0960-0779/93$6.(K) + 00 0 19Y3PcrgamonPress Ltd
~‘hou,. .so/rro,r, & froc,rr,, Vol.3.No. 5.pp.563-573. lYY3 Pr,ntcd I"GreatBr,t;t,n
Fluctuons -11.
Electromagnetism
MICHAEL CONRAD Department
of Computer
Science,
Wayne
State University.
(Receit,ed 2 Februury
Detriot,
MI 48202, USA
1993)
Abstract-The fluctuon principle is used to formulate the electromagnelic interaction. Virtual and real photons are interpreted as chains of transient electron-positron pairs. Electromagnetic waves are interpreted as compression-expansion waves of vacuum density. The fine structure constant is interpreted in terms of the probability of an interaction between a transient electron-positron pair and a charged absorber. The model eliminates the problem of infinite self-energies.
1. INTRODUCTION
The fluctuon model subsumes a sea of latent vacuum fermions. The gravitational interaction, since it is universal, is mediated by the full sea. A subset of particles (the latent electrons) mediates the electromagnetic interaction. From a strictly logical point of view it would be preferable to present a thorough treatment of the gravitational interaction first, and then turn to electromagnetic and other interactions. For the purposes of exposition, however, it is simpler to give a brief preliminary discussion of gravity at this point, sufficient to frame the discussion of electromagnetism, and to defer the more comprehensive discussion to Part III of this series.
2. PREVIEW
OF GRAVITATION
Recall (from Part I) that gravitons are interpreted as fluctuons propagating in the full sea of vacuum fermions. Two alternatives are possible: direct attractive interactions or direct repulsive interactions with indirect attractive interactions [l]. The direct repulsive model is simpler overall, partly because propagating fluctuons most naturally mediate repulsive interactions. The indirect attractive interaction is due to the feature that permanent positive energy masses are stronger repellers than trapped fluctuons. Consequently vacuum particles are repelled from regions of positive energy mass. The low mass regions of the universe will have the highest density of vacuum particles. Gravitational chains emanating from these low mass regions will push positive energy masses towards each other (see Fig. 1). The pushing effect follows because the vacuum is most depressed in the region between the positive energy masses and because two interacting masses each terminate some fraction of the chains that can push them towards each other. Blockage by mass 1 prevents such chains from exerting a repelling effect on mass 2, and similarly blockage by mass 2 prevents chains that could exert a repelling effect on mass 1 from reaching that mass. Consider two interacting positive energy masses in a vacuum that is unaffected by any other positive energy masses. For the present we ignore the interactions of the particles with trapped fluctuons and the consequent attractive interaction. We suppose, as an idealization, that the absorbers are stationary and consider only the direct repelling effects 563
564
Fig. 1. Schematic representation pushed together by propagating
of gravitational interaction. Positive masses fluctuons emanating from trapped fluctuons the universe.
(represented by blackened circles) are in high density (low mass) regions ot
arising from the exchange of momentum through the chain propagation mechanism. This abstracted interaction should follow the inverse square law [equation (17). Part I], yielding Fgrav
=
G,m’ e
Agrav
r*
where for simplicity we take the two masses as equal. The proportionality constant G, reflects the degree of coupling between the masses. The interaction is repulsive, due to the omission of the interactions with trapped fluctuons that are responsible for the net attractive interaction. The latter are individually less important, since trapped fluctuons are only temporary absorbers, but collectively dominant due to their large numbers. The universal gravitational constants, G, should reflect this net effect. Eliminating r from equation (1) yields
where we take the positive root, since manifest masses should be positive. The only quantity in the equation that could vary significantly when m either increases or decreases affected only by extreme variations in m). The is perav Cc and Agrav are significantly presence of mass must therefore decrease the density of surrounding vacuum fermions, in agreement with the repulsive interaction between positive energy masses and trapped fluctuons described above. The spatial dependence of this depression is not represented, to however. As m goes to zero it may seem that pgrav should go to infinity (corresponding the fact that the force between two particles decreases as vacuum density increases). This is clearly a pathology, since the mass of two objects could not create an infinite vacuum density. The pathology is due to the fact that picturing two interacting objects in isolation is an idealization. In reality, the vacuum is full of masses, due to the existence of trapped fluctuons. Consequently masses are always present that ensure a finite vacuum density, and as the vacuum density increases the density of such trapped masses increases as well. The Newtonian inverse square attraction between positive energy masses obviously cannot directly follow equation (1). All the masses in the universe, both latent and manifest, contribute to the interaction. The distribution of particle mass (or energy) and gravitational field can be identified with the density structure of the vacuum. In Part III we will show that density distribution is isomorphic to the space curvature of general relativity, and that the inverse square law therefore follows from the model as an approximation (since it is an approximation to the general relativistic law of gravitation). Gravitons are ordinarily supposed to have a spin 2 character. corresponding to the tensor character of the gravitational field. The fluctuon model allows for spin 1 gravitons, however, since the tensor character arises from an indirect interaction.
565
Fluctuons-II
3. COULOMB
LAW
Now let us consider the subset of particles that mediate the electromagnetic interaction (i.e. the vacuum fermions that have the potentiality for exhibiting electric charge). Let us in terms of denote the density of this subset by &,,,. The inverse square law expressed vacuum densities (equation (17), Part I) then becomes the propagating fluctuon version of the Coulomb law F Cod
A =
em
C2h2py.
Virtual photons mediating the Coulomb interaction are thus interpreted as chains of transiently excited electron-positron pairs. Since photons are spin (or helicity) 1 particles, the spins of the electron and positron (l/2 each) must be parallel. Equating equation (3) to the usual form of the Coulomb law F Coul
=
9?= r2
where q is electric charge, and for simplicity interacting objects is the same. Eliminating Y 4 =_ +-
A
em
(4)
c2h2p~~r2 we assume
that
the
charge
on
the
two
A iI2 em ciip,'l," .
(5)
The only two quantities in this equation that are reasonably free to vary are q (the total decrease the density charge on an object) and pem. The presence of charge must therefore of surrounding vacuum electrons, just as the presence of positive energy mass decreases the density of the full sea of vacuum fermions. As in the case of mass [equation (2)] the spatial dependence of this depression is not represented. Similarly as q goes to zero it may seem since the removal of charge that pem should go to infinity. Again this would be a pathology, from two objects could not create an infinite vacuum density. In reality the vacuum is full of charges, just as it is full of masses, due to the existence of trapped fluctuons. Charges are therefore always present that ensure that pem is finite. Again in analogy to mass, the density of such trapped charges increases as pem increases. Charge cannot exert its depressing effect on pem in precisely the same manner as mass as a whole are electromagnetically exerts its depressing effect on pgrav, since fluctuons neutral but never gravitationally neutral. The fluctuon model nevertheless provides a simple mechanistic interpretation for the depressing effect of charge: charged permanent particles repel vacuum electrons by acting on them in their short lived (but nonfluctuonic) positive energy states and attract holes (positrons) by pushing electrons into them. Nonfluctuonic (or ordinary) pair production processes occur whenever the fluctuation energy does not match the vacuum density, either initially or as a result of trapping through collision with a propagating chain. The electron and positron will generally find a partner to recombine with in a short amount of time, and must do so in a time allowed by the time-energy uncertainty principle if the pair arises as a result of an energy fluctuation rather than as a result of a collision process. During this time the electron and positron will exhibit non neutral behavior, due to the fact that they are not tied together in the manner of a fluctuon. Consequently, they will be repelled by like charges, leading to the depression suggested by equation (5). Fluctuons, whether propagating or trapped, also interact with the electromagnetic field (i.e. with propagating electromagnetic fluctuons) by virtue of being constructed from an electron and a positron, but these interactions cannot produce a charge induced depression due to the overall charge neutrality and bound nature of the pair.
S66
M. CONRAD
Equation (5) allows for two roots, one corresponding to positive and one to negative charge. However, all interactions between the charges are repulsive (in fact all direct interactions in the chain propagation model can be interpreted as repulsive). The choice of the negative root (i.e. the negative root of A,,) corresponds to the usual convention of assigning negative charge to positive energy electrons. Charge induced depression means a relative absence of charge, and hence a positive vacuum polarization. Permanent positive charges may be interpreted as permanent deficits of negative charge. These will have a concentrating effect on the surrounding vacuum electrons, due to the fact that transiently excited (but nonfluctuonic) electrons will be repelled into the neighborhood of the permanently nonrepelling hole. To use equation (5) to describe the influence of positive charge on vacuum density, but at the same time retaining both the negative root of A,,,, and the physical picture that all chain mediated interactions are repulsive, it is only necessary to redefine pem as the density of vacuum holes. Positive charge may then be viewed as depressing the density of vacuum holes, but of course this is fully equivalent to concentrating the density of vacuum electrons (negative polarization). Similarly, the attraction between negatively and positively charged particles is due to the repulsion of short lived electrons into the neighborhood of the latter.
4. MAXWELL’S
EQUATIONS
The Coulomb law is a specialized case of the electromagnetic field in which the interacting particles are stationary. However, we can call on the principle of relativity, already implied by the chain propagation model, to deduce the more general structure. If the two interacting particles are not at rest in some frame L* they will obey the force law obtained by transforming them to L” from a frame L in which they are at rest and in which they obey an inverse square law. The above procedure, along with an assumption of charge conservation and some technical assumptions limiting the effects of a moving source charge, may be used to extract magnetic forces and more generally Maxwell’s equations from Coulomb’s law [l-4]. The structure of the gravitational field cannot be obtained in this way, since it depends on the density structure of the whole vacuum, and therefore on the entire distribution of mass in the universe. Consequently, interacting positive energy absorbers are always accelerating relative to masses that contribute to the net attractive interaction between them. The connection between gravity and the global distribution of mass links the fluctuon model to Mach’s principle. The gravitational field affects electromagnetic phenomena, since mass depresses the density of all vacuum particles. including those that mediate the electromagnetic field. The electromagnetic field can have no measurable reverse effect on the gravitational field, however. This is because charge affects only the subsea of vacuum particles that mediate the electromagnetic interaction. This subsea constitutes a negligible fraction of the full sea of vacuum particles (since pK ~1 Fe’;‘, Fgrav> p,,), and consequently alterations in its density are invisible so far as gravity is concerned. The ‘two sea’ feature of the fluctuon model eliminates the conceptual dissonance between the geometrical description of force suggested by the relativistic theory of gravitation and the virtual particle mechanism suggested by quantum field theory. The but is describable in geometrical gravitational field is mediated by propagating fluctuons, terms because space curvature corresponds to the density structure of the full sea of vacuum fermions. The electromagnetic field can cohabit the same space because it is mediated by a subsea whose density is too low for alterations in it to have any impact on the density of the full sea. but which must itself undergo alterations commensurate with
Fluctuons-III
567
those that occur in the full sea. Geometrical descriptions concocted for it would be incompatible with the relativistic theory of gravitation from the geometrical point of view, since it is not possible for the same space to share different geometrical descriptions on intersecting spatial scales, but this does not preclude their being compatible from the point of view of density structures (since two density structures can share the same space).
5. INTERPRETATION
OF WAVE PROPERTIES
The fluctuon model requires that the virtual photons that mediate the l/r2 field be accompanied by vacuum density waves in pem. It also requires that real photons that mediate the l/y radiation field be accompanied by density waves in pem and provides an account of the energy-frequency relationship in terms of the proportional effect of energy on the density structure of pern and pera”. The density waves in &,,, provide a ‘mechanistic’ interpretation for the wave aspect of electromagnetism inherent in Maxwell’s equations. The interpretation follows directly from the spatial variation of vacuum density induced by charge. This implies that propagating fluctuons must be accompanied by compression-expansion waves of vacuum density. The fluctuation energy required for pair production increases as the fluctuon propagates away from a positive energy electron and decreases as it propagates away from a positive energy positron. However, once initial pair production occurs, it fixes the fluctuation energy, Al?,,,,, required for the transient electron-positron pair production processes that comprise the fluctuon. Thus as the chain of transient electron-positron pairs propagates away from an initiating electron its fluctuation energy would become too small to meet the conservation law and uncertainty principle requirements. If it enters a region of more depressed vacuum density (due to the presence of more positive energy charges) the fluctuation energy would become too small. In either case the energy requirements for pair production become inconsistent with the vacuum density. In order to satisfy the conservation and uncertainty principles it is necessary for the propagating chain to alter (expand or compress) the vacuum density in the neighborhood surrounding it in such a manner as to mask any vacuum inhomogeneities. The chain must therefore be accompanied by compression-expansion waves of vacuum density. As noted in Part I (Section 5), transverse motions of this type accompany chain propagation even in a homogeneous vacuum. Chain propagation processes originating in the vicinity of trapped fluctuons must also be accompanied by compression-expansion waves, since these chains will generally have to pass through regions of space whose density structure is altered by positive energy charges. Also, note that the chains referred to in the above discussion originate from electrons and not from positrons. This is consistent with the convention that electrons represent charge and positrons represent the absence of charge. Positrons can, however, be a source of chains that mediate the gravitational field. Compression-expansion waves allow propagating fluctuons to carry real energy. The energy depends on the frequency of the wave, according to the usual relation, E = hv = hc/&,. The momentum depends on the wavelength, according to the usual relation p = h/h,,. In an ideally homogeneous vacuum il,, would extend to infinity, corresponding to a pure virtual particle that carries no real energy. As the acceleration of the source charge increases (in some frame) or as the spatial variation of vacuum density of the full sea increases, A,, decreases (in the limit decreasing to Y,,). The formulae, E = hc/)\ and p = h/A, inherit their applicability to the electromagnetic field from their applicability to the gravitational field. This must be the case since mass,
and therefore energy and momentum, would have no meaning apart from the gravitational interaction. The key point is that chains of transient electron-positron pairs gravitate (i.e. photons gravitate). Consequently all such chains produce a depression in the full sea of vacuum fermions. The greater the energy and momentum of the chain the greater the depression in the full sea and therefore the shorter the de Broglie wavelength associated with this depression (recall that increased mass is identified with increased depression in pB”,\, in the fluctuon model of gravity). The chain of electron-positron pairs, since it is electrically neutral as a whole. alters the density of the subsea (i.e. of p,,) in proportion to its altering effect on the density of the full sea (i.e. of pgrav). A fortiori the length of the (transversely structured) density wave in pern is inversely proportional to the depression in pgravr and therefore inversely proportional to the energy it carries. Virtual photons are the special case in which the compression-expansion wave in the full sea entails no net expansion (since expansion is equivalent to depression). No net energy is then transferred. The distinction between virtual and real photons is not absolute, however, since a virtual photon in an inertial coordinate system will appear real to an observer in an accelerating coordinate system. The latter will be unable to correlate propagating chains with a source particle in the same way that an inertial observer can, since chains that carry real energy appear and disappear depending on the acceleration of the source particle. Consequently momentum appears to become detached from the source particle from the point of view of the accelerating observer. Actually all virtual particle exchanges involve some acceleration on the part of the interacting absorbers, and consequently it is never possible to find a coordinate system in which every photon exchanged between two absorbers is purely virtual (i.e. carries no real energy). The phase relation between the pair creation-annihilation event and the density wave is determined by the relative position on the wave at which the vacuum density matches the of the initial fluctuation energy. The length of the pe,,, wave depends on the acceleration of the initiating absorber fixes source charge, whereas the value of &,, in the neighborhood the initial fluctuation energy (i.e. fixes p,,). In the absence of gravitational effects these subsea two factors control the phase relation, since the density, &,, of the electromagnetic will then be constant except in the local vicinity of positive energy charges. This is due to the fact that positively charged regions will attract negative charges and negatively charged Gravitational effects. since they are concomitant to regions will repel negative charges. large scale inhomogeneities in peril\, exert a slight altering effect on the wavelength of the velocity of the excitation chain. As the wave passes P elll wave and also on the propagation field) to a region of high ppla\ (low from a region of low Ppra, (high gravitational and therefore its energy will decrease gravitational field) its density will increase, (gravitational red shift). The phase relation between the propagating excitation and the wavelength will also change (this is possible because of the effect of gravity on light velocity).
6. NONEXISTENCE
OF SELF-ENERGIES
An attractive feature of the fluctuon model is that it eliminates problems of infinite self-energy. Suppose that a charged particle can be treated as a sphere or radius r with a uniform distribution of surface charge. The classical expression for the self energy is (only the proportionality constant would differ if the volume distribution were E self = q2/r uniform). If r goes to zero. ESelf g oes to infinity. In the fluctuon picture the particle is an absorber. The fluctuon density expression for charge [equation (5)) should be substituted
FItmuons-
for q in the expression fluctuation lengths, is
for
E,,if.
The
resulting
E self =
569
equation,
written
writen
in terms
of
&llC3, ~c2ii2N ’
No interaction can occur unless N 2 1. Thus EIelf must remain finite as long as Y,, remains finite. Furthermore, the amount of momentum transferred in a virtual exchange process are genera1 features, independent of any remains finite as long as r,, > 0. These assumption about the size or structure of the absorber. The reason why self-energies disappear in the fluctuon mode1 is that self-interactions do not occur, since a true self-interaction would correspond to the excluded case of N = 0. This case is excluded because the absorber obtains its fluctuation momentum from a vacuum particle that is a distance rem away, either by virtue of its serving as an initiator or terminator of a propagating fluctuon. (The absence of a self-interaction does not preclude a Lamb shift, since atomic electrons undergo a fluctuating motion by virtue of their acting as chain initiators and terminators.) 7. FINE STRUCTURE
CONSTANT
The strength with which a charged positive energy particle (a charged absorber) is coupled to the electromagnetic field should depend on the probability that it either initiates or terminates a propagating chain of transient electron-positron pairs. The fine structure constant is a measure of this coupling strength, and consequently the fluctuon mode1 can be used to build up a picture of the manner in which the fine structure constant arises. Consider an absorber and an adjacent vacuum electron. The vacuum electron undergoes either spontaneously or because it is an uncertainty fluctuation of energy AE = h/r,,, activated by a propagating chain. In the former case it is possible for a propagative chain (or fluctuon) to be initiated. In the latter case it is possible for a chain to be terminated. the probability that the absorber The fine structure constant, LY= l/137, should represent and excited vacuum electron interact either to initiate or terminate chain propagation. The probability, hence w, can be written notationally as
where S denotes the spatial range of the fluctuon at any given point in time (i.e. the spread of the transient excitation) and R denotes the average distance between the fluctuon and absorber. We can use the position-momentum uncertainty principle to estimate S. Recall that we treated the interaction between the transiently excited pair and the absorber as a scattering process [equation (lo), Part I]. The coupling between the pair and the absorber can be mediated either by its constituent electron or positron. If the electron, positron, and absorber were classical objects, scattering and some consequent transfer of momentum would occur as soon as the electron or positron makes contact with the absorber. In the fluctuon model all of the momentum of the pair must be transmitted to the absorber if any such ‘contact’ is made, since transfer of momentum entails annihilation of the pair. Also, contact must be defined quantum mechanically, in terms of the positional uncertainties of both the electron and positron. The spatial range of the pair may accordingly be taken as the sum of the positional uncertainties for the electron and positron, each denoted by x,. Thus
M. COYHAD
570
where the mass assignments (m, to both the electron and positron) are required for consistency with the fluctuation energy, AE,, = 2m,c2. The energy-momentum relation, AE = cp, is applicable since the fluctuon propagates with velocity c, hence the electron and positron must have zero rest mass in their fluctuonic mode (recall that the rest mass assigned to the electron in its unmanifest state is a formal property). The range 2Ax, is equivalent to the sum of the Compton wavelengths of the electron and positron (i.e. 2Ax r = 2& = 2h/m,c). Here il, is the portion of de Broglie wavelength (or equivalently, of the Compton wavelength expressed as & = h/m,c) that can contribute significantly to the probability of scattering. The average distance, R, between the absorber and the transiently excited pair depends on the radius of the pair. This is because the maximum likely distance between the absorber and either constituent of the pair is equal to its diameter, while the minimum distance is zero (since the absorber and vacuum electron are neighbors). The radius of the pair can be taken as the radius of the lowest energy state of positronium, giving 2h’
R = r,,(,\ = ~ m,e2
(9
’ The rationale for this identification is that the electron and positron composing a transiently excited pair are absorbers during their transient existence. The chance of initiating or terminating a chain during this short amount of time is small. The chain itself, however, comprises a sequence of pairs that as a whole will act as an initiating or terminating absorber for as many other chains as a positive energy electron and positron would over the same period of time. Consequently, the transiently excited electron and positron should on the average behave as if they have the same charge as a permanently promoted electron and positron. The average radius of the pairs should then equal the radius of positronium in its lowest energy state, as specified in equation (9). The fine structure constant is now easily obtained by substituting equation (8) and (9) into equation (7): PAX, e2 1 S a=__=_Z_Z_ (10) 137’ hC R r roa The above derivation is based on the fluctuon model, and on the scattering picture that led to the vacuum density form of the Coulomb law [equation (3)] and by implication to Maxwell’s equations (through relativistic transformations). It did not, however, utilize these equations and therefore did not directly utilize the main feature of the theory, namely that of coupling the fluctuation length rem is also a measure of field strength. The dependence strength on cr must therefore be consistent with its dependence on Y,,,. To investigate this question we can set 4 equal to e and use equation (5) and (10) to calculate
(11) This agrees with the fourth power dependence of the Coulomb force (or field strength) on fluctuation length that is explicitly exhibited in equation (16) of Part I. Thus the fine structure constant can be interpreted as a measure of fluctuation length in the electromagnetically active subsea. The r& dependence of N appears to disagree with the rz, dependence of the probability of chain initiation or termination exhibited by equation (11) of Part 1. To elicit the reason for this we can write, from equations (10) and (ll), e=_
: Akzrz,, hc
.
(12)
Fluctuons-II
571
This is the same as equation (5), except for the replacement of q by e and the conversion from fluctuation density to fluctuation length. These replacements make it clear that charge scales as the square of fluctuation length. From this it immediately follows that cr scales as since it scales as e2 and e scales as rf,. Our model of the fine structure constant rZmr actually entailed the interaction of three charges: the transiently excited electron, the transiently excited positron, and the absorber. This implies a probability of coupling that scales as r-2, rather than r&. However, the l/r2 character of the electromagnetic field attached to the absorber attentuates this coupling by l/r& (since l/r = l/N*r&,). The r#&, dependence of the probability of chain initiation or termination should thus be interpreted as the dependence of electron charge on fluctuation length. The contribution of the third charge and the inverse square falloff are for this reason requisite to a consistent analysis of the fine structure constant, despite the fact that they are self-canceling. Expressing Let us now consider what happens to CYif r,, either increases or decreases. the radius of positronium [equation (9)] in terms of the fluctuation length expression for e [equation (12)], 2h4C2 rpos =
(13) Aemmerzm’
in accordance with l/m,r&,, as expected. If the Thus rpos decreases as rem increases increased, the diameter of the electromagnetic average value of rem in the universe fluctuon (i.e. of the excitation chain corresponding to the photon) would decrease rapidly. The fine structure constant would increase with equal rapidity, due to the fact that the average distance between the absorber and either the transiently excited electron or positron would decrease. This would be the case even if m, increased, which would happen to an increase in rgrav. Such an increase in m, if the increase in r,, were concomitant would not affect the fine structure constant, since it would have an equal shrinking effect to our earlier observation on both rpos and the spatial range, S. This can be compared (Part I, end of Section 9) that the probability that two absorbers interact increases with an increase in the average fluctuation length while the momentum carried by the propagating fluctuons that mediate the interaction decreases, and does so in a manner that allows the This is consistent with the fact that the fine prefactor A em to be treated as a constant. structure constant is independent of m,. The actual diameter of positronium (2r,,,) is larger than the distance between two vacuum particles as defined by the fluctuation energy required for pair production. Recall that the fluctuation energy for an electron-positron pair is AE,, = 2m,c’. From equations (la) and (3) of Part I, we can also write AE,, = h/t,, = k/r,,. Equating these two expressions for A E,, yields r,, = h@m,c, and hence the ratio 2r POS z
rem
8hC
_
e2
=
8a-l
’
(14)
The diameter of positronium is thus about 1096 times larger than the fluctuation length as defined by the vacuum density. This disparity is connected with the compression-expansion waves that compensate for the variations in vacuum density induced by mass and charge. The average diameter of the transient positronium that carries the excitation energy extends over many vacuum electrons, as does the wavelength of the compression-expansion wave. The uncertainty and conservation principles require that the positronium structure, whether viewed in particulate or wave terms, collapse back into a vacuum state in such a way as to transmit its energy to an immediate neighboring particle, where neighboring is defined by the fluctuation length rem = CT,,.
572
M. CONRAD
The neighboring particle can either be a vacuum electron or a charged absorber. The fine structure constant plays no role in the absence of an absorber, since the transfer of energy involves no element of probability (its nonoccurrence would violate the uncertainty and conservation principles). This is not the case if an absorber is present, since the absorber will compete with vacuum particles for absorbing the excitation energy. The fine structure constant then controls the coupling, since the probability that the absorber ‘wins’ depends on the spatial range of the transiently excited electron and positron relative to their average distance from the absorbing particle. If the absorber ‘loses’, the fluctuation energy must be transferred to a neighboring vacuum particle within a time interval equal to the initial fluctuation time (assuming the chain was initiated elsewhere). For this to occur the neighboring vacuum particles in the immediate region of the absorber must undergo sufficient interval compression or expansion to compensate for the fact that the absorber is a lacuna in the vacuum sea that is either depressed or elevated (depending on the sign of the charge). The depression or elevation corresponds to field, and the absorber to a discontinuity in the vacuum density. The coupling between the absorber and spontaneous excitation of neighboring vacuum particles is also controlled by the fine structure constant. The competitive processes in this case are initiation of chain propagation and disappearance of the vacuum fluctuation without initiation of chain propagation.
8. CONSTANCY
OF CHARGE
Finally, we must consider the justification for treating electron charge as a constant [i.e. replacing q by e to obtain equation (ll)]. If pe,,, could change on the average, the assumption of constancy would break down since the force between two electrons depends on the intervening density of vacuum electrons (i.e. since e depends in a definite way on Y,,). The neutralizing effect of positive and negative charge prevents this from happening. This ensures that the local depressions or elevations in pen, induced by charge are a characteristic feature of the universe. Consequently, the average value of pe,,, is also a characteristic feature of the universe. This will be the case as long as charge does not build up in an asymmetric manner in the way that mass does. Such buildup is also inhibited by the existence of positive and negative charge, since this ensures overall charge neutrality. Very high masses have a depressing effect on all vacuum particles. This would be associated with an increase in e, due to the attenuating effect of mass on c and to the decrease in pern [cf. equation (12)]. The decrease in pern is very much less than the decrease since it is largely corrected by charge neutralization. All else constant, in Pgrav, however, such an increase in e would increase the attractive and repulsive forces associated with electrons and positrons in the depressed region. All else is not constant, though, since this region would contain fewer vacuum electrons than the surrounding vacuum. Electrons would therefore be repelled into the region from the outside (if electrons were repelled the vacuum structure would clearly be from the lower to the higher density region, unstable). This exerts the corrective effect on the decrease of pern in the high mass region, without affecting the depression in pgrav, and by so doing pushes the electron charge back towards its standard value. The correction is not complete, however, since it is limited by the balance struck between the repelling effect of mass on vacuum electrons and the much stronger corrective effect of charge neutralization. It is also limited by the fact that pgrah controls c, which enters into the expressions for both e [equation (12)] and N [equation (ll)]. At this stage it is not possible to analyze the effect of high mass on e and (Y in a since this requires a general analysis of the relations between the precise manner,
Fluctuons-III
electromagnetic and strong self-regulatory
gravitational interactions. properties of the fluctuon
Acknowledgemenf-The preparation National Science Foundation.
of this paper
573
But again vacuum.
was supported
in part
we have
by Grant
an intimation
ECS-9109860
from
of the
the US
REFERENCES 1. M. Conrad, Force, measurement and life, in Toward CITheory of Models for Living Systems, edited by J. Casti and A. Karlquist. pp. 121-200. Birkhauser, Boston (1989). 2. W. G. V. Rosser, The Special Theory of Relativity. Butterworth, London (1964). 3. J. Tessman, Maxwell-out of Newton, Coulomb, Einstein, Am. J. Phys. 34, 1048-1055 (1966). 4. D. Frisch and L. Wilets, Development of the Maxwell-Lorentz equations from special relativity and Gauss’s law. Am. .I. Phys. 24, 574-579 (1969).
