Eq 1
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The Evolving Quantum Equation
As historically established and empirically useful concepts, we have partial differential equations in the second order. Laplace's Equation,
Eq 1
is closely related to the Poisson equation. It is the basis and starting point for a dazzling realm of mathematics, including descriptions of waves, charge, mass and other phenomena in ordinary three dimensions, in the four dimensions of Cartesian coordinates with time, in Reimannian and relativistic space-time in which the curvature of space by gravitational masses occurs, and many other applications. It is also the basis for the diffusion equation used in heat, concentration, electrical, and many other diffusion processes.
The inception began with the realization that at least three second order partial differential equations -- vorticity, diffusion and wave equations -- occur not only throughout the universe but also throughout a vast range of time and distance. The number of equations creates an induction on the exponents within the equations, leading to a kind of emergent systems phenomenon among equations themselves. This began my speculation (moderated by many atomic spectra) that the naive photoelectric field wave equation is in fact too simple-minded, that possibly we do not know the exponents of that equation completely, and that a more general form of the equation is necessary to describe light waves. The conclusion I came to is that the diffusion equation should be superposed on or fitted to the wave equation. In addition, vorticity should have something to do with spin but that will not be discussed here.
The first derivative term already exists in some formulations of the wave equation. Dettman describes the unbounded wave as:
Eq 2
This equation is intended to suggest a potentially infinite number of terms but these are sufficient now.
It is necessary to be explicit about the structure of the equation because the decay
process being described is based on the idea that the decay process is intrinsic to the
photon wave, not to its origin or to the motion of its origin. This means it is necessary
to merge the two equations:
Eq 3, and 
Eq 4
The equation on the left is the naive wave equation. It is naive because it postulates that all photons are immortal. In the diffusion equation on the right, the quantity a is the as-yet-unspecified diffusivity. Determining that is one of the objectives of this paper.
Diffusion fits neatly into the unbounded wave Equation 2 described as engineering mathematics by Dettman. It contains material not needed here, at least not yet. The two Equations 3 and 4 above are thus simply parts of that most general form. Equation 5, below, shows only the terms important to both the wave motion and the decay process.
Eq 5
It appears to be true of every photon everywhere in the entire universe, and does not force any distortion upon our concept of celestial photons. Light, it appears, has two diffusions going on at the same time--the very rapid diffusion of the wavefront of a disturbance at the speed of light in which the permittivity is equivalent to the diffusivity, and the very slow diffusion described here, in which the action quantum is equivalent to the diffusivity. There remains the question of the constants involved in the various terms of Equation 2, equivalent to determining the correct form of the f in the second term of the right hand side of equation 5.
The only invariant constants which exist in the interstellar photon are the speed of light c which is the product of wavelength and frequency, and the action quantum h which is the ratio of the wave energy to frequency. Thus, one of the constants is c, already well known from many applications in radio, television and optics. This results in a more narrowly defined general form:
Eq 6
In this canonical form, a little wool has crept in, for it shows more possible terms than are actually necessary. c and p do not actually appear in the quantity in parenthesis,
We also need the decay function to be logarithmic, since the photon is required by this hypothesis to be similar to the decay of radioactive materials. At this point there are several possible ways to go. One might be to arbitrarily assign a series expansion of the logarithm function to the diffusion term, and see what that decay constant has to be to result in a half-life on the order of 10 to 100 million years. Another would be to look at the nearer time scale, say on the distance from here to the moon, since we now have a socially sensible -- empirical -- concept of that. On considering those tactics, a better solution becomes manifest.
The action quantum is the diffusivity
I like the following solution. It does not require the complicated series expansion, and goes better with one's feel of the nature of things. The rate of decay, per unit time, must depend on the state of the photon, and any of the variables can be used to influence that rate of decay.
The energy and momentum are not quantized eigenfunctions in the wave because the wave is an eigenfunction of a system (the medium of the vacuum of space itself) which has only sufficient constraints to determine the velocity c and action h. Thus, the wave cannot commute (in fact, never does) with its next cycle. That is the entropic order of time. The first term on the right is the diffusion term, and the second term on the right is the wave term.
Eq 7
The energy and momentum are Markovian in the sense that the energy and momentum of each wave depend on the state of the wave in the past. The wave energy and momentum were truly quantized only at the instant when the wave was first emitted from a quantized system -- usually an atom or molecule. As an aside, there are no measuring rods inside photons.
This expression is simple. It contains no more than the necessary information. It touches on known relativity in the form of the parameter of the first order derivative term, for the quantity hc appears often in the physical constants and other relativistic mathematics. It goes in the right direction, which is to say, that for a new, energetic photon (say in the X-rays or ultraviolet) the diffusion term and thus the decay rate is stronger, while for an ancient, decayed photon (infrared, microwave or broadcasting) the diffusion term is weak. Finally, the expression harkens to the Gaussian coordinate system in which c=1, which is the sort of scale in which photon decay must be comprehended. Gauss saw the distance light travels in one second a century and more before footsteps appeared on the moon, and most people had to wait until satellite communications before encountering such distances.
Each of the two constants, c and h, is manifested in several ways in the nature of light. The speed of light c determines (or is) the velocity of the wave, the velocity of the diffusion in the permittivity, and the product of wavetime and frequency. The action quantum h determines (or is) the rate of decay, the magnitude of the action, and the product of momentum and wavelength.
Does it fit the facts? It has to predict a decay time consistent with the observed Red Shift data. At first it seems the diffusion term shown here will be too strong, and would create a decay rate much too rapid. As we shall see, it turns out that the action quantum is just right for the diffusivity, and results in a predicted half life which is very close to the observed Red Shift. The results of Sandage and others who observe the Red Shift were converted to an estimate of half-life from their velocity-based measure.
