Instantaneous Quantum Entanglement, Vacuum Processes, and Calculation of Cosmological Constant
Richard Bradford
Jul 17, 2017
I want to thank Gordon Rogers for enlightening conversations
Abstract
“Spooky action at a distance” as said by A. Einstein. Bell type experiments have eliminated various loopholes for deterministic hidden variable influence and established that quantum entanglement of light and particles is nonlocal. Experiments have shown that increasing separation distance between detectors and using faster and greater precision processing electronics, the lower limit of the speed of entanglement, all outside the lightcone, increases. There is an implication that the quantum influence is instantaneous. It is proposed that instantaneous nonlocal processes are supported by forward time positive and backward time negative energy connected electromagnetic and if applicable transient dipole ground states determined by the fine structure constant. The energytime relation is determined by a quantized harmonic oscillator ground state solution of Maxwell’s equation. The positive and negative ground states evolve according to forward and backward time and spatial quantum clocks based on concepts of distinguishability and indistinguishability developed in the HilgevoordUffink uncertainty relations. Positive plus negative energy ground states support opposite time currents corresponding to relativistic invariant equations having positive and negative frequency components. The description of transient dipoles using the StüecklebergFeynman interpretation determines the correct total displacement, time of existence of the dipole, and density of transient dipoles in the vacuum. Nonelastic and elastic electronpositron or electronelectron scattering is proposed using ground state processes. The issue of the large vacuum energy is addressed and the cosmological constant is calculated from frequency of occurrence of transient dipoles based upon the fine structure constant.
Discussion
Let there be two identical measuring devices, A and B, equidistant from a source emitting correlated light quanta or particles in some rest frame, which measure the polarization attribute of light quanta or spin orientation of particles, simultaneously. At A or B, it is found that measurements have random results but, when compared, the results are correlated with one another, violating Bell’s inequality and agreeing with the predictions of Quantum Mechanics. Experiments discussed below performed with light quanta have closed in separate experiments nearly all deterministic hidden variable loopholes with the conclusion the quantum influence between A and B is in fact nonlocal.
The freedom of choice loophole has been closed. Nondeterministic polarizer settings are determined by random quantum generators at A and B each outside the light cone of the emission point so not to be influenced by the source. Any hidden variable theory, then, can only specify a probability distribution and deterministic outcome theory would then pose no fundamental problem since the dependence is on stochastic settings.
The locality loophole has been closed. The results of measurements of the photon polarization at A and B are correlated when done outside of each other’s light cone establishing the nonlocal nature of the entanglement i.e. no influence of the measurement outcomes between A and B occurs at or less than the speed of light. Precise clocks, previously synchronized, running at A and B record the time, random polarizer settings, and outcomes. The results are later compared and found to violate Bell’s inequality. Closing the freedom of choice and locality loopholes have been done in one experiment.
The fair assumption loophole, which is photons detected are an accurate account of the emitted photons, has been closed by counting and taking into account the lost photons with newly available highly efficient and highly sensitive detectors.
A single experiment closing all these loopholes have yet to be done. Following Bell, “there is no reason to think that if the loopholes are closed in separate experiments, there is a conspiracy that the local hidden variable theories would suddenly be valid if all closing of loopholes can be done in one experiment”.
It should be mentioned that super deterministic or hidden variables influencing the experiment from the past light cone has not been ruled out as of yet. This may be achieved by using two quasars separated by a large angle in the sky so not to have interacted with each other yet. The detectors would be configured by the quasars and provide for independence of quantum measurement. If hidden variable interactions exist, then the detectors configured via the quasars would produce different results than detectors configured by ordinary random process. This experiment has not yet been performed.
Also, experiments involving increasing the distances between A and B have been performed that set larger lower bounds on the speed of the quantum influence, which are all greater than the speed of light. One such experiment established a lower bound with the detectors at a distance of 18 km apart on the earth’s surface where measurements are nearly, as set by technological standards, simultaneous with respect to a stationary observer on earth. Their result was that the lower bound was equal to 10,000 and the upper bound was 200,000 times the speed of light depending on the angle between detectors A and B relative to EastWest given that the earth has a velocity, or β = 10^{3}, with respect to a universal reference frame fixed to the cosmic background radiation. Other experiments having distances of separation of 144km to 300km have set lower bounds even greater.
The descriptions of the entanglement connection between the detectors are fourfold:

The entanglement connection can be explained with a timelike or nulllike velocity.

The entanglement connection is a finite superluminal velocity.

The entanglement connection is diminished over distance.

The entanglement connection is an instantaneous connection.
Closing the locality loophole rules out the entanglement connections described by condition 1. The entanglement connection described by condition 2 within terrestrial experiments give no indication of a finite speed, but due to present day technological limitations, instantaneity is not ruled out. Experimental increases of the lower bound of the connection speed leads to the possibility that a maximum speed may not exist and the connection is instantaneous. The technological limits for measurement of spacetime resolution is quantum uncertainty, which has a spatial order of atomic dimensions in the detectors and a time based upon the speed of light crossing that distance. Then, as the speed increases, the maximum distance increases. Any distance beyond the maximum distance for a given speed leads to measurement reduction from either detector, which if equidistant from the source in a rest frame as a special case, to not reach the other during its measurement reduction resulting in a loss in correlated results.
In condition 3, the entanglement connection diminishes with distance by an absorbing field until no correlation occurs. The entanglement connection field carries no energy but may be diminished by expansion of the universe if the attenuation is consistent with redshift of electromagnetic waves. Both effects may be present together but both or either one have not been detected within terrestrial distances of separation. An entanglement connection not diminishing with distance has not been ruled out.
To summarize. The entanglement connection requires the existence of a field since it cannot be supported by nothing. The connection supported by a field either has a finite speed or is instantaneous and the characteristics of the field are that it has absorption qualities and/or is affected by the expansion of the universe or is nonattenuating. Other than simply naming these qualities, having physics of the field supporting and explaining these qualities provides for predictions and experimental verification, which would increase its viability and depth of strength.
There exists a field having these physical properties. If entanglement is supported by the existing vacuum, which is physical with specific describable properties, then there is no reason to assume existence of a field, which suffers from a lack of describable field structure to explain the above mentioned qualities. The vacuum has components that support and establish the qualities of the entanglement connection. Propagation of electromagnetic radiation shows that the vacuum has existed in spacetime for billions of light years in an expanding universe since the big bang. The ground state parameters in the vacuum supporting the entanglement connection do not change due to redshift from an expanding universe since if the frequency of the ground state modes are changed by redshift, then the maximum ground state Planck frequency, corresponding to spacetime itself, is redshifted. This implies that the fundamental constants, G, ħ, c, composing the Planck frequency, are changing over long times and distances in which there is no strong evidence of this occurring. Thus, the entanglement connection is not affected by expansion is explained. The vacuum, then, exists at all locations in an expanding universe and beyond the observable horizon since the vacuum or real objects cannot be interfaced with a region of nonvacuum or nothingness. The Casimir, Lamb effect, and polarization illustrate that the vacuum is a dynamic entity.
The following two postulates are then introduced.

The entanglement connection between correlated objects is undiminished and instantaneous.

Dynamic vacuum ground states is the field that supports the entanglement connection.
1) Structure, processes, and dynamics of electromagnetic and dipole ground states in the vacuum.
A) Positive and negative energy and time in equations, propagator, dynamic vacuum and distributions.
A scattering event such as an electron and positron is described with two probability amplitudes where M_{1} ≈ ∫d^{4}x_{1}d^{4}x_{2}[expip_{1}.x_{1}][expip_{2}.x_{2}](ig)[d^{4}q/(2π)^{4}][iexp(iq.(x_{1} – x_{2})g^{μν}/q^{2}](ig)exp(ip_{3}.x_{1})exp(ip_{4}.x_{2})(spinors are not present) and amplitude M_{2} is similar. The coupling g^{2} = α is the fine structure constant. The photon Feynman propagator is D_{F}(x_{1} – x_{2}) = ∫d^{4}q/(2π)^{4}iexp(iq.(x_{1} – x_{2})g^{μν})/[q^{2 }+ iε] and contains aspects of elastic and inelastic scattering. The fourmomentum vector of the virtual photon is q and 1/q^{2} is the amplitude for a virtual particle existing. The relation q^{2} = 0 describes an onshell photon. These aspects are shown by the Fourier decomposition of the photon propagator given as D_{F}(x) = (1/(2π)^{4})∫[PP/q^{2} – iπδ(q^{2})]exp(ikx) = D + D_{I}. D contains the real principle term, PP/q^{2}, and is a time symmetric bound field, which describes an off shell virtual photon +q^{2} ≠ 0 exchange of momenta in energymomentum conserving elastic interactions, where q^{2} is spacelike elastic scattering. +q^{2} is timelike, ee+ annihilation and creation with amplitude α. The imaginary part D_{I}, where δ(q^{2}) implies q^{2} = 0, describes the inelastic interaction with emission of an electromagnetic wave packet with an amplitude α. The Feynman prescription of integrating the propagator is integrating along the real axis from negative to positive q^{0} and counterclockwise (clockwise) in the positive (negative) complex plane at infinity circling the pole q^{0} = lql + i0 (q^{0} = +lql – i0) giving t’ > t (t’ < t). The Feynman virtual photon propagator is the sum: iD_{F}_{μν}(xy) = <0ΙT(Aμ(x)Aν(y)Ι0> = <0ΙAμ(x)Aν(y)Ι0>Θ(t(x)t(y)) + <0ΙAν(y)Aμ(x)Ι0>Θ(t(y)t(x)), where Θ is the Heaviside step function, where Θ(t(x)t(y)) = 1 if t(x)t(y) ≥ 0 and Θ(t(x)t(y)) = 0 if t(x)t(y) < 0 and similarly for Θ(t(y)t(x)).
The relativistic equations describe onshell spin 0, spin ½, and spin 1 particles all with a wave nature. They are the KleinGordon equation for spin 0 particles, (∂^{2}/∂t^{2} – ∂^{2}/x^{2})φ(x,t) + m^{2}φ(x,t) = 0, the Dirac equation i∂ψ(x,t) = iα*∂ψ(x,t) + βmψ(x,t) describing up and down spin ½ fermions, and the vector potential, A, in electromagnetic theory describing spin 1 photons. These equations have positive and negative energyfrequency components describing forward and backward time particles or waves. If time, t, in the negative energy state becomes –t, then the negative energy state looks a positive energy state, which is the basis of the StücklebergFeynman interpretation. Introducing Lorentz invariance to quantum mechanics leads to a local and causal description of fields in spacetime, antiparticles, and have the correct spin statistics. Since all physical measurements have local field operators that commute at space like separation, the negative energy states must be included and leads to positive and negative frequency states travelling forward and backward in time.
Maxwell’s equations for electromagnetic fields can be expressed in terms of the vector potential, A. In space, free of electrical charges and currents, the wave equation for the electromagnetic vector potential modes is (1/c^{2})∂^{2}A/∂t^{2} – Laplacian A = 0. The general solution is a sum over all modes of single mode solutions where a single mode solution is A_{kλ}(r,t) = A_{kλ}(t)e^{ikr} + A^{*}_{kλ}(t)e^{ikr} and kλ labels a distinct mode. Placing the single mode solution into the wave equation leads to the harmonic oscillator equation ∂^{2}A_{kλ}(t)/∂t^{2} + ω_{k}^{2}A_{kλ}(t) = 0. The two solutions are the positive frequency and forward time solution given as A_{kλ}(t) = A_{kλ}exp(iω_{k}t) and negative frequency backward time solution, A_{kλ}(t) = A_{kλ}exp(+iω_{k}t). Both energy solutions are taken into account for the energy levels associated with the harmonic oscillator. In quantum mechanics the harmonic oscillator is quantized and is described by discrete energy levels. The energy level of a quantum harmonic oscillator for a mode is determined from an eigenvalue equation +Hꞁn> = nꞁn> and discretized with n, where n = 0, 1, 2 … The positive and negative energy levels are +E_{n} = +ħω(n + ½), where n > 0. In particular, n = 0, is the energy of the positive and negative energy ground state modes as +E_{0} = +(ħω_{0})/2. Due to negative energy and reversed time, the negative energy levels mirror those of the positive ones. Then, a negative energy excited state (n ≠ 0) corresponds to a real negative energy and a time reversed entity. The positive and negative frequency solutions of the relativistic equations for n ≥ 1 excited forwardbackward time entities and the time ordering associated with the photon propagator are assumed to apply to, n = 0, ground states in the vacuum. There is then no fundamental dichotomy between onshell and ground states in the vacuum without adding an extraneous theory of explanation.
In a static vacuum, there are sets of ground state positivenegative energy │E_{0}>, │E_{0}>* and momentum llP_{0}l>, llP_{0}l>*, eigenstates, with infinite spatial and temporal single frequency oscillations in space. From +E_{n} = +ħω/2 = +hf/2, where 2πf/2 = ω/2, the frequency has an associated time defined +T = +(f/2)^{1} = +h/E. Momentum +lPl and spatial displacement +lQl relate to period and energy given as +E = +clPl and +Tc = +lQl. In a dynamic vacuum, processes occur with finite time and spatial intervals and ground states, ꞁψ>_{E,P} and ꞁψ>_{E,P}*, are no longer eigenstates but have energy, frequency, period, momentum, and spatial displacement distributions, +ΔE, +Δω, +ΔT, +lΔPl, and +lΔQl having relations of, +ΔT = h/(+ΔE) +ΔE = ħ(+Δω/2) = +clΔPl, and +lΔQl = c(+ΔT). +ΔT and +lΔQl are a measure of the temporalspatial dynamics occurring in the vacuum. The speed of light, c, associated with electromagnetic ground states is determined by the vacuum in the same as for onshell electromagnetic waves.
B) HilgevoordUffink (HU) timeenergy and spatial displacementmomentum relations, distributions.
Let electromagnetic ground state positive and negative wavepackets be described by │ψ_{t}>, │ψ_{t}>*. Time states │ψ_{t}> and │ψ_{t+τ}> may be written as <ψ_{t}│U(τ)│ψ_{t}>, and │ψ_{t}>* and │ψ_{tτ}>* as *<ψ_{t}│U(τ)│ψ_{t}>*, where
U(t) = exp(+iHt/ħ) is the unitary operator of the time evolution and H is the Hamiltonian. For the positive time case, define τ_{ρ} as the smallest time when the absolute value of the integral │<ψ│U(τ_{ρ})│ψ>│ has decreased to 1 – ρ or │<ψ│U(τ_{ρ})│ψ>│ = 1 – ρ, with the condition 0 ≤ ρ ≤ 1. │<ψ│U(τ_{ρ})│ψ>│ is the transition amplitude of the states and its square is the probability of finding the state ꞁψ_{t+ρ}> if it has been prepared in state ꞁψ_{t}> . The parameter ρ is the reliability that the states ꞁψ> and U(τ_{ρ})ꞁψ> can be distinguished. When states coincide the reliability is 0, ρ = 0, │<ψ│U(0)│ψ>│ = 1, and when orthogonal or distinguished the reliability is 1, ρ = 1, │<ψ│U(τ_{1})│ψ>│ = 0, where τ_{1} = +ΔT is the minimum time for orthogonality. The argument is verbatim for the negative energy and time states, τ_{ρ}. There is a Fourier transform between τ_{ρ} and probability density of the energy distribution │<E│ψ>│^{2}, or <ψ│U(τ)│ψ> = ∫ │<Eꞁψ>│^{2} e^{iτE/ħ }dE. A timeenergy relation between τ_{ρ} and the width of the energy distribution can be derived. Let +W_{α}^{E} be the smallest energy width W such that │+∫_{W}│<+Eꞁψ*(0)>│^{2} dE│ = α, where α <≈ 1. If α = 0.9, then 90% of the energy distribution is contained in the interval.
The HU relation is given as τ_{ρ}W_{α}^{E} ≥ 2ħ arccos[(2 – α – ρ)/α] with ρ ≥ 2(1 – α) as derived from the MandelstamTamm relation. It can be shown that the constant on the R.H.S. is the best possible for any state │ψ> having a complete set of energy ground states here with energies from E_{min} to the Planck energy E_{P}. Define the time for orthogonality as τ_{1} = +ΔT. Orthogonality implies │<ψ│U(+ΔT)│ψ>│ = │<ψ_{t}│ψ_{t+ΔT}>│ = 0, and for negative energy states│*<ψꞁU(ΔT)ꞁψ>*│ = │*<ψ_{t}ꞁψ_{tΔT}>*│ = 0, ρ = +1. The HU timeenergy relation becomes +ΔT(+W_{α}^{E}) ≥ 2ħ arccos[(1 – α)/α]. Inserting the relation, +ΔT = h/+ΔE, gives the energy width +W_{α}^{E} ≥ [(+ΔE)/π][arccos[(1 – α)/α]. Having α ≈ 1, arccos[(1 – α)/α] ≈ π/2 giving │+W_{α}^{E}│≥ │+ΔE/2│ and + W_{α}^{E} and W_{α}^{E} or +ΔE/2 are nearly contained in the states, │ψ_{n}> or │ψ_{n}>*.
Minimum resolution of position depends on the displacement, +X, for distributions of x at positions x, and x + X to be distinguishable or │ψ_{x}>, │ψ_{x + X}> and │ψ_{x}>*, │ψ_{x – X}>* are orthogonal. U(x) = exp(+iPx) is a unitary operator of displacement and P is the operator of total momentum. Looking at the positive case, the overlap integral between a state and its spatially translated state is <ψ│U(x)│ψ>. Reliability of │<ψ│U(x)│ψ>│ = 1 – p. Distinguishability or orthogonality ρ = 1 is defined with +lΔQl = +X_{,} as │(<ψ│U(+lΔQl)│ψ>│ = 0, and verbatim for the negative case, lΔQl = X. Indistinguishability is given by the relation │(<ψ│U(0)│ψ>│ = 1, ρ = 0. The temporalspatial states are related by, +ΔT = +clΔQl, they pass through distinguishability and indistinguishability together on the quantum clock.
The Fourier transform relationship between X and probability density of momentum distribution │<Pꞁψ>│^{2} is │<ψ│U(X)│ψ>│ =∫│<Pꞁψ>│^{2 }e^{ixP}dP. The width +W^{P}_{α} of the momentum distribution is the smallest interval W such that │+∫_{W} ꞁ<P│ψ>│^{2} dPl = α. Making the substitutions, +lΔQl = c(+ΔT) = ch/+ΔE, p = 1, +ΔE = c(+lΔPl), in the HU relation +lΔQl(+W^{P}_{α}) ≥ 2ħ arccos[(2 – α – ρ)/α] gives the minimum momentum distribution +W^{P}_{α} ≥ {2ħ arccos[(1 – α)/α]}/ch/(+ΔE) giving W^{P}_{α} ≥ {lΔPl/π arccos[(1 – α)/α]}. With α ≈1, arccos[(1 – α)/α] ≈ π/2, and the momentum distribution │+W^{P}_{α}│ ≥ │+lΔPl/2│, and nearly contained in the ground states, │ψ_{n}> and │ψ_{n}>*.
The HU relations are relativistic invariant when the Hamiltonian and total momentum operators are combined into a four vector. Time direction and energy are connected and share the same sign i.e. negative (positive) energy cannot move forward (backward) in time since there are no such solutions in the electromagnetic equations. Ground states in the negative realm with distributions, ΔT, ΔQ, –ΔP, and ΔE operate oppositely to those in the positive realm, +ΔT, +ΔQ, +ΔP_{, }and +ΔE but the spatial and momentum threevectors allow for propagation of positive and negative states in any direction. Energy and time are proportional to momentum and spatial displacement where the constant of proportionality is given as +lΔQl/+ΔT = c = +ΔE/+lΔPl, where c is the constant speed of light. The energy and momentum along with displacement and time distributions have a constant ratio with the speed of light c and as a result pass through indistinguishability and distinguishability together.
C) Quantum clock and evolution of states. Relative orientations of positivenegative energy distributions.
Positive and negative temporal and spatial quantum clocks are defined as systems, where in the course of evolution, governed by positive or negative temporal and spatial parameters, a positive state │ψ_{n}> evolves to a distinguishable state │ψ_{n+1}>… at times and positions t_{n}, t_{n+1} and q_{n}, q_{n+1} …, where t_{n+1} > t_{n}, and q_{n+1} > q_{n} and a negative state │ψ_{n}>* evolves to a distinguishable state │ψ_{n1}>*, where t_{n1} < t_{n}, q_{n1} < q_{n,} where the temporal and spatial resolution of distinguishable displacements are, Δt_{+} = +(t_{n+1} – t_{n}) = +ΔT and Δq_{+} = +(q_{n+1} – q_{n}) = +lΔQl. Positive and negative states │ψ_{n}>, │ψ_{n}>* have equal magnitude energy distributions.
The transition amplitudes │(*)<ψ│U(+τ_{ρ})│ψ>(*)│ = 1 – ρ and│(*)<ψ│U(+lΔQl)│ψ>(*)│ = 1 – ρ, where the negative sign corresponds to (*), describe evolution of forward and backward temporal and spatial quantum clocks. Evolution of forward time ground states is described as the transition │<ψ_{t=0}│ψ_{t=0}>│ = 1 to │<ψ_{t=0}│ψ_{t=+ΔT}>│ = 0 and backward time states as │*<ψ_{t=0}│ψ_{t=0}>*│ = 1 to │*<ψ_{t=0}│ψ_{t=ΔT}>*│ = 0. The transitions of spatial states occur in lockstep, where +ΔT becomes +lQl. Let │<ψ_{n}│ψ_{n}>│ = │<ψ_{t=0}│ψ_{t=0}>│ and │*<ψ_{n}ꞁψ_{n}>*│ = │*<ψ_{t=0}│ψ_{t=0}>*│ = 1, be defined as time n, which then evolves to │<ψ_{n}│ψ_{n + 1}>│ and │*<ψ_{n}│ψ_{n – 1}>*│ = 0 in time +ΔT, becoming distinguished and at that time are in a condition of indistinguishability with states │ψ_{n+1}>, │ψ_{n1}>* or │<ψ_{n + 1}│ψ_{n + 1}>│ and │*<ψ_{n – 1}│ψ_{n – 1}>*│ = 1 both defined as time n + 1 for forward and n – 1 for backward time. The states, │ψ_{n+1}>, │ψ_{n1}>*, during time intervals +ΔT and –ΔT evolve from indistinguishability to distinguishability or │<ψ_{n + 1}│ψ_{n + 2}>│ and │*<ψ_{n – 1}│ψ_{n – 2}>*│ = 0, where there is indistinguishability of states │ψ_{n + 2}>, │ψ_{n – 2}>* or │<ψ_{n + 2}│ψ_{n + 2}>│ and │*<ψ_{n – 2}│ψ_{n – 2}>*│ = 1, both defined as n + 2 and n – 2 and so forth and so on.
Evolution of states, │ψ_{n}>, │ψ_{n + 1}>, │ψ_{n + 2}>… and │ψ_{n}>*, │ψ_{n – 1}>*, │ψ_{n – 2}>*… from an undistinguished to a distinguished condition as above all occur during time interval +ΔT. Thus, none of the previous or future temporal and spatial states are relinquished to nonexistence and all states are consecutively connected and ordered throughout the vacuum. During the condition │<ψ_{t = 0}│ψ_{t = +tρ}>│ = 1 – ρ, where ρ ≠ 0 or 1, time n and time n + 1 or n – 1 cannot be determined with certainty. An analogy is a set of clock second hands representing all states moving on a clock without the numbers on the face, which only serves as a visual aid to help with telling time from the position of the hand relative to the face of numbers. The evolution of time and position can be surmised by the evolution of the hand relative to itself in time.
Consider negative backward time states │ψ_{n + 1}>*, │ψ_{n}>*, │ψ_{n – 1}>*, │ψ_{n – 2}>* and positive forward time states │ψ_{n + 1}>, │ψ_{n}>, │ψ_{n – 1}>, │ψ_{n – 2}> propagating at the speed of light in opposite directions reflecting their opposite time nature, where at one point in spacetime during their relative evolution, the states are aligned. The spatial states │ψ_{m + 1}>*, │ψ_{m}>*, │ψ_{m – 1}>*, │ψ_{m – 2}>* and │ψ_{m + 1}>, │ψ_{m}>, │ψ_{m – 1}>, │ψ_{m – 2}>, are also aligned. In both cases they are defined as indistinguishable, │**<ψ_{n}│ψ_{n}>│ = 1, where │ψ_{n}>** represents time reversal of a negative energy state, │ψ_{n}>*. If time reversal is performed on a negative energy backward time distribution, or │ψ_{n}>* goes to │ψ_{n}>** = │ψ_{n}>, then the energy distribution is positive and forward time.
Let the backward spatial states propagate left while evolving from a time of n + 2, to n + 1, n, n – 1, n – 2, and the forward spatial states propagate right while evolving from a time of n – 2 to n – 1, n, n + 1, n + 2. Holding the forward time states stationary, the negative time state n + 1, evolves and propagates to time state n, while n evolves and propagates to time state n – 1 relative to the forward time state n, where the negative state evolving from n to n – 1 changes from indistinguishability, │**<ψ_{n}│ψ_{n}>│ = 1, to distinguishability, │**<ψ_{n – 1}│ψ_{n}>│ = 0 with respect to the positive state at n. Holding negative states stationary, the positive time state n – 1 evolves and propagates to time state n, while n evolves and propagates to time state n + 1 relative to the negative time state n undergoing indistinguishability to distinguishability. Then, as the positive and negative states evolve and propagate past each other in opposite directions, the positive time state n – 1 evolves and propagates to time state n during time +ΔT, while the negative time state n + 1 evolves and propagates to time state n during time –ΔT. Thus, a positive energy state evolving from the past to the future will correspond with a negative energy state evolving from the future to the past.
The positive and negative energy distributions +ΔE and –ΔE of ground states │ψ_{n}>, │ψ_{n}>* are linear functions of the frequency distributions +Δω or +Δf given as +ΔE = +ħ(Δω/2) = +h(Δf/2) with a slopes ħ or h. Graphically, one end point of the energy distribution is +│E_{MAX}│ at +│ω_{MAX}│/2 or +│f_{MAX}│/2 and the other endpoint is +│E_{MIN}│ at +│ω_{MIN}│/2 or +│f_{MIN}│/2. The relative orientation of the positive and negative energy distributions result from a time reversal symmetry. Due to time reversal, the negative and positive energy distributions are in tandem at the same quantum time, have +│E_{MAX}│ corresponding to │E_{MIN}│ and │E_{MAX}│ corresponding to +│E_{MIN}│ or an indistinguishable positive and negative state, defined as │**<ψ_{n}│ψ_{n}>│ = 1. As they propagate oppositely in spacetime, the positive and negative energy distributions do not cancel but superimpose into positive and negative energy waveforms with varying durations, where time cannot be determined with certainty defined as │**<ψ_{n + 1}│ψ_{n}>│ = 1 – ρ, where ρ has condition 0 < ρ < 1, until distinguishability, │**<ψ_{n}│ψ_{n + – 1}>│ or │**<ψ_{n + 1}│ψ_{n}>│ = 0 occurs. Momentum distributions are similar and behave with the same relationships given above since energy and momentum are related as +ΔE = +lΔPlc.
D) Proving StücklebergFeynman interpretation using thermodynamic irreversibility and acceleration in a gravity field. Quantum fluctuations
A positive energyforward time positively charged electron or positron (antielectron) is a negative energy backward time electron by the StücklebergFeynman interpretation of positrons. Behavior such as a positron moving forward in time and direction losing energy and slowing when passing through lead seem to disprove the StücklebergFeynman interpretation because a negative energy backward time electron moving in a reversed direction must speed up when passing through the lead gaining energy contrary to the thermodynamic behavior of solids. A positron reduces positive energy or equivalently absorbs negative energy given by +E_{FINAL} < +E_{INITIAL} and +E_{FINAL} – (+E_{INITIAL}) = ΔE and slows in a forward time and space direction. The loss in positive energy to or taking negative energy from the solid increases its temperature. The negative energy electron, has equal but opposite energy to the positron and passes through the solid in a reversed time and direction having the negative energy relation –E_{FINAL }< E_{INITIAL} and E_{FINAL} – (E_{INITIAL}) = ΔE, where the past state of the negative energy electron coincides with the future state of the positron so that +E_{FINAL} of the positron corresponds to –E_{INITIAL} of the negative energy electron. The future state of the negative energy electron corresponds to the past state of the positron or –E_{FINAL} corresponds to +E_{INITIAL}. The negative energy electron loses positive energy or equivalently absorbs negative energy causing it to speed up and increase the temperature of the solid. The forward time positron and the backward time electron meet each other at all increments in time since as the positron moves in time increments from its past state to its future state, the negative energy electron moves from its past state, which is the future state of the positron, to its future state, which is the past state of the positron meeting the positron at all time increments in its past. The same holds for future time states of the positron. Thus, thermodynamic behavior is not compromised since positive energy is given up in both cases warming the solid.
A positron moving forward in time and direction accelerates toward a gravitational body and increases its positive kinetic energy or equivalently decreasing negative energy as shown in the relations, given as +E_{FINAL} > +E_{INITIAL} and +E_{FINAL} – (+E_{INITIAL}) = +ΔE, and the negative energy electron moving backward in time and direction have the relations E_{INITIAL }< E_{FINAL} and E_{FINAL} – (E_{INITIAL}) = +ΔE, where the initial and final labels are interchanged as above, increasing positive energy, which is equivalent to reducing its negative kinetic energy, implying that negative energy electron decelerates while moving away from the gravitational body backward in time and direction in tandem with positron as explained above. Thus, the interpretation in terms of positive energy positron and direction is consistent and identical to the interpretation in terms of a negative energy electron.
Quantum fluctuations can be described not as positive energy appearing and disappearing from nothing or the vacuum governed by some energytime uncertainty principle, which technically does violate strict conservation of energy even for a short time but can be explained naturally as a spontaneous transfer of energy from a negative ground state to a positive ground state rendering both in a positive and negative excited energy state and then relax back to the ground state. The time interval of excitation existence is determined by the energy of the particular ground state involved by a relation similar to the uncertainty relation. The spontaneous excited ground states interact with onshell entities in such a way that energy is imparted and removed or vice versa in such a way that energy is conserved without any net transfer of energy from the vacuum unless there is an input of external onshell energy.
E) Vacuum field processes of dipole creation and annihilation analyzed with Feynman and quantum clock process and their comparison.
From above, the scattering event of an electron and positron has two probability amplitudes where M_{1} ≈ ∫d^{4}x_{1}d^{4}x_{2}[expip_{1}.x_{1}][expip_{2}.x_{2}](ig)[d^{4}q/(2π)^{4}][iexp(iq.(x_{1} – x_{2})g^{μν}/q^{2}](ig)exp(ip_{3}.x_{1})exp(ip_{4}.x_{2})(spinors are not present) and M_{2} is similar. The coupling g^{2} = α is the fine structure constant. The photon Feynman propagator is D_{F}(x_{1} – x_{2}) = ∫d^{4}q/(2π)^{4}iexp(iq.(x_{1} – x_{2})g^{μν})/[q^{2 }+ iε] and the exchange of momenta, +q^{2}, in the energymomentum conserving elastic interaction, where +q^{2} is timelike, describing electronpositron annihilation and creation with amplitude α.
Figure 4 diagrams the process of creation and annihilation of an positive electron and negative electron dipole using QED with time, t, and position, q, as reference markers needed for comparison to the process described by the quantum clock in Figure 5. Forward time is right and backward time is left.
The QED description in Feynman propagator theory that is charge neutral is a virtual positive frequency electron emerges from a scattering event with a virtual photon and propagates forward in time from vertex 1 to 2, where it is scattered from a virtual photon into a virtual negative energy electron, where it propagates backward in time to destroy itself at vertex 1. If the virtual negative frequency electron upon arriving at vertex 1 destroys the virtual positive frequency electron, then the dipole would not exist. Instead of annihilation of the positive electron at vertex 1, the virtual negative electron propagating from vertex 2 to 1 may not destroy itself by an unknown process but is scattered by the virtual photon into a virtual positive electron as shown in Figure 4, which is then a symmetric process. The process is charge neutral in that the positive electron at each moment in its future to its past position from vertex 2 to 1 is in tandem with the negative electron as it propagates backward in time or as the negative electron at each moment in time from vertex 1 to 2 it is in tandem with the positive electron as it propagates forward in time. The transient nature of the dipole would be due to the transient nature of the entire process. The Feynman propagator describes the emission (absorption) of a negative electron as equivalent to an absorption (emission) of a positive frequency forward time positron (antielectron) so that, equivalently, the process for a positive electron and positron is creation of an electronpositron dipole pair from a virtual wave packet at vertex 1, where they propagate to vertex 2 and annihilate creating a virtual wave packet. The process consisting of two virtual wavepackets and a dipole is an isolated, and elementary process, in which these building blocks may connect to processes occurring in the future or past.
Figure 5 illustrate the quantum clock process of creationannihilation of electromagnetic wave packets, positive electron and negative electron ground states. At t = 2, q = 2, an incoming positive energy ground state wave packet evolves to t = 1, q = 1 and annihilates creating a positive energy electron. The positive electron evolves to t = 0, q = 0 annihilates creating a positive wavepacket that evolves to t = +1, q = +1. Concurrent with positive evolution, a negative wavepacket evolves from t = +1, q = +1 to t = 0 and q = 0 and annihilates creating a negative energy antielectron, which evolves to t = 1 and q = 1 and annihilates creating an outgoing negative wave packet evolving to t = 2, q = 2. The positive and negative electromagnetic ground states evolve until creation of another dipole reflected by the amplitude α, where the potential of a positive or negative electron is an enabler for the dipole process to exist concurrently.
The dipole structure is summarized by the following, which is similar to the QED description of dipole creationannihilation. At t = 1, q = 1 a negative energy electron annihilates and a positive energy electron creates similar to the process at vertex 1 and at t = 0, q = 0 a positive energy electron annihilates and a negative energy negative energy antielectron creates similar to vertex 2. The dipole pair exists between t = 1, q = 1 and t = 0, q = 0. Distinguishability and indistinguishability applies with respect to the positivenegative ground states at t, q = 2, 1, 0, +1 i.e. creation of the electron at t = 1 occurs at distinguishability with the wavepacket when t = 2 and indistinguishable with the annihilation of the wavepacket when time t = 1 or another is creation of a negative energy electron when t = 0 occurs at distinguishability with the negative energy wavepacket when t = +1 and indistinguishable with its annihilation when t = 0.
If in QED there were backwardforward time channels in the virtual photons to remove and add positive and negative time instead of a scattering process description, then the QED process is similar to the quantum clock process, which allows positive and negative forwardbackward time channels in the ground state wave packets allowing for a removal and addition of forwardbackward time to positivenegative electrons. A forward and backward time ground state wavepacket combination is equivalent to one virtual photon in QED. The QED description would then be negative time is transferred from a scattering event from the negative electron removing negative time while adding positive time to a positive electron and viceversa. The quantum clock dipole processes is also an elementary building block with possible connections to past and future evolving electromagnetic ground states.
F) Definition of total charge, energy, time, and displacement. Time of existence and displacement parameters associated with the dipole and electromagnetic ground states.
Definition and conservation of total charge, energy, time, and spatial displacement during the quantum clock creationannihilation process uses the StücklebergFeynman interpretation. At t = 1, creation of an electron lowers charge, e, while the annihilation of the negatively charged antielectron raises charge, +e for zero net charge. At t = 0, annihilation of the electron raises charge while creation of the antielectron lowers charge for zero net charge. The separated charges form a dipole between t = 1 and 0 that is charge neutral as in the Feynman propagator description of the dipole.
At t = 1, creation of a positive energy electron and annihilation of a negative energy electron raises energy by +ΔE_{e} – (ΔE_{e}) = +2ΔE_{e}, while at t = 0, annihilation of a positive energy and creation of a negative energy electron lowers energy by –ΔE_{e} – (+ΔE_{e}) = 2ΔE_{e} conserving energy, where ΔE_{e} = m_{e}c^{2}. Total energy, +2ΔE_{e}, bifurcates into +ΔE_{e} and –ΔE_{e} between t = 1 and 0.
Ground state time and energy relate as +ΔT = h/(+ΔE). At t = 1, creation of an electron and annihilation of an antielectron raises time +h/ΔE_{e} – (h/ΔE_{e}) = +2h/ΔE_{e} or +ΔT – (ΔT) = +2ΔT. At t = 0, electron annihilation and antielectron creation lowers time h/ΔE_{e} – (+h/ΔE_{e}) = 2h/ΔE_{e} or –ΔT – (+ΔT) = 2ΔT conserving time. Total time, +2ΔT, is bifurcated into +ΔT and –ΔT between t = 1 and 0.
Displacement relates to time as +ΔTc = +lΔQl. At t = 1, creation of an electron and annihilation of an antielectron raises displacement by +lΔQl – (lΔQl) = +2lΔQl. At t = 0, annihilation of an electron and creation of an antielectron lowers displacement by lΔQl – (+lΔQl) = 2lΔQl conserving displacement. Total displacement, +2lΔQl, bifurcates into +lΔQl and lΔQl between t = 1 and 0.
The electron and antielectron each have a fundamental measure of displacement, which is the Compton wavelength or +λ_{C} = +h/m_{e}c, having an associated momentum term, +m_{e}c, unrelated to any external momentum vector. The two Compton regions, +λ_{C} and λ_{C}, corresponds to the electron and antielectron, respectfully, with total displacement +λ_{C} – (+λ_{C}) = +2λ_{C}. At t = 1, creation of an electron and annihilation of an antielectron raises spatial displacement by +λ_{C} – (λ_{C}) = +2λ_{C} and at t = 0, annihilation of an electron and creation of an antielectron lowers displacement by λ_{C} – (+λ_{C}) = 2λ_{C} conserving total displacement. A net external energymomentum input would separate the electron and antielectron to a distance greater than +2λ_{C} to where they are resolved and become onshell.
The ground state time, +ΔT = +h/ΔE_{e} = +h/m_{e}c^{2}, is related to the displacement given by ΔT = +2λ_{C}/c = +h/m_{e}c^{2}. Total positive time of dipole existence is +ΔT – (ΔT) = +2ΔT or +h/Δm_{e}c^{2} – (h/Δm_{e}c^{2}) = +2h/Δm_{e}c^{2} = +2λ_{C}/c. Thus, the total distance travelled at the speed of light in time 2ΔT is the total displacement of the dipole 2λ_{C}. The electron is associated with a Compton wavelength. Thus, the rest energy of the electron, E_{RELATIVISTIC} = m_{e}c^{2}, includes the momentum term, m_{e}c, in the Compton wavelength.
In order to create a positive (negative) energy electron the electromagnetic ground state wavepacket must carry positive (negative) energy at least equal to the rest energy of the positive (negative) electron. The energy of the electromagnetic ground states in the vacuum are not confined to the energy of the electron but encompasses an entire spectrum of energies near zero minimum to Planck energy. The positive or negative energy of the ground state, +ΔE, relates to time of its existence as +ΔT = h/(+ΔE) since the electromagnetic ground state wave packet has a positive and negative energy channel the associated total time is +ΔT_{TOT} = +2ΔT. The total displacement is +2ΔTc = +2lΔQl or +ΔT_{TOT}c = + lΔQl_{TOT}. If the electromagnetic ground state carries energy +m_{e}c^{2}, then total associated time, +T_{TOT} = +2h/ m_{e}c^{2} and the total displacement equals +T_{TOT}c = +2h/m_{e}c = +2λ_{C}, equal to the transient ground state dipole.
2) Inelastic and elastic scattering supported by vacuum field processes.
A) Inelastic interaction involving ground states.
During a positive energy charged particle inelastic interaction, an onshell electromagnetic wavepacket is emitted with probability α. The necessary condition satisfying inelastic interaction is having a positive energy distribution +ΔE equaling its corresponding threemomentum energy distribution as +ΔE = +lΔPlc giving the onshell condition ΔE^{2} – lΔPl^{2}c^{2} = q^{2} = 0, which has nonrest mass propagation. The inelastic process of emission and absorption satisfies conservation of energymomentum without net transfer of vacuum to real energy. ΔE^{2} – lΔPl^{2}c^{2} = 0 also applies to the corresponding positive energy excited ground state. The electric field between the particles creates a potential in the vacuum, which depends upon their proximity, where with probability α, energy in the amount of ħΔω is transferred from the negative energy ground state to the positive energy ground state producing a positive and negative excited ground states with energy +(3ħ/2)Δω. There are then two onshell wavepackets having energy +ħΔω (n = 1), where the positive energy propagates from the interaction region and the positive state deexcites to the ground state, +(3ħ/2)Δω – (ħ/2)Δω = +(ħ/2)Δω, while a potential from the negative energy excited state absorbs kinetic energy +ħΔω from an outgoing particle deexciting its state to the ground state, (3ħ/2)Δω + ħΔω = (ħ/2)Δω. Proximity of the interacting particles and the interaction energy change is related to ΔT and ΔE by the HU relation.
B) Elastic interaction involving ground states.
If there is no energy emission and absorption from the interaction region, then the interaction is elastic with probability distribution in 1 – α. The energy distribution +ΔE and magnitude of three momentum distribution +lΔPlc is now related as ΔE^{2} – lΔPl^{2}c^{2} = +q^{2} ≠ 0, where there is internal exchange of energy and momentum. The exchange is weighted with 1/q^{2} so that the farther (closer) offshell a ground state exchange is the less (greater) the probability of emission.
Having the electronpositron or electronelectron elastic interaction in the center of mass frame, where the mass of an electron equals the positron and the direction of approach is colinear with equal but opposite velocities, gives for the momentum P_{1} = –P_{2}. The energy and magnitudes of threemomentum for each incoming and outgoing particle are equal or E_{1} = E_{2} = E_{1}’ = E_{2}’ and lP_{1}l = lP_{2}l = lP_{1}’l = lP_{2}’l, conserving those quantities. There is a change in direction between the outgoing and incoming particles or P_{1} ≠ P_{1}’ and P_{2} ≠ P_{2}’. With P_{1} = –P_{2}, P_{1}’ = –P_{2}’, an e e scattering has fourmomentum q_{s} = p_{1} – p_{1}’ = (0, p_{1} – p_{1}’) and for the exchange scattering it is q_{s} = p_{1} – p_{2}’ = (0, p_{1 }– p_{2}’), where q_{s} is spacelike. For e e+ scattering q_{s} = p_{1} – p_{1}’ = (0, p_{1} – p_{1}’) and for the exchange interaction, or e e+ annihilation, it is q_{t} = p_{1 }+ p_{2} = (2E, p_{1} + p_{2}) = (2E, p_{1} –p_{1}) = (2E, 0), where q_{t} is timelike. The momentum change transfer distribution, Δp, and energy transfer distribution ΔE are found in energymomentum ground states and determined by incoming energymomentum and proximity of the particles. The annihilation process is an energy exchange where there is no potential for threemomentum exchange, since Δq_{t} = (2ΔE, 0) and Δp = 0. In e e or e+ e+ nonexchange scattering, only momentum, for example Δq_{s} = (0, Δp), weighted 1/Δp^{2}, is exchanged. There is no potential for an energy distribution exchange since ΔE is zero.
The proximity of the interacting particles for a given input energy determines the angle of scattering and introduces a field energy density, which creates a potential in the vacuum. In the case of e e or e e+ nonexchange scattering, no energy distribution exchange is involved and the scattering momentum change is onedimensional, Δp, which is an intrinsic opposite momentum change between the interacting particles. Momentum components of Δp depend on arbitrary coordinate systems and are not involved since ground state involvement is intrinsic. Momentum –Δp = ħΔk from a negative ground state momentum distribution –Δp = (ħ/2)Δk is transferred to a positive ground state momentum distribution +Δp = +(ħ/2)Δk placing both distributions in a positive and negative excited state +Δp = +(3ħ/2)Δk, where the distributions have a relation lΔQllΔpl = πħ with particle proximity, lΔQl. One incoming particle receives positive momentum change in the amount of +ħΔk while the other gives up momentum +ħΔk to the negative momentum potential receiving an equal but opposite negative momentum change. Both momentum distributions deexcite to ground states. The positive momentum ground state then propagates forward in time away from the interaction site while the negative momentum distribution propagates away backward in time. Since momentum is intrinsic and coordinate independent, there is radial superposition of directions to absorbers. Direction of detection is conveyed back to the interaction site by a backward time energymomentum ground state from the absorber when an outgoing particle is detected reducing the scattering probability wave form there. The particle travels on a timelike path whereas the energymomentum ground state travel on a lightlike path. The potential of the particle interacts with the potential of the backward time ground state in its immediate vicinity altering its propagation path from one backward light cone to a complementary (perpendicular) one and so on along backward light cones following the path of the particle. The process occurs within the confines of the Compton wavelength of the particle and the wavepacket of the energymomentum distribution so trajectories are directly untraceable. The process for forward time ground states is alternating paths on forward light cones. The particle undergoes a jittering motion as a result.
Potentials associated with greater field densities from closer proximity affects ground state momentum distributions corresponding to particle reversal. Within the largest momentum distribution are subsets of distributions that obey the analysis above (inelastic scattering is similar). Indistinguishable e e scattering involve two momentum distribution subsets, which are complements, where the sum of the difference between maximum and minimum momentum values in each subset equals the difference in the maximum distribution. As one subset increases the other decreases and when the two subsets pass through equal difference the designation as to which subset belongs to which outgoing particle and scattering angle is indistinguishable reflecting the quantum property. The e e+ nonexchange scattering does not carry quantum indistinguishability and as a result there is only one subset. The probability from small scattering angle to the maximum angle decreases. The relation lΔQllΔpl = πħ or lΔQl = πħ/lΔpl shows that as the momentum spread lΔpl associated with larger momentum increases, the proximity distance lΔQl decreases as 1/lΔpl similar to the amplitude and probability (1/lΔpl)^{2} = 1/lql^{2} and ΔE^{2} – lΔpl^{2}c^{2} determines the sign. For energy exchange, in e e+ annihilation lΔTllΔEl = πħ and ΔT is the time associated with the interaction for either the annihilation or creation of one particle. Annihilation of a positive energy and creation of a negative energy electron gives an energy transfer +ΔE – (ΔE) = 2ΔE. Then, 1/(2ΔE)^{2} = +1/q^{2}. The exchange of momentum by excited ground states during an elastic interaction is described by D and is a time symmetric bound field due to no emission of wavepackets or particles from the interaction site.
The positive energy deexcited ground state travels forward in time along with the onshell wavepacket while the negative energy deexcited ground state propagates backward in time from the interaction site. Direction of detection is relayed by a backward time ground state from the absorber when the onshell wavepacket arrives back to the time of emission reducing the emission probability wave form. Despite having backward time ground states, emissionabsorption events involving onshell wave packets are time ordered. The ground state processes involved with transfer of energy from the outgoing particle to the outgoing onshell wavepacket via excited ground states is time symmetric but the time ordering of emission and absorption coincides with D_{I} since there is onshell emission.
3) Entanglement from vacuum field ground state processes.
The positive and negative ground states evolve temporally and propagate spatially until another bifurcation of charge occurs with probability α. Then there are 137 evolving positive and negative electromagnetic ground states from annihilation of a dipole to creation of the next reflecting the probability α = 1/137. Consider a dipole creationannihilation event. The second event associated with the reference dipole occurs again after 137 time periods and at position 137 times the electromagnetic ground state total displacement in a particular direction, where one time period is the total period of the dipole 2h/m_{e}c^{2} and the total displacement of the ground state is equal to (c)2h/m_{e}c^{2} = 2λ_{C}.
The positive and negative ground states evolve in time and propagate spatially possibly jumping from one set of connected ground states to another until another bifurcation of charge occurs with probability α.The evolutionary nature of electromagnetic and dipole ground states according to the quantum clock, which comprises the vacuum, forms the basis for the entanglement process in the vacuum. Due to positive and negative energy, a potential of binding between connected sets of positive and negative ground states, allows them to propagate and evolve oppositely along a common path. The vacuum is composed of positive and negative tandem connected sets of ground states randomly oriented in space, where two intersecting connected sets do not affect one another. The vacuum is a homogeneous mixture of connected sets with random temporal alignments and spatial directions. Positive and negative time and spatial states in the vacuum carry no absolute time or position and so only intrinsic ordering with arbitrary time and position is relevant in the vacuum. There is symmetry between a ground state and nonground state forward and backward time and spatial processes.
As stated in postulate 2 on page 3, the vacuum processes support quantum entanglement of real particles and electromagnetic wavepackets. Entanglement of two electromagnetic wavepackets occurs during rapid emissions due to a cascade process of two decaying energy levels in an atom or spontaneous down conversion in a nonlinear transparent crystal to name two. Real particles such as electrons can be entangled in solid state processes currently being researched for quantum computing. Support for entanglement utilizes backward and forward time components of ground states, which satisfy locality so there is no need for a spacelike connection. Polarization reduction and detection for a given orientation of the polarizer occurs at a precise time for the onshell wavepacket, positive and a set of negative ground states with various frequency intervals. The polarization reduction of the forwardbackward time ground states is possible by requiring the positive energy onshell wavepacket state to have a potential. Without an external potential, the vacuum is homogeneous with states randomly jumping to other connected set of states and the potentials associated with ground states have no tendency to sustain an increase in density. The onshell wavepacket operating on the ground states introduces a nonhomogeneity with a cylindrical grouping of all ground states without random jumping along a common segment on the forwardbackward light cones defined as a bundle. The onshell wavepacket is absorbed after measurement whereas the ground states are not. The positive energy ground state propagates onward into the future with a reduced polarization. The reduced state of the forward time ground state is not unreduced by another ground state with a polarization superposition since they contain one polarization state which matches. The matching polarization state then has predominate influence over the unmatched states. If the reduced positive ground state incidents with a reduced ground state with a different or matching single orientation, then both states are unaffected and in the case of a match the states are indistinguishable. The reduced set of negative ground states propagate from time of measurement to emission. Backward time ground states from the absorber are in the vicinity of the backward time displaced onshell wavepacket and ground state at all instants from measurement and detection to emission. The negative and positive ground states along with the onshell wavepacket reduce their mutual polarizations. The backward ground states then propagate onward from the emission site into the past and affect the polarization of other ground states in the same way. The onshell wavepacket anywhere on its path will have the corresponding reduced polarization outcome at the time of measurement regardless of when the polarizer setting is changed. Emissionabsorption have a pastfuture relation at any relative location and distance where one reduced negative state arrives at the interaction site during emission, while those reduced earlier or later will not arrive at emission time. If there is no absorption and measurement, then there is no backward time reduced state and propagation probability reduction. The reduced set of negative ground states propagate from time of measurement to emission. Backward time ground states from the absorber are in the vicinity of the backward time displaced onshell wavepacket and ground state at all instants from measurement and detection to emission. The negative and positive ground states along with the onshell wavepacket reduce their mutual polarizations. The backward ground states then propagate onward from the emission site into the past and affect the polarization of other ground states in the same way. The onshell wavepacket anywhere on its path will have the corresponding reduced polarization outcome at the time of measurement regardless of when the polarizer setting is changed. Emissionabsorption have a pastfuture relation at any relative location and distance where one reduced negative state arrives at the interaction site during emission, while those reduced earlier or later will not arrive at emission time. If there is no absorption and measurement, then there is no backward time reduced state and propagation probability reduction. Nonbundled ground states on different light cones crossing the bundle perpendicularly are reduced by a mutual tipping of light cones when in the vicinity of the nonhomogeneous potential allowing components of each other polarization states to interact in the manner described above. When two bundles cross, polarization interaction occurs in the same way. When polarizations match there is indistinguishability between the two bundles and emissionabsorption sites provided the onshell wavepackets are similar. Positive and negative energy ground states travel along lightlike world lines in spacetime on forward and backward light cones. The particle having nonzero rest mass travels on timelike world lines with slopes contained within the light cones. As opposed to the inelastic interaction, there is nonzero rest mass propagation. To unify the process with the inelastic process having cohesiveness and uniqueness with a nonzero rest mass particle and backwardforward time ground states, the particle must have a potential. The particle at a position backward in time by a small increment, t_{1}, from the absorber is not yet measured and its superposition state is not reduced. The particle at –t_{1} is not on the backward light cone of the absorber but a backward time ground state reduced at the absorber is in the vicinity of the particle at –t_{1}. The particle at t_{1} is not on the forward light cone of its position further back in time by an additional small increment at –t_{2}. A nonreduced forward time ground state is then in the vicinity of the particle at –t_{1}. The forward and backward light cones are on nonintersecting paths with the particle at –t_{1}. In three dimensional space at one time, the vacuum is distorted or nonhomogeneous due to its potential with associated tipping of the forwardbackward light cones inward around the particle. Ground states on complementary (right angle) forwardbackward light cones crossing the nonintersecting light cones in the vicinity of the particle potential also have their light cones tipped. The complementary light cones passing through the particles are not tipped. In all cases due to tipping of light cones there is a component of polarization of the nonintersecting light cones in the complementary forwardbackward light cones. Reduction of a superposed state is described by the mechanism developed in the inelastic case. The reduced backward time ground state from measurement at a particular time reduces the backward time superposed ground state on the complementary backward light cone which then reduces the particle at –t_{1}. The positive nonreduced ground state from the particle position at –t_{2} passes a nonreduced state to a nonreduced positive ground state on the complementary forward light cone which moves to the vicinity of the particle at –t_{1} and becomes reduced by the backward time ground state. All ground states on their forwardbackward light cones involved in the reduction process at –t_{1} propagate onward away from the vicinity of the reduced particle and its potential and affect other ground states described above in the inelastic case. The particle and positive ground state reduction at all times in the past from the absorber are in tandem with the reduced backward time ground state on the nonintersecting and complementary light cones path mechanism from absorber to interaction site. Due to the external potential of the particle and its associated integrity of the backwardforward time of ground states, the particle anywhere on its path has the corresponding reduced polarization outcome precisely at the time of measurement regardless of when the polarizer setting are changed. (Figure 3). Higher positive and lower negative energy states with a relatively shorter time periods and a smaller wavepacket respond to a greater degree and are closer to the particle than smaller magnitude distributions with longer periods and large wavepackets. Due to the potential of the particle there is a change of the quantum time distribution from a homogeneous condition in empty vacuum to inhomogeneous condition in the vicinity of the particle or wavepacket. The relation of the vacuum wavepackets appear to be one where they are redshifted farther away and blueshifted when closer to the particle potential.
As discussed earlier, the quantum entanglement connection was demonstrated to be a nonlocal spacelike connection. An instantaneous connection was not ruled out and became the view taken. Because of this, the question of causality comes into view where a cause must always precede all of its effects in any frame of reference ruling out spacelike transmission of macroscopic, microscopic communication, correlated quantum influence, and closed time loops. In terms of quantum correlation, there is no spacelike transmission of information in the sense of radio, television, computer code, etc. Quantum correlation is not a form of communication since the measurements at each detector are random and in order to compare and find correlation the results of detections must be carried by nulllike or timelike means that obey causality. Due to the nonlocal quantum connection causality is violated. This is the form of strong causality. Weak causality is similar in that cause precedes an effect in all reference frames but it supports microscopic backward time processes.
In the quantum entanglement connection developed below, the vacuum contains electromagnetic ground states, which have components travelling forward and backward in time on forward and backward light cones due to the presence of positive and negative frequency components in the equation for the vector potential. The forwardbackward time components, which are not directly traceable, where the forward time ground states link the emitter of both correlated quanta to the detectors and the backward time ground states link the measurements of the detectors back to the emitter when the quanta were emitted to each the other through the source The forward and backward time connections from source to detector are then nulllike and there is no field which supports spacelike transmission. The nonlocal and instantaneous nature of the entanglement connection can be shown mathematically as the resultant vector resulting from the addition of forward and backward time vectors. Thus, causality is upheld since there is physically no spacelike transmission of quantum results by any field. The transmission of quantum results are through nulllike channels.
4) Derivation of vacuum energy density and cosmological constant.
A) Mechanics of charges in the Compton wavelength, charge separation and transient dipole as a dipole antenna. Average emitted energy.
Position and motion on various trajectories of constituent particles comprising a transient dipole is unknown and unmeasurable as well as the relative configuration of their electromagnetic fields. As a result, electromagnetic emissions cannot be determined. It is also unknown as to the dynamics of dipole creation and annihilation concomitant with their Compton wavelengths. The only facts are, there is creation and annihilation of charge separation associated with the dipole having a fundamental total time of existence, 2h/ΔE_{e}, and, also, associated with the dipole, a total Compton wavelength, 2λ_{C}/c, composed of fundamental constants, h, m_{e}, c, within which there is a constant momentum term, m_{e}c, assigned to each charged constituent particle implying constant velocity generating no electromagnetic emission.
Constant momentum is independent of transient charge separation. Charge separation also occurs in a dipole antenna with emission of electromagnetic radiation. Assume that the creation and annihilation of separated charge in a transient dipole is represented by separation and recombination of oppositely charged currents in a dipole antenna having dimensions proportional to the total Compton wavelength of the dipole. Then, the transient dipole emits electromagnetic radiation.
According to the StücklebergFeynman interpretation a negative energy backward time antielectron with negative charge can be represented as a positive energy forward time antielectron with positive charge. Then, all negative sign operations are changed to positive sign operations. Dipole charge separation is then the separation and recombination of negative and positive charge, which is used for comparison to the forward time observed behavior of a dipole antenna.
Separation and recombination of opposite charge in the transient dipole is assumed to be equivalent to onehalf cycle of a sinusoidal alternating current in the antenna, which emits an electromagnetic wave (the same is true if backward time antielectrons interpretation is used, however). Parameters of the dipole antenna are determined only by fundamental relations used in the transient dipole, which are total Compton wavelength with its constant momentum term, total time determined from the fundamental ground state energy and frequency relation, along with the vacuum impedance.
Average radiative power and energy from the dipole, relation to the fine structure constant.
The average radiative power of an infinitesimal dipole antenna is given by:
P_{RAD} = η_{0}(π/3)(I_{0}L/λ)^{2}, where η_{0} is the electromagnetic wave impedance of the vacuum, I_{0} is electric current, L is the length of the dipole antenna, and λ is the emitted wave length.
P_{RAD} = (1/2)I_{0}^{2}R_{RAD} and so R_{RAD} = 2P_{RAD}/I_{0}^{2} = η_{0}(2π/3)(L/λ)^{2} where R_{RAD} is the total antenna resistance. The value of η_{0} is 120π = 1/ε_{0}c = 377 ohm, where ε_{0} is the permittivity of the vacuum and c is the speed of light. Using 120π, R_{RAD} =120π(2π/3)(L/λ)^{2} = 80π^{2}(L/λ)^{2}. So P_{RAD} = I_{0}^{2}(40π^{2})L^{2}/λ^{2}, where 40π^{2} = 394 ohm.
The antenna when compared to the transient dipole contains no physical wire implying no resistance to electric current. The resistance associated with the antenna is then due to the impedance of the vacuum to the electromagnetic wave and the ratio of the length of the dipole antenna to the wavelength emitted, which is (377ohm)(L/λ)^{2}. Then, 40π^{2} is replaced by 1/cε_{0}. Using the vacuum impedance of 1/ε_{0}c, P_{RAD} = I_{0}^{2}/ε_{0}c(L/λ)^{2} and with (1/2π)(2π/λ) = (1/2π)k = (1/2π)ω/c, the relations for the average power of the infinitesimal antenna is given as:
P_{RAD} = I_{0}^{2}/ε_{0}c(L/λ)^{2} or equivalently as [I_{0}^{2}L^{2}ω^{2}]/[4π^{2}ε_{0}c^{3}].
The current I_{0} is defined as the (number of positive or negative charges per volume)X(number of charges passing through an area)X(the speed of the charge, v)X(the unit of charge, q). The speed of the electric charge is determined by the momentum term, m_{e}c, in the Compton wavelength relation, λ_{C} = h/m_{e}c, where m_{e} is the rest mass of the electron and is set equal to an equivalent relation involving the relativistic mass of the electron multiplied by a speed, v, which is less than the speed of light.
Then m_{e}c = m_{e}γv and c = v/(1 – v^{2}/c^{2})^{1/2} implying c^{2} = v^{2}c^{2}/(c^{2} – v^{2}) or 1 = v^{2}/(c^{2} – v^{2}), which gives for the speed of the electric charge v = c/(2)^{1/2}. There is one electric charge per volume of λ_{C}^{3} or 1/λ_{C}^{3}. The charge passes through an area of λ_{C}^{2}, and the unit of charge is q = 1.6X10^{19}C. Then, the electric current I_{0} is given as (1/λ_{C}^{3})(λ_{C}^{2})(c/(2)^{1/2})(q) = (c/(2)^{1/2})(q)/λ_{C} = [(3X10^{8}m/s)/(2)^{1/2}](1.6X10^{19}C)/(2.42X10^{12}m) = 14.0 Ampere.
The dipole antenna length, L, equals the total dipole displacement of 2λ_{C} = 4.84X10^{12}m and the angular frequency is ω = 2πf. The frequency, f, equals ½(2T)^{1} = 1/4T, where 2T = 2h/m_{e}c^{2} is the total time associated with the lifetime of the dipole. The lifetime of the dipole is onehalf of a cycle of the dipole antenna. A full cycle of the antenna is then 4T = 4h/m_{e}c^{2} with angular frequency ω = 2πf = 2π/4T = π/2T, equaling π/2(8.1X10^{21}s) = 1.94X10^{20}/s. The emitted wavelength, λ = c/f = 4cT = 4(3X10^{8}m/s)(8.1X10^{21}s) = 9.72X10^{12}m. L = 2λ_{C} = 4.84X10^{12}m. Thus, L/λ ≈ ½.
The quantities, I_{0}, ε_{0}, L, c, ω, λ, in the relation for the average power of the infinitesimal antenna are:
I_{0} = 14.0 Amp ε_{0} = 8.85X10^{12}C^{2}/Nm^{2} L = 4.84X10^{12}m c = 3.0X10^{8}m/s ω = 1.94X10^{20}/s λ = 9.72X10^{12}m
Then the average power of electromagnetic wave emission using [I_{0}^{2}L^{2}ω^{2}]/[4π^{2}ε_{0}c^{3}] is given by:
P_{RAD} = [(14.0Amp)^{2}(4.84X10^{12}m)^{2}(1.94X10^{20}/s)^{2}]/[4π^{2}(8.85X10^{12}C^{2}/Nm^{2})( 3.0X10^{8}m/s)^{3}] = 1.83X10^{4}J/s.
P_{RAD} = 1.83X10^{4}J/s.
Average radiated energy E_{RAD} = P_{RAD}(2T) = (1.83X10^{4}J/s)(1.62X10^{20}s) = 2.96X10^{16}J.
1) E_{RAD} = 2.96X10^{16}J.
The frequency of the dipole antenna is given by f = 1/4T = ¼(8.1X10^{21}s)^{1} = 3.09X10^{19}/s. Let the equivalent energy of the prevalent associated quanta be E = hf = (6.626X10^{34}Js)(3.09X10^{19}/s) = 2.05X10^{14}J. The emitted energy quanta has 69.3 times the average emitted energy from the dipole antenna. The product of a probability of emission and energy value in an ensemble gives the average energy. If the value of the prevalent probability is consistently 1/69.3 = 0.0144 (≈2α = 0.0146, α is the fine structure constant) along with the energy of the quanta of 2.05X10^{14}J, then the average energy is 0.0144(2.05X10^{14}J) = 2.96X10^{16}J. Equivalently, the dipole antenna can be described as emitting a quanta having energy 2.96X10^{16}J with probability one. This particular average energy occurs for single dipole antenna emissions since the presence of many other dipoles with electromagnetic wave emissions interfere with random phases in the vacuum reducing the electromagnetic energy or quantum mechanical probability of finding dipole antenna average energy quanta as discussed next.
As a curious aside, the equivalent energy of a quanta having the average radiative energy of the dipole antenna, which is E_{RAD} = 2.96X10^{16}J, gives a frequency of E_{RAD}/h = 4.47X10^{17}/s using the energy quanta relation E = hf. Time of existence of the transient dipole is 2T = 1.62X10^{20}s. The product of the frequency of quanta associated with the average energy emitted by the dipole antenna to the time of existence of the transient dipole, (f_{AVERAD})(2T) = (4.47X10^{17}/s)(1.62X10^{20}s) = 7.24X10^{3}. The value of the dimensionless fine structure constant α ≈ 1/137 = 7.29X10^{3}. The percentage error is 0.69%. The fine structure constant may have relation to the total time of dipole existence and its equivalent dipole antenna average energy.
B) Energy or probability reduction of emitted electromagnetic wave.
The classical dipole antenna emits electromagnetic waves, where the amplitude squared is related to energy and its radiative pattern. In quantum mechanics, the electromagnetic wave amplitude squared is not energy, which depends upon frequency, but the probability of finding energy quanta at any particular location. The vacuum contains a large number of electromagnetic waves or probability amplitudes emitted from dipole antennas that superimpose with random phases everywhere in the vacuum. The result of the superposition is destructive interference with reduction of electromagnetic wave or probability amplitude. The reduction of the electromagnetic amplitude reduces the probability of finding emitted energy quanta at the dipole antenna and in intervening vacuum regions.
The wavelength of the emitted electromagnetic wave is 4λ_{C}. The destructive interference has an interval of uncertainty associated with the positive and negative electrical or probability amplitudes and transition point assumed to be given by the total spatial displacement of a dipole or 2λ_{C}. The maximum and minimum uncertainty of the resultant electrical amplitudes are assumed proportional to the two dimensionless spatial displacement components +λ_{C}/m derived from the total spatial displacement of the dipole given as 2λ_{C} = +λ_{C} – (λ_{C}), where m is the unit meter. The reduced amplitudes, +E_{AMPRED}, are given by +E_{AMPRED} = E_{0}(+λ_{C}/m), where E_{0} is the undiminished amplitude of emission. Energy or probability reduction is proportional to the amplitude squared or E_{AMPRED}^{2}. The energy or probability reduction P_{RED} is defined P_{RED} = E_{AMPRED}^{2}/E_{0}^{2} = [E_{0}^{2}(+λ_{C}/m)^{2}]/E_{0}^{2} = (λ_{C}/m)^{2} = 5.86X10^{24}. Superimposed waves having equal amplitudes emitted and traveling undiminished from dipole antennas result in the probability reduction P_{RED}.
Emitted electromagnetic waves from a dipole antenna spread over an increasing spherical area due to increasing radius with positive time reducing energy or probability of finding an energy quanta per constant cross sectional area covering the surface. P_{RED} is then further reduced by a factor F_{RED}, which is a fraction, dependent upon spherical surface area and the dipole density reduction factor, α. The cross section of a central dipole antenna, proportional to the total uncertainty, (2λ_{C})^{2}, is linearly linked through the emitted expanding wave to all cross sections, (2λ_{C})^{2}, of electromagnetic ground states and dipoles in the neighboring threedimensional vacuum, which are projected onto a particular spherical surface in the neighboring vacuum reflecting the average reduction of wave energy or probability amplitude to each cross section on the surface.
Dipole antennas are created and annihilated and recreated after 137 space and time steps, 2λ_{C} and 2T, of the connected set of electromagnetic ground states. Then, each dipole created, emits an electromagnetic wave that propagates out 137 spatial steps in 137 time steps with a reduction in amplitude before a subsequent emission. At any particular time, there are neighboring dipoles anywhere from 1 to 137 spatial steps of the electromagnetic ground states from any dipole antenna undergoing similar process, governed by α ≈ 1/137, which defines the extent of the local neighborhood to any central dipole.
In the local neighborhood, dipoles and electromagnetic ground states have distances of separation from (2λ_{C}) to 137(2λ_{C}) from a central dipole. The average radial distance of separation from an emitting dipole antenna to its local neighboring dipole antennas and intervening vacuum is (2λ_{C})(1 + 137)/2 = 69(2λ_{C}) defining the spherical surface area onto which the electromagnetic ground states and dipoles in the neighboring threedimensional vacuum are projected reflecting the location for the average reduction of wave energy or probability amplitude to each cross sectional area in the neighboring volume.
The reduction of P_{RED} is equivalent to having a smaller uncertainty interval determining amplitude. The least reduction in energy or probability amplitude occurs in the local neighborhood. Then, P_{RED} is reduced by the largest fractional factor and the probability amplitudes associated with larger diminished amplitudes emitted from greater distances are reduced by a smaller fraction. Thus, the associated uncertainty is within the dominating uncertainty of the local neighborhood and nonmeasurable. Then, F_{RED}, of the local neighborhood along with dipoles with their local neighborhoods establish the probability reduction factor, F_{RED}, throughout the vacuum.
F_{RED}, due to the distance of the spherical surface of projection in the local neighborhood from a central dipole antenna and 1/α is defined as the ratio of the central dipole cross section, to the number of cross sections covering the spherical area having the average radius 69(2λ_{C}) or 2λ_{C}^{2}/4π69(2λ_{C}^{2}) = 1/4π(69)^{2}.
Due to the dipole antenna emission pattern, the total spherical area of coverage is approximately ½. Then F_{RED} = 1/4π69^{2}/2 = 2/4π69^{2} = 1/2π(69)^{2} = 3.34X10^{5}.
The total energy or probability reduction, P_{TOT}, due to the fundamental uncertainty of 2λ_{C} and to the link distance is given as, P_{TOT} = (P_{RED})(F_{RED}) = (λ_{C}/m)^{2}/2π(69)^{2} = (2.42X10^{12})^{2}(3.34X10^{5}) = 1.96X10^{28}.
2) P_{RED} = 1.96X10^{28}.
The vacuum density of dipole antennas or #/m^{3} is equal to [(2λ_{C})^{3}]^{1}/137 = [137(2λ_{C})^{3}]^{1}, where 2λ_{C} is the total displacement of a dipole. The density of dipole antennas, given as #/m^{3} = [137(2(2.42X10^{12}m))^{3}]^{1} = 6.44X10^{31}/m^{3} (see appendix for details).
3) #/m^{3} = 6.44X10^{31}/m^{3}.
C) The vacuum energy density depends only upon the charge of a constituent particle in the transient dipole and is independent of its mass.
Average emitted energy from a dipole antenna is invariant to mass.
Assume that the probability of creation and annihilation of an electron and antielectron pair is one and that the relative probability of formation of other dipoles, where the constituent particles have mass nm_{e} is 1/n. Both electrical charges are equal.
Then, λ_{n} = λ_{e}/n, T_{n } = T_{e}/n, ω_{n} = nω_{e}, and I_{n} = nI_{e}
Energy, E_{n} = [I_{n}^{2}L_{n}^{2}ω_{n}^{2}]/[4πε_{0}c^{3}]2T_{n} = [(n^{2}I_{e}^{2})(L_{e}^{2}/n^{2})(n^{2}ω_{e}^{2})]/[4πε_{0}c^{3}]2T_{e}/n = nE_{e}. Then, E_{n} = nE_{e}.
The relative probability of the dipole energy emission with particle mass nm_{e} is 1/n. So E_{n}(1/n) = nE_{e}/n = E_{e}. The net result is that the occurrence of greater energy, E_{n} = nE_{e} occurs 1/n as often and its average is equal to the emission due to the electron, n = 1, and thus is invariant to the mass in the dipole antenna.
The density of dipole antenna emitters is invariant to mass.
#/m^{3}_{e} = [(2λ_{e})^{3}137]^{1} and #/m^{3}_{n} = [(2λ_{n})^{3}137]^{1} = [(2λ_{e}/n)^{3}137]^{1}.
Having a relative probability of 1/n of occurrence for particles of mass nm_{e}, implies that there are n times of occurrences with probability 1/n, before the appearance of the particle with probability one. Then 2λ_{n} becomes 2nλ_{n} and #/m^{3} = [(2nλ_{n})^{3}137]^{1} = [(2nλ_{e}/n)^{3}137]^{1} = [(2λ_{e})^{3}137]^{1} = #/m^{3}_{e} and the density of dipole antennas is invariant to mass of the constituent particles in the dipole. The size of the dipole having constituent particles with mass nm_{e} is (2λ_{e}/n)^{3} and leads to greater density of dipoles but the probability decreases the density in such a way that it takes n constituent particles to equal 2λ_{e} and density equality.
Energy or probability reduction is effectively invariant to mass.
E_{RED} depends upon the uncertainty determined by 2λ_{e} multiplied by the reduction factor F_{RED} that is independent of dipole mass, that is, 2λ_{e}^{2}/4π69(2λ_{e}^{2}) = 2λ_{n}^{2}/4π69(2λ_{n}^{2}) = 1/4π69^{2}. Having a mass of nm_{e} would imply that the uncertainty is determined by 2λ_{n} = 2λ_{e}/n, where 2λ_{n} < 2λ_{e}/n. Then, the uncertainty 2λ_{n} is contained within the uncertainty 2λ_{e}, which cannot be meaningfully measured and would be irrelevant. E_{RED} is then effectively invariant to mass of the constituent particles in the dipole.
D) Calculation of Cosmological Constant.
Then, the product, (E_{n})(E_{RED n})(#/m^{3}_{n}) = (E_{e})(E_{RED e})(#/m^{3}_{e}), the vacuum energy density, is independent of the dipole mass dependent only on the charge determining the energy emitted from the dipole antenna.
As determined from the tables of leptons, mesons, and baryons given by the Particle Data Group, there are 3 types of single charged leptons, 76 types of single charged mesons, 37 types of single charged baryons, and 7 types of double charged baryons. It is assumed that these particles have antiparticles and form transient dipoles. Charge enters into the calculated energy emission from a dipole antenna as charge squared. Then, double charged baryons enter into the calculated energy emission and vacuum energy density as four times the energy emission from a single charge. The effect of 7 double charged baryons contribute 7(2^{2}) = 28 times. Adding the total effectiveness of the 116 types of single charged particles and 7 types of double charged baryons, 28, gives a total of 144.
The total vacuum energy density, ρ_{VAC}, is given by [(I_{0}^{2}L^{2}ω^{2})/4πε_{0}c^{3}][2T][(λ_{C}/m)^{2}/2π(69)^{2}][(2λ_{C})^{3}137]^{1}[144] = (2.96X10^{16}J)(1.96X10^{28})(6.44X10^{31}/m^{3})(144) = 5.37X10^{10}J/m^{3}, where [(I_{0}^{2}L^{2}ω^{2})/4πε_{0}c^{3}][2T] is the average unhindered energy or probability wave emission of the dipole antenna, [(λ_{C}/m)^{2}/2π(69)^{2}] is the reducing factor to the emitted electromagnetic energy or probability of finding an emitted energy quanta at a particular spacetime location, and [(2λ_{C})^{3}137]^{1} is the density of dipole antennas, and 144 is the effect on vacuum energy density due to the charged particle and antiparticle dipoles.
4) Total vacuum energy density is 5.37X10^{10}J/m^{3}.
The cosmological constant Λ = 8πρ_{VAC} = 8π(5.37X10^{10}J/m^{3}) = 1.35X10^{8}J/m^{3}. The best measured value from satellite data is 1.351X10^{8}J/m^{3}, which equals the calculated value to 3 significant figures.
5) Cosmological constant is 1.35X10^{8}J/m^{3}.
The result supports the concept total time and total displacement using the StücklebergFeynman interpretation as well as supporting the number of evolving electromagnetic ground states in spacetime governed by a quantum clock from dipole annihilation to dipole creation determined by the fine structure constant. This in turn supports the structure of the quantum entanglement connection consisting of the evolution of positive energyforward time and negative energybackward time electromagnetic states governed by a quantum clock of the quantum entanglement connection whether or not there is dipole creation and annihilation involved.
E) Appendix with examples.
Given a number of states where one state is a dipole and the others are nondipole states at each position on a three dimensional lattice, the probability of any number of dipoles on the three dimensional lattice from 0 to the number of positions, developed by the author (Bradford) is:
Probability of number of dipoles a or P_{a} = [(S – 1)^{n – a}(n!/a!(n – a)!]/S^{n}, and the mean number of dipoles is n/S.
 n is the number of positions in three dimensional space or number in one dimension cubed.
 a is the number of dipoles.
 (n – a) is the number of nondipole positions.
 S is the total number of states, which is 1 dipole state plus nondipole states S – 1.
Example one is 3 positions in 1D or 3^{3} = 27 positions in 3D and 4 states per position. The probability is given as P_{a} = [3^{27 – a}(1.0889X10^{28}/a!(27 – a)!]/1.8014X10^{16}.
# dipoles a probability P_{a} # dipoles a probability P_{a}
27 5.55X10^{17 } 13 5.326X10^{3}
26 4.4964X10^{15} 12 0.0138
25 1.75359X10^{13} 11 0.0311
24 4.38399X10^{12} 10 0.0605
23 7.8912X10^{11} 9 0.1008
22 1.0890X10^{9} 8 0.143
21 1.1979X10^{8} 7 0.172
20 1.0781X10^{7} 6 0.172
19 8.086X10^{7} 5 0.141
18 5.1209X10^{6} 4 0.091
17 2.765X10^{5} 3 0.0459
16 1.282X10^{4} 2 0.0165
15 5.128X10^{4} 1 3.81X10^{3}
14 1.775X10^{3} 0 4.23X10^{4}
The mean calculated by ∑aP_{a} ≈ 6.7506 and the mean given by n/S where n = 27 and S = 4 is 27/4 = 6.75.
Example two is n = 8 and S = 10 then P_{a} = 9^{8a}8!/a!(8 – a)!/10^{8}
# dipoles a probability P_{a}
8 1X10^{8}
7 7.2X10^{7}
6 2.27X10^{5}
5 4.082X10^{4}
4 4.593X10^{3}
3 0.033
2 0.149
1 0.383
0 0.43
Mean = ∑aP_{a} ≈ 0.8006 Mean n/S = 8/10 = 0.8
The mean n/S is used in the case where there are dipole as well as electromagnetic ground nondipole states. The effect is a lowering of the density of dipoles according to the number S – 1 of nondipole states plus 1 dipole state, where the total is S – 1 + 1 = S when a condition of all positions n in 3D space at one time are filled with dipoles by definition of space filling dipole density.
Dipole as well as nondipole states evolve in spacetime according to the quantum clock. Consider the evolution where after annihilation of a dipole there is a spacetime succession of S – 1 nondipole states until on the S^{th} succession a dipole is created. Given this, what is the reduced density of dipoles if all positions n in 3dimensional space at one time were occupied with dipoles? The solution is n/S.
As an example let there be n = 27 (3^{3}) positions in 3D space with volume, 1 m^{3}, at one time. Let there be S = 9 states. Then, after annihilation of each dipole, there will be a spacetime evolution of 8 nondipole states and creation of a dipole on the 9^{th} step. If there are no nondipole states, then after each time increment of the quantum clock, there would be dipoles at all 27 positions in 3 space. Introducing nondipole states into the evolution will lower the density of dipoles at any one time. The density with nondipole states added is (n/m^{3})/S = (27/m^{3})/9 = 3 dipoles/m^{3}.
The problem at hand, is calculating the density of dipoles when the probability of dipole formation and annihilation is the fine structure constant, α ≈ 1/137. It was assumed that after annihilation of a dipole there is a spacetime evolution according to the quantum clock of 136 nondipole electromagnetic ground states until a dipole state is created on the 137^{th} evolution step, S = 137, to achieve a probability of 1 for dipole creation. The number of positions, n, in 3 space at any one time is 137^{3}. According to the general formula, the probability of a dipoles in 3 space is P_{a} = 136Exp(137^{3} –a)137^{3}!/a!(137^{3} – a)!/137Exp(137^{3}), which is a formidable equation to calculate and find the mean value. The number of dipoles without nondipole states is 137^{3}. The mean number of dipoles with 136 nondipole states and 1 dipole state, S = 137, is n/S = 137^{3}/137 = 137^{2}. In term of dipoles with total spatial displacement 2λ_{C} the number of dipoles present per m^{3} assuming that there are no nondipole states is [(2λ_{C})^{3}]^{1} = 8.82X10^{33}dipoles/m^{3}. Having 136 nondipole states plus 1 dipole state for a total of 137 states reduces the density of dipoles to [(2λ_{C})^{3}]^{1}/137 = 6.44X10^{31}dipoles/m^{3}.
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Amazingly succinct article, I think. It reminds me about all the mathematics I took in college and thought I’d never see again. Now, if I could only interpret what I see. Perhaps a bit more studying…