### On Permittivity, Permeability, the Speed of Light, Transit time Fluctuations, and Wave Harmonics in the Vacuum

Richard Bradford and Gordon Rogers

Revised May 4, 2024 from Jan 1 2016, further revisions possible in light of current work

Abstract

Given transient dipoles immersed in low–energy or slowly varying electromagnetic fields of light, vacuum permittivity, permeability, the speed and the transient time fluctuations of light propagation in the vacuum are derived semi-classically. There are no introduced experimental parameters in the derivation, which need to be compared to experimental values. These characterizations are in accordance with the Heisenberg Uncertainty Relations, the Cauchy-Lorentz distribution and the exponential distribution for positronium. The semi-classical structure and characteristics of transient dipoles are revealed. A natural wave structure of harmonics describing parameters of dipoles and other vacuum parameters are found. The wave harmonic structure of parameters in the semi-classical vacuum ultimately depends only on the mean energy vacuum fluctuation of the transient dipole.

Introduction

The derivations of the permittivity, permeability, speed of light, and transit time fluctuations due to the properties of transient dipoles with electric and magnetic moments in the vacuum are based on the analysis of references 1, 2, and 3. Concerning vacuum permittivity, the authors of references 1 and 2 used solely the transient intrinsic electric dipole moment and its relative orientation to an external nearly constant weak electromagnetic field of light while the author of reference 3 used only the induced electrical dipole moment. The proper analysis would utilize both of these contributions. As opposed to references 1 and 2, the vacuum permeability is determined by the relative orientation to the external field of the total magnetic moment of the transient dipole, which has contributions from electrical currents and the coupled spins of its constituent Fermi particles. The transient dipoles have electrical currents and angular momentum characteristics that differ from real positronium. The algorithm for the calculation of the speed of light due to the vacuum is similar to that of the authors in references 1 and 2 but with important differences. Deriving the transit time fluctuations follows a different line of reasoning utilizing the Heisenberg Uncertainty Principle which is a divergence from reference 2. As opposed to references 1 and 2 there are no introduced parameters that need to be set by comparison with experimentally measured values. Four required mean values are established from the Cauchy-Lorentz energy-resonance and the Exponential distribution. The exponential distribution is the ground state of positronium. These values lead to the calculated vacuum permittivity, permeability to 0.7% of the experimentally excepted values, and the calculated speed of light to 1%. The same mean graphical values with uncertainties added are used in the formulation of the transit time fluctuation variance. Here an experiment needs to be undertaken to verify the result of the authors in reference 2 or the model of the present paper. It is also shown in the semi-classical vacuum that the derived parameters of a transient dipole, Compton wavelength, mean free paths, and distance traveled by constituent particles of the dipole and the transient wave packet are ultimately determined by only the mean value energy fluctuation of a transient dipole along with the harmonics of the spectral distribution of the transient wave packet.

Permittivity of the vacuum

The electric dipoles are of the form di = eQiCƛCi, where ƛCi = ħ/(mic), is the Compton wavelength of a fermion of mass, mi, in the transient dipole. C is a constant ≤ 1. If C > 1 the pair becomes real since the particles become distinguishable outside each other’s Compton wavelengths. Qi is a fraction between -1 and 1 inclusive. Qie is the charge for a particular species of particles e.g. -1e, Qi = -1 for the electron, muon, tau or +2/3e, -1/3e for Qi = +2/3 and -1/3 for up and down quarks. The vacuum permittivity and permeability results from a contribution of all transient particle-antiparticle dipoles. Needed in the derivations is the sum ∑iQi2 over all species of charged particles in the Standard Model. One generation of particles is e-, electron, +2/3e, up quark, and -1/3e, down quark. The sum is (-1)2 + 3X((+2/3)2 + (-1/3)2) = 8/3. The three is for the color charges of the quarks. Since there are three generations, then ΣiQi2 is 8/3 X 3 = 8. The sum of mass of the various dipole species turns out to not be required since the mass terms cancel out.

In a small volume of dipoles, a slowly varying or low energy real external electromagnetic field of light is taken as approximately constant. There are two components contributing to the vacuum permittivity from the transient electric dipoles immersed in the field, which are the average intrinsic electric dipole moment and the induced dipole moment.

The intrinsic dipole moments in the vacuum without an external electromagnetic field are random and the net vacuum dipole moment is zero. Given an external electromagnetic field, E, the net dipole moment in the vacuum is non-zero. The energy of orientation of the electrical transient dipole, d, to the electric field is d*E or d*E*cosθ, where θ is the angle between the dipole vector and the vector of the external electric field. The Heisenberg Uncertainty Relation for transient dipole energy-time is Ƭ = ħ/2ε, where ε is the energy of the dipole and Ƭ is the time of its existence. Adding orientation energy, the uncertainty in the time of existence of the dipole is modified to Ƭ(θ) = ħ/2(ε – d*E). The time of existence of the transient dipole is then asymmetric with the result that longer life transient dipoles dominate over ones with shorter life with the result of a net electric dipole established in the vacuum. The transient time Ƭ (θ) = ħ/2(ε – d*E) = ħ/2(1 – ηcosθ), where η = d*E/ε. The average dipole moment, Di, aligned with the external field E is determined by an integration weighted by the transient time as a function of orientation angle τ (θ) given as:

Di = ∫0πd*cosθƬ(θ)2πsinθdθ/∫0πƬ(θ)2πsinθdθ. The integral in the numerator is evaluated as follows: The constants in the numerator and denominator cancel, which are ħ/2 in Ƭ(θ) and the factor 2π. The integral in the numerator is written in the form d*∫-11udu/(1 – ηu) with the substitution of u = cosθ and du = -sinθdθ. The integration results in the evaluation of d*ꟾ-11-u/η – 1/η2lnꟾ1 – ηuꟾ. This equals d*[-2/η 1/η2ln(ꟾη + 1ꟾ/ꟾ1 – ηꟾ)]. The integral in the denominator becomes, after substitution u = -cosθ and du = sinθ, ∫-11du/(1 + ηu) = ꟾ-111/ηlnꟾηu + 1ꟾ = 1/ηln(ꟾη + 1ꟾ/1 – ηꟾ).

The series expansion of the numerator is d*(-2/η + 1/η2*2η + 1/η2*2/3*η3 + … and the expansion of the denominator is 1/η*2η + 1/η*2/3η3 + ….This is simplified to [d*(2η/3 + 2/5η3 + …]/[2 + 2/3η2 + 2/5η4 + …]. The factor η contains ƛ = ħ/mic and is of the order of 10-13m for the lightest charged fermion, the electron. Terms in the sum greater than η such as η3 and the quantity η2 compared to 2 are insignificant by 26 orders of magnitude and can be ignored. Thus, to first order in the numerator and denominator it equals [d*2η/3]/2 or Di = (d2/3ε)E.

The induced polarization onto the vacuum transient electrical dipoles from the external electromagnetic field of light is given by p = eQx, where p is the induced dipole moment of the dipole, eQ is the charge of the dipole, and x is the amount of displacement of the charges in the dipole. The transient dipole is considered a harmonic oscillator described by mω02x = eQE, where m is the mass, ω0 = energy/ħ, which is energy associated with a quantum transition from a ground state with energy at or above 2mic2. The maximum energy associated with a quantum transition, Egapmax, is the energy of transition, which would create two transient dipoles at rest, 4mic2, from minimum ground state energy of 2mic2. The upper limit of the creation of two dipoles from sufficient energy is not considered. The minimum energy of transition, Egapmin, is determined by a transition from an energy state of the transient dipole, determined below, to 4mic2. The polarization of the dipole is p = [e2Q2/(mω02)]E and the vacuum polarization, P = [e2Q2Ni/(mω02)]E, where Ni is the dipole density in the vacuum.

The calculated permittivity is ε’ and the polarization P = ε’E, so ε’ = P/E = Nidi2/3ε + e2Q2Ni/mω02.

Thus, the quantities needed are: Ni, di, ε, ω02

These are determined by using two statistical distributions. One is the Cauchy-Lorentz (CL) distribution and the other a special case of the Gamma distribution or Exponential (e) distribution, which is also the wave function of positronium in a zero angular momentum state, Ψ100. The CL-distribution is used to determine the quantities ε, the dipole energy, and ω02, the square of the transitional frequency, and the e-distribution is used to determine the size of the dipole or equivalently the constant C in CƛC. Given CƛC the dipole density N = [CƛC]-3 follows.

A CL-distribution neglecting the 1/π factor is pictured in Figure 1 on page 3 and will be used to establish the relationships between the minimum and maximum energy of the dipole. The energy ε ranges from the rest energy of the dipole, 2mic2 to a maximum of 4mic2 where two dipoles at rest may be created by the Heisenberg Indetermination of vacuum energy, ∆ε ≤ ħ/2∆Ƭ. The mean value of those energies are determined by finding the mean, not on the horizontal axis of energy as is traditionally done but on the vertical axis where the function f(E) = y is matched up with a probability distribution that allows a uniform probability along the vertical axis. Then the result can be solved for the energy E. The maximum width at half height is set to the maximum dipole energy of 4mic2 and is equal to Γ. The equation describing the curve is then f(E) = Γ/2/(E2 + (Γ/2)2) and without the mic2 factors is f(E) = 2/(E2 + 4) = y. The maximum height of the curve at E = 0 is f(0) = ½. E = 2 corresponds to 2mic2 at half height, f(2) = ¼. E = 1 corresponds to mic2 and f(1) = 2/5. Note: The actual total value of energy is twice that of 2mic2 and mic2, since the distribution is symmetric around E = 0. The uniform probability that y falls between ¼ and 2/5 is given by the normalized distribution 6.67∫2/51/4dy = 1. Thus, the mean f(E) = y is 6.67∫2/51/4ydy = 6.67*ꟾ2/51/4y2/2 = 6.67(0.08 – 0.03125) = 0.325 and corresponds to an E given by 0.325 = 2/(E2 + 4) or E = 1.46mic2 with the energy factor inserted. The total energy width is then 2(1.46mic2) = 2.92mic2 = ε.

The CL-distribution shown in figure 2 on page 3 is also used to set the value of ω02. The ground state of the transient dipole is equal to 2mic2 and its maximum excited state is 4mic2 so Egapmax equals 4mic2 – 2mic2 = 2mic2. The full width at half height is (Egapmax)2 = 4mi2c4 and is equal to Γ2. Egapmin with ε taken from above is 4mic2 – 2.92mic2 = 1.1mic2 and Egapmin2 = 1.2mi2c4. The equation describing the curve without the mi2c4 factor is f(E2) = (Γ2/2)/((E2)2 – (Γ2/2)2) = 2/((E2)2 + 4) with the maximum height at E = 0 giving f(0) = 1/2. One-half height, f(E2) = ¼, that gives Γ2 = 4mi2c4 occurs at E2 = 2 or 2mi2c4. The E2 that corresponds to 1.2mi2c4 is (1.2 mi2c4)/2 = 0.6mi2c4 or without the energy square factor is 0.6 giving f(0.6) = 0.46. Using the same statistical analysis to determine the mean along the vertical axis gives 4.76∫0.460.25ydy = 4.76ꟾ0.460.25y2/2 = 0.355. The E2 that corresponds to f(E2) = 0.355 is 1.26mi2c4. The full width is then 2(1.26mi2c4) = 2.52mi2c4. Thus, ω02 = 2.52mi2c4/ħ2.

Figure 3 on page 4 illustrates the exponential distribution and as mentioned above it is also similar to the wave function of positronium in the Ψ100 state. It is used to set the value of C in CƛCi, which determines the size of the dipole and the density of dipoles in the vacuum. It should be mentioned that the transient dipole is not equivalent to positronium in every way as will be shown semi-classically. This exponential distribution uses the parameter, ƛC to establish the mean size of the dipole and the exponential distribution curve with parameter 1/ƛC is described by the normalized distribution f(r) = (ƛCi)exp(-r/ƛCi) where ƛCi = ħ/mic. The value of f(r) at r = 0 is ƛCi and its value at r = ∞ is 0. The exponential curve used to set the dipole size has r = 0 at the position of one of the constituent particles of the dipole. This enables the size of the dipole to be determined from one constituent particle to the other. The mean value of f(r) is determined again along the vertical axis by letting f(r) = y and has equal probability of attaining any value between 0 to ƛC. The normalized probability distribution is 1/ƛC∫0ƛCdy = 1 so the mean of f(r) = y is then 1/ƛC∫0ƛCydy = 1/ƛCꟾ0ƛCy2/2 = 1/ƛC*ƛC2/2 = ƛC/2. The value r in the distribution f(r) that equals ƛC/2 or ƛCiexp(-r/ƛCi) = ƛCi/2 is ln1/2 = -r/ƛC = -0.69 and r = 0.69ƛCi, giving C the value of 0.69. This reflects that there is a probability of ½ that dipoles are larger or smaller than 0.69ƛCi. The dipole then is di= eQi0.69ƛCi and the density of dipoles, Ni, is then given by Ni≈1/(0.69ƛCi)3 = 3.00/ƛCi3 = 3.00mi3c3/ħ3.

To summarize: ε = 2.92mic2, ω02 = 2.52mi2c4/ħ2, di = eQ0.69ƛCi and Ni = 3.00/ƛCi.

The Pauli Exclusion Principle excludes two dipoles, each having a constituent particle with the same quantum numbers from forming at the same place at one time. The authors of reference 2 using the analysis of Fermi energies in a solid state determine that the separation of the identical particles is ∆xi = 2πƛCi/(KW – 1)1/2 where KW originated as a parameter in determining the transient time of the dipole Ƭi = ħ/KW4mic2. KW = 31.9 was determined by comparison with the experimentally determined value of the vacuum permeability and gives the value ∆x = 0.2ƛCi. The density of dipoles is Ni = 1/∆xi3 = 130/ƛCi3. The authors in reference 1 state that the minimum distance of separation of two identical particles due to the Pauli Principle is ƛCi/2 or 0.5ƛCi and Ni = 8/ƛCi3. The density found in the derivation here is 3/ƛCi3, and the maximum distance of separation of identical fermions in the same state is 0.69ƛCi and is outside the two criteria above. If two independent dipoles form at the same time and the density is 3/ƛCi3, then the dipoles are adjacent within the overlap of each other’s Compton wavelength and the identical particles are diametrically opposite. They cannot form at the same location within one fermion Compton wavelength, but in this case, the Compton wavelengths which contain the two dipoles can be distinguished, however. The dipoles are then oriented in such a way that the oppositely charged particles are closer together than the identical particles and a mutual dipole orientation occurs.

The calculated permittivity can now be done.

ε’ = (Nd2)/(3ε) + (Ne2Q2)/(mω02), which the total contribution from the intrinsic and induced electric dipoles in an external electromagnetic field, respectively. Using the values found and the analysis above the calculated permittivity is:

ε’ = Σi[3.00mi3c3 (0.69)2e2Qi2ħ2]/[ħ3mi2c2*3*2.92mic2] + [3.00mi3c3e2Qi2ħ2]/[ħ3mi*2.52mi2c4] =

(1.30 + 9.53)e2/ħc = 10.83*8.12X10-13F/m = 8.79X10-12F/m. The measured value is ε0 = 8.85X10-12F/m.

Figure 3.

Permeability of the vacuum.

Assume that in vacuum dynamics a transient dipole is created from one transient photon of sufficient energy annihilates into one transient photon. One transient photon, which has a zero net electric charge creates two charges of opposite electrical polarity in forming the transient electrical dipole. Conservation laws of momentum and energy are violated due to a short time period, which allows the one transient photon process whereas this process is forbidden on the mass shell where the conservation laws manifest over a long duration of time for measurement.

A spin one of magnitude ħ transient photon creates a transient dipole. Then the transient dipole must assume a spin one status. The total angular momentum of the particle and anti-particle according to the exponential distribution similar to the state Ψ100 requires that the angular momentum is zero. Since the orbital angular momentum is zero, the spin coupling in the transient dipole must have size ħ or spin one. The spin state that is spin equal one and where the particle and anti-particle have opposite spins is ꟾ1 0>. Then the transient photon creates a transient dipole with spin state ꟾ1 0> and decays to a transient photon with spin 1. The charge conjugation of the e-, e+ dipole is equivalent to an exchange of spin labels giving a sign change of – (-1)S and parity along with interchanging the electrical charges. The eigenvalues of charge conjugation are (-1)l+S. The dipole with total angular momentum l = 0 and total spin s = 1 has odd charge conjugation parity or (-1)0+1 = -1. The interchanging of the positive and negative charges reverses the direction of polarization of the electric field so the charge conjugation of a photon is -1. A system of n photons has charge conjugation parity of (-1)n. With one photon, n = 1, the odd charge conjugation parity of the transient photons is conserved during creation and upon annihilation of the transient dipole. Charge conjugation and spin one is conserved.

In Figure 4 on page 6, it is shown in a semi-classical way, the transient dipole created and existing for a brief period of time, ∆Ƭ = ħ/2*2.92mic2 = 0.17ħ/mic2 from a mean uncertainty of energy ∆ε = 2.92mic2. The dipole is created with its constituent particles separated by 0.69ƛC. This is possible as shown below due to the spectrum of the wave packets of transient photons. An e- that represents a charged particle and an e+, that represents the oppositely charged anti-particle, when created, travel a distance limited by the speed of light or c∆Ƭ = 0.17ƛC in the same direction on segments of an orbit around their center of mass until annihilation, which conserves linear momentum from the transient photon. The total linear momentum of the constituent particles in the transient dipole is calculated from ∆x = c∆Ƭ = ħ/[2(2.92mic)] = 0.17ħ/mic and ∆p∆x = ħ/2, so ∆p = ħ/[2(0.17ħ/mic)] = 2.94mic. The momentum of the dominate wave in the transient photon is p = ħ/ƛ = ħk = 2.92mic and linear momentum is conserved.

The magnetic moment, angular momentum, and linear momentum vectors μL, L, and p, respectfully, are related as μL = q/2mL = q/2mrXp, where q is electric charge. In the case of angular momentum, the constituent particles have the same velocity, v1 = v2, ꟾv1ꟾ = ꟾv2ꟾ and linear momentum p1 = p2, ꟾp1ꟾ = ꟾp2ꟾ, but their radius vectors r are oriented oppositely since they are on opposite sides of their center of mass, r2 = -r1 and ꟾr1ꟾ = ꟾr2ꟾ. The angular momentums of the constituent particles, L1 = r1Xp1 and L2 = r2Xp2 = – r1Xp1 lead to -L1 = L2 and L1 + L2 = L1 – L1 = 0 for the total angular momentum for the system. The magnetic moments of the constituent particles are μL1 = -q1/2mL1 and μL2 = +q2/2mL2. Now, L2 = -L1. q of the particle is –q1 and for the antiparticle it is +q2. Then, μL2 = +q2/2mL2 = +q2/2m(-L1) = -q1/2mL1 = μL1. Since ꟾq1ꟾ = ꟾq2ꟾ, μL2 + μL1 = 2μL2. The result is that even though the total angular momentum is zero, there exists a momentary magnet moment contrary to real quantum systems in particular positronium.

Given the radius r, velocity v, and revolution frequency ν = v/(2πr), the current is I = qv around a loop. The magnitude of the magnetic moment, ꟾμꟾ = IA = qv/(2πr)*πr2 = ½*qrv = ½*qꟾrXvꟾ, where A is area containing the moving charge. The average momentum of both constituent particles derived above is 2.93mic = 5.86mic/2 for a reduced mass m2/m + m = m/2 on a partial orbit in a potential at the center of mass that conserves total linear momentum. Momentum takes the form of a mass multiplied by a velocity, so in the case of momentum equaling 5.86mc the total mass could be 5.86 multiplying a fundamental mass m0 to equal 5.86m0 multiplying the speed of light c, or equivalently the fundamental mass m0 multiplying 5.86 times the speed of light c. The result numerically is the same. Taking v = 5.86c, ꟾμꟾ = ½*q5.86c(0.69/2)ƛC = 1.01qc*ħ/mic = (2/2)eħ/m = 2μB. Note: the velocities c cancel, so only the quantity 5.88 in that term is important. The mean size of the dipole 0.69ƛC together with 2.92mic2 together gives the Bohr magneton. This also supports the derived quantities. A magnetic moment of the transient dipole exists even without a total angular momentum and has a momentary existence with an orientation to the external constant electromagnetic field.

The orbital magnetic moment couples to the external homogeneous electromagnetic field, which has an energy associated with the orientation of the total orbital magnetic moment to the magnetic field of –μ*B = -2μBBcosθ. The magnetic moment in a homogeneous field undergoes a small precession with a constant angle of orientation to the external electromagnetic field.

The intrinsic spins of the two components of the dipole are oriented in the opposite directions and due to their opposite charges contribute 2μB to the total magnetic moment. The coupling of the magnetic moment from orbital and intrinsic spin to the external real electromagnetic constant field of light then has a total orientation energy of -4μBBcosθ.

As with the analysis for the permittivity, the transient time of existence of the dipole is modified by the total magnetic moment to the magnetic field coupling of -4μBcosθ to give τ(θ) = ħ/[ε – 4μBBcosθ]. Again there are dipoles with a longer transient time and some shorter. There is then an average non-zero vacuum magnetic moment of the dipole given by

To lowest order

Figure 4.

Speed of light, c.

The speed of light following references 1 and 2 is developed as follows. The Thomson cross section, σ, was used to describe absorption and emission of photons (hυ < < 2mc2) by transient dipoles in the vacuum without altering the photon beam structure. A red long lasting laser stream or white light from the sun streaming photons is obviously preserved when propagating a long distance in the vacuum. As assumed by reference 1 and 2, the speed of the photons between emission and absorption by the transient dipoles is infinite and the interaction of these photons with dipoles in the vacuum leads to the observed finite velocity of light.
In the Thomson cross section, σThom = (8π/3)α2ƛCi2, one fine structure constant α is the probability for absorption and the other α is the probability for emission as stated in references 1 and 2.The fine structure constant measures the strength of the electromagnetic interaction between electrons and photons and for fermions in general contains a factor of Qi2 for each α since the fine structure depends on the square of the charge. Since the dipole is transient, its absorbed photon is emitted with probability of α = 1 upon annihilation. Then, the emission of the photon is independent of the dipole charge and one Qi2 is eliminated. The probability of absorption, the other α has a Qi2 factor and the Thomson cross section, σThom = (8π/3)αQi2ƛCi2 = (8π/3)(8/137)ƛCi2 = 0.49 ƛCi2. Based on the algorithm of derivation in references 1 and 2, the mean free path between interactions is Λ =1/(σThomNi) and when travelling a distance L, the average number of stops is Nstopave = L/Λ. The photon will possibly encounter a transient dipole during its average time of its existence or Ƭ/2. The total mean time to cross the distance L is Tave = NstopaveƬ/2. Then, the average photon speed is cave =L/Ƭave = 1/(σiNiƬ/2).
A static interpretation of the density of dipole wave functions places all of them centered in their Compton regions adjacently touching with their centers separated by 0.69ƛCi. The Compton wavelength is determined by the constituent fermion in the dipole. The dimensions of the dipoles are less than their Compton wavelengths and dipole wave functions are best packed hexagonally together in 3-space with any dipole considered a center. Then, there are 12 overlapping spherical Compton regions with the center region each described as a lens. An overlapping Compton region implies that there is a certain probability that an adjacent dipole in its wave function finds itself in the overlapping region with the center and increases the Thomson cross section for absorbing the photon there. The spatial structure of the 12 clustering dipoles are 6 surrounding the center in one plane, 3 dipoles surrounding the center from above and below the center plane, thus 12 total overlapping lenses. The probability of the dipole being in its own Compton region is 1 and the probability of finding itself in any of 12 overlapping adjacent regions is 1/12 = 0.083. The 6 of 12 lenses with the center dipole in the middle plane of the hexagonal packing are illustrated in Figure 5. The probabilities of 12 to 1 dipoles being in the overlapping region with the center are:
Number Probability Number Probability Number Probability Number Probability
12 1.1X10-13 9 1.9X10-10 6 3.3X10-7 3 5.8X10-4
11 1.3X10-12 8 2.3X10-9 5 4.0X10-6 2 6.9X10-3
10 1.6X10-11 7 2.8X10-8 4 4.8X10-5 1 8.3X10-2
Figure 5
The system is not static, however, but dynamic in that there are fluctuations over time in the appearance of the number of adjacent Compton regions and the number of adjacent dipoles found in the center region. The fluctuations in the number of adjacent regions can vary from 12 to 0 with a mean of 6 due to the averaging of the transient times of adjacent dipoles compared to the time of existence of the center dipole. This would present fluctuations in the probabilities of adjacent dipoles being in the center region.
Probabilities of 6 to 1 dipoles being in the overlapping region with the center are:
Number Probability Number Probability
6 2.1X10-5 3 4.6X10-3
5 1.3X10-4 2 2.8X10-2
4 7.7X10-4 1 1.7X10-1
It is seen that the change in fluctuations of probabilities over time for a mean of 6 dipoles is a factor of 100. Based on the probabilities above for 12 and 6 adjacent regions, it is mostly favored that there are one or two adjacent dipoles in the center Compton region. When there are one or two additional adjacent dipoles in the center region then those adjacent regions have a smaller probability of having one additional dipole since two dipoles are required from their adjacent regions. An adjacent dipole considered a center region carries the same argument but it, also, may be the region having a greater or smaller probability of an additional dipole. The process continues from center region to center region in space. Also, an adjacent dipole in the center region is equivalent to a center dipole in that adjacent region leading to no net change in the number of dipoles in the center region. Thus, there are spatial fluctuations in the number of dipoles in center regions. In regards to time, the fluctuations in the number of adjacent dipoles finding themselves in the center region leads to similar arguments above. With large numbers of dipoles these space-time fluctuation effects are averaged with the result of there being one dipole in each Compton region and six adjacent dipoles. Also, to have the dipole enclosed in its Compton region its center is confined to a spherical region with a radius of (1 – 0.69)ƛC/2 = 0.15 to 0.16ƛC and with large numbers of dipoles the mean position is at the center of the Compton region.
Considering low energy electromagnetic fields, the dominate wavelength of the middle of the visible spectrum is 550nm and the diameter of the dipole is 0.69ƛC = 0.69ħ/mc = 2.6X10-13 m for an electron and is smaller for larger mass dipoles but the argument below is unchanged. The dominate wavelength of the visible light is transversely spread across a two dimensional area of (550nm)2. Then, in the transverse area, there are (550nm)2/(2.6X10-13m)2 = 4.4X1012 dipoles. The mean number of adjacent dipoles is 6 instead of the static number of 12 giving a mean value of 2.2X1012 dipoles. Electromagnetic waves are not confined to a single wavelength, however, but have a spectrum of wavelengths. This implies that the electromagnetic waves contain classical wave packets or photons, which have spatial extent in the direction of travel adding a third dimension to the transverse area. The cross section for single dipoles as given above is σ = (8π/3)αQi2ƛCi2 but the actual absorption area for the photon, which causes a change in the energy of an existing dipole inside is (0.69)2ƛC2. Then, with this absorption, area the cross section is given as σ = (8π/3)αQi2(0.69)2ƛCi2. The mean free path is Λ = 1/[(8π/3)*(1/137)*8*(0.69)2ƛC2*3.0/ƛC3] = 1.43ƛC for a local sample.
The dominate wavelength of 550nm equals the wave number 1.1X107/m. and is placed at the center position of a standard Gaussian curve. The spread in wave numbers are located on various points on the Gaussian curve. Given a Δx = 1/(2Δk) = 1.43ƛC or the mean free path, the spread of wave numbers required to produce a wave packet of this length with the dominate wavelength at 550nm is 9.1X1011/m – 1.1X107/m = 1.9X1011/m, where Δx = 1.43ƛC corresponds to a k value 9.1X1011/m with reduced wavelength, ƛ, of 2.86ƛC. The probability of having a comparably large k number spread is very small. The dipole created at the mean energy of 2.92mc2 has a corresponding ƛ of 0.34ƛC and k of 7.6X1012/m with Δx = 0.17ƛC and if the dipoles were created with minimum energy of 2mc2, then ƛ = 0.5ƛC and k = 5.2X1012/m with Δx = 0.25ƛC. The spread in k values which creates a minimal energy dipole is a factor of 27 of the spread of k values to span 1.43ƛC. This amount of spread in wavelengths or wave numbers may lead with the high energy components to the phenomenon to a changing the status of transient dipoles to the creation of vacuum dipoles or emission of real particles and antiparticles. For the higher energy components of the wave packet a different analysis is required such as using a different cross section and quantum electrodynamic techniques. A large reduction in the number of short wavelengths in the spread avoids any interaction with the vacuum transient dipoles and the Thomson cross section has greater validity. Also, the actual wave packets must maintain the premise of the paper, which is having a spread of wavelengths that meets the requirement of nearly constant electromagnetic fields compared to the size and transient times of dipoles. This implies that wave packets having a dominate wavelength of 550nm have a small spread in wavelengths and the wave packet, then, extends well beyond the mean free path of 1.43ƛC. For instance, the spread in wavelengths from 700nm to 400nm with k values 9.0X106/m and 1.6X107/m, respectively, centered on 550nm has a wave packet that extends 7.1X10-8 m = 1.8X105ƛC or 1.2X105 mean free paths, clearly exceeding 1 mean free path. The electromagnetic photon wave packet randomly interacts with any of the 2.2X1012 dipoles in the transverse region and anywhere along 1.2X105 mean free paths each of 1.43ƛC in the 3 dimensional region.
Consider a static configuration of hexagonally best packed dipoles as discussed above. The probability of absorption for any given dipole is αQi2 and equals 8/137 = 0.058. The statistical number of dipoles in the 2-D transverse area or plane out of the total, 2.2X1012 dipoles available absorbing the photon is 0.058*2.2X1012 = 1.3X1011 dipoles. In 2-D, then, 0.058 measures a type of porosity. In 3-D, however, there are dipoles packed together beyond the 2-D plane and there is an associated mean free path. Even if the light wave packet mentioned above is extended along a 1-D line, covering 1.2X105 mean free paths, and given that a probability of absorption 0.058 at a location where the 1-D line intersects the 2-D plane at each mean free path, then the photon will be absorbed within 0.058*1.2X105 mean free paths = 7X103 mean paths. Thus, in 3-D space, even when taking into account the probability of absorption and not just counting mean free paths the photon will be absorbed with certainty. Since there are temporal fluctuations of dipoles with mean 6 instead of the static number of 12 dipoles surrounding a center dipole and spatial fluctuations in the number of adjacent dipoles in the center Compton region or the center dipole in an adjacent region, the vacuum composed of transient dipoles is somewhat more porous.
Given this global analysis having large number of existing dipoles and their associated mean free paths corresponding to lower frequency and long wavelength electromagnetic waves, the absorption probability in the 3 dimensional region is nearly 1. The photon absorption area is (0.69)2ƛC2. The cross section having a probability αQi2 = 1 as seen by the visible light is σ = (8π/3)0.692ƛCi2.
With these values and each factor at 2 to 3 significant figures, Cave = 1/[[(8π/3)(0.69)2ƛCi2*3/ƛCi3]ħ/[4*2.92mic2]] = 1/[8π*(0.69)2ħ2*3mi3c3ħ)/(3.0mi2c2ħ3*4*2.92mic2)] = 1/[8*3.14*0.47*3/(3*4*2.92c)] = (1/1.01)c = 0.99c. The mass terms, mi, cancel. If the probability of absorption is less than 1 by a small amount, then the value of the speed of light becomes closer to 1c. For example, if the probability of absorption is 0.99, then Cave = 1/[(8π)*(0.99)*(0.47)*3/(3*4*2.92c)] = (1/1.00)c = 1.00c.
The result of cave = 0.99c or larger is a result of a large 2 dimensional planar sample of Compton regions and the extension into the third dimension of the wave packet over many mean free paths. The speed of light analyzed this way is a global speed much the same as measuring its speed over large distances and times. A small sample size of dipoles and short measurement times leads to a clocal, which may change from 0.99c due relatively greater amounts of local spatial and temporal fluctuations in smaller regions. For a static global large sample, there is an associated mean free path of Λ = 1/[(8π/3)(0.69)2ƛCi2*3/ƛCi3] = 0.085ƛC attributed to the probability of absorption being 1 and the absorption area for a photon of (0.69)2ƛC2.
Transit time fluctuations.
The transit time fluctuations is a variance of propagation time σƬ of a photon over a distance L. The main influence is due to the leading term consisting of transient electron-positron dipoles according to reference 2 and is used here.
As quoted σt2 = σNstop2(τ/2)2 + Nstopaveστ2. στ2 = τ2/12 and σNstop2 = Nstop. Nstop is variance of the number of interactions and σT2 = (1/3)Nstopaveτ2 = (L/3)σNτ2 = (2L/3)(σNτ/2)τ = (2L/3)(1/c)τ. The variance in the number NStop is given by fixed quantities and no differences in σ or N.
The variance in the transit time is related to the spread in 1/c, ∆1/c, together with the spread in possible positions of the photon during its propagation from emission to absorption. The spread of positions of where the photon is located in the mean free path is ∆Λ = σΛ, and the spread ∆1/c = σ1/c. These are determined using the uncertainty principle generating high and low values. It makes sense to define a local low sample variance in 1/c and Λ, since there are approximately 11-12 mean free paths in a distance of ƛC or approximately 1330-144 in a cubic volume of ƛC3. The cross section is local not global so σ = σ = (8π/3)*(1/137)*8*(0.69+-)2ƛCi2. This implies that a small region can have a local density, some deviations in cross sections, transient time fluctuations, and possible positions of the photon in Λ. A natural multiplier to quantify the number of these small volumes traversed with distance, L, is the non-unit ratio L/ƛC. The accumulation of variance of transit time fluctuations then occur with an increasing L.
Then the variance in transit time fluctuations, σT2 = (L/ƛC)*(σ1/cσΛ)2.
The uncertainty between the minimum and maximum values of the dipole is ∆ε = 2mc2. Then the uncertainty in the time of existence of the dipole is ∆Ƭ = ħ/2∆ε = ħ/(2*2mc2) = 0.25ħ/mc2. The change in the time of existence of the dipole using values above is ∆Ƭ = Ƭ2– Ƭ1, where Ƭ2 = ħ/[2(2.92 – 0.25)mc2] = 0.19ħ/mc2 and Ƭ1 = ħ/[2(2.92 + 0.25)mc2] = 0.16ħ/mc2. Then ∆Ƭ = 0.03ħ/mc2.
From the relativistic energy-momentum relation, the minimum value of the momentum of the dipole is (2mc2)2 = (2mc2)2 + p2c2 that leads to p = 0 or a dipole created at rest. The maximum value of the momentum is governed by the maximum energy before creating two dipoles at rest or (4mc2)2 = (2mc2)2 + p2c2 where p = √12mc. Then ∆p = √12mc and ∆x = ħ/2√12mc = 0.14ħ/mc = 0.14ƛC.
Based on ∆x, the change in the cross section is ∆σ = σ2 – σ1, where σ2 = (8π/3)*(1/137)*8*(0.69 + 0.14)2ƛC2 = 0.34ƛC2 and σ1 = (8π/3)*(1/137)*8*(0.69 – 0.14)2ƛC2 = 0.15ƛC2. Then, ∆σ = 0.19ƛC2. Also based on ∆x, ∆N = N2 – N1. N2 = 1/[(0.69 – 0.14)3ƛC3] = 6.0/ƛC3 and N1 = 1/[(0.69 + 0.14)3ƛC3] = 1.7/ƛC3. ∆N = 4.3/ƛC3.
The lower value of 1/c = σNƬ = 0.15ƛC21.7/ƛC30.16ħ/mc2 = 0.041/c. The high value is 0.34ƛC26.0/ƛC30.19ħ/mc2 = 0.39/c. The upper value of Λ = 1/σN = [0.15C21.7/ƛC3]-1 = 3.92ƛC and the low value is [0.34C26.0/ƛC3]-1 = 0.49ƛC. The extreme points pertaining to the speed of light and mean path lengths are the highest speed of light over the longest mean free path giving the highest speed with fewer absorptions denoted X1, and the other is the lowest speed of light over the shortest mean free path giving the lowest speed with greater absorptions denoted X2. The upper value of the speed of light corresponds to 0.041/c = or c equaling 24 and the lower value corresponds to 0.39/c or c= 0.2.6. The results with all values, σ, N, Ƭ within the uncertainties calculated above centered on their mean values used as independent variables in two functions 1/c and Λ are mapped onto two 1-dimensional regions where the extreme values are the endpoints. All values of (1/c)Λ are then located as points in a 2-dimensionsal region, which is a product of points in the two 1-dimensional regions above. The value X1 is then is 0.041/c*3.92ƛC = 0.16ƛC/c and the value X2 is 0.39/c*0.49ƛC = 0.19ƛC/c. The difference is X2 – X1 = 0.19ƛC/c – 0.16ƛC/c = 0.03ƛC/c. The same value can be also arrived at by noting that 1/c*Λ = σ1N1Ƭ1/σ2N2 = Ƭ1, where the extreme values of c and Λ together lead to σ1 = σ2, N1 = N2 Then X1 = Ƭ1 = 0.16ħ/mc2 and the same for the other extremes, where X2 = Ƭ2 = 0.19ħmc2. X2 – X1 = 0.19ħ/mc2 - 0.16ħ/mc2 = 0.030ƛC/c. The transient time fluctuation variance depends only on the mean value and uncertainty of the energy of the dipoles.
The variance in transit time, σT2 is equal to the variance of 1/c and Λ together, (σ1/cσΛ)2 multiplied by the natural non-unit number L/ƛC. The derivation of the speed of light and of the transit time fluctuations assumes that the photon travels on a forward momentum path due to an overall conservation of linear momentum when traveling a large distance L. Traveling in a deviant random non-straight path after each emission sets a random walk situation where the probability is low that the photon will cross the base line straight path as the number of absorptions and emissions increase with time with the result that the photon will with overwhelming odds may not reach the target as is not observed physically.
Then, the variance of the transit time fluctuations, σT2 = L/ƛC(0.030ƛC/c)2 = 0.00090(L/ƛC)(ƛC2/c2) = 0.00090LƛC/c2. The transit time fluctuation, σT = √(0.00090)√L√ƛC/c = (√L)*0.030*2.06fs/m1/2 =0.062fs/m1/2.
The authors in reference 2 with references propose and want to fund an experiment where an initial 9fs pulse is being reflected back and forth travelling a total distance of 3km, broadening to a 13fs pulse with a difference of 4fs. The authors result for σT was 0.05fs/m1/2. Then 0.05fs/m1/2√3000m ≈ 2.7fs. With the value calculated in this paper is 0.062fs/m1/2 giving the result of 0.062fs/m1/2√3000m = 3.4fs.
Percent deviation analysis for vacuum permittivity, permeability, speed of light, and transit time fluctuations.
Using the values: ε = 2.92mic2, ω02 = 2.52mi2c4/ħ2, di = eQ0.69ƛCi, and Ni = 3.00/ƛCi, the vacuum permittivity is calculated at 8.79X10-12. The percent deviation between the calculated value and the measured value of permittivity is given by [8.79X10-12F/m - measured value 8.854X10-12F/m]/ 8.854X10-12F/m = -0.72% and for the permeability the percent deviation is [12.48X10-7N/A - measured value 12.57X10-7N/A2]/12.57X10-7N/A2 = -0.72%. Using the mean dipole diameter of 0.69ƛC and probability of 1 to calculate the average speed of light above, the percent deviation of the speed of light is (0.99c – 1c)/1c = -1%. If the probability is less than 1, which is most likely the case, the percent deviation will be less than 1%. If the probability of global absorption is 0.996 due to a porous vacuum of transient dipoles, the percent deviation is also -0.72%. The values of the percentage deviations are consistent in that the variation between them is small with no exaggerated outliers. Comparison with experimentally measured values, the parameters using graphical mean values and the analysis with 2 to 3 significant figures along with semi-classical arguments in this paper are remarkably close. Then, semi-classicality, a model of the vacuum can be visualized. Using 1/√εcalμcalc gives 1.02c. The transit time fluctuations of 0.062fs/m1/2 compares to the author’s in reference 2 result of 0.05fs/m1/2. The result calculated here predicts a broadening of 3.4fs, which is 85% of the 4fs broadening of a 9fs laser pulse over 3000m whereas reference 2 has a value of 2.7fs or 68%. An experiment is waiting to be done.
Nature of the wave structure of the vacuum and development of the semi-classical quantum vacuum.
In the vacuum, it is assumed that a spin one of magnitude ħ transient photon wave packet composed of a spectrum of frequencies creates a transient dipole and the transient dipole annihilates into a similar transient photon. The life time of the transient photon and dipole is governed by the energy-time uncertainty relation of ∆t∆ε = ħ/2 where ∆ε = 2.92mc2, where 2.92mc2 is the mean energy of the dipole. Because the photon in the wave packet is spin one, the dipole when created assumes a spin one status. The total orbital angular momentum of the particle and anti-particle pair in the dipole is zero and so the spin coupling of the constituents in the transient dipole must have size ħ or spin one. The spin state that is spin one and where the particle and anti-particle have opposite intrinsic spins is ꟾ1 0>. The dipole is then created with a spin coupling state of ꟾ1 0> and annihilates from the same spin state to create a transient photon wave packet of spin one. The process repeats. The following argument supports the one photon creation process. The charge conjugation of the q-, q+ dipole is equivalent to an exchange of spin labels, giving a sign change of – (-1)S, parity reversal, and interchanging the electrical charges. The eigenvalues of charge conjugation are (-1)l+S. The transient dipole with total angular momentum l = 0 and total spin s = 1 has an odd charge conjugation parity given by (-1)0+1 = -1. The interchanging of the positive and negative charges reverses the direction of polarization of the electric field in the dipole so the charge conjugation of a photon is -1. A system of n photons has charge conjugation parity of (-1)n. With one photon, n = 1, and the odd charge conjugation parity of the transient photon is conserved during creation and upon annihilation of the transient dipole. Charge conjugation and spin one is conserved during the cyclical process of transient photon to transient dipole and back to transient photon. The short transient nature in time and space allows the one transient photon process, whereas when long measurement times and distances are involved, this one photon process is prohibited.

Due to the limiting spatial displacement via the time-energy uncertainty principle, c∆t = ħ/[2*2.92mc2] = 0.17ƛC, the transient dipole cannot be created at one point to annihilate at another point diametrically opposed, semi-classically, at a distance of 0.69ƛC. They are created as explained below in a structure similar to Figure 4 as opposed to positronium. Also, the transient dipoles are not similar to real positronium where the electron and positron move in orbits around each other diametrically in opposite directions and decay to n = 1 and l = 0 states with no orbital magnetic moment before annihilation. In the transient dipole, semi-classically, the particle and antiparticle would move diametrically in the same direction on a partial orbit each in the state l = 1 to give a total angular momentum of l + -l = 0, but their electrical currents move in the same direction to give a net magnetic moment of 2μB. This was demonstrated above in that that an external electromagnetic field is influenced by a required partial contribution of the orbital magnetic moment of the transient dipole in the derivation of the vacuum permeability. The other contribution was the magnetic moment from the coupled spin one state where the spins were oriented oppositely. The transient dipole creates and annihilates in this state.

Derived quantities describing the transient dipoles, wave packets, and other parameters of the vacuum, using the vacuum fluctuation of energy, ∆ε = 2.92mc2, the constants ħ, m, c, the Cauchy-Lorentz and exponential distributions are the Compton wavelength, ƛC = ħ/mc, the diameter of the dipole, Ld = 0.69ƛC, the mean energy of a transient dipole, the spatial displacement of the transient photon as well as of the constituent particles in the dipole on their orbital arcs after creation, c∆t = ħ/[2*2.92mc2] = 0.17ƛC, the global mean free path length for visible light interacting with a large sample of dipoles, Λ = 1/[σN] = 1/[(8π/3)*(0.69)2ƛC2*3.0/ƛC3] = 0.084ƛC, and the local mean free path for a visible light photon, which changes the energy of an existing dipole having an absorption area of (0.69)2ƛC2 leads to Λ = 1/[(8π/3)*(1/137)*8*(0.69)2ƛC2*3.0/ƛC3] = 1.43ƛC both already discussed in more depth above in the speed of light.

The relationship between the dominate wavelength of the transient photon to the energy of the transient dipole is ħω = 2.92mc2 or ƛ0 = 0.342ƛC, which is the reduced wavelength of hν = 2.92mc2 where λ0 = 2.14ƛC.

Due to the spatial displacement limitations on the transient wave packet, ∆x, the wave packet has a spectrum, ∆k, of wavelengths centered on the dominate wavelength, ƛ0, given by the relationship, ∆k∆x = ½.The harmonics 4ƛ0 = 1.37ƛC compares to the local mean free path of a real photon, which causes a change in the energy of an existing dipole or 1.43ƛC with 4% deviation, 3ƛ0 = 1.02ƛC compares to one Compton wavelength, ƛC, with 2% deviation, 2ƛ0 = 0.684ƛC, compares to the dipole diameter and mean distance of separation between their centers reflected by the density, which is 0.69ƛC with 0.9% 1ƛ0 = 0.342ƛC corresponds to the mean energy of the dipole, 2.92mc2, precisely by definition, ½ ƛ0 = 0.167ƛC compares to, ∆x = 0.17ƛC, the maximum spatial displacement limited by the uncertainty relation for energy-time, with 1.8% deviation, 1/3ƛ = 0.11ƛC, corresponds to the mean free path given by the mean number of surrounding adjacent dipoles in their Compton regions due to temporal fluctuations all influencing the cross section of the center Compton region giving 12/2 = 6σ leading to Λ = 1/[(8π/3)*(1/137)*8*6*ƛC2*3.0/ƛC3] = 0.11ƛC, with deviation of 0%, and ¼ ƛ0 = 0.086ƛC compares to the global mean free path of 0.084ƛC with 2.3% deviation. The mean deviation is 1.6%. The values 4ƛ0, 3ƛ0, 2ƛ0, 1ƛ0, ½ ƛ0, 1/3ƛ0, and ¼ ƛ0 are harmonics of 1ƛ0 or the mean energy of the dipole. These values of the nearly precise harmonics in the wave packet appear to be reinforced while other wavelengths are diminished. Some relations are shown in figure 6.

The dipole diameter and Compton wavelength corresponding to the harmonics of two and three times the dominate wavelength, which represents the energy of the transient dipole in the transient wave packet spectrum, allows the dipole to create at a mean separation distance of 0.69ƛC concurrent with the formation of the Compton wavelength, ƛC, during the annihilation of the transient photon. During the time of existence of the transient dipole, the constituent particles move on partial orbits of distance Δx as shown in Figure 4 and the dipole properties, such as charge separation, angular momentum, magnetic moment, and spin coupling are manifested as shown in the derivation of vacuum permittivity and permeability. The concurrent creation of a relationship between the Compton region and dipole from a transient photon, utilizes the cross section, σ, with the entire Compton region of diameter ƛC and not 0.69ƛC. As seen, σ = (8π/3)*(1/137)*8*ƛC2 = 0.49ƛC2. N is the mean density of dipoles = 3.0/ƛC3 and the mean free path length is, then, equal to Λ =1/[σN] = 1/[(8π/3)*(1/137)*8*ƛC2*3.0/ƛC3] = 0.68ƛC, which is the diameter of the dipole to 1.5%. The visible light then interacts with an existing dipole with diameter 0.69ƛC.The mean free path divided by the spatial displacement of the transient photon or 0.68ƛC/0.17ƛC = 4.

Figure 6.

The wave packet has a spectrum, ∆k, of wavelengths due to its spatial displacement of Δx given by the relationship, ∆k∆x = ½. The distribution of inverse wave lengths is modeled on a standard Gaussian curve where the dominate wave length, ƛ0 = 0.342ƛC, is taken to be the standard deviation with the curve centered at 0. The relation ∆k∆x = ½ gives ∆k = 1/(2∆x) or 1/(2*0.17ƛC) = 1/(0.34ƛC) = 2.94/ƛC. The harmonics for the various physical quantities above are illustrated as multiples or fractions of the standard deviation on the standard Gaussian curve as shown in Figure 7.

As shown in Figure 7, 4 standard deviations renders the global free path 0.085ƛC, 3 standard deviations, ƛ = 0.11ƛC, corresponds to the mean free path given by the mean number of surrounding adjacent dipoles influencing the center region, 2 standard deviations corresponds to the spatial displacement of a transient photon and the constituents particles in the transient dipole 0.17ƛC, 1 standard deviation is the reduced wavelength of the wavelength in the relation ħν = 2.92mc2 or 0.34ƛC, 1/2 standard deviation corresponds to the size of the dipole 0.69ƛC, 1/3 standard deviation corresponds to one Compton wave length ƛC, and 1/4 standard deviation corresponds to the local mean free path, 1.43ƛC, where the absorption area in σ is (0.69)2ƛC. Values <1/4Δk correspond to larger distances. The probability of 1/4 to 4 standard deviations under the standard Gaussian distribution is 0.80 or 80% under the curve. The other 20% corresponds to larger distances.
Figure 7.
Conclusion:
The consideration in the paper was real photons of relatively low frequency and long wavelength electromagnetic radiation interacting with transient nature of dipoles in the vacuum. The results for the permittivity, permeability, speed of light, and the transit time fluctuations are sensitive to small changes in the values of the means obtained from the Cauchy-Lorentz and exponential distributions. It is interesting to note that the transit time fluctuations depended only upon the mean value of the energy of the transient dipole along with its associated uncertainty in energy. Many significant figures are not supported by a paper of this semi-classical type but the quantities derived supported 2 to 3 significant figures. Despite the semi-classical nature and 2 to 3 significant figures, the calculation of the vacuum permittivity, permeability, and the speed of light were 0.72% to 1% of the experimental values, which indicate good agreement and consistency with each other. The chosen distributions above are based on the real positronium wave function or exponential distribution and the Lorentz energy distribution. The results of the analysis using those distributions producing good agreement suggests a deeper understanding of the significance of these distributions and the method used. Also, it is again surprising that the harmonic wave relationships and the transient dipole and vacuum parameters above are closely derived from the mean energy of the transient dipole. The mean energy and uncertainty in energy of the transient dipole is, then, a fundamental quantity and ultimately supported by ħ. This provides for a deeper understanding of the semi-classical quantum structure of the vacuum.
References
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