Gauge Transformation
For the rest of this paper the properties of inductance and capacitance are collectively referred to as the polarizability of the manifold at the point P. They are functions determined by the density and distribution of polarizable wave functions on the manifold. Transformations of the polarizability from point to point induce phase shifts in the propagator analogous to impedance transformations along a wave transmission line. On a transmission line, such transformations can lead to reflections and standing waves. This is equivalent to adding a background electromagnetic gauge potential to the existing gauge potential.
In what follows the reader must be careful to distinguish between the affects of a Lorentz transformation or coordinate transformation,. and that of a Gauge transformation. Both are discussed in order to illustrate the connection between Lorentz invariance and gauge invariance.
In a Minkowski space-time the gauge potentials of the Maxwell field provide the equations,
(20)
Where 
is the field strength and 
is the current
density at the sources. The polarizability added to empty
space-time by the presence of matter changes the gauge potential
through the interaction of the field with matter. The
polarization matrix 
, also known as the Moments tensor is added to
the field strength.
(21)
The field 
is taken to be that of a background
potential relative to a test particle due to the presence of
matter in its neighborhood. This is the effective potential
resulting from the polarizability of the manifold. Lorentz
transformations of the components of the Moments tensor can then
be seen immediately as transformations in the values of
inductance and capacitance.
Quantum mechanically we can impose the loose
restriction that the expectation values of 
equal zero,
and also that there are no nearby sources in free space. The
expectation values of the field are then normalized to the vacuum
field at that point.
(22)
The field 
then becomes simply the Zero Point
electromagnetic field (ZPF). Since it has no contribution to
Maxwell’s equations, and leaves the expected field strength 
invariant under
this transformation, it is essentially a gauge transformation.
Gauge transformations affect the phase of the
wave function. Since the charge density 
is held
gauge invariant, then the addition of the gauge
transformation changes the polarizability along an open path from
the point 
.
(23)
Note that the inductance and capacitance are
both changed equally in such a way as to preserve the constant
ratios of permeability, permittivity and the speed of light. This
necessitates a transformation of the coordinates. In particular
note that the spectral energy density 
of the ZPF is
Lorentz invariant [2]. Therefore there will not be any frequency
dependence in the coordinate transformation induced by the gauge
transformation. Therefore one would not expect there to be any
chromatic dispersion, or colorful prismatic effects caused by the
ZPF.