New Gauge Field from Extension of Space Time Parallel Transport of Vector Spaces to the Underlying Number Systems

Abstract

One way of describing gauge theories in physics is to assign a vector space $V_{x}$ to each space time point $x.$ For each $x$ the field $\psi$ takes values $\psi(x)$ in $V_{x}.$ The freedom to choose a basis in each $V_{x}$ introduces gauge group operators and their Lie algebra representations to define parallel transformations between vector spaces. This paper is an exploration of the extension of these ideas to include the underlying scalar complex number fields. Here a Hilbert space, $\bar{H}_{x},$ as an example of $V_{x},$ and a complex number field, $C_{x},$ are associated with each space time point. The freedom to choose a basis in $H_{x}$ is expanded to include the freedom to choose complex number fields. This expansion is based on the discovery that there exist representations of complex (and other) number systems that differ by arbitrary scale factors. Compensating changes must be made in the basic field operations so that the relevant axioms are satisfied. This results in the presence of a new real valued gauge field $\vec{A}(x).$ Inclusion of $\vec{A}(x)$ into covariant derivatives in Lagrangians results in the description of $\vec{A}(x)$ as a gauge boson that can have mass. The great accuracy of QED suggests that the coupling constant of $\vec{A}(x)$ to matter fields is very small compared to the fine structure constant. Other physical properties of $\vec{A}(x)$ are not known at present.