Spinor Geometry and Signal Transmission in Three-Space

摘要

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 ε = m·c~2 is derived from the geometry of the principal bundle. On SU(2) a relation between spin representations and Schrb'dinger representations is established. The link between the spin 1/2―model and the Schrb'dinger representations yields a connection between a microscopic and a macroscopic viewpoint.