  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.
