Invariance of Total Probability
The electromagnetic background can be thought of as the probability of an interaction between a test particle and the field in its immediate environment. The interaction can be defined as an exchange of some quantity of information in the form of energy, momentum , angular momentum, polarization or charge, mediated by real or virtual photons. The wave function of the test particle is that of equation (1) , a solution of Dirac’s equation. The probability density for this particle is found by multiplying the probability amplitude by its complex conjugate probability wave,
(24)
The total probability is,
(25)
The total probability normalized to 1 implies
that the particle must be somewhere along 
. Now if a
Lorentz transformation is done on the coordinate 
, then in the
new coordinates 
the total probability is unaffected, meaning it is
Lorentz invariant. The probability density however is not
Lorentz invariant and must change to preserve the total
probability!
(26)
Speculate for a moment that the conjugate wave
function 
is
actually the conjugate reflection of the wave function 
itself. There
is some coefficient of reflection 
that reverses the sign of the imaginary
exponential and may change the amplitude of the wave,
thereby changing the probability density. This function could for
instance be determined by the spectral energy density 
of the ZPF
resulting from the interaction of the test particle with the
matter and energy in the environment. Simply put, the greater the
density of matter and energy the greater the probability of an
interaction.
The conjugate wave is space-time reversed. Its phase advances in the opposite direction in both space and time relative to the wave function of the test particle. The interaction between the test particle and its conjugate reflection must depend on the relative refractive index as a function of the coordinates through which it is passing.
In the Probability Wave Dispersion Interpretation the wave function is reflected by interactions with the background field as the wave propagates along a path. In this manner gradients in the refractive index cause reflection and absorption which change the probability density and lead to transformations of the coordinates. This is identical to what was described above when the coordinates were intentionally assumed to have been Lorentz transformed.
(27)
There is no reason to suppose that the
coefficient of reflection in each direction, or the spectral
energy density of the potential 
are constants at all coordinates. In fact
one needs to suppose that they would depend on the density of
matter and energy generating the background field 
at the point
. The gauge
transformation is the gradient of a scalar function, and that
gradient corresponds to the gradient in the probability of an
interaction. Thus by means of a variable index of refraction
resulting from the absorption and emission of photons the
probability density is transformed into a function of the local
density of matter and energy, and so are the coordinates!