THE QUANTIZATION OF

INTRODUCTIONIn this paper it will be shown that there is an electromagnetic saturation constant in nature. Maxwell’s equations will be evaluated to derive this constant. It will also be shown that electric and magnetic change is quantized in nature. This is realized when Maxwell’s displacement current is viewed as the actual photon of energy. The constant charge for and electromagnetic wave is shown to be the curvature of space around its electric amplitude and equivalent to Planck’s constant. A mathematical derivative for frequency is established and a photon model is introduced. Also, Maxwell’s equations are modified to incorporate single photons and the quantum phenomenon. 
Maxwell’s EquationsThe climax of electromagnetic theory in the nineteenth century was the prediction and the experimental verification, that waves of electromagnetic fields could travel through space. The theoretical prediction of electromagnetic waves was the work of the Scottish physicist James Clerk Maxwell (1831  1879), who unified, in one theory, all the phenomena of electricity and magnetism. The concept of fields was not generally used until Maxwell showed that all electric and magnetic phenomena could be described using only four equations involving electric and magnetic fields. In the absence of dielectric or magnetic materials, Maxwell’s equations are:
Equation Image is from wikipedia.
Maxwell’s equations in the integral form are summarized in words: Eq. 1.1 is a generalized form of Coulomb’s law relating the electric field to its sources; electric charge (Gauss’s law). Eq. 1.3 is the same for magnetic fields except that there are no magnetic monopoles or magnetic sources. The magnetic field lines are continuous; they do not begin or end as electric field lines do on charges. Eq. 1.5 is Faraday’s law, where an electric field is produced by a changing magnetic field. Eq. 1.7 is Ampere’s law (as extended by Maxwell) where a magnetic field is produced by an electric current or by a changing electric field. 
According to Maxwell, a changing magnetic field will be produced in empty space if there is a changing electric field. From this, Maxwell derived another startling conclusion. If a changing magnetic field produces an electric field, the electric field itself will be changing. This changing electric field will in turn produce a magnetic field. The latter will be changing and so will produce a changing electric field; and so on. When Maxwell manipulated his equations, he found that the net result of these interacting, and changing fields was an electromagnetic wave that propagates through space at a constant speed, c » 3 * 10^{8} m/s. Today these changing fields are called photons. 
The Brown Theory of Planck’s ‘Action’In the Brown photon theory, Brown postulates that there is a finite electromagnetic amplitude at which all photons exist. According to Brown; if photons consist only of the fields and Einstein’s photon postulate, e = hv, holds, then the electromagnetic amplitude of a photon can not be a variable. This is because Maxwell’s equations clearly state that the rate of change of electromagnetic fields determines the energy, and Planck’s constant clearly states that frequency alone determines the energy. Thus, frequency is the rate of change of the electromagnetic fields. Furthermore if frequency alone determines the energy, then the peak electromagnetic amplitude of the photon’s sinewave must be constant. Any variation in amplitude would cause a variation in Planck’s constant. According to Brown’s theory, this peak amplitude of photons is a constant of nature from which Planck’s constant derives. 
The Brown Photon Theory, further states that Planck’s constant being 6.626176 * 10^{34} Joule Seconds, has the units associated with energy * time. Planck’s constant is the amount of potential energy available that could be applied over a period of time. However; Planck's action is not the same as work, which is the product of expending energy over a period of time. If the energy were actually applied, the quantity would be, power * time = Watt Seconds. Since a photon’s energy is never applied until it is absorbed, it must exist only as potential energy; although, its potential is always Planck’s exact value of energy * time. The energy is only potential energy because in order to become energy, it must cease to exist in its potential state; when it is absorbed. 
If a photons electromagnetic amplitude were a variable, then Planck’s constant energy * time could not exist. Since the only two variables that produce Planck’s constant are electromagnetic change and time, the electromagnetic amplitude reached by the change must be a constant. Hence, Planck’s constant is caused by the peak amplitude which is something real, rather that to suspect that peak amplitude is caused by Planck’s constant, something abstract. 
Brown’s understanding of Planck’s constant completely describes the potential work content (action) of every photon. However; if the electromagnetic amplitude of a photon was a variable; not only would Planck’s constant be non existent, but the electromagnetic energy of a photon could not be quantized. 
The Photon ModelMaxwell’s equations describe the effect of large numbers of photons traveling together. Maxwell’s equations do not directly describe the behavior of a single photon propagating in space. Therefore, Maxwell’s equations need to be modified to describe the phenomena of a single photon. A new model needs to be put forth. Brown states, in his paper How Come The Quantum that a photon model would consist of a field of electric charge occupying a circular plane and a field of magnetic lines of force occupying a similar plane opposed from the electric plane by 90 degrees. However; I do not agree that Brown’s points in a photon are observed as particles. An electromagnetic wave does not have particles. The photon’s inertia serves as the particle nature of the photon. 
Every photon must exist as a moving wave in a volume of space. According to Maxwell’s model, the electric and magnetic fields at any point are perpendicular to each other, and to the direction of motion. The electric field and magnetic fields are in phase with each other, except that they are rotated about their line of motion by 90 degrees. 
Being in phase means that the magnetic and electric fields are zero at the same points and they reach their maximum amplitudes in space at the same points. 
The new photon model is such that the wavelength l
of a photon behaves like a standing wave that maintains a constant
second harmonic (n = 2) in space. 
Figure 1 New photon Model 
An electromagnetic wave is a traverse wave that resembles a standing wave, where the end points are fixed. Electromagnetic waves are waves of fields, not waves of material particles as are waves on water or a rope. The photon has a sinusoidal shape solely due to the way that it was produced. Electromagnetic waves are produced by oscillating or accelerated charges. Thus, electromagnetic waves are a product of cause and effect. When a charge is oscillating, or accelerating, it reaches a maximum upper potential, stops then swings to a maximum lower potential. During the stop and starts or accelerations of the charges, the electromagnetic fields are sent propagating outward in space with a discrete amount of energy associated with the rate of electromagnetic change (frequency). 
The
energy that is being propagated in space, due to the accelerations
of charged bodies is potential energy. In the medium of free space
the potential energy moves away from its source in a straight line
at a constant velocity (c = 3 * 10^{8} m/s). The
standing wave pattern of the electromagnetic wave is governed by a
constant of nature from which Planck’s constant derives. The peak
maximum amplitude points and the zero points are produced when the
changing electric and magnetic fields are sinusoidally trying to
approach a peak amplitude or maximum potential. According to
Maxwell’s equations this should naturally happen. However, there
is no ceiling in Maxwell’s equations that prevent the changing
peak electric and magnetic fields from peaking to an infinite
potential.

The electromagnetic change that photons' experience can not be just any amount of electromagnetic change. The electromagnetic change is quantized or fixed and governed by a constant that couples the changing electric and magnetic fields together. That constant is called the electromagnetic saturation constant. This is because the electric and magnetic change saturates to a quantized value within the wavelength of the photon. Electromagnetic saturation is the coupling of changing electric and magnetic fields, and the fixed quantity of energy within the wavelength of a photon. Electromagnetic saturation occurs when the electric and magnetic change reaches their maximum potential; thus allowing there to exist a constant amplitude. The constant amplitude manifests itself as Planck’s constant and electric charge. Planck’s constant is the potential limit of the electromagnetic saturation of a point in space. The electric charge constant is neutral being the curved space surrounding the electric amplitude in the positive and negative direction of space, in each half cycle of the photon. 
As the wave amplitude approaches the finite electromagnetic amplitude potential in a finite period of time, there appears Planck’s constant. The constant area surrounding this amplitude is charge. The maximum amplitude is saturated because the quantized energy of the fields must be conserved during electromagnetic change (changing electric and magnetic fields). The fields saturate to a maximum in space at the Brown amplitude of the photon which, is (r_{c} = l/2p). The sinusoidal amplitude of the photon is given by (r = r_{c}×sin(wt)). The electric and magnetic fields diminish with the square of the distance away from the sinusoidal amplitude of the photon. 
Thus the
electromagnetic saturation constant is the coupling of the changing
fields and Planck’s constant is the potential limit due to the
saturation of the fields. The electric charge constant is the space
that curve around the electromagnetic amplitude in the medium of
free space. Hence, the quantized energy and the saturation of
electromagnetic change are what causes Planck’s constant and the
electric charge constant. 
Planck’s value is a constant whose quantity is the measure of energy * time and has the units of Joule Seconds. Therefore, it can be concluded that for every electromagnetic wave (photon) the ratio of the electromagnetic change potential energy, to the rate of electromagnetic change, which is frequency, n = 1/T, must be a constant. The result is constant electromagnetic amplitude of finite points in space. Planck’s constant should be called the amplitude potential. 
(1.11)
h =
= e * T = 6.626176 * 10^{34}
J * s.

The Rate Of Electromagnetic ChangeIt is known from simple wave motion that T, is the time period that is takes a wave to experience one complete cycle of oscillation. Thus, a number of complete time periods would add up to the total time traveled by a wave. The equation is given by, (1.12) t = N * T. Time
Figure 2 Cycles of wave oscillation
The symbol t, denotes generalized time. The symbol N, denotes: 1. The number of complete cycles in a given amount of time. 2. The amount of oscillation in a given phase of angle measure. Thus, N is any number; it can be fractional or an integer. This is shown by using the generalized radian to degree measure,
(1.13)
q =
q^{°}. Radian to Degree.
(1.14) N = . Dividing Eq. 1.12 by the time period T, and equating with Eq. 1.14 yields, Thus, the generalized time at any point in the wavelength of the wave or the total time traversed by the wave is given by, (1.15) t = T. Time

The rate
at which the electromagnetic wave experiences electromagnetic change
is termed frequency. Thus, rearranging Eq. 1.12 it can be expressed
as the frequency of the wave

The definition for the instantaneous frequency at any given moment is
the limiting value of the average frequency, as Dt
approaches zero.

Furthermore, the angular frequency or angular rate or EM change of the wave is easily derived by substituting Eq. 1.14 into Eq. 1.18 which yields,
n =
n = .
Hence,
the angular frequency is given by,

Now that
a solid definition for the rate of electromagnetic change or
frequency has been established, Einstein’s photon postulate Î
= hn, can be expressed with the
definition of the rate of electromagnetic change.

The Displacement/Photon Current
Maxwell,
in is endeavor to prove that light was indeed electromagnetic waves
of changing electric and magnetic fields, used classical concepts in
electromagnetism already known at the time to prove his argument.
His greatest accomplishment was his modification to Ampere’s law,
where he introduced a quantity called the displacement current.

To
derive the quantum of current concept, Maxwell’s argument for the
displacement current must first be derived. Starting with Gauss’s
law it can be shown that the relation between the electric flux
through a closed surface and the net charge Q, within that surface
is given by,

Figure 3 Electric Flux diagram The
symbol A is the projection area on the surface perpendicular to E
as shown in figure 3. The area of a surface can be represented by
a vector
whose magnitude is A and whose direction is perpendicular to the
surface. The angle q is the angle
between
and
so the electric flux can also be written,

To complete deriving Maxwell’s displacement/ photon current, Maxwell
discovered that there was some inconsistency in Ampere’s law.
Maxwell’s argument begins by looking at Ampere’s law and
applying it to the field around a straight wire.

Now,
lets consider a parallel plate capacitor that is being discharged.
When a capacitor is discharging, Ampere’s law works for the
current leaving the capacitor plate and flowing through the wire.
This current is called the conduction current. Before Maxwell’s
great insight tackled the problem, no one was able to determine how
the current got from one side of the plate to the other plate.
Faraday had demonstrated
The electric field between the plates is changing in time. that a changing magnetic field induces an electric field, but the converse, the induction of a magnetic field by a changing electric field, had neither been observed nor suggested before Maxwell’s hypothesis. All the while it was known that there was a magnetic field due to the electric field between the plates, when the current was flowing to or away from the capacitor plates. Maxwell resolved the problem by stating that the changing electric field between the plates is equivalent to an electric current. He called this current the displacement current. Ampere’s law can now be written as, 
(1.25) The above equation is Maxwell’s modification of Ampere’s law. Where, I is the conduction current, and I_{Î}, is known as the displacement current, that is the current between the parallel plates. The displacement current exists only when the electric field is changing.
The
displacement current I_{Î}
can be expressed in terms of the changing electric field between the
capacitor plates as in Fig. 5. The charge Q on a capacitor of
capacitance C is,
The
symbol V is the potential difference between the plates whose units
is given in Joules/Coulomb (J/C), Coulomb/Farad (C/F), or Volt. The
symbol C denotes the capacitance between the plates. The potential
difference V, is V = E×d,
where d is the small separation between the plates. The symbol, E
denotes the electric field strength between the parallel plates;
neglecting any fringing of the electric field at the ends of the
plates. Thus the potential difference is given by,
The capacitance between a parallel plate capacitor is given by, (1.28) C = e_{r}×e_{o}. Capacitance 
The unit
of measure for capacitance is the Farad, F or Coulomb/Volt (C/Volt).
The symbol, A is the Area of each parallel plate. The symbol e_{r},
denotes the relative permittivity of the material or media of
transmission. In the vacuum of free space the relative permittivity
equals one, e_{r} = 1.
Next, combining Eq. 1.27 and Eq. 1.28 and substituting into Eq. 1.26
yields,

The
equation for the net charge in the vacuum of free space is given by,
Furthermore,
the net charge on the plate changes at a rate dQ/dt, the electric
field changes at a proportional rate. Assuming the medium of free
space where e_{r} = 1,
then differentiating Eq. 1.29 yields,

Although,
the name is not very popular today. The above Eq 1.32 can also be
expressed in terms of the changing electric flux. From Eq. 1.23 the
electric flux is given by, F_{E}
= E×A; thus, substituting
into Eq. 1.32 yields,

Ampere’s
law with Maxwell’s modification can now be expressed as,

Quantization Of The Photon CurrentMaxwell’s displacement current which will be known as the photon current is quantized. When the electric field is changing across the capacitor plates an electromagnetic wave is carrying the current from one end of the plate to the other. The energy of this electromagnetic wave is quantized. Therefore; the displacement current which is equivalent to the changing electric and magnetic fields would also be equal to the electromagnetic energy of the photon. Hence the displacement current is a photon which is an electromagnetic wave. 
The
instantaneous current is obtained by the limiting process of the
average current, and is given by,

In Eq.
1.37 (dN/dt) is the electromagnetic change frequency or rate of
charge flow. Thus, using Eq. 1.18 and substituting into Eq. 1.37,
yields,

The electronic charge constant e, of a photon in Eq. 1.38 does not manifest itself in the same way that charge is revealed for an electron, proton, or positron. The charge of a photon manifests itself as the curvature of electromagnetic amplitude in the positive and negative direction of the medium through which the electromagnetic wave is propagating. Thus, the ratio of the photon current (energy) to the rate of electromagnetic change (frequency) is always a constant value (e_{r} × e = I_{Î}/n). That constant value is the quantized charge. 
The
charge that is manifested on the electron, positron, or proton is
the result of a photon curled into a stable resonating orbit. The
static electric field that results from that charge is radiated from
the circumference of that orbit.

The above photon current equation is a generalized equation for all electromagnetic waves. Although, the photon current was derived for a special case, this equation is valid in general. 
Maxwell’s displacement current is equivalent to the changing electric and magnetic fields, which is a photon. Thus, the displacement current is naturally dependent on the wavelength and frequency of the electromagnetic traveling wave. For gammaray frequencies the displacement current will be fairly large, and for visible light the displacement current will be very small. However, this result was not looked for because there was not a direct relation between wavelength and current. The photon current is not the same current as that of a moving charged particle. The photon’s current is that of changing fields. Hence, the displacement current should be viewed as a quantum of current equivalent to energy. 
The Electromagnetic Saturation ConstantThroughout the history of electromagnetic phenomena there has always been a mystery as to why the electric and magnetic fields behave such that they are intertwined in nature. The moving electric creates the magnetic and the moving magnetic creates the electric and so on. Well, the mystery is revealed in a constant of nature termed the Electromagnetic Saturation Constant (ESC). 
The Electromagnetic Saturation Constant is a constant of nature that is a result of the coupling between the changing electric and magnetic fields and the quantized energy of the fields. The energy of the electromagnetic wave is conserved. This means that the Law Of Conservation of energy is valid. Saturation Conatant “The total amount of energy is neither increased or decreased in any process. Energy can be transformed from one form to another, and transferred from one body to another, but the total amount remains the same.” 
Thus, when the electric and magnetic fields are changing it must result in a fixed or quantized amount of energy. If that did not occur then there would be a violation in the conservation of energy; and that does not occur! This conservation of the fields produces a constant amplitude or saturation in the fields known as the (ESC). Furthermore, this coupling of the fields constant is what causes Planck’s constant. The electromagnetic saturation constant is nature’s way of preventing the magnetic field from dominating over the electric field and the electric field from dominating over the magnetic field. The (ESC) is natures way of keeping electromagnetic harmony! 
The
electromagnetic saturation constant is easily derived, knowing that
there is a direct relationship between the photon current,
electromagnetic change, and the electromagnetic energy of the
photon. This is shown simply by equating Eq. 1.38 and Eq. 1.33,
which yields,

Next
using Einstein’s photon postulate Î
= hn, for an electromagnetic
wave, solving for the frequency and substituting into Eq. 1.39, then
yields,
Y_{Î} = 3.661827631´10^{26}
or

The units of the electromagnetic saturation constant are Joule × seconds per Volt × meter, or electron Volt × seconds per Volt × meter. The ESC is defined as: The electromagnetic saturation constant is the amount of energy available in a given period of time per quantum of electromotive force in a given wavelength. 
The
electromagnetic saturation constant is just as universal as Planck’s
constant h, or the Universal Gravitational constant G. The
electromagnetic saturation constant is a constant of nature that has
yet to be verified. The electromagnetic saturation constant is not
just a mathematical construct, but is a requirement of nature. This
constant has to exist, so that energy can be quantized during
electromagnetic change (changing electric and magnetic fields). The
electromagnetic saturation constant is the coupling of
electromagnetic change, as well as the link to the quantization of
Maxwell’s equations.

Revealing that charge for a photon is a result of the electric and magnetic change saturating to a maximum amplitude in space. Hence, when the electromagnetic change saturates, Planck’s constant is manifest as the amplitude potential in any medium, and the electric charge manifest as curved space around that amplitude; thereby making Planck’s constant and charge equivalent. This also reveals that charge is caused by something dynamic, rather than something static! 
The Quantization of Maxwell’s EquationsMaxwell’s equations must be modified to incorporate the quantum effects of electromagnetic phenomena. The quantization of Maxwell’s equations would have to show that the electric and magnetic change is directly equivalent to the quantum energy of the photon. 
The
equivalency between the electromagnetic energy and the electric
change is shown by simply substituting the ESC Eq. 1.41 into Eq.
1.40 which yields,
The electric change is equivalent to electromagnetic energy and is defined as the saturation to a fixed amount of changing electric field or flux with time in the wavelength of an electromagnetic wave. 
It can
be shown that the electric change of a photon is quantized and can
be predicted, with just the knowledge of the wavelength or frequency
of the electromagnetic wave. This is shown by rearranging Eq. 1.42
and expressing the photon energy in terms of its wavelength Î
= h × c/l,
which yields,

The
displacement current or photon current is also shown to be
equivalent to the energy of the photon; thereby making the
displacement current the actual photon. This is shown by
substituting Eq. 1.43 into Eq. 1.33 and dividing by the permittivity
to get,

There
also is an equivalency in nature between the magnetic change of a
photon and the photon’s electromagnetic energy and momentum.
Classically, it is known that,
Where the electric field is in the ydirection (), and the magnetic field is in the xdirection (). The above equation can now be expressed, . Next, multiplying and dividing the above equation by the area quantity A, yields, . Thus, the electric flux is given by F_{E} = E × A, and the magnetic flux is given by F_{B} = B × A. The above equation now becomes, (1.45) F_{E} = c×F_{B}. Differentiating Eq. 1.45 with respect to time and rearranging it gives, (1.46) . Magnetic Change The symbol c is given by (c = 1/). The above Eq. 1.46 is interesting because is reveals that the electric change is magnetic change propagating through a medium governed by the speed of propagation in that medium. 
Thus,
going back to Eq. 1.43 and rearranging it such that it can be
expressed,

From Einstein’s photon postulate the photon momentum is given by p_{Î}
= Î/c = h/l.
Therefore, substituting Eq. 1.46 into Eq. 1.47 gives the photon
momentum equation in terms of magnetic change.
The magnetic change is equivalent to electromagnetic energy and momentum and is defined as the saturation to a fixed amount of changing magnetic field or flux with time in the wavelength of an electromagnetic wave. 
The
magnetic change of a photon is quantized and can also be predicted,
with just the knowledge of the wavelength or frequency of the
electromagnetic wave. This is shown by rearranging Eq. 1.48 which
yields,

The two new additions to Maxwell’s equation, Eq. 1.42 and Eq. 1.48 give insight to things about electromagnetic waves that were known but were not tied together. The electromagnetic amplitude constant as hypothesized by Vernon Brown is the fundamental link that was needed to complete the quantization of Maxwell’s equations. The electromagnetic saturation constant allows the photon energy and momentum to be equivalent to the magnetic change and electric change of the photon. 
(1.51) Î = Y_{Î}_{}. Î = c×Y_{Î}_{}. Electromagnetic Energy 
The
modifications to Maxwell’s equations in free space for time
varying fields in the integral form are,


ConclusionThis is just what Brown predicted in his photon theory. Brown predicted that the electromagnetic amplitude or saturation constant would be a constant from which Planck’s constant derives. This is saying that the electromagnetic saturation causes Planck’s constant. The changing fields are what propagate through space, therefore, a constant that would cause Planck’s constant would govern the changing fields. This new constant also unifies the quantum concept to Maxwell’s equations. Hence, it was shown that electromagnetic saturation produces Planck’s constant which in turn produces electronic charge. The electronic charge is the curvature of the medium surrounding the amplitudes. It is also proven that the electromagnetic change is quantized and equivalent to electromagnetic energy. Hence, Maxwell’s equations are quantized. In concluding, this is only the beginning. The fun starts when the equations for quantized electromagnetic change and energy are analyzed in different media. 
REFERENCES
1. Modern Physics for Scientist and Engineering, M.L. Burns, Harcourt Brance Jovanowich, Publishers, New York 1988.
2. The Feynman Lectures on Physics, Volume 2, R. P Feynman, R.B. Leighton, M. Sands, AddisonWesley Publeishing Company, Reading Massachusetts 1964
3. Fundamentals of Photonics, B.E.A. Saleh, M.C. Teich, WileyInterscience Publication, New York 1991
4. Photon Theory, V.E. Brown, Photonics, P.O. Box 1351, Cabot, Arkansas 1994
5. How Come The Quantum, V.E. Brown, Photonics, P.O. Box 1351, Cabot, Arkansas, January 1994
6. An Introduction to the Meaning and Structure of Physics, L.L. Cooper, Harper & Row, Publishers, New York 1968
7. A brief history of Time, S.W. Hawking, Bantam Books, New York 1988

1 Vernon E. Brown, How Come The Quantum, Photonics (Cabot, AR.: Photonics, P.O. Box 1351, January 1994), 912.