Quantum Matter "Beyond The Point Particle - A Wave Structure For The Electron" Part II

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Figure 1.  The Dynamic Waves of a Space Resonance. 

The resonance is composed of a spherical IN wave which converges to the center and an OUT wave which diverges from the center.  Their separate amplitudes are infinite at the centers.  When combined, the two waves form a standingwave which has a finite amplitude at the center. The standing wave is the structure of the electron. The inward and outward waves provide communication with other matter of the universe. Spin of the electron is a result of the reversal  of the IN wave at the center to become the OUT wave.

SECTION III  -  THEORY OF THE NEW ELECTRON

Three assumptions about the properties of space determine the space resonances.  In return for this  investment, the theory obtains a physical and mathematical origin for natural laws plus relationships between particles and cosmology.

A.  Assumption I  -  The  Wave Equation.

Because it must be compatible with quantum theory, a scalar  wave equation is needed to describe the structure of natural electrons.  Spherically symmetric solutions are required because charged particles have spherical symmetry.  Quantum theory requires the frequency of the waves to be proportional to the mass according to the formula f = mc2/h.  Two solutions of the wave equation shown in Figure 1 describe the physical structure of the electron.

Equation (2) below shows that an electron is comprised of two spherical scalar waves traveling in space with velocity c;  one inward to a center and the other outward.  The two superimposed waves form a standing wave, termed a space resonance (SR).  The center of the  wave structure is the nominal location of the electron.  These space resonances are perpetual spherical oscillators.  Each resonance extends throughout space and interacts with other resonances so that the natural laws result from the properties of the waves and the medium they travel in, 'space' or the ether.

 The Wave-equation  for the electron, in spherical coordinates, is:

d2(AMP)/dr 2  + 2/r{d(AMP)/dr} - 1/c2x d2(AMP)/dt 2= 0                          (1)

where AMP is a continuous scalar amplitude with values everywhere in space and c is the propagation speed.  This equation has two spherical wave solutions  for the amplitude AMP:  One of them is a converging IN wave and the other is a diverging OUT wave, shown in Figure 1,

AMPIN    = (1/r) Ao e(iwt + ikr)                      [1]

AMPOUT  = (1/r) Ao e(iwt - ikr)

The IN and OUT waves combine to form a standing wave.  w is the frequency characteristic of an electron proposed by deBroglie and Schroedinger. k the wave constant. The amplitude of the continuous waves is a scalar number, not an electromagnetic vector.  At the center the standing wave amplitude is finite not infinite, in agreement with the observed electron.

A standing wave results by combining them with their amplitudes opposing at r = 0 , to get

AMPSTANDING =  AMPIN -   AMPOUT  =  (1/r) Ao e(iwt + ikr)- (1/r) Ao e(iwt - ikr)        [2]

The equation, Energy = mc2 = hw ,  converts units of energy into units of frequency.  Thus mass is proportional to frequency of the electron's space resonance oscillator:  w = 2 pi mc2 /h.  All the  waves of all the charged particles in the universe have this same frequency because the frequency is a property of the wave medium -  space itself, the ether. This frequency is the universal cosmic clock which regulates the laws of nature and our sense of time.  The velocity c is also a universal property of the ether which we observe as the speed of energy exchange (light).

This equation becomes clearer when changed to a simpler exponential  function,

AMPSTANDING= Ao e (iwt) sin(kr)/r

The exponential factor is an oscillator.  The sine function modulates the  rapid oscillator  waves with a standing wave of wavelength 1/k which surprisingly is the Compton wavelength of the electron.  The intensity is the envelope of  AMP*AMP, which decreases as 1/r2   away from its centers This equation is simulated in the animation of the electron  on this web-site.

The amplitude, AMP STANDING corresponds to the electric potential of the electron.  The  amplitude at the center is obtained by taking the limit as r --> 0 in  sin(kr)/r  in the  equation above and is equal to Ao.  This finite amplitude explains why 'renormalization' in QED theory works.   Renormalization boils down to an arbitrary cut-off of the Coulomb electric potential to avoid an unwanted infinity at the center when r --> 0.  Avoiding  the infinity because it is was annoying was Dirac's complaint.  Although the reason was contrived, the cut-off worked. Now, since experiments show extremely accurate verification of the cut-off, one can regard the cut-off as an observed correction to the electron potential, which elsewhere is the well-known 1/r2 .  The space resonance structure correctly  shows the origin of the correction, the finite center  amplitude.

Predicted Properties from the Wave Equation.

From only the first of three assumptions, several properties of the electron are already observable:

1. There are two kinds of SR electrons as a result of two ways to superimpose the IN and OUT waves.  One  combination has a negative IN-wave amplitude at the center and corresponds to the electron.  The other  has a negative OUT-wave at the center forming an anti-resonance which is the positron.   If the anti- resonance  is superimposed upon the resonance, they annihilate  like electron and positron. This is seen from the equations.

2. They obey Feynman's Rule: A positron is an electron going backward in time.  To see this, replace the variable t with a -t in the function for an electron resonance, Equation (2).    Replacing the  time exchanges the IN and OUT waves and you obtain the equation for a positron as Feynman said.

3. The Origin of Conservation of Energy.  Energy is exchanged in nature by two resonances (oscillators) interacting with each other. For all oscillator pairs known in nature,  significant coupling occurs only if both have the same resonant frequency.  If one oscillator changes frequency upward, the other changes frequency downward.  Thus, the frequency (energy) changes of interacting space resonances are equal and opposite.   This is exactly the content of the Conservation of Energy law.

B.  Assumption II  -  Establishing the Density of Space

 The wave equation provided a structure which possesses some of the electron's characteristics  but a means for the SRs to interact and exchange energy is also  needed.  Unfortunately, since waves in a homogeneous medium pass through each other,  the medium has no means for interaction.  To find the means of interaction, we  recognize that space is not homogeneous everywhere.  For example, it has been observed that a star  will bend the path of light which goes near it.  A similar behavior occurs at the center of a charged particle.

To examine this requirement we first make a quantitative  assumption, similar to Mach's Principle, which establishes the density of space (ether or vacuum).  Then we will examine the density formula seeking a means of intereaction.   

The Space Density assumption is :  Assume that the mass (wave frequency) and propagation speed of an SR wave in space depends on the sum of all SR wave intensities in that space;  a superposition of the  intensities of  waves from all particles inside the Hubble  (H) Sphereof radius  R = c/H,  including  the  intensity  of  a  particle's  own  waves.

   

mc2= hw= k'  SUM OF:{(AMPn)2x (1/rn2)}           (4)

In other words, the frequency w or mass m of a particle depends on the sum of amplitudes squared of all waves AMPn, from the N particles in the universe, whose intensities decrease inversely with range squared.  That is, waves from all particles in the universe combine their intensities to form the total density of 'space'. This density  determines the electron's wave frequency.  This space corresponds to Einstein's "aether" or quantum theory's "vacuum."

 Now examine the homnogeneity of the space.  The universe contains so many particles that the density of space is nearly constant everywhere.  But close to the center of an electron, the amplitude of an electron's own waves following the 1/r2 rule, is larger, producing a "lump" in space density.   This lump at the center of the electron causes wave interactions.  It is the way energy is transferred and what we call "charge".  Its correctness is tested below.

Energy Transfer Mechanism of the Space Resonance.

 How does the charge mechanism operate?   It is well-known that a-c signals flowing through a non-linear element in a circuit will mix.  That is, if there is a two-signal input:

INPUT = Acos(w1t )+ Bcos(w2t)

then the output will be:

OUTPUT = AB[ cos(w1t + w2t) + cos(w1t - w2 t ) ] /2 + other components

The non-linear element produces sum and difference frequencies of the original w1 and w2.

Similarly in space, different waves passing through the dense, non-linear region at the particle center will mix.  If an input frequency and a particle frequency are similar, resonance can occur.  An example of this is a tuned radio receiver.  An energy (frequency) exchange between resonances behaves like two coupled oscillators in a circuit, or like two pendulums joined with a spring.

A Test of Assumption II . 

If an electron's own waves can create a  denser region near its center, then the intensity I of those waves at some radius of non-linearity ro , must be comparable to the intensity of waves from all other N particles in the Universe.  This requirement  is written:   

Intensity  I   =  AMPo2/ro2 = SUM { AMPn2/rn2  }              

                     = N/V x INTEGRAL OF:{ AMPo/ro}2 4 pi r 2dr

where V is the volume inside the Hubble Sphere and R its radius.  The integral, from r = 0 to R = cT = c/H, extends over a sphere whose expanding radius R depends on the age T of the particle.  Thus T is  the maximum range of the particle's spherical waves. This reduces to    

ro2 = R 2/3N                                                                         (5)

Inserting values from astronomy measures, R = 1026 meters and N = 1080 particles, the critical radius ro equals 6 x 10-15 meter.  If the assumption is right, this  should approximate the classical radius rc = e2 /mc2of an electron, which is 2.8 x 10-15 meters.  The two values almost match, so the prediction is verified.  Apparently  dense  centers  do exist, and

e2 / mc2   =  R / SQUARE ROOT OF: {3N}                           (6)

Equation (5) is a relation between the size ro of an electron and the size R of the Hubble Universe. 

It is termed the Equation of the Cosmos. 

Observations on the  non-linearity Properties of Space Density

The large density of an electron's own waves at the center are the causes of 'charge' effects, wave-coupling,  and energy exchange between particles. 

1. Charge and mass interactions occur at the center (lump).  

The electron resonance extends throughout space  but energy exchanges take place in the non-linear bump at the center.  Thus the SR "looks" like a point particle  but no mass or charge substance is required to produce this experimental appearance.  It is all waves.

2. Modulation of the waves behaves like a photon. 

When two resonances exchange energy (shift frequency  - Section H below),  the IN/OUT waves traveling between them are modulated with the frequency-shift information.  This modulation travels at velocity c, like a photon.   But the only events we observe are two energy shifts,  one at the source and one at the absorber.  This corresponds exactly with experimental observation of the photon.

C.  Assumption III - The Minimum Amplitude Principle.

Assumptions I and II  describe the electron's structure, its energy exchange mechanism, conservation and electric force.  But there has to be a law to determine whether two particles should move together or apart, or whether their frequencies will change up or down.  One more assumption is needed that governs the behavior of energy exchanges within a group of particles.  A Minimum Amplitude Principle  (MAP) is found, described by        

INTEGRAL OF:  {AMP1+AMP2+ AMP3 +... AMPn }2 dx dy dz = a minimum                   (7)        

or,  The total amplitude of particle waves in space always seeks a minimum.

In other words, all the waves of  the total number n of particles inside the Hubble Sphere adjust themselves at each point to make total amplitude a minimum.  To accomplish this, energy (frequency) exchanges take place, or wave-centers move in order to minimize the total amplitude.  This principle is very powerful and predicts many observations. For example, waves of two electrons close together will have a higher intensity than electrons farther apart.  Therefore two electrons must repel in order to satisfy the MAP. A positron and an electron will attract. It  also creates the  Pauli Exclusion Principle, forces between atomic nuclei, and gravitation.

Observations on the Minimum Amplitude Principle.

1. The Pauli Exclusion Principle is one result. This is because MAP prevents two identical resonances (fermions) from occupying the same state since their total amplitude would be a maximum rather than a minimum.

2. The electric charge force between two resonances is  F = k/r2, where k = e2/4 pi eo .  It is the  same as Coulomb force everywhere except at the center.  This force arises as a result of the Minimum Amplitude Principle which attempts to minimize wave amplitudes near the resonances.  The 1/r 2 factor is the result of the 3D geometry of ordinary space.  The electric constant  k is a measured parameter which can be approximated from Equation (6) which shows it to be a property of space. Thus only one value of charge occurs in nature.The complex amplitude AMP can be regarded as the electric potential of the electron.