Doebner-Goldin Equation for Electrodynamic Particle. The Implied Applications
aa r X i v : . [ phy s i c s . g e n - ph ] F e b Doebner-Goldin Equation for ElectrodynamicParticle. The Implied Applications ∗ With Appendix: Dirac Equation for Electrodynamic Particles ∗∗ J.X. Zheng-Johansson † , ‡ † . Institute of Fundamental Physics Research, 611 93 Nyk¨oping, Sweden. ‡ . In affiliation with the Swedish Institute of Space Physics, Kiruna, SwedenSeptember 30, 2007; updated January 28-29, 2008 Abstract.
We set up the Maxwell’s equations and subsequently the classical waveequations for the electromagnetic waves which together with the generating source,a traveling oscillatory charge of zero rest mass, comprise a particle traveling in theforce field of an usual conservative potential and an additional frictional force f ; thesefurther lead to a classical wave equation for the total wave of the particle. At thede Broglie wavelength scale and in the classic-velocity limit, the equation decomposesinto a component equation describing the particle kinetic motion, which for f = 0identifies with the usual linear Schr¨odinger equation as we showed previously. The f -dependent probability density presents generally an observable diffusion current ofa real diffusion constant; this and the particle’s usual quantum diffusion current as awhole are under adiabatic condition conserved and obey the Fokker-Planck equation.The corresponding extra, f -dependent term in the Hamiltonian operator identifieswith that obtained by H.-D. Doebner and G.A. Goldin. The friction produces to theparticle’s wave amplitude a damping that can describe well the effect due to a radiation(de)polarization field, which is always by-produced by the particle’s oscillatory chargein a (nonpolar) dielectric medium; such a friction and the resulting observable diffusionas intrinsically accompanying the particle motion was strikingly conjectured in theDoebner and Goldin original discussion. The radiation depolarization field in adielectric vacuum has two separate significances: it participates to exert on anotherparticle an attractive, depolarization radiation force which resembles in overall respectsNewton’s universal gravity as we showed earlier, and it exerts on the particle itselfan attractive, self depolarization radiation force whose time rate gives directly thefrictional force f . ∗ , ∗∗ Submitted to journal for publication. ∗∗ Included temporarily as Appendix I, pp. 20–36; to be submitted as a separate paperin near future.
1. Introduction
While the usual linear Schr¨odinger equation has demonstrated to be adequate forthe common nonrelativistic quantum systems, L. de Broglie suggested [1] in the 50s–60s that the quantum mechanical wave equation may be more generally nonlinear.Various forms of nonlinear equations have been proposed and investigated subsequently, oebner-Goldin Equation for Electrodynamic Particle & Applications/ JXZJ a simple, single-charged particle is constituted of an oscillatory point charge q of a zero rest mass andthe resulting electromagnetic waves propagating at the speed of light c . In so far as theway the mechanism of the model operates, q can be of arbitrary quantity; q is to begiven as an input (out of two sole input data, the other is the total energy of the charge)to yield the actual material particles. [For examples, of the elementary particles, for theisolatable charged one where the multiple- or ”neutral-” charged particles are viewed asachieved by integration processes n ← p + e + ν e and N ← p + n , clearly then | q | = e ;for the, as of today, nonisolatable charged ones, quarks, | q | = 1 / , / oebner-Goldin Equation for Electrodynamic Particle & Applications/ JXZJ oebner-Goldin Equation for Electrodynamic Particle & Applications/ JXZJ
2. Particle model with electrodynamical internal processes
We consider an IED model particle is traveling at a velocity υ as its oscillatory charge q does, for simplicity in a one-dimensional box along X -axis. The oscillation of the chargeis associated with a total energy ε q , ε q being smallest at υ = 0, ε q | υ =0 = E q . E q or ε q may be endowed e.g. in a pair production in the vacuum. E q describes the groundstate and therefore cannot be dissipated or detached from the charge except in a pairannihilation.The charge will owing to its oscillation generate electromagnetic waves, with theradiation electric field E j and magnetic field B j (of the j th component) governed by theMaxwell’s equations given in a medium of dielectric constant κ in zero external fields as ∇∇∇ · E j = ρ jq ǫ , ∇∇∇ · B j = 0 , ∇∇∇ × B j = µ j jq + 1 c ∂ E j ∂t , ∇∇∇ × E j = − ∂ B j ∂t . (41)Where ρ jq is the density and j jq the current of the charge q of the particle, assuming noother charges and currents present; ǫ = κǫ and µ ≃ µ , with ǫ and µ the permittivityand permeability of the vacuum, and c the velocity of light in the medium. (Untilthe specific application oriented denotations in Sec. 6, unless explicitly specified therespective media for conveying the particle and for the reference of measurement willnot be explicitly specified in writing the electromagnetic variables.) Considering onlyregions sufficiently away from the source charge so that ρ jq = j jq = 0, making someotherwise standard algebra of the equations (41), and replacing the field variables by amore general dimensionless displacement ϕ j , with E j = Aϕ j , B j = E j /c = Aϕ j /c and A a conversion constant, we obtain the corresponding classical wave equation for eachcomponent electromagnetic wave ϕ j : ∂ ϕ j ∂T = c ∇ ϕ j (42)Here, in view of a Doppler effect to result from the source motion (to express explicitlylater), we distinguish by the superscript j the component wave generated in the directionparallel with the source velocity υ , denoted by j = † , and the one in the directionantiparallel with υ , denoted by j = ‡ ; within walls in stationary state there mustalso simultaneously prevail their reflected components and we may regard these as ifbeing generated by a virtual charge (compared to the presently actual charge) which isreflected and traveling in − X -direction, denoting by j = vir † and j = vir ‡ . Apparently,the total wave given by the sum of all of the component waves, P j ϕ j = Y ∅ , describesthe particle.Based on the general results of electrodynamics applied here to the particle’sinternal processes as governed by the basic equations (41)–(42), basic properties ofa given particle can be predicted; in the remainder of this section we outline twodirectly relevant ones of these. The first is the total energy of the wave and accordinglythe particle. As a general result of classical electrodynamics based on solution to theMaxwell’s equations, the (41) here, combined with Lorentz force law, an electromagnetic oebner-Goldin Equation for Electrodynamic Particle & Applications/ JXZJ j ) transmits at the speed of light c a wave energy ε j and a linear momentum p j = ε j /c . In virtue of the stochastic nature of the electromagnetic waves, itstotal dynamical quantities, the total wave energy and linear momentum here, areappropriately the geometric means, ε = √ ε † ε ‡ , p = p p † p ‡ ; √ ε † ε ‡ = p p † p ‡ c or ε = pc. (49)From the underlining laws afore-used, mathematically the amplitudes of E j , B j , ϕ j , etc,and accordingly ε j , p j , E and p are permitted to take on continuous values.Following M. Planck’s discovery of quantum theory in 1901, it has been additionallyunderstood that the amplitudes of these quantities are in nature quantized; the totalwave energy of an electromagnetic wave of frequency ω/ π is ε = n ~ ω , that is, ε consists in general of n momentum-space quanta, or photons, each of an energy ~ ω .The electromagnetic wave comprising our basic particle, like an electron, positron, etc.,has, based on experimental indications especially the pair processes, a ”single energyquantum”, n = 1; the Planck energy equation for the total wave of the particle thereforeis ε = ~ ω. (4)This total wave of a single energy quantum here has in a one-dimensional box twocomponents, ϕ † and ϕ ‡ , a situation no different from discussed after (42). Their wavefrequencies are Doppler displaced to ω † and ω ‡ as a result of the source motion (to expressexplicitly below); and similarly as (4), ω † = ε † / ~ , ω ‡ = ε ‡ / ~ . Further from (49) we have ω = √ ω † ω ‡ . For the total wave comprising the particle, ε represents therefore the totalenergy of the particle. It has been proven and formally expressed especially throughquantum electrodynamics that, the Maxwell’s equations and naturally the derivativeclassical wave equation continue to hold; and the quantization of the fields and the waveenergy etc. formally is the result of subjecting the corresponding canonical displacementand momentum, corresponding to the u (= aϕ ) and ˙ u here, to the quantum commutationrelation [ u, ˙ u ] = i ~ . In this generalized framework, clearly the classical solution of acontinuous amplitude for say ε merely is an approximation when n is large.The second property is the inertial mass of the wave and thus the particle. Thetwo components of the electromagnetic wave, ( E j , B j ) or ϕ j , rapidly oscillating at ageometric mean frequency ω/ π and wavelength λ = c/ ( ω/ π ), viewed at some distanceand ignoring the detail of the oscillations will appear as if being two rigid objects,wavetrains, traveling at the speed of light c ; the two trains of the component wavestogether make a total wavetrain. In view that its speed of travel, c , is finite as contrastedto infinite, the total wavetrain has inevitably a finite inertia mass, denoting this by m .This mechanical representation of the total wave, as a rigid ”wavetrain”, permits us atonce to express according to Newtonian mechanics the linear momentum of the wavetrainto be p = mc . Combining this with the classical electrodynamics result ε +0 = pc of (49)gives the kinetic energy of the wavetrain ε = mc ; this is just the Einstein’s mass-energy oebner-Goldin Equation for Electrodynamic Particle & Applications/ JXZJ m = ~ ω/c . ( A. m of the total wavetrain comprising the particle naturally represents the massof the particle, here acquired dynamically through the total motion of the waves of ageometric mean frequency ω/ π . m is dependent on the particle velocity and is thusrelativistic, see further after equation (14) below.
3. Wave equation of total motion of particle in external fields
To the particle we now apply a Coulomb force F = −∇ V owing to a conservative scalarpotential V , and in addition a viscous force f ; these give a total applied force F ′ = F + f .We express the f as follows. Suppose out of the total oscillation of the charge, afractional displacement u q only produces radiation and is defined here to be equal tothe wave displacement u = a Y , Y being the dimensionless total wave displacementin the field of applied potentials; this in zero applied potential field is the Y ∅ earlier.Making direct analogy to the viscous force of ordinary mechanics, we can write downthe frictional force opposing the total motion of the particle as f = P n b n ( a Y ) n ( d ( a Y ) dT ) n inunits of N. That is, in general f is a function of the time rate of the total displacementof the charge or alternatively of the resulting wave displacement in the medium; section7 will give a concrete representation of such a force. Assuming d ( a Y ) /dT is small, so toa good approximation f = b Y L ϕ d Y dT = b Y L ϕ ( ∂ Y ∂T + ∇ Y ∂X∂T ), where b is a constant in unitsof Nms and is real; f is in units of N and is generally imaginary (pointed out by D.Schuch) for Y being generally complex. This may rewrite as f = − mD Y L ϕ ( ∂ Y ∂T + ∂ Y ∂X W )where D = − b m , (6) W = − iβ ~ υ obs mD − W ∗ = iβ ~ ( ∇ Y ∗ ) Y m | Y | . (7) W ( ≡ ∂X∂T = ∂ω/∂k ) is the wave speed of Y , and W ∗ (= − ∂X/∂T = ∂ω ′ /∂k ′ ) ofthat of the imaginary Y ∗ . The expressions in (7) follow firstly from the requirementthat W is in direct proportion with the velocity υ obs of the current j obs (= υ obs ρ )of the probability density ρ (= | Y | ) in order to ensure the continuity of current ina non-absorbing medium. That is, W = Iυ obs − W ∗ , where the imaginary W ∗ issubtracted from the generally complex Iυ obs . The current j obs = υ obs ρ of ρ with auniform translation at velocity υ alternatively is according to Fick’s law the diffusionof a varying ρ in a viscous medium, j obs = − D ∇ ρ , with D the diffusion constant. So, υ obs = j obs ρ = − D ∇ ρρ = − Dρ [( ∇ Y ∗ ) Y + Y ∗ ∇ Y ]; here we have taken D to be as defined in oebner-Goldin Equation for Electrodynamic Particle & Applications/ JXZJ j obs in (29). The aboveleads explicitly to the first and second expressions in (7) once we put the proportionalityconstant as I = − iβ ~ mD , where β is a parameter yet to be determined [by the equation(29) below], and the other constants are inserted so that β will have the simple solutionvalue 1.In virtue of the electrodynamic origin of F and inevitably also f which areempirically established for point particles, extending to the extensive IED particle herethe two applied force and thus their total F ′ act apparently directly on the point charge.We now want to map this F ′ into a force directly interacting with the E j , B j or ϕ j . Wenotice that by its mathematical form equation (42) represents just a classical waveequation for the electromagnetic wave ϕ j , aϕ j therefore a mechanical wave propagatedin an elastic medium, and F ′ interacts with aϕ j by an effective force F ′ med acting ondirectly on this apparent medium. On the basis of this direct correspondence, buttaken as a heuristic means only in this paper (so that we need not to firstly introducewith sufficient justifications at any detail the structure of this elastic medium), weshall below map the force F ′ into F ′ med . Now, while F ′ drives the charge, of a mass m of the particle, into an acceleration ∂ Y /∂T , F ′ med drives the medium of mass M ϕ (effectively) into acceleration ∂ Y med /∂T in X -direction. Supposing the charge and themedium oscillate at a fixed phase difference if not in phase, the two accelerations mustbe equal, we thus have F ′ med = M ϕ m ( F + f ) = − ρ l (cid:20) Vm + 2 D Y (cid:18) ∂ Y ∂T + W ∇ Y (cid:19)(cid:21) . (8)Where M ϕ = L ϕ ρ l , with ρ l the linear mass density of the medium along the wave pathof a total effective length L ϕ = J L ( Y winds in J loops about the box side L ).We below further implement the force F ′ med in wave equation (42) similarly usingthe heuristic approach by applying directly Newton’s laws to the apparent elasticmedium. In the apparent medium Y corresponds to a physical, transverse displacement u = a Y = aC P j ϕ j as produced by the disturbance of the charge oscillation, with a aconversion constant of length dimension. The deformed elastic medium is consequentlysubject to a tensile force F R = ρ l c , with c the velocity of light at which ϕ j propagates.This force F R and the applied force F ′ med together give the total force acting on theparticle through directly acting on the apparent elastic medium F ′ R = F R − F ′ med = ρ l (cid:20) c + Vm + 2 D Y ∂ Y ∂T + β Di ~ m | Y | |∇ Y | i , (9)where the minus sign in front of F ′ med represents that this force tends to contract thechain. Consider on the linear chain of the medium a segment ∆ L at ( X, X + ∆ X ) isupon deformation tilted from its equilibrium ∆ X an angle ϑ ( X ) and ϑ + ∆ ϑ ( X + ∆ X );assuming Y is small, F R will be uniform across the entire wave path L ϕ . Thetransverse ( Z -) component force acting on it is ∆ F ′ R t = F ′ R [sin( ϑ + ∆ ϑ ) − sin ϑ ], with[sin( ϑ + ∆ ϑ ) − sin ϑ ] = [1 + O ( ϑ )]∆ ϑ ≃ ∆ ϑ = ∇ ( a Y )∆ R ; O ( ϑ ) collects the higher oebner-Goldin Equation for Electrodynamic Particle & Applications/ JXZJ F ′ R we have∆ F ′ R t = F ′ R ∇ ( a Y )∆ R = aρ l (cid:20) c + Vm + 2 D Y ∂ Y ∂T + β Di ~ mρ |∇ Y | (cid:21) ∇ Y ∆ R. (10)Applying Newton’s second law to the segment of a mass ∆ M ϕ = ρ l ∆ L , ≃ ρ l ∆ X , wehave ρ l ∆ X ∂ ( a Y ) ∂T = ∆ F ′ R t . Placing (10) in it, dividing aρ l ∆ X , we get the equation ofmotion for per unit length per unit mass density of the elastic chain of medium at X ,or equivalently the classical wave equation for the total (electromagnetic) wave of theparticle: ∂ Y ∂T = (cid:20) c + Vm + 2 D Y ∂ Y ∂T + β Di ~ mρ |∇ Y | (cid:21) ∇ Y . (11)In summary, (11) has a basic part ∂ Y ∅ ∂T = c ∇ Y ∅ which one will get from summing overall j values the wave equations (42) given earlier directly from the Maxwell’s equations inzero applied potential, and it has an additional part describing the effect of the appliedforce F ′ med , derived with the help of the ”heuristic elastic medium” approach.Concerning the solution of (11), for the present we only consider explicitly the caseof D = 0. So (11) reduces to ∂ Y ∅ ∂T = (cid:2) c + Vm (cid:3) ∇ Y ∅ ; this being linear, thus Y ∅ = P j ϕ j and ∂ ϕ j ∂T = (cid:20) c + Vm (cid:21) ∇ ϕ j (12)Supposing also V is a constant, V c , equation (12) can be immediately solved to consistof plane waves, ϕ † = C exp[ i ( k † X − ω † T + α )], ϕ ‡ = − C exp[ i ( k ‡ X + ω ‡ T − α )]. Where k j = γ j K are the Doppler-displaced wavevectors for the wave generated parallel withthe source velocity υ ( j = † ) and antiparallel with υ ( j = ‡ ), and ω j = γ j Ω are thecorresponding angular frequencies, with γ † = 1 / (1 − υ/c ), γ ‡ = 1 / (1 + υ/c ). K and Ω = Kc are the values of k j and ω j at υ = 0, c being the velocity of light as before.The explicit superposition of the incident waves ϕ † , ϕ ‡ and their reflected ones ϕ vir † , ϕ vir ‡ give a standing wave (for a systematic representation see [3a-c,e]): Y ∅ = X j ϕ j = Ce i [( K + k d ) X ] e − iωT , (13)where k d = p ( k † − K )( K − k ‡ ) = γK d , K d = (cid:16) υc (cid:17) K ; ω = √ ω † ω ‡ = γΩ ; (14) γ = √ γ † γ ‡ = 1 / p − υ /c . Canceling ω between (14) and (A.1) gives further m = γM ,with M = ~ Ω/c the classic-velocity limit ( υ /c →
0) of m , i.e. the rest mass of theparticle. An explicit inspection of (13) will readily show that k d is the de Brogliewavevector (for an existing elucidation see e.g. in [3c,i]), K d being its value at the limit oebner-Goldin Equation for Electrodynamic Particle & Applications/ JXZJ υ /c →
0. We shall later (see after equation 23) generalize the representation to thecase where V may be arbitrarily varying in L ; until then we shall proceed the followingdiscussion for the constant V , V c . Substitution of Y ∅ in the total wave equation givenfrom the linear sum of the wave equations (12) over all j , i.e. with D = 0 in (11),directly gives the expected relativistic energy-momentum relation for the particle[3b],which gives an additional check that Y ∅ of (13) is the correct solution to the total waveequation.For the solution of wave equation (11) with D finite we shall use the trial function: Y = ZY ∅ , where Z = e iQ , Q = Q + iQ . (15) Z represents a damping factor; Q and Q are real variables and are in general functionsof X, T . We shall restrict ourselves to the case where D is small and accordingly | iQ | << | i ( K + k d ) X − iωT | . Under such a condition, for the derivation of a nonlinearSchr¨odinger equation in question below, until the context of equation (33), an explicitsolution form of Q needs not be known.
4. Transformation to wave equation for kinetic motion of particle
At the classic-velocity limit υ /c →
0, the total wave function Y ∅ reduces to (see [3a-c])lim υ /c → Y ∅ = Ce i ( K d X − Ω − d T ) with Ω − d (= Ω ( υc ) ) = K d υ + V c , which is equivalentto the solution for Schr¨odinger equation for an identical system as described by thewave equation (11) in the case of D = 0 and V = V c . Therefore, as we noted in [3a-c],equation (11) must inevitably have a direct correspondence to the Schr¨odinger equation,and the remaining question mainly then was to identify a physically justifiable procedureto transform (11) to a form of the Schr¨odinger equation. Such a formal procedure waselaborated in detail in [3a-c] by a back-substitution of the explicit function Y ∅ in waveequation (11) in the case of D = 0; by use of the Fourier theorem the procedure furtherled to a Schr¨odinger equation for V arbitrarily varying and also, by a straightforwardextension, for three dimensions. For the present case of D being in general finite, webelow similarly first reduce and simplify wave equation (11) at the classic-velocity limit υ /c → K - and K d - processes are separable, by means ofback-substitution of the formal function Y (15) where the function Y ∅ is explicitly knownand Q assumed small.We first prepare for the separation of the K - and K d -processes in three aspects, thefirst two being similar as for the case D = 0 [3a,b]: (i) We observe that (11) containsthe derivative ∂ Y ∂T which relates to the acceleration of the particle and, as such, the K -and K d - processes are not separable; but the two processes are separable for the firstderivative ∂ Y ∂T which relates to the total energy (for a detailed analysis see [3b]). Thissuggests us to lower the time derivative one order as ∂ Y ∂T ≃ ∂∂T [( ∂ Y ∅ ∂T ) e iQ + Y ∅ e iQ i ∂Q∂T ] ≃ ∂∂T [ − iω Y + 0] = − iω ∂ Y ∂T . (ii) In the two terms Vm ∇ Y and β Di ~ mρ |∇ Y | ∇ Y in (11), thecoefficients in front of ∇ Y , being approximately the scale of quadratic thermal velocity υ or lesser, are relatively small for V and D being small; and also these are constant. So oebner-Goldin Equation for Electrodynamic Particle & Applications/ JXZJ υ /c → ∇ Y can to goodapproximation be replaced by its computed value: ∇ Y = ∇ [( ∇ Y ∅ ) e iQ + Y ∅ e iQ i ∇ Q ] ≃ i ( K + k d ) ∇ Y + 0 = − ( K + k d ) Y , where in going to the second last expression wedropped the cross-term products between the mutually orthogonal ∇ e iKX and ∇ e ik d X whose contribution to the final expectation value is in general zero (for an explicit proofsee [3a,b]). Using the identity relation γ = 1+ γ υ c , the above rewrites ∇ Y = − γ K Y .(iii) In the two terms D Y ∂ Y ∂T ∇ Y and c ∇ Y in (11), the coefficients ( ∂ Y /∂T ) / Y ∝ − iω and c are large, with ω being the scale of the particle’s total energy. So, in these the ∇ Y ought to be kept in functional form. But in the first of the two terms the large ∂ Y /∂T itself effectively will be unaffected by the small V and D , and can therefore be replacedby its computed value − iω Y (used the small Q assumption), thus D Y ∂ Y ∂T = − i Dω .Substituting with the reduced forms of (i)–(iii) for the respective ∂ Y ∂T , ∇ Y , and D Y ∂ Y ∂T in (11), simplifying using the basic relation K γ c = ω and the relation mc = ~ ω given in (A.1), multiplying the resulting equation by − ~ ω , (11) finally reduces to i ~ ∂ Y ∂T = − ~ m ∇ Y + V c Y + i D ~ ∇ Y + iβ D ~ |∇ Y | | Y | Y . (16)We next proceed to separate in wave equation (16) the K - and the K d - processes,which are inexplicitly contained in a factor γ in each term as we will see explicitly below,and based on this we further reduce the equation at the classic-velocity limit. To thisend, with Y formally given in (15), we first compute each derivative in (16) explicitly,and expand the γ factor ( γ = 1 + υ c + υ c + . . . ) in each: ∂ Y ∂T = − iγΩ Y = − i [ Ω + Ω − d (1 + 34 υ c + . . . )] Y , where Ω − d = 12 Ω d , Ω d = (cid:16) υc (cid:17) Ω, m ∇ Y = − γ K Y γM = − γK Y M = [ − K M − K d M (1 + 34 υ c + . . . )] Y , ∇ Y = i ( K + γK d ) Y , |∇ Y | = ( ∇ Y ∗ )( ∇ Y ) = ( K + γK d ) | Y | . (17)On equal footing as the above, Y expands in its exponent as Y = Ce i [( K + γK d ) X − ( Ω + Ω − d (1+ υ c + ... )) T + Q ] . (18)The condition υ /c → K >> K d , Ω >>> Ω d . So, on the scales of K d and Ω d , the harmonic functions e iKX and e − iΩT oscillate so rapidly that they presentto any external observation effectively constants. Hence, e − iΩT ≃ e iKX ≃
1; andlim υ /c → Y = Ce i [ K d X − Ω − d T + Q ] = Z Ψ ∅ ≡ Ψ, Ψ ∅ = Ce i [ K d X − Ω − d T ] . (19)Taking accordingly the classic-velocity limit of the relations of (17), substituting in theresulting relations with (19) for Ψ and its derivatives ( ∇ Ψ = − K d Ψ , ∂Ψ∂T = − i Ω − d Ψ , ∇ Ψ = iK d Ψ , ∇ Ψ ∗ = − iK d Ψ ∗ , |∇ Ψ | = K d | Ψ | for small Q assumption as earlier) for oebner-Goldin Equation for Electrodynamic Particle & Applications/ JXZJ K d -, Ω − d - terms while keeping the K -, Ω -terms as computed values which are largeand will be unaffected for V and D being assumed to be relatively small, we havelim υ /c → ∂ Y ∂T = − iΩΨ + ∂Ψ∂T , lim υ /c → ∇ Y m = − K ΨM + ∇ Ψ M , lim υ /c → ∇ Y = iKΨ + ∇ Ψ, lim υ /c → ∇ Y ∗ = − iKΨ + ∇ Ψ ∗ , lim υ /c → |∇ Y | = K | Ψ | + |∇ Ψ | . (20)We dropped the cross-term products in the last relation of (20) for similar considerationas earlier. Finally, subjecting wave equation (16) to the classic-velocity limit andsubstituting in the resulting equation with the expressions of (20) we have ~ ΩΨ + i ~ ∂Ψ∂T = ~ K M Ψ − ~ M ∇ Ψ + V c Ψ − i D ~ K Ψ + iD ~ ∇ Ψ + iβ D ~ | Ψ | (cid:2) K γ | Ψ | + | Ψ | |∇ Ψ | (cid:3) Ψ. (21)Equation (21) multiplied by Ψ contains a component equation ~ Ω = ~ K M − i D ~ K + iβ D ~ K (22)for a monochromatic electromagnetic wave produced by the given source but at zerovelocity, and is not of our direct interest here. This equation holds always true for agiven particle of a fixed rest mass and can be subtracted from Ψ × (21); multiplying Ψ back to the resulting equation from left, we obtain i ~ ∂Ψ∂T = − ~ M ∇ Ψ + V c Ψ + iD ~ ∇ Ψ + iβ D ~ |∇ Ψ | ρ Ψ. (23)If V varies arbitrarily with X , thus V = V ( X, T ), ϕ † and ϕ ‡ are in general nolonger plane waves. On the other hand, assuming V ( X, T ) is well behaved, we candivide L into a large, N number of small divisions of width ∆ X each. In each smalldivision, ( X j , X j + ∆ X ), the potential, V ( X j , T ) = V cj , continues to be approximatelyconstant and is exactly so in the limit ∆ X = 0, and here the above plane wave methodholds valid. Elsewhere, V ( X j , T ) = 0. Going through therefore the foregoing proceduresimilarly for each division, j , with j = 1 , . . . , N , we obtain equations of identical formsas (19), (23), etc., except with Ψ , K d , Ω − d etc. denoted by Ψ Kdj , K dj , Ω − d Kdj , etc. The { Ψ Kdj ( R , T ) } ’s are mutually orthogonal and form a complete set. So the total wavefunction is the sum Ψ ( X, T ) = 1 √ N X Kdj A Kdj Ψ Kdj ( X, T ) = 1 √ N X K dj A Kdj Ce − iΩ − dj T + iQ · e iK dj R ; (24)or , Ψ = Z Ψ ∅ , Ψ ∅ = ξe − iΩ − d T , Z = e iQ − Q , ξ = 1 √ N X K dj A Kdj Ce iK dj X − i ( Ω − dj − Ω − d ) T , (24) ′ with A Kdj Ce − iΩ − dj T + iQ = 2 π P Ns =1 Ψ ( X s , T ) e − iK dj · X s the Fourier transform of Ψ ( X s , T ). oebner-Goldin Equation for Electrodynamic Particle & Applications/ JXZJ √ N A Kdj through the corresponding equation of (23) for Ψ j , summingthe equations over all j values we have i ~ ∂ √ N P j A Kdj Ψ j ∂T = − ~ M ∇ √ N X j A Kdj Ψ j + X j V cj √ N X j A Kdj Ψ j + iD ~ ∇ √ N X j A Kdj Ψ j + iβ D ~ ( ∇ √ N P j A Kdj Ψ j ) ∗ ( ∇ √ N P j A Kdj Ψ j )( √ N ) P A Kdj ρ j √ N X j A Kdj Ψ j . (25)Where, P j V cj = . . . + 0 · V ( X j − , T ) + 1 · V ( X j , T ) + 0 · V ( X j +1 , T ) + . . . = V ( X j , T ); A ∗ Kdj = A Kdj since the amplitude of the physical displacement Ψ must be real; ρ = P j P j A Kdj ρ j = P j A ∗ Kdj Ψ ∗ j P j ′ A Kdj ′ Ψ j ′ for Ψ ∗ j and Ψ j ′ mutually orthogonal and A ∗ Kdj real; and P j P j |∇ A Kdj Ψ j | = ∇ P j ( A Kdj Ψ j ) ∗ ∇ P j A Kdj Ψ j for the two factors mutuallyorthogonal.Substituting (24) in (25) we obtain a generalized result of (23), a wave equationdescribing the kinetic motion of the particle in an arbitrarily varying, well-behavedpotential V : i ~ ∂Ψ∂T = − ~ M ∇ Ψ + V Ψ + iD ~ ∇ Ψ + iβ D ~ |∇ Ψ | ρ Ψ. (26)Equation (26) is seen to represent an ordinary Schr¨odinger equation except for the extra,nonlinear term iD ~ ∇ Ψ + iβ D ~ |∇ Ψ | ρ Ψ due directly to the frictional force f .
5. Diffusion currents. Continuity equation. Doebner-Goldin Equation
Making some standard algebra to equation (26) and its complex counterpart leads toan equation for the total current j tot = j qm + j obs of the probability density ρ = | Ψ | : ∂ρ∂T + ∇ ( j qm + j obs ) + ( β − D |∇ Ψ | ρ | Ψ | = 0 . (27)Where j qm = ~ M i [( ∇ Ψ ∗ ) Ψ − Ψ ∗ ∇ Ψ ] , j obs = − D ∇ ρ = b M [( ∇ Ψ ∗ ) Ψ + Ψ ∗ ∇ Ψ ] (28)with j qm the usual quantum diffusion current and j obs the observable diffusion currentas earlier except now expressed in terms of the classic-velocity limit function Ψ . Thefirst quantity, j qm , has an imaginary diffusion constant D qm = i ~ M and this we know isto an external observer non-observable.Suppose there are no ”sinks” in the medium nor external reservoir in contactto the medium that trap or conduct the total current j tot . So the particle and the(continuous) medium as a whole is adiabatic—a condition having an equal footing withthe ”unitary representation of vector field (of diffeomorphisms group)” employed in [2]. oebner-Goldin Equation for Electrodynamic Particle & Applications/ JXZJ Ψ governed by wave equation (26), needs to conform to the continuity equationwhich, for D being real and j obs being observable, is of the Fokker-Planck type: ∂ρ∂T + ∇ ( j qm + j obs ) = 0 . (29)Comparison of this with (29) suggests the third term in (27) must vanish; so β = 1.With the β value in (7) we find: W + W ∗ = − i ~ υ obs mD , W = + i ~ mρ ( ∇ Ψ ∗ ) Ψ , and W ∗ = + i ~ mρ ( ∇ Ψ ) Ψ ∗ . With the β value in turn directly in wave equation (26), wefinally have i ~ ∂Ψ∂t = − ~ M ∇ Ψ + V Ψ + iD ~ ∇ Ψ + iD ~ ( |∇ Ψ | | Ψ | ) Ψ, (30)or i ~ ∂Ψ∂t = H ′ Ψ, H ′ = H + iD ~ G, H = − ~ M ∇ + V, G = ∇ Ψ + ( |∇ Ψ | | Ψ | ) . (30 a )Equation (30) is seen to be exactly the Doebner-Goldin form of nonlinear Schr¨odingerequation, the Doebner-Goldin equation, introduced in [2]. In view of their respectivemeanings, the unitary representation of vector fields in [2] and the probability densityconservation in an adiabatic total system here are apparently two alternative butequivalent conditions. It is thus natural that the use of the latter here has led tothe same result as based on the former in [2].As was well appreciated in [2] (1994), the nonlinear total Hamiltonian H ′ as of(30a) is in general complex and not Hermitian as would be required by the usuallinear Schr¨odinger equation; a complex nonhermitian Hamiltonian is today a topic ofincreasingly many studies. In this regard the foregoing derivation of equation (30) basedon the IED particle model additionally points to that, underlining the complex H ′ andits imaginary part iD ~ G respectively are a complex total force F ′ med = F med + i | f med | and an imaginary frictional force f med , a property drawn the author’s attention by D.Schuch at the SNMP conference, Kiev, 2007. As is suggested by the mathematical form,we may comprehend the imaginary f med as a physical variable orthogonal to the real F med . As such, a measurement of the total force F ′ med would then inform the modulus ofit, | F ′ med | = p F + f , and not the direct addition of two scalar component forcesand also not an ordinary vector sum F ′ med = F med + f med .Concerning the solution for the Doebner-Goldin equation (30) we shall later onlyrefer to an interesting and also relevant case treated by H.-D. Doebner and G.A.Goldin in [2]. Starting with the denotations specified in (24) ′ we have the moregeneral expressions: ρ = | ξ | e − Q , ∂ρ∂T = − | ξ | e − Q ∂Q ∂T , j obs = b M [( ∇ ξ ∗ ) ξ + ξ ∗ ∇ ξ +2( ∇ Q ) | ξ | ] e − Q , j qm = ~ Mi [( ∇ ξ ∗ ) ξ − ξ ∗ ∇ ξ − i ( ∇ Q ) | ξ | ] e − Q . Following[2] we put Q = 0; the foregoing then become: ρ = | ξ | , ∂ρ∂T = ∂ | ξ | ∂T ; (31) j qm = ~ M i [( ∇ ξ ∗ ) ξ − ξ ∗ ∇ ξ − i ( ∇ Q ) | ξ | ] , j obs = b M [( ∇ ξ ∗ ) ξ + ξ ∗ ∇ ξ ] . (32) oebner-Goldin Equation for Electrodynamic Particle & Applications/ JXZJ ∂ρ∂T = 0. (Since for Q = 0, ρ of (31) does not contain D explicitly,so the ”stationary state” here is not a small- D approximation but is exact as long as(30) holds. But we derived (30) based on a small D condition, which agrees with thesmall D requirement in [2].) Then ρ = | ξ | of (31) is independent of time. Combiningthis with (27) follows ∇ ( j qm + j obs ) = 0. Or, j qm = D ∇ ρ + B with B a constant.Substituting in this last equation with (32) for j qm , restricting ∇ [( ∇ ξ ∗ ) ξ − ξ ∗ ( ∇ ξ )] = B to be independent of time which ensures that when D = 0, ξ is a solution to theusual Schr¨odinger equation, and further with the specific choice B = 0, one gets − ~ Mi ( − iρ ∇ Q ) = − D ∇ ρ . Or, ∇ Q = − Γ ∇ ρρ where Γ = mD/ ~ . Integrating gives: Q = − Γ ln | ξ | . Substituting in (24) ′ with this solution for Q and the Q = 0 earlier,one gets: Ψ = Z Ψ ∅ , Ψ ∅ = ξe − iΩ − d T , Z = e − iΓ ln | ξ | . (33)Paper [2] also discussed that the other of the two possible descriptions of the stationarystate to be j qm = 0. This will also find a significant application in the examples later.Different forms of the nonlinear term would imply other boundary conditions thanan adiabatic one, or other applied forces than of the form here. In recent years, interms of group theoretical approach H.-D. Doebner and G.A. Goldin [2] (1994) andA.G. Nikitin and A.G. Nikitin and R.O. Popovych [6] gave classifications of nonlinearSchr¨odinger equations in association with diffeomorphism group representations and ingeneral terms. D. Schuch gave an insightful review [7] on the nonlinear Schr¨odingerequations proposed by different authors with analysis regarding the quantum physicaljustifiability of solutions, and introduced an interesting logarithmic form of nonlinearSchr¨odinger equation. H.-D. Doebner, A. Kopp and R. Zhdanov generalized in [6]nonlinearity to Dirac systems. Corresponding representations of these and beyond basedon the IED particle model in the future can be similarly of value for test of the model andfor gaining insight into the corresponding mechanical nature of nonlinearity of quantumsystems.
6. Damping in dielectric media as generic application of theDoebner-Goldin equation
In most applications the motion of a macroscopic object will in general be dissipatedor more restrictively, damped, to a greater and lesser degree. The dissipation istypically known in the form of heat exchange with the environment and manifestingas an observable diffusion current. But such a description for a single quantum particlesystem, as described by (30) being in stationary state, needs be taken in an effective,average way only. This directly follows from the circumstances that heat reflects ingeneral an energy current composed of many random collisions of a large populationof individual (quantum) particles, during which the particles in general deviate fromstationary state. Apart from its possible applications in the aforesaid effective way, it oebner-Goldin Equation for Electrodynamic Particle & Applications/ JXZJ n and the penetrating vacuum taken here literally to be dielectric, of a totaldielectric constant κ as measured against a true empty space, the space after removal ofthe dielectric vacuum. An explicit knowledge of the structure of the dielectric vacuum ‡ is not needed for the dielectric relations given in this paper. The total medium and theparticle are as a whole evidently adiabatic. Measured against the true empty space, theparticle’s component radiation electric field propagated in the total dielectric medium is E , and would be E ∅ if ”propagated”[8] in the empty space. (In this section we shall forconciseness drop the superscript j , either because this is not directly of concern or thevariables actually may represent the geometric mean quantities.) When measured in theusual way against the vacuum with the vacuum regarded as effectively ”non dielectric”,the field E in the total medium would be E ; E and E represent the same force (notethat the charge involved apparently causes no effect, for an original discussion see [31e, g]) acting on the same medium as measured in the same inertial frame and musttherefore be equal, E ≡ E ; this is irrespective of against which medium the force ismeasured and represented. Supposing for simplicity the material medium is nonpolar,with the vacuum being naturally nonpolar, so the total dielectric medium is nonpolarand the charge produces in it a depolarization field E p . The corresponding dimensionlesswave displacements accordingly are: Y ( ≡ Y ) = E /A , Y ∅ = E ∅ /A , and P = E p /A .Applying the standard dielectric theory for ordinary materials to the generalizeddielectric system of an ordinary material and the vacuum here we can write downthe following relations (for a systematic treatment see [3g,e]): E ( ≡ E = E κ n ) = E ∅ κ , E p = − χE , E = E ∅ + E p , with κ = κ n κ , χ = κ − χ n + 1)( χ + 1) − . (34)Where, χ is the susceptibility of the total dielectric medium and χ that of the puredielectric vacuum each measured against the empty space; κ n is the dielectric constantand χ n the susceptibility of the ordinary material medium n measured in the usual wayagainst a ”non-dielectric” vacuum; ǫ , ( ≡ ǫ ) = κ ǫ ∅ , is the permittivity of vacuum and ǫ ∅ the permittivity of the empty space. Multiplying by 1 /A , taking the classic-velocitylimit as in (18), the dielectric relations for the fields in the above become then Ψ = Ψ ∅ /κ, P = − χΨ, Ψ = Ψ ∅ + P . (35) ‡ There exist today various propositions for the contents and structure of the vacuum as held in differentfields like in QED, QCD, etc., or by individual authors including the ”vacuuonic vacuum structure”proposed by the present author [3g-h,e]; there appears to exist no direct experimental informationregarding the explicit structure of the vacuum. oebner-Goldin Equation for Electrodynamic Particle & Applications/ JXZJ Ψ = Z Ψ ∅ of (19) or more generally (24) ′ , we have Z = 1 /κ. (36)This states that, the damping factor Z corresponds rather generally to the inverse ofthe dielectric constant κ of the medium in which the particle resides. In the case where ρ is independent of time, as specified by the Q = 0 and small D conditions, Ψ and P are described by the specific solutions (33), which combined with (36) gives thecorresponding expressions for the two dielectric parameters κ = 1 / Z = e iΓ ln | ξ | , χ = e iΓ ln | ξ | − . (37)In the specific case when no ordinary material presents, we have a pure dielectricvacuum, thus κ n = 1, χ n = 0; (34) and (35) reduce to κ = κ , χ = χ = κ − Ψ = Ψ ∅ κ = Ψ ∅ + P , P = − χ Ψ ; and (36) and (37) reduce to Z = 1 /κ . Z and κ are for a specified particle here evidently universal constants, given that the vacuum isubiquitous, isotropic and uniform throughout the space to as far as we know all of thetime. As a consequence, the wave function Ψ ∅ appears to have never directly manifesteditself in our present day’s detections which are commonly based on the variation ofthe wave amplitude of a particle as a function of location and time; our only directknowledge of the particle wave appears to be the Ψ ( ≡ Ψ ) of which the Z or κ is aninseparable component. Despite this, we see that first of all there presents a completeagreement between the prediction from the Doebner-Goldin equation, applied to theIED particle, that the electromagnetic waves ”inside” (or comprising) a particle canin general admit damping but without decaying with time in amplitude, and the factthat electromagnetic waves ”outside” (i.e. detached from) a particle, becoming directlyobservable, factually essentially do not decay with time in amplitude in the vacuum.Further, the dielectric vacuum, hence the E jp field of a charge in it and accordinglythe Ψ ∅ wave in the empty space, has an indirect yet profound manifestation according toa recent theoretical prediction [3j,f] by the author with coauthors. Namely, the E p fieldparticipates to produce an attractive depolarization radiation force acted universallybetween two particles 1 , M and M and charges q and q , separated at adistance R . This force is as the result of the Lorentz force in their mutual E p , B ( ≡ B )fields say in the case of a pure vacuum: F ii ′ = q j ∆ T q i ′ E pi B i M i ′ , i, i ′ = 1 ,
2. The geometricmean of the mutual forces is F g = p | < F >< F > | = CM M R , where <> representstime average, | q i | , | q i ′ | = e , C = πχ e /ǫ h ρ l , e is the elementary charge and the otherconstants are as specified earlier; this force F g was elucidated in [3j,f] to resemble in allrespects Newton’s universal gravity. To this application of the E p field, the present studyadds that the E pij field of particle i producing the depolarization radiation force leadsdirectly to a damping in the particle’s Schr¨odinger wave Ψ ∅ i , by a factor Z , and theassociated extra Hamiltonian term is a Doebner-Goldin nonlinear term added to that ofthe usual Schr¨odinger equation. In Sec. 7 we shall explicitly express the force directly oebner-Goldin Equation for Electrodynamic Particle & Applications/ JXZJ j qm = 0 solution mentioned earliermay be a case where the particle wave and thus j qm is shielded, say by a material wall.And on the other side of the wall j qm = 0; here only the Doebner-Goldin observablediffusion current j im prevails. This directly corresponds to the property of the gravitywhich can not be shielded by any materials and on the other side of the wall as here itwill propagate alone.In another specific case when an ordinary dielectric medium presents and werepresent the vacuum in the usual way as non-dielectric which thus effectively playsthe role of an empty space in the dielectric relations, the total dielectric medium thusreduces to the ordinary dielectric material medium alone, thus κ = 1, χ = 0. Therelations of (34)–(35) now reduce to κ = κ n , κ n − χ n , Ψ n = Ψ /κ n , etc., and (36)reduces to 1 /κ n = Z n . Finally, with ρ = | ξ | , (37) reduces to κ n = 1 / Z n = e iΓ ln | ξ | , χ n = e iΓ ln | ξ | −
1. That is, Z n represents now a damping of the wave function Ψ one would measure in a pure vacuum medium, into Ψ n one will measure in the ordinarymaterial medium of a dielectric constant κ n . This is an usual representation of a particlein a material medium; this is directly comparable to the familiar fact that a radiated(detached) electromagnetic wave in an optical material in general experiences a complexdielectric constant and susceptibility.The above two specific situations would in general simultaneously enter in ourmaterial world, where a material particle commonly is more or less surrounded byother material particles or substances and which, in the extreme case when all ordinarymaterial substances are absent, is left to be the dielectric vacuum itself. Besides, amaterial particle is electromagnetic in nature (in the sense that such a particle invariablycontains an electric charge), which is a direct observational fact and is irrespectiveof the specific IED particle model employed in this study (whereas the IED particlemodel only is essential in leading to a formal relationship between the wave function,the electromagnetic field of the charge and the corresponding depolarization field ofthe particle). These two universality features of the material world determine thata depolarization (radiation) field presents intrinsically universally with a particle. Ittherefore follows that a Doebner-Goldin damping, identified here with a depolarizationradiation field, is an intrinsic phenomenon presenting always to a material particle. Sucha prospect that the nonlinear process could be intrinsically universally accompanied withthe particle process as elucidated in the above two examples was strikingly conjecturedin the Doebner-Goldin original paper[2].
7. Self depolarization radiation force: Gravity from the dielectric medium
We now give a concrete expression for the frictional force due to a total dielectricmedium, of dielectric constant κ , against a particle moving in it. In the medium theparticle’s component radiation fields are E j , B j (= E j /c ), j = † for the fields propagatedin the direction parallel with υ and j = ‡ for fields antiparallel with υ similarly as earlier.These fields are the results after damping in the medium, from the un-damped E j ∅ , B j ∅ oebner-Goldin Equation for Electrodynamic Particle & Applications/ JXZJ E jp = − ( E j ∅ − E j ) = − χE j and a corresponding B jp = − ( B j ∅ − B j ) = − χ m B j due to the presence of the medium,where χ m = √ κ −
1, and c = c ∅ / √ κ (the permeability is assumed to be 1 here). Thecharge, due to the E jp -, B jp - fields induced by itself, is acted by a magnetic force accordingto the Lorentz force law: F jm.p = q v jp × B jp = (sign) χχ m q E j M c ˆ X, (38)where v p = qE p /M ; sign= + for j = † and = − for j = ‡ . Equation (38) expresses that,irrespective of the sign of the charge and of the momentary directions of the alternatingfields generated by the charge to its right ( E † , ± B † with a velocity c ) and to its left( E ‡ , ∓ B ‡ with a velocity − c ), F jm.p is always a pull to the charge on either side fromthe medium. In this connection, F m.p refers to a self depolarization radiation force onthe particle; and this represents a gravitational force on the particle from the dielectricmedium.While the particle is in stationary state, its (oscillatory) charge is constantlytraveling in an alternating + X - and − X - directions. The charge when traveling inthe − X - direction is similarly acted by a pull on either side, but now the ”sign” = +for j = ‡ (wave generated opposite to the charge motion direction) and ”sign” = − for j = † . So, on average the charge is acted from either side by a pulling force, anattraction, given by the geometric mean of the two Doppler-displaced forces: F m.p = q F † m.p F ‡ m.p = (sign) χχ m q E M c ˆ X, (39)where E = √ E † E ‡ . This mapped to the medium is similarly a pull to the deformedsegment in question by its surrounding in the medium, corresponding to a reduceddisplacement of the medium, this is opposite to the tensile force associated in generalwith the usual E j field.We are here mainly interested in the resistance produced by F m.p against thetotal motion of the particle, that is the time rate of F m.p as measured over a certaintime interval ∆ T : R ∆ T ( dF m.p /dT ) dT ≃ ∆ T dF m.p /dT . With this, the frictional forcefollowing the usual definition is: f m.p = ∆ T Y dF m.p dT = ∆ T ( E/A ) ( ∂F m.p ∂T + ∂F m.p ∂X ∂X∂T )= ∆ T χχ m q A M c ( ∂ Y ∂T + ∂ Y ∂X W ) (40)where E = A Y and W = ∂X/∂T as earlier. Similarly as F m.p , f m.p is always opposite indirection to the tensile force F R . We see that indeed, in both its acting as a resistanceagainst the particle total motion and in its functional form, f m.p of (40) resembles fullythe frictional force f expressed formally in (8) earlier. oebner-Goldin Equation for Electrodynamic Particle & Applications/ JXZJ Acknowledgements
The author would like to thank scientist P.-I. Johansson for his continued moral andfunding support of the research, and the Swedish Research Council for granting a TravelGrant that enables the author to travel to the 7th Int. Conf. on Symmetry in NonlinearMathematical Physics at Kiev for presenting this work and the Swedish Institute ofSpace Physics for administrating the grant. The author would like to thank ProfessorD. Schuch for pointing out the frictional force is imaginary, and for the subsequentuseful discussion regarding this and other related aspects from Professor D. Schuch,Professor H.-D. Doebner, Professor Bender, and Professor Nikitin. The author wouldlike to thank Professor H.-D. Doebner, Professor J. Goldin and Professor Dobrev forvaluable discussion and suggestions for a more informative and adaptable introductionto the particle model, and thank several distinguished Professors in Sweden for valuablereading of this and the related papers. [1] L. de Broglie, ”On the casual and non-linear interpretation of wave mechanics”, Comptes RendusHebdomadires des Seances de l’Academie des Sciences , 441-444 (1953);
Nonlinear WaveMechanics (Elsevier, Amsterdam, 1960).[2] H.-D. Doebner and G.A. Goldin, ”On a general nonlinear Schr¨odinger equation admitting diffusioncurrents,” Phys. Lett.
A 162 , 397-401 (1992); ”Properties of nonlinear Schroedinger equationassociated with diffeomophism group representation,” J. Phys.
A 27 , 1771-80 (1994).[3] J. X. Zheng-Johansson and P-I. Johansson, (a): ”Inference of Schr¨odinger equation from classicalmechanics solution,”
Quantum Theory and Symmetries IV , ed. V.K. Dobrev, Heron Press, , 2006, pp.763-770; arxiv:phyiscs/0411134v5; (b) ”Motivation for Schr¨odinger equation froman electrodynamic internal process”, submitted for publication; (c): ”Developing de Brogliewave,” ibid. , , 32-35 (2006); (d): ”Mass and mass–energy equation from classical-mechanicssolution,” Phys. Essay , , nr. 4 (2006); arxiv:phyiscs/0501037; (e): Unification of Classical,Quantum and Relativistic Mechanics and of the Four Forces , Foreword by R. Lundin, (NovaSci. Pub. Inc., N.Y., 2nd. print, 2006); (f):
Inference of Basic Laws of Classical, Quantum andRelativistic Mechanics from First-Principles Classical-Mechanics Solutions (Nova Sci. Pub., Inc.,NY, 2006); (g): J. X. Zheng-Johansson, ”Dielectric theory of vacuum”, arxiv:physics/0612096;(h): J. X. Zheng-Johansson, ”Vacuum structure and potential”, arxiv:physics/0704.0131; (i): J.X. Zheng-Johansson, ”Derivation of Dirac Equation for Electrodynamic Particles”, presentationat the 5th Int. Symp. Quantum Theory and Symmetries, (Valladolid, 2007); submitted forpublication (2007); (j): ”Depolarization Radiation Force in a Dielectric Medium. Its Analogywith Gravity,” ed. V.K. Dobrev, Heron Press, , 2006, pp.763-770, pp. 771-779, with R. Lundin;arxiv:phyiscs/0411245v3.[4] H.-D. Doebner, Lecture at the 4th International Symposium on Quantum Theory and Symmetries(Varna, 2005).[5] H.-D. Doebner, A. Kopp and R. Zhdanov, ”Extension of Physical Theories: Quantum Mechanicswith Internal Degrees of Freedom and Nonlinear Transformations”, in Symmetry in NonlinearMathematical Physics, Proceedings of Institute of Mathematics of NAS of Ukraine, , part 2,700-7077 (2004).[6] A.G. Nikitin, ”Group classification of systems of non-linear reaction-diffusion equations withgeneral diffusion matrix. I. Generalized Ginzburg-Landau equations”, J. Math Anal. Appl. ,615-628 (2006); with R.O. Popovych, ”Group classification of nonlinear Schr¨odinger equations”,to be published (2007).[7] D. Schuch, ”A logarithmic nonlinear Schr¨odinger equation and similar approaches for dissipativesystems”, in ”Nonlinear deformed and irreversible quantum systems”, ed. H.-D. Doebner, V.K.Dobrev, and P. Natthermann (World Scientific, Singapore, 1995); Refs. therein.8] Mechanical wave can not be propagated in empty space. Here the ” E ∅ ∗ propagated in empty space”should be understood as the field E propagated in the dielectric vacuum but after compensatedfor the damping in amplitude by the dielectric medium. Appendix I
Dirac Equation for Electrodynamic Particles
J.X. Zheng-Johansson
1. Institute of Fundamental Physics Research, 611 93 Nyk¨oping, Sweden; inaffiliation with the Swedish Institute of Space Physics, Kiruna, Sweden
Abstract.
We set up the Maxwell’s equations and subsequently the classical waveequations for the electromagnetic waves which together with their generating source, anoscillatory charge of zero rest mass in general travelling, make up a particle travellingsimilarly as the source at velocity υ in the field of an external scalar and vectorpotentials. The direct solutions in constant external field are Doppler-displaced planewaves propagating at the velocity of light c ; at the de Broglie wavelength scale andexpressed in terms of the dynamically equivalent and appropriate geometric meanwave variables, these render as functions identical to the space-time functions of acorresponding Dirac spinor, and in turn to de Broglie phase waves previously obtainedfrom explicit superposition. For two spin-half particles of a common set of space-time functions constrained with antisymmetric spin functions as follows the Pauliprinciple for same charges and as separately indirectly induced based on experimentfor opposite charges, the complete wave functions are identical to the Dirac spinor.The back-substitution of the so explicitly determined complete wave functions inthe corresponding classical wave equations of the two particles, subjected furtherto reductions appropriate for the stationary-state particle motion and to rotationinvariance when in three dimensions, give a Dirac equation set; the procedure andconclusion are directly extendible to arbitrarily varying potentials by use of the Furioustheorem and to three dimensions by virtue of the characteristics of de Broglie particlemotion. Through the derivation of the Dirac equation, the study hopes to lendinsight into the connections between the Dirac wave functions and the electrodynamiccomponents of simple particles under the government by the well established basic lawsof electrodynamics. irac Equation for Electrodynamic Particles /JXZJ
1. Introduction
P.A.M. Dirac established in [1] a relativistic quantum mechanical wave equation,Dirac equation, for a point electron based on the relativistic energy-momentumrelation subjected to Lorentz transformation under rotation. In [1] P.A.M. Dirac alsotheoretically predicted for the electron the existence of an internal oscillation state,a magnetic moment, and by interpretation of the negative energy solution, an anti-particle state known today as the positron. The Dirac equation has proven to be anaccurate equation of motion for (two) spin-half quantum particles at high velocities;most notably, Dirac predicted based on his equation the relativistic intensities of Zeemancomponents of spectral lines and the frequency differences [1 (1928b)] in exact agreementwith experiment. Up to the present however it has remained an open question thatwhat is waving with the Dirac wave functions, or Dirac spinor, a similar question asfor the Schr¨odinger wave functions and the de Broglie waves ? In addition, the Diractheory meets with a few its own open questions. What is the nature of a Dirac internaloscillation? How are the Dirac space-time functions explicitly connected with the spinorientations, the signs of charges, the signs of the energies, and in the extreme situationwhen an electron and position annihilate, the emitted two gamma rays and conversely?What is the symmetry of the total spin of an electron and positron? Also, in the case ofan isolated single electron or positron in zero external field where the spin orientationis of no consequence, it would be desirable to have a way to directly write down thecorresponding Dirac equation without involving the Pauli matrices. These as well asvarious other not fully addressed questions relating to fundamental physics seem toconsistently point to the inadequacy of the point particle picture of today and the needfor a representation of the internal processes of the particles.Recently, using overall experimental observations as input data the author proposedan internally electrodynamic (IED) particle model [2a] (with coauthor P.-I. Johansson)or sometimes termed a basic particle formation (BPF) scheme, which states that asimple (basic) particle like an electron and positron, etc., briefly, is constituted of anoscillatory point-like (elementary) charge with a specified sign and a zero rest mass,and the resulting electromagnetic waves in the vacuum.
As a broad test of the IEDparticle model and also as an endeavour of understanding the various puzzles relating tofundamental physics, in terms of solutions for the electrodynamic processes of the modelparticle with its charge’s sign and total energy as two sole input data, the author hasfurther achieved with coauthor(s) derivations/predictions of a range of basic propertiesand relations of the simple particles [2a-j] including the relativistic mass, de Broglie wave,de Broglie relations, Schr¨odinger equation, Einstein energy-mass relation, Newton’s lawof gravity and Doebner-Goldin equation, among others. As to the Schr¨odinger wavefunction specifically relevant here, the solution[2a,c] showed that it is the (envelope ofthe) standing wave, superposed from the Doppler-differentiated electromagnetic wavesgenerated by the particle’s travelling source charge, that is waving.As previously shown e.g. in [2c], the direct solutions for the classical wave equations, irac Equation for Electrodynamic Particles /JXZJ
2. Wave equations for the electromagnetic waves of particle. Solutions
We consider an IED particle, here an electron or positron, is as its source charge q (= e or − e ) travelling at a velocity υυυ in + z -direction for the present along a one-dimensionalbox of side L in the vacuum. The charge q of the particle has an oscillation associatedwith a total energy ε q , which is minimum at υ = 0, denoted by E q ; E q may be endowede.g. in a pair production in the vacuum. In virtue that it describes the ground state, E q cannot be dissipated or detached from the charge except in a pair annihilation.The charge q of the particle generates owing to its oscillation electromagneticwaves of radiation electric fields E j ’s and magnetic fields B j ’s described in zero appliedpotential field by the Maxwell’s equations as: ∇∇∇ · E j = ρ jq /ǫ , ∇∇∇ · B j = 0 , ∇∇∇ × B j = µ j jq + (1 /c ) ∂ t E j , ∇∇∇ × E j = − ∂ t B j . (41)Where ρ jq is the density and j jq the current of the particle’s charge, assuming no othercharges and currents present; ǫ is the permittivity and µ the permeability of thevacuum, and c is the velocity of light; ∂ t ≡ ∂/∂t . Expressing the j th fields generallyby a dimensionless displacement ϕ j , E j = Dϕ j , thus B j = E j /c = Dϕ j /c , with D aconversion constant, considering regions sufficiently away from the source only so that ρ jq = j jq = 0, and with some otherwise standard algebra of the Maxwell’s equations (41),we obtain the corresponding classical wave equations for the electromagnetic waves ϕ j ’s c ∇ ϕ j = ∂ t ϕ j , (42)with ∂ t ≡ ∂ /∂t . In the above, j = † labels the component wave generated in thedirection parallel with + υ , and j = ‡ the wave parallel with − υ ; within walls there irac Equation for Electrodynamic Particles /JXZJ − z -direction, labelled by j = vir † and j = vir ‡ . j is to distinguish a Doppler effectowing to the source motion to be expressed in (47) below. In Appendix A we outline inrelevance to the particle model a few further standard relations of classical and quantumelectrodynamics for the electromagnetic waves, and a derivation of the particle’s massgiven by the author previously[2a,e] (with P.-I. Johansson).To the particle we now apply an electromagnetic force F = F e + F m , with F e = − q ∇∇∇ φ a the Coulomb force in z - direction and F m = − q v × ∇∇∇ × A a the Lorentzforce due to an external scalar potential φ a and vector potential A a , expressed in SIunits as for all other quantities in this paper. F m may be simplified using the BAC-CABrule as F m = − q [ ∇∇∇ ( υυυ · A a ) − A a ( υυυ · ∇∇∇ )]. In the applications below A a is constant in L or in each small division in question (see end of Sec. 3), and υ is constant in L for theparticle being in stationary state and also is parallel with ∇∇∇ and z , so ∇∇∇ ( υυυ · A a ) = 0and F m = q A a ( υυυ · ∇∇∇ ) = q A a υ ∇ . Thus, F = − q ∇∇∇ φ a + q A a υ ∇ . The formula of F is inthe usual usage established for a point particle; so when extending to the extensive IEDparticle here, F apparently directly acts on the particle’s point charge.We need to map the F to a force directly interacting with the internal fields E j , B j ,or ϕ j of the particle. We observe that, in virtue of its form, (42) represents just a classicalwave equation for a mechanical wave of a transverse displacement aϕ j propagated in anapparent elastic medium , a being a conversion factor of length dimension and apparentlybeing cancelled in (42). On grounds of this direct correspondence, but taken as aheuristic means only in this paper (so that we here need not involve the details of thiselastic medium), F therefore interacts with the internal fields through a force F j med directly acting on this apparent medium. We can think of the medium to be composedof coupled dipole charges which do not move along the z -axis but the F j med propagatesacross the dipoles at the wave speed c . If viewing in a frame where F j med is at rest, theneffectively the dipole charges are travelling at the speed c ; so as a first step of mapping,the Lorentz force on the medium ought to scale as F j ′ m = ( ± c/υ ) F m = ± q A a c ∇ with+ , − for the j = † , ‡ waves; thus F j ′ = F e + F j ′ m . Under the actions of the respectiveforces, the acceleration F j ′ /m j of the particle’s charge of a dynamical mass m j (dueto the charge’s total motion and equivalently the ϕ j motion, see further AppendixA), and that of the medium of a dynamical mass M jϕ , F j med / M jϕ must equal, i.e. F j ′ /m j = F j med / M jϕ . Thus F j med = M jϕ m j F j ′ = M jϕ qm j L jϕ ( −∇∇∇ φ a ± A a c ∇ ).By its pure mechanical virtue the force F j med acting on the continuous medium isnonlocal and will be transmitted uniformly across the medium here along the z -axisof effective lengths L † ϕ , L ‡ ϕ for the j = † , ‡ waves ( ϕ j winds J j loops about L ). Usingthe geometric mean L ϕ = p L † ϕ L ‡ ϕ , thus ∇∇∇ φ a = ± ( φ a /L ϕ )ˆ z and ∇ = ± /L ϕ . Withthese, putting M jϕ = ρ l L jϕ where ρ l is the (geometric mean) linear mass density of themedium, writing for conciseness ∇∇∇ φ a and also the final F j med in scalar forms and keeping irac Equation for Electrodynamic Particles /JXZJ A a in vector form only, F j med becomes F j med = − ρ l V j /m j , V † = qφ a − q A a c, V ‡ = − qφ a − q A a c. (43) F j med can be implemented in (42) by directly establishing the corresponding waveequation for the apparent elastic medium acted by F j med . If without F j med , the elasticmedium would be deformed owing to the disturbance of the oscillation of the sourcecharge alone, by a total displacement u = a P j ϕ jυ ′ , and be thus subject to a tensileforce F R = ρ l c . The applied F j med and F R add up to a total force acting on the particlethrough acting directly on the medium F j R ′ = F R − F j med = ρ l (cid:2) c + V j /m j (cid:3) . (44)Where, the minus sign of F j med is because this force tends to contract the chain.Assuming ϕ j is relatively small which in general is the case in practical applications, F j R ′ is thus uniform across the L . A segment ∆ L of the medium along the box, ofmass ∆ M ϕ = ∆ M jϕ /J j = ρ l ∆ L ≃ ρ l ∆ z , will upon deformation be tilted from itsequilibrium position z -axis an angle ϑ j and ϑ j + ∆ ϑ j at z and z + ∆ z . The transverse( y -) component force acting on ∆ M ϕ is ∆ F j ′ R t = F j ′ R [sin( ϑ j + ∆ ϑ j ) − sin ϑ j ] = F j ′ R ∇ ( aϕ j )∆ z = aρ l h c + V j m j i ∇ ϕ j ∆ z . Newton’s second law for the mass ∆ M ϕ writes ρ l ∆ z∂ t ( aϕ j ) = ∆ F j ′ R t . The two last equations give the equations of motion,on dividing aρ l ∆ z , for per unit length per unit linear mass density of the medium at z or equivalently the classical wave equations for the electromagnetic waves ϕ j ’s in thefields of the applied potentials φ a , A a : (cid:2) c + q ( φ a − A a c ) /m † (cid:3) ∇ ϕ † = ∂ t ϕ † , [ c − q ( φ a + A a c ) /m ‡ ] ∇ ϕ ‡ = ∂ t ϕ ‡ . (45)This for φ a = A a = 0 reduces to (42) given directly from the Maxwell’s equations earlier.Assuming for the present φ a , A a are constant and also A a is small such that theparticle motion effectively deviates not from the linear path, so the solution of (45)consists of plane waves ϕ † = C f † and ϕ ‡ = C f ‡ (Figure 1a, solid and dotted curves)generated in + z - and − z - directions and initially also travelling in these directions atspeed ω j /k j = c , with f † = Ce i [ k † d z − ω † t + α ] , f ‡ = − Ce i [ − k ‡ d z + ω ‡ t − α ] (46)(Figure 1 a-b, single-dot-dashed and triple-dot-dashed curves), C = e iKz , and C (= 4 C / √ L ) a normalisation constant. Where, k † = K/ (1 − υ/c ) = γ † K, k ‡ = K/ (1 + υ/c ) = γ ‡ K and ω † = γ † Ω, ω ‡ = γ ‡ Ω (47)are the source-motion resultant Doppler-displaced wavevectors and angular frequencies; γ † = − υ/c , γ ‡ = υ/c ; K, Ω = Kc are values of k j , ω j at υ = 0. (47) further gives k † d = k † − K = γ † K d , k ‡ d = K − k ‡ = γ ‡ K d where K d = ( υ/c ) K. (48) irac Equation for Electrodynamic Particles /JXZJ υ << c (yet υ /c may be large so that dynamically the γ factor in (49) belowcan be different from 1) and accordingly the de Broglie wavelength ( λ d = 2 π/ ( γK d ))later will be much greater than the electromagnetic wavelength ( Λ = πK ), so at the scaleof λ d the rapid variation of C is to an external observer no different from the constant1, that is lim υ< 3. Wave equation for total motion of particle For single particle or for many particles without regarding the spins, the functions e ψ and e ψ vir , or equivalently the f r and f ℓ of (55) below, are seen to be identical to the usualsolutions to the Dirac equation, c.f. Appendix C. So their wave equations, originallythe (45), evidently must have a direct correspondence with the Dirac equation. Theremainder of the task mainly will be to identify a physically justifiable procedure totransform (45) to a form of the Dirac equation under corresponding considerations. irac Equation for Electrodynamic Particles /JXZJ ∅ † , W ‡ t=t'+ t /4, a =- p /2 j † , j ‡; f † , f ‡ ; u (a) c f † f ‡ f r f u - W W(b)1/8 t t'= 2/8 t t'= t t'= L= l d z z L= l d Figure 1. (a) shows an IED model electron constituted of an oscillatory charge − e of zero rest mass ( ⊖ ), travelling at velocity υ , and the resulting Doppler-differentiatedelectromagnetic waves ϕ † and ϕ ‡ (solid and dotted curves) of a angular frequencies ω † , ω ‡ generated in + z - and − z - directions, plotted for the real parts in a timeinterval (3 / τ in a one-dimensional box of side L = λ d ; τ = 2 π/ω , ω = √ ω † ω ‡ , λ d = 2 π (( υ/c ) ω ) being the de Broglie wavelength. f † and f ‡ (single-dot-dashed andtriple-dot-dashed curves) are the corresponding external-effective waves, shown in both(a) and (b). f r and f ℓ (solid and dashed curves) in (b) are the dynamically equivalentmean-variable wave functions; these resemble directly the (opposite-travelling) deBroglie phase waves and are equivalent to the space-time functions of Dirac spinor. First, similarly as Dirac (or as alternatively but compatibly argued in [2c]) we wantthe eventual wave functions of the particle, and thus immediately the f j or the original ϕ j to be linear in ~ ω and thus ∂ t f j here, so that f j at any initial time determines itsvalue at any future time; and we want similarly for the linear momentum here. Weshall thus transform the second order differential equations (45) to first order ones andin the end take the limit for c >> υ as follows. For the c ∇ ϕ j terms of (45), startingwith the full wave functions ϕ j = C f j we first lower the spatial derivative one orderas ∇ ϕ j = ∇ [ iγ j K C f j ] = iγ j K [( ∇ C ) f j + C ∇ f j ]. Restricting to υ << c , we thus canreplace ∇ C by its computed value iK C and in turn put C ˙=1 for each term; this gives ∇ ϕ j | C ˙=1 = [ − γ j K C f j + iγ j K C ∇ f j ] C ˙=1 = − γ j K f j + iγ j K ∇ f j , j = † , ‡ . (50)Next, assuming φ a , A a relatively small as typically is true in applications, so the resultingforce constant (i.e. force per unit displacement) on the particle does not vary across L ; we can thus replace the ∇ ϕ j in the φ a , A a terms of (45) by its computed value as irac Equation for Electrodynamic Particles /JXZJ ∇ ϕ j | C =1 = − γ j K f j , thus q ( φ a − A a c ) m † ∇ ϕ † | C =1 = q ( − φ a + A a c ) γ † Ωf † ~ , − q ( φ a + A a c ) m ‡ ∇ ϕ ‡ | C =1 = q ( φ a + A a c ) γ ‡ Ωf † ~ (51)For the final expression we used Kc = Ω as earlier, and m j = γ j M and M c = ~ Ω given after (A.1) and (49). Finally, the ∂ t ϕ j ’s of (45) lower one order as ∂ t ϕ † | C =1 = − iγ † Ω∂ t f † , ∂ t ϕ ‡ | C =1 = iγ ‡ Ω∂ t f ‡ . Substituting these and equations (50)–(51) in waveequations (45), multiplying the first resulting equation by − ~ cKγ † and the second by ~ cKγ ‡ ,with cK = Ω and ~ K = M c as before and after (A.1), we eventually obtain the waveequations for the electromagnetic waves of the particle expressed by f † , f ‡ :[ M c + qφ a − c ( i ~ ∇ + q A a )] f † = i ~ ∂ t f † , [ − M c + qφ a + c ( i ~ ∇ + q A a )] f ‡ = i ~ ∂ t f ‡ . (52)For the particle dynamics in question we want to further transform (52) to beexpressed by the particle wave variables, i.e. the k d and ω defined in (49), and thecorresponding wave functions, the f r , f ℓ to obtain below. We shall below obtain suchfunctions through a dynamic equivalence transformation directly from the f † , f ‡ ; theseought to be and will show to be functions identical to the e ψ, e ψ vir obtained in a physicallymore transparent way earlier; the present approach below will advantageously preservea direct tractable connection with the original f † , f ‡ , thus also ϕ † , ϕ ‡ , whose waveequations (52) or (45) give the relativistic energy-momentum relation exactly based onthe Doppler equations (47), see (B.2) of Appendix B. What (accordingly) is in questionin the transformation mainly is to maintain an equivalence to the quadratic equation(B.2); this corresponds to the equations ∂f j ∂z ν ∂f j ′ ∂z ν = ∂f µ ∂z νn ∂f µ ′ ∂z ν , etc., with j, j ′ = † , ‡ , µ, µ ′ = r, l , z ν = t, z ( ν = 0 , f µ ∂f µ ′ ∂z = 0. The equivalence condition requiresin particular the transformed quadratic to be ∂f r ∂z ∂f l ∂z = k d , that is, it has a plus sign infront and its cross-term product with M c (i.e. the O discussed after 57) is absent. Thiscan be achieved if we introduce a wavevector being the imaginary of (thus orthogonalto) k d : ¯ k d = ( γ/iγ † ) k † d , ¯ k d = ( γ/iγ ‡ ) k ‡ d ; thus k † d k ‡ d = (1 /i )¯ k d (53)(compare ¯ k d with the operator p υ.op = ~ i ∇ later). (53) alternatively can be expressed by(a) : k † d ( − k ‡ d ) = ¯ k d ¯ k d or (b) : ( − k † d ) k ‡ d = ¯ k d ¯ k d . (54)We now first transform the Doppler-differentiated f j (; k jd , ω j )’s (as short hand notationsof f j ( z, t ; k jd , ω j )’s) to a pair of mean (wave)-variable functions f µ (; ¯ k d , ω )’s (denoting f µ ( z, t ; ¯ k d , ω )’s) by, say, satisfying (a) of (54) and ordinarily ω = √ ω † ω ‡ of (49): f † (; k † d , ω † ) → f r (; ¯ k d , ω ) = C r e i [¯ k d z − ωt + α ] , f ‡ (; k ‡ d , ω ‡ ) → f ℓ (; ¯ k d , ω ) = C ℓ e i [¯ k d z + ωt + α ] ; (55)see these functions plotted in Figure 1b. The transformed f r , f ℓ indeed are desirablyidentical functions to the original f † , f ‡ if disregarding the high-order differences inthe coefficients γ † , γ ‡ and γ in the wave variables and the reversed travel direction irac Equation for Electrodynamic Particles /JXZJ f ℓ from f ‡ . The f r , f ℓ , being identical functions to the e ψ , e ψ vir earlier, indeed aretherefore the pertinent space-time functions of the particle; these are each functionsof the source motion and the total (electromagnetic) wave oscillation and accordinglydirectly resemble the de Broglie phase waves; and these are equivalent to Dirac’s space-time functions.To entail that in the matrix representation later (Sec. 5) a cross-term product O discussed after (57) is similarly absent, for transformation of the first derivatives weare compelled to satisfy the alternative condition (b) of (54). The use of (54b) andordinarily the ω = √ ω † ω ‡ of (49) first directly leads to the intermediate transformationsfor the f j ’s given in the left column below: f † −−−−−−→ k † d →− ¯ k d α →− α ′ f ∗ ℓ −−→ (a1) f ℓ , ∇ f † = ik † d f † −−−−−−→ k † d →− ¯ k d ,f † → f ℓ i ( − ¯ k d ) f ℓ = −∇ f ℓ ,f ‡ −−−−−→ k ‡ d → ¯ k d α → α ′ − f ∗ r −−→ (a2) − f r , ∇ f ‡ = − ik ‡ d f ‡ −−−−−→ k ‡ d → ¯ k d ,f ‡ →− f r − i ¯ k d ( − f r ) = ∇ f r (56)where, f ∗ ℓ = C ℓ e i [ − ¯ k d z − ωt − α ′ ] , f ∗ r = C r e i [ − ¯ k d z + ωt + α ′ ] , α ′ = − α ; the transformations (a1)and (a2) in (56) finally naturally lead to the same f ℓ , f r as in (55), which indeed alsorepresent the original f ∗ ℓ , f ∗ r in all aspects (having the same phase velocities and waveforms) except the opposite rotating phases on the complex plane and the opposite signsof α ′ and α that altogether are dynamically inconsequential. The relations in the leftcolumn and the use of (54b) again then lead to the results in the right column which isactually in question in respect to dynamical equivalence here.Substituting in wave equations (52) the transformation relations (55) and (56) gives( M c + qφ a ) f r + c ( i ~ ∇− q A a ) f ℓ = i ~ ∂ t f r , ( − M c + qφ a ) f ℓ + c ( i ~ ∇ + q A a ) f r = i ~ ∂ t f ℓ . (57)As a check, placing in (57) the f r , f ℓ of (55), multiplying the first and the negativeof the second resulting equations, dividing f r f ℓ , putting for simplicity φ a = A a = 0for the problem mainly of concern here, we correctly obtain the same result as (B.2): M c − ~ ¯ k d c + O = ~ ω where ¯ k d = − k d following (53) and (49); O = 0. 4. Two-particle system: spins, charges, and time-arrows Consider two spin-half particles 1,2 having identical sets of { f r , f ℓ } ’s tend to occupythe same location z or more precisely the same region in (0 , L ); suppose these arenoninteracting (a finite particle-particle interaction can in principle be included in V j and will not affect the general conclusions below). In virtue of the statisticalnature of the electromagnetic displacements, the probability of finding a portion ofparticles 1 and 2 at locations z and z is proportional to the product f µ ( z , t ) f µ ′ ( z , t ).Since these have identical space-time function sets, the corresponding total space-time function is evidently symmetric, thus f s , here in the only form f s ( z , z ) = √ [ f r ( z , t ) f ℓ ( z , t ) + f ℓ ( z , t ) f r ( z , t )] to be compatible with the antisymmetric totalspin function later. irac Equation for Electrodynamic Particles /JXZJ χ a = √ [ α (1) β (2) − α ′ (2) β ′ (1)]). Apparently it is in general also relevant that weintroduce charge functions Q - (1) , Q -(2) to reflect the sings of charges 1,2; these for thetwo like charges are trivial identities and lead to a trivial symmetric total charge function, Q s . The above functions together define an antisymmetric two-electron function (Figure2a, right graph), ψ a = f s Q s χ a , yielding as expected a total probability independent ofhow we sample the two stationary-state identical, indistinguishable (as result of beingidentically extensively distributed in L ) particles.In the case of an electron and positron (Figure 2, panel b), the total two-particlefunction is symmetric (right graph), thus ψ s , as follows from the observational fact thattwo such particles can approach each other arbitrarily close and, in an extreme case ∅ S j ‡ Particle 2: Positron j † S Particle 2: Electron j † j ‡ Particle 1: Electron S j † j ‡ z E ‡ E † W-WW-W - W y (2) + y (1) + y (2) - y (1) - (a)(b) z z ' a ' b r ff l r ff l r ff l a y (2) + y (2) - z v z † E ‡ E W Q (2)= - Q (1) is explicitly contained in y (n)'s in the graph, not in the text. m + - b ' ab --- --- Figure 2. Two spin-half IED particles 1 and 2 described by identical, thus mutuallysymmetric sets of Doppler-displaced space-time functions { ϕ † , ϕ ‡ } ’s or effectively { f r , f ℓ } ’s tend to occupy the same location z ; panel (a) shows two electrons and (b) anelectron and positron. Particle 1 has a spin S in + z - direction, denoted spin up, α (1),and is parallel with the generating direction of the f r wave; particle 2 has spin S in − z -direction, denoted spin down β (2), and is parallel with f ℓ ; their opposite counterparts(termed virtual spins) β ′ (1) and α ′ (2) are parallel with f ℓ and f r . Right graphs showthe complete wave functions { ψ + (1) , ψ - (1) } and { ψ + (2) , ψ - (2) } of particles 1 and 2.In panel (a), ψ ν (1) and ψ ν (2) ( ν = + , -) are antisymmetric due to the antisymmetricspin functions and same charges. In (b), these are symmetric (as shown in the graph)due to antisymmetric spins and also opposite charges ( Q + (2) = − Q - (1)), the latter ofwhich leads to the radiation electric fields E j ’s are opposite in direction (in the text the Q + (2) , Q - (1) are not explicitly regarded, rendering the ψ ν (1) and ψ ν (2) for the electronand positron to be antisymmetric and the same as for two electrons). irac Equation for Electrodynamic Particles /JXZJ Q a . (If imagine thewave displacement ϕ j in the medium is executed by a chain of dipole charges then itis immediately clear that the corresponding radiation electric field E j , with | E j | ∝ ϕ j ,produced by the positive charge is reversed from that by the negative charge, see leftgraph in Figure 2b.) Placing the two known functions in ψ s = Q a f s χ Sym gives thereforeSym=antisymmetric and thus χ Sym = χ a for the electron-positron system.Our particles contain each the electromagnetic waves ϕ † , ϕ ‡ , or effectively f r , f ℓ ,generated in the specified + z - and − z - directions which assume definite relationshipswith the spin orientations that turn out in a measurement: Given, say, particle 1 ismeasured to be spin up, α (1), with f r being parallel with it, then its other internalprocess, f ℓ , is parallel with a virtual (indicated by a prime) spin-down state, β ′ (1).Similarly for particle 2, being then actually spin down, thus f ℓ parallel with β (2), its f r is parallel with a virtual spin-up state, α ′ (2). From the foregoing antisymmetric spinrequirement follow the relations for the spin functions: α ′ (2) = − α (1) , β ′ (1) = − β (2); with β ′ (1) = − α (1) , α ′ (2) = − β (2) (58)following the opposite signs as is meant by the ”virtual” spins. Through (58), the virtualspin vector of particle 1, S ′ (1) (virtual spin down) and the actual spin vector of particle2, S (2) (actual spin down), pointed each in the − z - direction, are now each representedas scalar quantities with minus signs.If disregarding the signs of charges explicitly, the electron-electron and the electron-positron are two equivalent systems of identical, spin-half particles, each described bythe space-time functions f µ (; ¯ k d , ω )’s with µ = r, ℓ , which joined together with the spinfunctions give the complete wave functions: ψ ν ( n ; ¯ k d , ω ) = f µ (; ¯ k d , ω ) α ν ( n ) with ν = +,-, n = 1 , α ν = α, β . Now as a further step to conform our wave equation later to matrixform, we hereafter require that the ψ ν ( n ) functions are elements of a matrix of onecolumn, ψψψ . The matrix wave equation itself will entail the two desired features discussedafter equation (57), that is, (i) the product of − k d and k d in the quadratic equation ispositive: k d (entailed by the situation that these in the matrix form are offdiagonalelements, see (62) or (C.1), and (ii) the total cross-term product O = 0 (entailed by thecharacteristics that in matrix equation the ψ ν ( n )’s are explicitly mutually orthogonal).The first of these two features which we have up to now enforced by use of ¯ k d for k d ,should no longer be used in the matrix form to avoid a dual accounting. The space-timefunctions accordingly write f r (; k d , ω ) = C r e i [ k d z − ωt ] , f ℓ (; k d , ω ) = C ℓ e i [ k d z + ωt ] with k d theordinary scalar quantity and related with k † d , k ‡ d through (49). Accordingly, ψ + (1) = α (1) f r (; k d , ω ) = α (1) C r e i [ k d z − ωt ] , ψ - (2) = β (2) f r (; k d , ω ) = β (2) C r e i [ k d z − ωt ] , (59 a ) ψ - (1) = β ′ (1) f ℓ (; k d , ω ) = β ′ (1) C ℓ e i [ k d z + ωt ] ; ψ + (2) = α ′ (2) f ℓ (; k d , ω ) = α ′ (2) C ℓ e i [ k d z + ωt ] . (59 b )The complete wave functions (59a)–(b) (Figure 2, right graphs) describe two identicalparticles of opposite oriented actual spins and accordingly opposite virtual spins, and irac Equation for Electrodynamic Particles /JXZJ ψ a = √ [ ψ + (1) ψ - (2) − ψ - (1) ψ + (2)] correctly leads tothat the probability of finding two identical particles at any location z in L is notaltered by interchanging the locations of the particles (the indistinguishability). Wecan also check that the same ψ a is given by the product of the separate total functions: ψ a ( z , z ) = f s ( z , z ) χ a (1 , f r ( z , t ) f ℓ ( z , t ) β ′ (1) α ′ (2) and f ℓ ( z , t ) f r ( z , t ) α (1) β (2) are zero sincethese do not describe the present reality.Lastly, the spin-up state of particle 1, α (1), is associated with an effectiveelectromagnetic wave f r travelling to the right, thus ∂ t f r /f r = − iω , while the spin-upstate of particle 2 with f ℓ travelling to the left, thus ∂ t f ℓ /f ℓ = iω ; the latter has as if areversed time arrow relative to the former. We may introduce the time arrow functionsdefined for particles 1 and 2 as T (1) = 1 , T (2) = − , such that the action of theseon the time derivatives project the wave propagations to be both in the + z -direction: T (1) ∂ t ψ ν (1) = ∂ t ψ ν (1), T (2) ∂ t ψ ν (2) = − ∂ t ψ ν (2). 5. Dirac equation For two identical, spin-half particles of identical sets of space-time functions f r , f ℓ described by wave equations (57) tending to occupy the same location z , we shall nowexpress the corresponding wave equations in terms of the complete wave functions ofSec. 4. For particle 1, we thus multiply the first equation of (57) by α (1) and the secondby β ′ (1), act T (1) in front of the time derivatives, denote its charge by q , and get( M c + q φ a ) f r α (1) + c ( i ~ ∇ − q A a ) f ℓ ( − β ′ (1)) = i ~ T (1) ∂ t f r α (1) , ( − M c + q φ a ) f ℓ β ′ (1) + c ( i ~ ∇ + q A a ) f r ( − α (1)) = i ~ T (1) ∂ t ( f ℓ β ′ (1)) . (60)For particle 2, instead we multiply the first equation of (57) by − β (2) and the secondby − α ′ (2), act both equations by T (2), denote its charge by q , and get( M c + q φ a ) f r ( − β (2)) + c ( i ~ ∇ − q A a ) f ℓ (+ α ′ (2)) = i ~ T (2) ∂ t f r ( − β (2)) , ( − M c + q φ a ) f ℓ ( − α ′ (2)) + c ( i ~ ∇ + q A a ) f r (+ β (2)) = i ~ T (2) ∂ t f ℓ ( − α ′ (2)) . (61)In the second terms in equations (60)–(61) we made the replacements α (1) → − β ′ (1)and β ′ (1) → − α (1), β (2) → − α ′ (2), α ′ (2) → − β (2) based on (58), to conform to thetransformed space-time functions earlier.Substituting in (60)–(61) with (59a)–(b) for the ψ ν ( n )’s and the T ( n )’s expressedearlier, and, to form a direct contrast between the actual spin directions of the twoparticles, re-arranging the resulting four equations in the order of spin-up states ofparticles 1 and 2 first and then spin-down states of particles 1 and 2, we finally obtain aset of four coupled linear first order partial differential equations governing the motions irac Equation for Electrodynamic Particles /JXZJ ψ ν ( n ):( M c + q φ a ) ψ + (1) − c ( i ~ ∇ − q A a ) ψ - (1) = i ~ ∂ t ψ + (1) (particle 1, spin up)( M c − q φ a ) ψ + (2) + c ( i ~ ∇ + q A a ) ψ - (2) = i ~ ∂ t ψ + (2) (particle 2, spin up)( − M c + q φ a ) ψ - (1) − c ( i ~ ∇ + q A a ) ψ + (1) = i ~ ∂ t ψ - (1) (particle 1, spin down)( − M c − q φ a ) ψ - (2) + c ( i ~ ∇ − q A a ) ψ + (2) = i ~ ∂ t ψ - (2) (particle 2, spin down) (62)From the discussion of Sec. 4 that the ψ ν ( n )’s and accordingly also their first derivativesare mutually orthogonal, it follows that the linear equations (62) are equivalent to amatrix equation. Supposing specifically the two particles are a positron and an electronand therefore q = q , q = − q , the matrix form of (62) is thus H op ψψψ = i ~ ∂ t ψψψ, with H op = b M c + qφ a + cααα ( p υ.op − q A a ) and p υ.op = − i ~ ∇ (63)being the relativistic total Hamiltonian and linear momentum operators. Where, b = I − I ! , ααα = σσσ z σσσ z ! , ψψψ = ψψψ + ψψψ - ! and I = ! ; σσσ z = − ! ψψψ + = ψ + (1) ψ + (2) ! ; ψψψ - = ψ - (1) ψ - (2) ! . (64)The off-diagonal elements of the matrix σσσ z , σ z (= 1) and σ z (= − 1) here correspondto the α (1) = 1 and α ′ (2) = − ψψψ is equivalent to a Dirac spinor, σ z the z -component of Pauli matrices, and as a whole, equation (63) is identical to theDirac equation for an electron-positron system equivalent to here. For the present caserotation transformation is trivial, so ~σ = σ z ˆ x .Suppose more generally the two particles’ spin angular momenta, S (= ~ σσσ )’s, arealong an axis n executing in general a precession about the z -axis at a fixed angle(arccos( S z S )). For each particle being in stationary state, its S (similarly its magneticmoment µµµ s (= − e S /m )) as a vector quantity when in small rotations about the z -axismust maintain invariant with respect to its projection on the z -axis, n · S = ± ~ ,and is Hermitian. In addition to the antisymmetric condition given by the σ z of (64)above, an infinitesimal rotation transformation as such needs be unitary. A specific setof transformation matrices having these properties are known to be the Pauli matrices, σ x , σ y and σ z of the standard expressions and σ z as expressed in (64), σσσ = σσσ x ˆ x + σσσ y ˆ y + σσσ z ˆ z .And, the unitary matrix I ˆ z about the z -axis naturally extends to a unitary matrix aboutthe new n - axis in three dimensions, given by ~I = I ˆ x + I ˆ y + I ˆ z . Substituting in (63)with σσσ and I for σσσ z and I gives a Dirac equation of the same form, now for spins inarbitrary directions. irac Equation for Electrodynamic Particles /JXZJ Appendixes IA-IC:Appendix A. Total energy and inertia of particle wave As a general result of classical electrodynamics based on solution to the Maxwell’sequations combined with Lorentz force law, an electromagnetic wave j transmits atthe speed of light c a wave energy ε j and a linear momentum p j = ε j /c . Here, theamplitudes of ε j , accordingly of p j , E j , B j and ϕ j , etc., are continuous values. FollowingM. Planck’s discovery of quantum theory in 1901, it has been additionally understoodthat these quantities are by nature quantized in amplitudes; an electromagnetic waveof frequency ω/ π has an energy ε = n ~ ω , consisting in general of n momentum-space quanta, or photons, each of an energy ~ ω ; and the classical continuous amplitudesolutions to these are only approximations when n is large. In the present problem,in conformity with experiments, especially the pair processes, the electromagnetic wavecomprising our basic particle has a ”single energy quantum”, n = 1; so ε = ~ ω . It hasbeen further proven especially through quantum electrodynamics that the Maxwell’sequations, and the subsequent classical wave equation (42) or (45), continue to hold,and the quantisation of the fields and wave energy etc. is the result of subjectingthe canonical displacement and momentum, the u (= aϕ ) and ˙ u here, to the quantumcommutation relation [ u, ˙ u ] = i ~ .The total wave of our particle of a single ”quantum energy level” ~ ω in a one-dimensional box has, following the solution to the Maxwell’s equations earlier [see after(42)], two components, ϕ † and ϕ ‡ , with their frequencies being Doppler-displaced to ω † and ω ‡ as a result of the source motion as given in (47), which are related to ω through(49). For the total wave comprising the particle, ε represents therefore a dynamicalvariable of the particle, here the total energy of the particle.The electromagnetic waves, E j , B j ’s or ϕ j ’s, rapidly oscillating at frequencies( ω j / π )’s, of a geometric mean frequency ω/ π and wavelength λ = c/ ( ω/ π ) will, whenignoring the detailed oscillation as will effectively manifest at some distance, appear as ifbeing two rigid objects, wavetrains, travelling at the speed of light c . In view that theirspeed of travel, c , is finite as contrasted to infinite, the wavetrains have inevitably each finite inertial masses, m j ’s, thus an inertial mass m = √ m † m ‡ for the total wavetrainand hence its resulting particle. This mechanical depiction of the total wave, as arigid ”wavetrain”, permits us at once to express according to Newtonian mechanics thelinear momentum of the wavetrain to be p = mc . Combining this with the classicalelectrodynamic result ε = pc above gives the kinetic energy of the wavetrain ε = mc ,being equivalent to the Einstein’s mass-energy relation. This energy and the Planckenergy earlier ought to equal, thus m = ~ ω/c ; or at υ = 0: M = ~ Ω/c ( A. M the rest mass of the particle; combining (A.1) with (49) gives m = γM .Combining (A.1) with p = mc further gives mc = ( ~ ω/c ) c = ~ k and accordingly M c = ~ K , with ω = kc , Ω = Kc and k = γK as earlier. irac Equation for Electrodynamic Particles /JXZJ Appendix B. Relativistic energy–momentum relation for theelectromagnetic waves of particle Consider first the simpler case of A a = 0. Placing in wave equations (52) with f † , f ‡ of(46), dividing the resulting first and second equations by f † and − f ‡ and sorting give M c + ~ k † d c = ~ ω † − qφ a , M c − ~ k ‡ d c = ~ ω ‡ + qφ a . Multiplying gives M c − ~ k † d k ‡ d c + Q = ~ ω † ω ‡ − q φ a ( B. k † d k ‡ d = k d and ω † ω ‡ = ω following (49); Q = M c c ~ ( k † d − k ‡ d ), with k † d − k ‡ d =2 k d ( υc ) γ and M c = ~ Kc , so Q = 2 ~ k d c . With these, putting ~ k d = ± p υ , ~ ω = ± ε where p υ , ε are here variables having positive and negative solutions and thus the righthand side of (B.1) reduces as p [( ~ ω − qφ a )( ~ ω + qφ a )] = p [ − ( ε − qφ a )( ε − qφ a )] =( ε − qφ a ) , then (B.1) reduces exactly to M c + p υ c = ( ε − qφ a ) . This, or this in themore familiar form for φ a = 0, M c + c p υ = ε , ( B. A a finite, denoting k † d ′ = ¯ k † d − q A a ~ , k ‡ d ′ = k ‡ d + q A a ~ ,the particular feature that (the effective portion of) A a is always perpendicular to k d z leads to k d ′ = k † d ′ k ‡ d ′ = k d − q A a / ~ , or, ( ± p ′ υ ) ≡ ( ± ~ k ′ d ) = ∓ ( ~ k d − q A a )( − ~ k d − q A a ) = [ ∓ ( p υ − q A a )] . (B.2) thus generalises to M c + c p υ ′ = ( ε − qφ a ) . Appendix C. Solution of Dirac equation from the standpoint of particleinternal process We shall here mainly discuss the choice of the solution forms of the Dirac equationfrom the standpoint of internal processes of the IED particle model for simplicity forspins along z -axis, in an otherwise basically standard procedure. The two equationsof (60) or (61) for particle n = 1 or 2 are coupled in ψ + ( n ) and ψ - ( n ) and can notbe solved separately as in Appendix B. We need to solve each two, or more generallythe four equations of the Dirac equation (60) together. Let the trial functions be: ψ ν ( n ) = C sn e i ~ [ p υ z − εt ] , s = + , − , n = 1 , 2. Placing these in (63) and rearranging give ε − M c − qφ a − [ p υ − q A a ] c ε − M c − qφ a p υ − q A a ] c − [ p υ − q A a ] c ε + M c − qφ a 00 [ p υ − q A a ] c ε + M c − qφ a ψ + (1) ψ + (2) ψ - (1) ψ - (2) = 0( C. ψ s,j ’s as four unknowns below; for these to have nontrivial solutions, the determinantfor the matrix of the coefficients of (C.1) needs be zero. This is det= [( ε − qφ a ) − irac Equation for Electrodynamic Particles /JXZJ M c ] − c p ′ = 0, with p ′ υ = p υ − q A a . This has two degenerate sets of square rootssolutions: ε − qφ a = ± p M c + p υ ′ c , which being identical to (B.2). In view that eachparticle has internal processes, we thus naturally assign symmetrically two of the foursolutions to particle 1, as ε − qφ a = ± p M c + p υ ′ c , and the other two for particle 2as ε − qφ a = ∓ p M c + p υ ′ c . These two distinct sets of square-roots solutions to thealgebraic equation above represent two (distinct, identical) particles, like an electronand a positron, which do not transit from one to the other, a point agreeing withreality and having been stressed by P.A.M. Dirac from the very beginning in [1]. Thesealgebraic solutions are in contrast to the usual problem of eigen values arising generallyfrom boundary conditions and being each possible states of same particle between whichtransitions generally can occur.With p υ and ε as known parameters, we further solve the four algebraic equations( ε − ( M c + qφ a )) ψ + (1) = p ′ υ cψ - (1) ( a ) , ( ε − ( M c + qφ a )) ψ + (2) = − p ′ υ cψ - (2) ( b )( ε +( M c − qφ a )) ψ - (1) = p ′ υ cψ + (1) ( c ) , ( ε +( M c − qφ a )) ψ - (2) = − p ′ υ cψ + (2) ( d )( C. ψ ∗ - (1) ψ - (1) = C - , ψ ∗ + (1) ψ + (1) = C + , ψ ∗ - (2) ψ - (2) = C - and ψ ∗ + (2) ψ + (2) = C + , we get( ε − ( M c + qφ a )) C + = ( ε + M c − qφ a ) C - , ( ε − ( M c + qφ a )) C + = ( ε + M c − qφ a ) C - These have two independent solutions, and in mathematical terms two of the four wavefunctions can thus be arbitrarily chosen. In view of the IED particle model by which eachparticle has internal, wave processes consisting of two components in the one-dimensionalbox, it is natural here that we choose the values for C + and C + symmetrically, in thesense also C + = − C + , with these in the two equations above the values for C - and C - then follow to be uniquely given as C +1 C +2 = ± (cid:16)q ε +( Mc − qφ a ) ε − ( Mc + qφ a ) (cid:17) / C, C - C - = ∓ (cid:16)q ε − ( Mc + qφ a ) ε +( Mc − qφ a ) (cid:17) / C where | C + C - | = C , | C + C - | = C . With the above in the trial functions, we get thecomplete solution for Dirac equation ψψψ = ψ + (1) ψ + (2) ψ - (1) ψ - (2) = (cid:16)q ε +( Mc − qφ a ) ε − ( Mc + qφ a ) (cid:17) / Ce i ( k d z − ωt ) − (cid:16)q ε +( Mc − qφ a ) ε − ( Mc + qφ a ) (cid:17) / Ce i ( k d z + ωt ) − (cid:16)q ε − ( Mc + qφ a ) ε +( Mc − qφ a ) (cid:17) / Ce i ( k d z + ωt ) (cid:16)q ε − ( Mc + qφ a ) ε +( Mc − qφ a ) (cid:17) / Ce i ( k d z − ωt ) . ( C. ψ + (1) , ψ - (1) aretwo opposite travelling component waves of particle 1, and ψ + (2) , ψ -(2) of particle 2; inthe meantime, the spin-up component waves of particles 1 and 2, ψ + (1) and ψ + (2) travelin opposite directions and similarly the spin-down component waves. irac Equation for Electrodynamic Particles /JXZJ References [1] P. A. M. Dirac, ”The Quantum Theory of the Electron,” Proc. Roy. Soc. A117 , 610—624 (1928a);”The Quantum Theory of the Electron. Pt II,” Proc. Roy. Soc. A118 , 351–361 (1928b).[2] J. X. Zheng-Johansson and P-I. Johansson, (a): Unification of Classical, Quantum and RelativisticMechanics and of the Four Forces , Foreword by R. Lundin, (Nova Sci. Pub. Inc., N.Y., 2ndprint, 2006) (see a very early sketch of the ideas in: arxiv:physics/0412168); (b): Inference ofBasic Laws of Classical, Quantum and Relativistic Mechanics from First-Principles Classical-Mechanics Solutions , (Nova Sci. Pub., Inc., N.Y., 2006); (c): ”Inference of Schr¨odinger Equationfrom Classical Mechanics Solution,” in Quantum Theory and Symmetries IV.2 , ed. V.K. Dobrev(Heron Press, Sofia, 2006), pp.763-770; arxiv:phyiscs/0411134 v5; ”Schr¨odinger Equation forElectrodynamic Model Particle,” submitted for publication; (d): ”Developing de Broglie Wave,” Prog. in Phys. , , 32-35 (2006); arxiv:phyiscs/0608265; (e): ”Mass and Mass–Energy Equationfrom Classical-Mechanics solution,” Physics Essays , , nr. 4 (2006); arxiv:phyiscs/0501037;(f): J.X. Zheng-Johansson, ”Spectral Emission of moving atom,” Prog. in Phys. , , 78-81(2006); arxiv:phyiscs/060616; (g): J. X. Zheng-Johansson, ”Vacuum Structure and Potential,”arxiv:physics/0704.0131; (h): J.X. Zheng-Johansson, ”Dielectric Theory of the Vacuum,”arxiv:physics/0612096; (i): ”Depolarisation Radiation Force in a Dielectric Medium. Its Analogywith Gravity,” with R. Lundin, in ” Quantum Theory and Symmetries IV.2 , ed. V.K. Dobrev(Heron Press, Sofia, 2006), pp. 771-779; arxiv:phyiscs/0411245; (j): J.X. Zheng-Johansson,”Doebner-Goldin Equation for Electrodynamic Model Particle. The Implied Applications,”