Fiber bundle description of number scaling in gauge theory and geometry

Abstract

This work uses fiber bundles as a framework to describe some effects of number scaling on gauge theory and some geometric quantities. A description of number scaling and fiber bundles over a flat space time manifold, M, is followed by a description of gauge theory. A fiber at point x of M contains a pair of scaled complex number and vector space structures, $C^{c}_{x}\times V^{c}_{x} $ for each c in GL(1,C). A space time dependent scalar field, g, determines, for each x, the scaling value of the vector space structures that contain the values of a vector valued matter field at x. The vertical components of connections between neighboring fibers are taken to be the gradient field A(x)+iB(x), of g. Abelian gauge theory for these fields gives the result that B is massless and no mass restrictions for A. Addition of an electromagnetic field dies not change these results. In the Mexican hat Higgs mechanism B combines with a Goldstone boson to create massive vector bosons, the photon field, and the Higgs field. For geometric quantities the fiber bundle is a tangent bundle with a fiber at point x containing scaled pairs, $R^{r}_{x}\times T^{r}_{x}$ of real number and tangent space structures for each x and and nonnegative real r. B is zero everywhere. The A field affects path lengths and the proper times of clocks along paths. It also appears in the geodesic equation. The lack of physical evidence for the gradient field, A(x)+iB(x) means that it either couples very weakly to matter fields or that it is close to zero for all x in a local region of cosmological space and time. It says nothing about the values outside the local region.