schrodinger representations is established. The link between the spin ½ model and the Schrodinger representations yields a connection between a microscopic and a macroscopic viewpoint. Quantization and its link to information is derived out of the Schrodinger representation.

Isomorphism classes, Chern classes and homotopy classes of singularity free vector fields in 3-space

For a singularity free gradient field in an open set of an oriented Euclidean space of dimension three we define a natural principal bundle out of an immanent complex line bundle. The elements of both bundles are called internal variables. Several other natural bundles are associated with the principal bundle and, in turn, determine the vector field. Two examples are given and it is shown that for a constant vector field circular polarized waves travelling along a field line can be considered as waves of internal variables. Einstein’s equation e = m ⋅ c2 is derived from the geometry of the principal bundle. On SU(2) a relation between spin representations and Schrodinger representations is established. The link between the spin 12‐model and the Schrodinger representations yields a connection between a microscopic and a macroscopic viewpoint.