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in the year 2000 the standard model was believed to be consistent with all known experimental facts ( except the possible existence of a neutrino mass ) . yet there are many aspects of the standard model which are assumed but not explained : the wave nature of matter , the quantization of the photon ( @xmath0 ) , and newton s second law of motion . in addition gravity is not usually considered to be part of the standard model , but a separate interaction . with the clear measurement of dark energy by the wmap collaboration@xcite , it became important to know what makes up the majority of our universe . the explanation apparently does not lie within the standard model . this paper investigates the possibility that dark energy is caused by the wave nature of matter . the resulting model apparently succeeds in explaining dark energy . it also appears to naturally explain the quantization of the photon and newton s second law , as well as provide an electromagnetic model of gravity .    according to the standard model @xcite electrons are believed to be point objects when examined on scales as small as @xmath1 m . recent limits  @xcite show that quarks are also not composites of other objects when examined on similar scales . however , we know that quarks have _ external _ structure . they are bound together by color flux tubes@xcite to form hadrons , so that free quarks can not exist . we here discuss the postulate that the electron is similarly bound to another object by a line of magnetic flux  a quantized vortex in the electromagnetic field . we suggest that this line may be key to understanding the full meaning of spin . since we model the structure of the electron , it is possible to attempt a calculation of the properties of the electron in terms of planck s constant @xmath2 , the electron charge @xmath3 , and the distribution of matter in the universe . this is a _ classical _ model in the sense that we attempt to follow the detailed motion of vortex - lines as a function of position in space and time . the detailed motions of lines are then averaged to find the behavior of waves and particles  normally discussed in quantum mechanics . we do not follow the copenhagen interpretation@xcite of quantum mechanics , but rather take the view , in sympathy with einstein@xcite , that it should be possible _ in principle _ ( though it would take a rather large computer ) to predict the future course of the universe given _ perfect _ knowledge of the positions and velocities of all lines ( and other objects  hadrons ... ) at a given time . thus , in this view , indeterminacy is not intrinsic but results from measurement uncertainties . some authors@xcite would call the positions and velocities of these lines _ hidden variables_. these quantities are difficult , but not impossible , to measure . of course , all measurement uncertainties are subject to limitations from the heisenberg uncertainty principle , but discussion of the motions of the lines is useful because it can provide underlying structure to relate and interpret more measureable quantities . this model is a classical model of a quantized superconducting object . hence it is intrinsically a quantized model in which e.g. newtonian gravity is obtained as an average over the line - pair distribution . the individual quantum fluctuations are automatically built in . the classical solution for the motion of electrons and lines appears much like in standard quantum theory . this model is based on the quantization of magnetic flux in free space ( discussed in section  [ ssec - fq ] , and which we claim was first derived by dirac@xcite ) , which follows from the requirement that wave functions be single valued . we take this result from quantum mechanics and use it as a general principle which reaches its most general form in equation  [ eq - quantization ] , which implies the relativistic london equations  [ eq - london ] for our superconducting fields . we make a major assumption in section  [ ssec - electroncharge ] about the origin of electric charge . aside from this quantization condition and our assumption about electric charge , this classical model of a quantized object is based on the principle of least action , or hamilton s principle . the principle of least action leads to the microscopic maxwell equations which we take to be valid at arbitrarily small length scales :    @xmath4    @xmath5    @xmath6    @xmath7    we generally follow the conventions of jackson@xcite . the electric and magnetic induction fields can be defined in terms of the vector and scalar potentials @xmath8 @xmath9 which we presume to satisfy the lorentz gauge condition @xmath10 so that the four - vector potential and current density take their standard form @xmath11 @xmath12 for @xmath13 . however , we deviate from the standard assumptions by asserting that gauge invariance is broken and the electromagnetic four - vector potential @xmath14 , at least sometimes , is directly measureable . gauge invariance is already asserted @xcite to be broken in superconductors , and this fact is quite relevant to our present dicussion . we prove below ( section  [ ssec - static ] ) that in a very important special case , our line of flux , the vector potential is unique . our default reference frame uses flat minkowski space referred to the local rest frame of the universe . this is defined to be the frame in the region of our solar systsem in which the 2.73k black - body radiation is isotropic@xcite . we claim that this is a classical model even though it starts with the dirac quantization condition  a quantum mechanical result which implies the structure of our line of flux . since electron wave functions in standard quantum mechanics have infinite extent , the only way in which they can be single - valued in the presence of magnetic flux is if the lines of flux are quantized and filamentary . this implies that the magnetic flux in the lines must be contained ( equation  [ eq - ampcirc ] ) . since normal charged particles are not sufficient to contain this flux in general , the most natural assumption we can make is that the flux is contained by a superconducting line , with charge carriers of a new type . if we take this model seriously , it pays great dividends . we presume that one end of a line is an electron , the other a positron ( or ends in a proton ) . thus space - time is crisscrossed by many such lines ( in pairs - section  [ ssec - overview ] ) . then , in turn , all fundamental quantization conditions  including the dirac quantization condition , the quantization of the photon , and the quantization of matter waves  can be understood from the fact that space - time is a multiply connected superconductor ( equation ( [ eq - multisup ] ) ) . this , in turn , implies quantum mechanics itself , when the interaction of a bare electron with the flux lines is taken into account ( section  [ sec - quantmech ] ) , and appropriate averages are taken . thus , the dirac quantization condition , our starting point , is understood at a deeper level and , in fact , implied by our classical model in a self - consistent way . the classical model implies quantum mechanics , but the reverse is not true since the _ classical _ model is at the deeper level . ( we , of course , take the value of @xmath2 from experiment and we define _ classical _ wave functions which are _ not _ continuous , before smoothing to find the quantum mechanical wave functions . ) we suggest that at the birth of quantum mechanics the problem which the founding fathers solved was not due to the inadequacy of classical mechanics . they had the wrong model of the electron . it s not a point particle , but rather the end of a string . since our method is totally new , it seems appropriate to summarize the highlights of our model to orient the reader  to let the reader know what we will claim :    1 . our model , many superconducting line pairs crisscrossing in space - time , naturally explains six dominant aspects of everyday physics which are either unexplained or questionable in the standard model : * quantization of the photon ( section  [ ssec - quantphoton ] ) and matter waves ( section  [ sec - quantmech ] ) : @xmath0 * wave - particle duality ( section  [ sec - wpduality ] ) and the wave nature of matter ( section  [ sec - quantmech ] ) * gravity ( section  [ sec - gravity ] ) * newton s @xmath15 law of motion ( section  [ sec - newton ] ) :  @xmath16 * mass of the electron ( section  [ ssec - masselectron ] ) * spin of the electron ( section  [ ssec - spinelectron ] ) and spin of the photon ( section  [ ssec - spinphoton ] ) 2 .   our model gives a natural explanation of dark energy ( section  [ ssec - darkenergy ] ) . the line pairs are under tension or negative pressure ( section  [ ssec - tensiondarkenergy ] ) . we have modeled the propagation of the photon as a helical wave on the two lines of a pair ( section  [ ssssec - helicalphoton ] ) . after applying the photon quantization condition and making a binding energy correction ( section  [ ssec - energyphoton ] ) , we calculate the spin of the photon ( section  [ ssec - spinphoton ] ) to be @xmath17 with no free parameters . our natural model of the bare electron ( end of a line of flux ) has the pole field of a dirac monopole plus a line field ( inside a superconducting sheath ) . after considering interaction with the surrounding line pairs , the motion of this bare electron is _ zitterbewegung _ ( section  [ ssec - motionbpp ] ) , very rapid motion near the speed of light in rapidly changing directions . the line pairs completely screen the pole field due to the quantization of magnetic flux so that the magnetic induction field reaching macroscopic distances is zero . 5 .   in order to agree with experiment we must add the electric charge of an electron by hand ( section  [ ssec - electroncharge ] ) to our model of the bare electron . this is consistent with the principle of least action and does not change the character of the zitterbewegung motion ( appendix  [ ssec - motionelectron ] ) . the unnatural part of this addition is that the electric charge and effective magnetic charge of the electron must have their relative separation _ fixed _ at a very small distance parallel to the line . this direction is the direction of the spin of the electron , and the magnitude of the spin is calculated to be @xmath18 . 6 .   using a toy version of our model ( which we hope to be accurate within a factor of 10 ) , we have calculated formulas for newton s constant @xmath19 ( equation ( [ eq - g ] ) ) and the mass of the electron ( equation ( [ eq - me ] ) ) , containing some parameters whose values we do not currently know . we have also calculated the dark energy density of the universe . we then combine measurements of the dark energy density , @xmath19 and @xmath20 to calculate the line pair density near the earth ( @xmath21 ) ( equation ( [ eq - fulldensity ] ) ) , which is consistent with known constraints , and a factor of about @xmath22 larger than would be expected without inflation . our model predicts that the gravitational force will reverse under conditions of very high density . at least heuristically , this could explain inflation and agn jets . we offer a new explanation of the pauli exclusion principle . we need to do this since our model ties a string on every electron in the universe , and therefore electrons are _ distinguishable_. the explanation occurs naturally in the model . we offer a new explanation of the quantum hall effects simply by suggesting that the flux lines seen in these effects are real . this model amounts to a classical field theory of interacting strings ( or lines ) . the assumption that lines will condense into pairs avoids having to solve the general case . we presume that intercommutation is not an issue since all excitations are quantized and excitations with a small wavelengh must have a high energy . since this model implies quantum mechaincs and @xmath16 for the electron , differences from the standard model should be hard to find . ( but see section  [ sec - relevantexperiments ] . ) in order to restrict the scope of our discussion , weak and strong interactions are essentially not discussed . we use the word @xmath23 in the title of this paper with some trepidation , because it is an anathema to most physicists today . if the reader prefers he / she can substitute a euphemism such as @xmath24 . but we have apparently succeeded in describing the photon propagating as a quantized electromagnetic wave on one of our line pairs  which was the original function of the ether . we have also apparently succeeded in modeling dark energy , a uniform energy density in space which is widely believed to be inconsistent with the standard model . this real , measureable energy density is not consistent with the idea of @xmath24 . in section  [ ssec - overview ] we address einstein s reasons for rejecting the ether concept .    but ( using the language of string theory ) , we may have found the  vacuum selection principle"@xcite relevant to our universe . it makes sense that only electromagnetic strings exist outside nuclei , because the nuclear force has a short range . our selection principle is that every neutron , or electron - proton combination , is the source of a pair of flux lines , probably stretching out to great distances in all directions . our model gives a reasonable description of dark energy , gravity and the generation of mass without scalar fields . newton s second law of motion , @xmath25 , is valid only in _ inertial _ frames . the fact that inertial frames at a given point all move with constant velocity relative to each other implies that acceleration has absolute significance in our universe , unlike velocity for which only relative motion has significance . with respect to what must acceleration be defined ? mach pointed out@xcite that the mass of the earth and the other celestial bodies must determine the inertial frames . inertial frames near the earth do not seem to accelerate with respect to the average mass of the universe . this paper was stimulated by the fact that somehow information on the the average local rest frame of the universe must be present _ at _ each object , to act as its absolute reference for acceleration . this information could not be carried by a smooth gravitational field . if we were far from any stars , the newtonian gravitational field would add to zero in a symmetric universe . somehow the gravitational field of the universe must have _ structure_. it must be quantized . following dirac@xcite we consider a region of space containing a wave function of a particle of charge @xmath26 . if a nodal line appears in that wave function , this nodal line can contain magnetic flux as long as the wave function remains single valued@xcite in the presence of the flux . this requires that    @xmath27    where the path of the integral includes the line . this implies the quantization of magnetic flux in the line    @xmath28    where @xmath29 is the quantum of flux@xcite and @xmath30 is an integer . this condition is usually@xcite taken to imply that charge is quantized if magnetic monopoles exist . we reason that electron wave functions fill the universe and therefore we take this condition to imply the quantization of magnetic flux in free space@xcite ( with _ q = e _ ) , which , as we will see , leads in a fairly natural way to the quantization of not only electric charge , but also spin angular momentum  even though true monopoles do not exist , at least in this model . ( the experimental situation on this point is discussed in section  [ ssec - existing ] . ) figure  [ fig - line ] shows the building block of our model  a line of flux@xcite  static , for the moment . we only consider the case @xmath31 in equation ( [ eq - fq ] ) . here we consider the case @xmath32 appropriate if the wave function is that of an electron ( charge @xmath3 ) . magnetic induction field enters through one pole ( labelled @xmath33 ) , continues through the line and exits through the other pole ( @xmath34 ) . the total flux exiting @xmath34 is @xmath35 , where @xmath36 is the strength of a dirac monopole@xcite . hence the line of flux is a dirac string . dirac introduced a singularity to exclude the line from physical reality so that the pole is a monpole . here we _ include _ the line in our picture of reality and attempt to interpret one end of the line as an _ electron _ and the other as a _ positron_. since we assert that gauge invariance is broken , the position of the line is observable@xcite . note that the total magnetic flux through the positive pole is zero , since we must include the line field as well as the pole field . we will speak of the total configuration as a _ vortex_. it consists of one line field and two pole fields , one of each sign . the _ line _ is a _ string _ and we will use both terms , depending on the context . figure  [ fig - vortex ] shows the vector potential associated with a static vortex / line of flux and indicates why we call it a vortex . the differential dipole magnetic moment of an element of the line is given@xcite by    @xmath37    where @xmath38 denotes a point on the line and @xmath39 points in the direction of the magnetic induction field along the line ( which we call @xmath40 ) so that the total field of the vortex at the field point @xmath41 is given by    @xmath42    taking the curl of @xmath43 we find the magnetic induction field of the vortex    @xmath44    to 2.6 in     to 2.0 in     to 1.5 in     where @xmath45 and @xmath46 are the positions of the negative and positive poles at the ends of the line . the two - dimensional dirac delta function operates in the plane perpendicular to @xmath47 . the line is presumed to have no sharp kinks , but it can bend smoothly in an arbitrary manner . in the particular case that the static line is straight , the vector potential from the vortex at the field point @xmath41 is given by    @xmath48}\ { \hat{\phi } } \label{eq - astline}\ ] ]    where @xmath49 and @xmath50 are the polar angles from the positive and negative poles to the point @xmath41 respectively , as shown in figure  [ fig - geom ] , where @xmath51 is the cylindrical radius to point @xmath41 from the line ( or extension of the line ) , and @xmath52 is the azimuthal unit vector . lamb@xcite has summarized the classic work on vortices in the 1800 s , principally by helmholtz@xcite and lord kelvin@xcite . most of this early work was stimulated by the circulation theorem of helmholtz , which showed that in a perfect ( nonviscous and incompressible ) fluid , the circulation of a vortex @xmath53 is an absolute constant , where * v * is the fluid velocity and the path of integration includes the vortex and moves with it . hence vortices retain individual identity so that kelvin proposed@xcite that  vortex atoms " might be the building blocks of matter ( 30 years before the discovery of the electron ) . the quantized vortices discussed in this paper bear strong resemblance to the classic vortices with the identification that the vector potential @xmath54 corresponds to the classical fluid velocity @xmath55 and that the magnetic induction field @xmath56 corresponds to the classic vorticity ( @xmath57 ) . the circulation around one of our vortices @xmath58 ( far from the ends to avoid flux returning from the poles ) is quantized , implying that they also retain a separate identity and are good candidates for building blocks . the classic vortices could not end , so that they either had to form rings or end on the walls of the container . our vortices , in the terminology of the classic works , actually do not end either . since the fields of our vortices satisfy the maxwell equation @xmath59 everywhere , technically our vortices do not end at the poles . their cross sections just become very large at the poles , and the magnetic field extends to infinity in all directions . there are two major differences , however , between our quantized vortices and the classic vortices . since our vortices are formed in the electromagnetic field , they have no rest mass and do not sense pressure in the normal way , except at the poles . ( see section  [ sec - pauli ] . ) consequently we must develop new techniques to follow the motions of our vortices .      since we wish to consider the energies of the lines of flux , the singular behavior of the magnetic induction field ( equation ( [ eq - singular ] ) ) in the filamentary model is unacceptable . we need to integrate @xmath60 . consequently we develop a model of a line of flux with a finite radius @xmath61 , called the _ line radius _ , which we take to be constant . for the moment we consider only a straight line far from its ends . we simply use a model of the abrikosov vortex  @xcite which has been successful@xcite in modeling the vortices in type ii superconductors @xmath62 where @xmath63 is the hyperbolic bessel function of the second kind@xcite of order @xmath64 . the line radius plays the role of the penetration depth in a type ii superconductor . since the magnetic flux is confined , there must be a current distribution associated with the line . in the static case , from equation ( [ eq - ampcirc ] ) we find @xmath65 where @xmath66 is the hyperbolic bessel function of the second kind of order 1 . we assume that this current density is _ superconducting_. it is associated with the magnetic field at zero voltage , as in a type ii superconductor . we assume the superconducting region around the line extends to a distance @xmath67 from the centerline , where @xmath68 , so that the flux is contained to any required accuracy . we will try to measure @xmath69 . we assume that the current density satisfies the london equations@xcite since , if a coherence length is relevant to this situation , we take it to be negligible . we treat this current density as a _ field _ , so we can take its charge carriers to be massless ( section  [ ssec - superconfield ] ) . hence the current density is a new concept since it can not be due to the motion of known charged parcicles ! these assumptions seem required by our main point , that , following ( [ eq - fq ] ) , magnetic flux is quantized in free space . we will see that every wave function contains such lines , and therefore all wave functions superconduct . this last statement is not surprising ; it is just diamagnetism@xcite . hence the surprising assumptions which we have been led to make in this section have one reasonable consequence . we need to see what else is implied .    with this magnetic induction field , or magnetic flux density , the total flux through the line at a given point on the centerline is @xmath70 as required . the self - energy per unit length of the line is given by @xmath71 where @xmath72 is the tension in the line . the differential magnetic moment of an element of the line is given by @xmath73 also as required . following the expected behavior@xcite in superconductor we assume a trial solution for the vector potential given by @xmath74 taking the curl of @xmath75 , we indeed find equation ( [ eq - bline ] ) . but our trial solution has the wrong behavior for large @xmath51 ( see equation ( [ eq - astline ] ) , for points far from the ends ) since @xmath76 for @xmath77 . in order to obtain the proper solution we must add a field with zero curl ( except at @xmath78 ) @xmath79 which gives the correct long - distance field of the line , so that the total field is given by @xmath80 the curl of ( [ eq - aunique ] ) gives the correct magnetic induction field of a static line ( [ eq - bline ] ) at all points@xcite ( not near the ends of the line ) . hence ( [ eq - aunique ] ) gives the _ unique _ solution for the vector potential of the line ( for n=1 , not near the ends ) . it is determined by the azimuthal symmetry of the line and the quantization condition ( [ eq - fq ] ) .      actually equation ( [ eq - singv ] ) is a special case . the requirement that a particle s wave function be single valued at a fixed time is stated more generally by @xmath81 where @xmath82 is the total momentum of the particle , and where we use a notation common in the case that the wave function represents superconducting pairs of electrons . if we extrapolate from the case of superconducting pairs of electrons @xmath83 ( where @xmath84 is the number of superconducting pairs per unit volume ) , to our line current density in terms of the penetration depth@xcite @xmath85 ( which has been identified with the line radius ) we find @xmath86 and hence @xmath87 from ( [ eq - aunique ] ) @xmath88 we can find the generalization of @xmath89 to our case @xmath90 which we call the _ phase field_@xcite of the vortex , because , in the case of our vortex , it does not behave like a momentum . it is independent of both @xmath91 and @xmath84 . we associate this phase field with the vortex supercurrent wave function . we say that this new type of current is due to _ aon _ superconductivity , in order to distinguish it from superconduction by cooper pairs . we will see that this new phenomenon is completely characterized by the line radius , which we call _ evidently , in our model of the core with a finite size , the phase field still has a filamentary curl , and in the general static case it is given by ( see equation ( [ eq - agenline ] ) ) @xmath92 which is just the law of biot and savart .    a supercurrent ( i.e. aon ) wave function will be single valued as long as @xmath93 throughout the region of interest . in general , using the line of reasoning for equation ( [ eq - singular ] ) , the curl of the phase field of a single static vortex not including the ends ( where @xmath89 is not smooth ) is given by @xmath94 the line field itself is quantized precisely so that its supercurrent wave function encircling the centerline is single - valued . since the curl of the phase field is filamentary , there is no region of space in which the wave function of one line is not single - valued and therefore two lines ( otherwise isolated ) can move through each other driven by their magnetic interaction  with no extra currents ( quantization currents ) needed to make their wavefunctions single - valued@xcite . we will see ( section  [ ssec - effectivemagcharge ] ) that this is _ not _ true for the ends of a line . the london equations@xcite may be expressed very simply in terms of the phase field @xmath95 @xmath96 where @xmath97 is the 0-component of the 4-vector @xmath98 and @xmath99 is the 4-velocity ( extrapolating from the case of superconducting pairs of electrons ) . the relativistic form of the london equations @xmath100 just implies that @xmath101 the phase field can be expressed as the gradient of a scalar ( @xmath17 times the phase ) . evidently the center of the vortex is outside the superconducting region , as in a type ii superconductor . this implies a phase transition in the center of the vortex . if we allow time variation , the requirement that a wave function be single - valued can be written in all generality @xmath102 so that @xmath103 the london equations imply that this requirement is automatically satisfied @xmath104 _ if _ the path and surface are totally in superconductor . ( we use stokes theorem in four dimensions and @xmath105 is an element of the area enclosed by the path . ) we consider the interaction of two static lines ( far from their ends ) , first in the case in which the magnetic induction fields of both lines are aligned with the @xmath106-axis . the magnetic energy per unit length is    @xmath107    @xmath108    @xmath109    where @xmath110 is the interaction energy . this interaction energy is easily calculated @xmath111 ( where @xmath112 locates the center of line @xmath113 relative to line @xmath114 ) using the expression obtainable from jackson@xcite    @xmath115i_m(\rho_</a)k_m(\rho_>/a )   \label{eq - k0fact}\ ] ]    for writing @xmath116 in completely factorized form . here @xmath117(@xmath118 ) is the greater ( lesser ) of @xmath51,@xmath119 . if @xmath120 and @xmath121 we obtain @xmath122\ \ .\ ] ] recognizing a wronskian determinant@xcite @xmath123\ = \ + { 1 \over { u^\prime } } \label{eq - wronskian}\ ] ] we find that the interaction energy is given by @xmath124 and the energy per unit length of two aligned lines is @xmath125                                          \ = \ { { g^2}\over{a^2}}\ \bigl [ 1+{{\rho^\prime}\over a}k_1({\rho^\prime}/a ) \bigr ] \ \ .\ ] ] since @xmath126 as @xmath127 , the total energy of two aligned lines is four times as great as the energy of one line ( see equation ( [ eq - t ] ) ) , as expected . hence two lines of like orientation _ repel _ each other .    from the above discussion , two lines with opposite orientation will attract each other . the interaction energy simply reverses sign . @xmath128                                          \ = \ { { g^2}\over{a^2}}\ \bigl [ 1-{{\rho^\prime}\over a}k_1({\rho^\prime}/a ) \bigr ] \ \ . \label{eq - twolinepot}\ ] ] the minimum of the twoline potential energy occurs at @xmath129 , and has the value @xmath64 . the two line self energies are cancelled by the interaction energy . since the lines are fields , the cancellation is complete  except , of course , for excitations of the twoline system . we note that the annihilation of two oppositely directed vortices has been observed@xcite in type ii superconductors . at the birth of the universe we envision a very large concentration of energy in the universe in accord with standard ideas of the hot big bang model@xcite . we envision that the energy is originally in the form of continuous quantized vortices ( lines ) which have no ends . when sufficient energy is available locally , some lines break  creating electron - positron pairs . as the universe continues to expand , oppositely directed lines are attracted to each other and eventually bind . initially excitations of the twoline systems are very large and lines frequently switch partners . but as the universe cools excitations become smaller and smaller , and in our current universe the line pairs are assumed to be condensed with an energy very close to zero ( equation ( [ eq - twolinepot ] ) ) . by this assumption we avoid having to solve the general case .    in order to limit the scope of our discussion , weak and strong interactions will not be discussed more than is absolutely necessary . presumably cooling processes occurred with these interactions , which could be similar to the processes which we will discuss . but , first we need to mention how protons and neutrons fit into our picture . noting the decays @xmath130 @xmath131 ( the latter occurring only in nuclei ) we presume that if a negative electric charge ( electron or antiproton ) is at one end of a line , any positive charge ( positron or proton ) can be at the other end . from ( [ eq - pton ] ) we infer that the proton must have the same type of line as the positron ( defined by positive charge ) , and from ( [ eq - ntop ] ) we infer that the neutron has _ both _ types of line , i.e. contains the end of a line pair . thus in today s universe , essentially all lines are bound in pairs , with the upper limit on the number of unbound lines set by the neutrality of matter . the stability of these line pairs is discussed in section  [ ssec - tensionpair ] . from ( [ eq - ntop ] ) we presume that one line pair can end at a hydrogen atom or at a neutron . the ends of the lines near the earth are bound together in atoms . in stars significant ionization occurs , but from the net neutrality of plasma matter in stars and energy considerations , it is reasonable to believe that the two ends of a line pair do not become greatly separated . the threshold for stringbreaking ( @xmath132 ) is barely approached in the cores of the most massive stars@xcite . our picture offers hope of explaining the evolution of the universe from the big bang to the present day@xcite , without adding any scalar fields . the earliest epoch is thought to be dominated by strings  which may correspond to our lines . the evolution of this model seems natural because the strings are still here ! we suggest that inflation was due to the reversal of the force of gravity under the very high densities of the early universe ( which we will discuss ) , when our line pairs were radiation dominated and pressures were naturally positive . in recent times , when the line pairs are cold , their tension ( negative pressure ) dominates and they become the source of dark energy which may explain the recent acceleration of the universe . photons are the differential excitations of a twoline system , and gravitational waves are common mode excitations of a twoline system ( differs from the standard model ) . the higgs field of the standard model is replaced by the field of line pairs . the concept of an ether was rejected in the early part of the @xmath133 century for many reasons . we quote einsteins@xcite reasons for rejecting the ether concept , along with our reply to each point for why we now think the concept is useful :    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  if light was to be interpreted as undulatory motion in an elastic body ( ether ) , this had to be a medium which permeates everything ; because of the transversality of the lightwaves in the main similar to a solid body , yet incompressible so that longitudinal waves did not exist . "    : in this model the ether does permeate essentially everything but it is not a solid body , but rather a mesh of lines . light waves are only transverse because they are waves on a string ; they are not bound longitudinally .     this ether had to lead a ghostly existence alongside the rest of matter , inasmuch as it seemed to offer no resistance whatever to the motion of  ponderable `` bodies . ''    : the ether is  ghostly " because it is made of magnetic fields . it does offer resistance to the motion of massive bodies , because we postulate that it is the origin of mass . it offers resistance to acceleration , but does not offer resistance to velocity in accord with newtons @xmath15 law , and special relativity .     in order to explain the refraction - indices of transparent bodies as well as the processes of emission and absorption of radiation , one would have had to assume complicated reciprocal actions between the two types of matter , something which was not even seriously tried , let alone achieved . "    : there is only one type of matter . electrons are literally connected to the ether in this model . if an electron accelerates it causes a wave on a string . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _    it should be noted that there is a great similarity between maxwell s _ molecular vortices_@xcite with which he constructed his model of the ether and the quantized vortices considered here . there is a big difference , however . following the prejudices of his day , maxwell attempted to explain electromagnetism in terms of mechanics . we , however , attempt to explain mechanics in terms of his electromagnetism . it is necessary to estimate the average length of a line pair , @xmath134 , in the universe today . in the early universe there were two competing processes that determined the length of the average line . during inflation the velocity of separation of the two ends of a line exceeded the velocity of light , perhaps greatly@xcite . this expansion competed with the breaking of lines  forming an electron - positron pair at each break . because of these two processes there will be a statistical distribution of line lengths about some mean , @xmath134 . in order to avoid dependence on a specific model of inflation we simply take , as a first guess , @xmath135 @xmath134 to be the distance to the horizon ( where @xmath136 is the time of the big bang ) . within a minute@xcite after the big bang both rapid inflation and string - breaking are expected to have stopped , so that the lines ( now in pairs ) have simply expanded with the universe since that time . thus we might expect our estimate for @xmath134 to be close in the absence of inflation . in this sense we view it as a _ minimum _ value . the value of @xmath134 is a major uncertainly in these considerations and we will attempt to measure it .      according to the picture we have developed , essentially all lines are paired and , at least near the earth , they start and end at atoms . there is on average throughout the universe about one hydrogen atom ( or neutron ) per cubic meter@xcite , so this minimum average length of line pairs per unit volume is @xmath137 we associate each atom with one - half of @xmath134 since a line pair has two ends . we assume all points in the universe are equivalent and we eliminate edge effects by taking the universe to be very large . taking the time since the big bang to be @xmath138 years we have @xmath139 and @xmath140 we refer to this quantity as the line - pair density of the ether . if the number of line pairs per unit area per unit solid angle is denoted by @xmath141  , it can be shown that @xmath142 if the distribution of line pairs is isotropic , and that @xmath143 where the solid angle of line pairs only changes from @xmath64 to @xmath144 to avoid double counting . ( we also refer to @xmath145 and @xmath146 as line pair densities . ) thus within the first bohr orbit of a hydrogen atom , with the above minimum value for @xmath134 @xmath147 there are about @xmath148 line pairs . however , within the radius of the proton ( @xmath149 as determined by electron scattering@xcite ) there are only @xmath150 about @xmath151 line pairs . ( in section  [ ssec - estimatelpd ] we will attempt to measure @xmath152 and tentatively show that it should be increased over our assumed value ( [ eq - lave ] ) by an estimated factor of @xmath22 , presumably as a result of inflation . but since this inflation factor is quite uncertain , we prefer to show these minimum estimates here . ) we note that the net flux of line pairs is always zero because no net direction is defined . if , however , the ether could be completely polarized ( elimintate lines with * b * pointing from right to left ) , the magnetic induction field would be tremendous ( with the above value of @xmath152 ) . @xmath153 @xmath154 @xmath155 this field indicates how cold ( condensed ) our universe has become . magnetic induction fields of this strength are thought to exist@xcite in neutron stars . a condensed line pair ( far from its ends ) has little energy density and no net magnetic field . how can it be detected ? it can respond to a test particle ( electron ) which does have an electromagnetic field , and cause it to have mass .    the quantum nature of this field of line pairs should be emphasized . essentially every one of these line pairs ends at a particular atom , or at least near a particular location , someplace in the universe . since the universe is homogeneous and isotropic on the large scale , the field of line pairs , reflecting the universe as a whole , should also be homogeneous and isotropic as well . but , for instance , if a star approaches the field point under our consideration , the line - pair field will vary both in time ( as the star approaches ) , and direction ( strongest toward the star ) . the star may be assumed to have one line pair per nucleon stretching off to great distances . as a first approximation , we assume the line pairs stretch out isotropically from a star . ( see section  [ sec - rotations ] for a discussion of rotation . ) the density of line pairs is obviously quantized , but with the above value for @xmath145 , the density of line pairs is sufficiently large for most purposes that we can average over the quantum fluctuations of this distribution , and treat it as a smooth field . our superconducting fields are _ aon _ superconductors . they become polarized in an electric or magnetic field but do not conduct electrons or cooper pairs  since the electric and magnetic fields of the electron can not exist inside the superconductor . ( we will show ( [ eq - poleforce ] ) that , at least near the earth , the pole field of the electron dominates at close range and repels superconductor . ) hence the line pairs do not short out all voltage differences in the universe . the density of line pairs in the ether is very large and we must remember that these pairs are _ superconducting_. hence the line pairs certainly have effects on each other . the line pairs cross and hence form closed superconducting circuits which cause the quantization of magnetic flux . we will show that these circuits also cause the quantization of the photon . _ because _ of these quantization effects it is reasonable to presume that cold , condensed line pairs can move freely through each other , i.e. intercommute freely , once this quantization is taken into account . we have seen that quantized lines can move through each other ( section  [ ssec - phase ] ) affected only by their magnetic interaction . excitation energies are quantized and low energy excitations ( such as photons ) have very long wavelengths , which do not couple well to objects as small as another line pair ( smaller than an atom ) . with this presumption we avoid a problem which would otherwise cause great difficulty@xcite , especially with the above density of line pairs ( equations ( [ eq - dldvnum ] ) and ( [ eq - dndanum ] ) ) . we presume that intercommutation was important only in the early universe , and perhaps in the interior of stars . we will show that _ all _ excitations of a line pair obey the photon quantization conditions . when these conditions are satisfied , as indicated above , the line pair can _ not _ be considered isolated . supercurrent flows around closed circuits as necessary to maintain the quantization conditions . we call these supercurrents _ quantization currents_. we will also show that all line pairs in the universe ( at the same time ) have the same tension if they do not have an excitation . in section  [ sec - introduction ] we defined the _ microscopic _ maxwell equations , and until now we have had no need to deal with macroscopic electromagnetic quantities , which jackson@xcite informs us involve a spatial average over a typical length of order @xmath156 , appropriate for standard atomic matter . in our case by _ microscopic _ we mean on a length scale @xmath157 , where @xmath158 is the typical separation of line pairs . we take _ macroscopic _ to mean on a length scale @xmath159 . after inflation is taken into account we will find that @xmath158 is very small . ( see table  [ table - lpdatearth ] . ) now we will need to consider macroscopic electromagnetic fields which we define as @xmath160 and @xmath161 in order to distinguish them from the microscopic fields @xmath162 and @xmath56 . the standard macroscopic fields are : @xmath161 - magnetic induction , @xmath163 - magnetic field , @xmath164 - magnetization , @xmath160 - electric field , @xmath165 - displacement vector , and @xmath166 - polarization . the fields @xmath163 , @xmath164 , @xmath165 , and @xmath166 are only defined as macroscopic fields . an electric field will certainly polarize the ether since dipole fields will be set up on the exterior of every line pair , ensuring that the net electric field falls to zero several penetration depths inside a line pair . hence , including this polarization @xmath166 , the ether has a displacement vector given by    @xmath167    where @xmath168 is the dielectric constant of the ether . a similar expression holds for the magnetic permeability @xmath169    @xmath170    we can achieve an exact analogy to the electric polarization by rearranging ( [ eq - bmac ] ) to find    @xmath171    since a superconductor is a perfect diamagnet , the magnitization @xmath164 is opposite to the applied field @xmath56 in order to cancel @xmath56 well within the superconductor . this follows because a magnetic dipole @xmath172 produces a @xmath56-field parallel to @xmath164 , but an electric dipole @xmath173 produces an @xmath162-field _ opposite _ to @xmath166 . hence with the negative sign in ( [ eq - h ] ) we have an exact analogy with equation ( [ eq - d ] ) . electric field * e * is excluded from each superconducting line pair in ( [ eq - d ] ) , and @xmath56 excluded from the same line pairs in ( [ eq - h ] ) . hence    @xmath174    so    @xmath175    for the superconducting ether .    we can achieve a fairly realistic calculation of the dielectric constant of the ether by considering the case of normal matter where atoms are modeled by superconducting spheres of radius @xmath176 , as originally proposed by mosotti@xcite . ( he called them perfect conductors before superconductivity was known to exist . ) in this case the local field at the atom in question is given by    @xmath177    and the atomic electric dipole moment is given by    @xmath178    where    @xmath179    if @xmath30 is the number of such atoms per unit volume then    @xmath180    where    @xmath181    is the fractional concentration of superconductor . we can approximate a line pair by a row of superconducting spheres , so we will simply take    @xmath182    to be an expression for the dielectric constant of the ether , where @xmath183 is the fractional concentration of superconducting fields . this expression approaches @xmath184 as @xmath185 as it must for a uniform superconductor . there is a major difference , however , between atomic spheres , which are subject to the pauli exclusion principle ( section  [ sec - pauli ] ) and line pairs , which can move through each other . because line pairs can move through each other , they can have significant overlap . as a result @xmath186 is lower than it would be if the line pairs were impenetrable . but if the density of line pairs becomes _ very _ large , e.g. near a black hole , we expect @xmath186 to approach infinity as the ether becomes a continuous superconductor . we presume that equation ( [ eq - epsminus1 ] ) is completely general and will apply it even if the ether is not homogeneous and isotropic . this equation is particularly useful because of its simplicity .      from the macroscopic maxwell equation@xcite    @xmath187    we see that the charge on the bare positron is _ not _ @xmath188 but    @xmath189    after correction for the dielectric constant of the ether . this implies that the flux quantum is actually    @xmath190    where    @xmath191    if we use the correct value of the bare electron s charge . standard methods of measuring the flux quantum@xcite use the measured charge of the electron , so will obtain @xmath192 . since the density of the ether ( [ eq - dndanum ] ) is large , @xmath186 could be large . we assume that @xmath193 and @xmath194 are fundamental constants but that @xmath188 is _ not _ , because the density of the ether changes as the universe expands . hence in all formulas to this point in this paper we should make the replacement    @xmath195    in particular the tension of a line becomes    @xmath196    the phase field ( [ eq - ps ] ) involves the product @xmath197 , so does not change , and the fundamental quantization condition ( [ eq - quantization ] ) is not affected . hence we recognize that the large magnetic field in ( [ eq - bunbalanced ] ) could be too large if @xmath198 . but it is the field which would be measured using current techniques . the classical solution for the motion of a filimentary light string has been illustrated by goddard , goldstone , rebbi , and thorn@xcite . following this reference we take the action for a light string proportional to the area of the surface swept out by the string between fixed times . if the string is parameterized as a function of @xmath199 , determining the position along the string , and @xmath200 determining the position in time , then a general position along the string is given by @xmath201 the action is given by @xmath202 where @xmath203 @xmath204 @xmath205 and @xmath72 is the tension ( [ eq - t0 ] ) , presumed constant . for some purposes it is convenient to consider this parameterization arbitrary@xcite . it is easy to see that the action is independent of the particular parameterization . in order to consider a detailed solution , it is convenient to take @xmath206 . in this case @xmath207    ( since we choose @xmath208 , @xmath209 is _ not _ a 4-vector . ) if @xmath210 ( a function of @xmath199 ) represents length along the string @xmath211 is a unit vector . if @xmath212 @xmath213 and @xmath214 the generalized momenta are @xmath215 and @xmath216 hence , from the variation of the action , the general equation of motion is @xmath217 with the boundary conditions @xmath218 where @xmath219 is the variation in @xmath220 , which is presumed to vanish at @xmath221 and @xmath222 . the boundary conditions are satisfied if @xmath223 the ends of the line are held fixed . for some purposes this might be a good approximation since we will see that the ends of many lines are bound in matter . but in general we can not require the ends of the lines to be fixed . in the more useful case that the ends of a line are free , the boundary conditions are satisfied if @xmath224 all four of these boundary conditions at each end point can be satisfied if @xmath225 and @xmath226 . this implies that the effective tension at the end of the line is zero and the end moves with the speed of light transverse to the line . we will interpret this motion as the _ zitterbewegung_@xcite of a bare electron familiar from the theory of the dirac equation . for the moment we simply assume that the boundary conditions are satisfied so that we can proceed to discuss the motion of a line . we give a fuller discussion of the boundary conditions in section  [ ssec - motionbpp ] . it probably comes as no surprise that any wave traveling at the speed of light is an exact solution to the equation of motion of the string ( [ eq - eqofmotion ] ) . we presume that the string stretches over space in the @xmath106-direction so that it is convenient to take @xmath227 as the parameterization along the string . then a solution of general shape is given by @xmath228 where @xmath229 is a smooth function of arbitrary shape . we have @xmath230 @xmath231 ( where @xmath232 is the derivative of @xmath229 with respect to its argument ) @xmath233 @xmath234 @xmath235 @xmath236 so that @xmath237 \biggr)\ ] ] @xmath238 note that the wave is only a function of its argument @xmath239 so that , when acting on @xmath232 @xmath240 and @xmath241 the equation of motion ( [ eq - eqofmotion ] ) becomes @xmath242 and we have a solution .    note that in our parameterization the wave is transverse to the @xmath243-direction , the average direction of the string , but it is _ not _ transverse to the instantaneous direction of the string as assumed in reference@xcite , because the string bends . indeed @xmath244 is always negative or zero . the only solution with @xmath245 everywhere requires @xmath246 , i.e. no wave . the authors of reference@xcite go on to discuss the quantum dynamics of the light string and claim that in order for a reasonable solution to exist , the dimension of spacetime must be 26 . this claim does not apply to our model because it is a classical model  though we will construct from it the elements of quantum mechanics . we work in four spacetime dimensions , but do not exclude the existence of hidden , compactified dimensions .    using the above methods , it can be shown that a slightly more general wave traveling at the speed of light @xmath247 is also a solution of ( [ eq - eqofmotion ] ) , where both @xmath248 and @xmath229 are smooth functions of @xmath249 with arbitrary shape .      since the electromagnetic string can interact with itself , we must consider interaction even when considering motion of a single line . it has been shown by barut and bornzin@xcite that the equation of motion for the filamentary dirac string with interactions is given by    @xmath250    where now @xmath251 is the mechanical string lagrangian density given by equation ( [ eq - lagrangiandensity ] ) and the total lagrangian density including interactions is given by    @xmath252    where @xmath253 is the dual of the electromagnetic field tensor at point @xmath254 . @xmath255    dirac@xcite showed that in the filamentary model , but otherwise in the general case , that the dual field of the line is given by    @xmath256    and therefore the electromagnetic field tensor of the line is given by    @xmath257    dirac s pole current formalism can be used ( section  [ ssec - diracpolecurrent ] ) to find the field at the ends of the line . using maxwell s equations the line current density is given by    @xmath258      we define a superconducting field as a region of spacetime within which the relevant phase field obeys the equation    @xmath259    for all closed paths enclosing areas that are totally within that region , so that the corresponding wave function is single - valued . in this case we see from ( [ eq - quantization0 ] ) that the london equations ( [ eq - london ] ) are exact . if the superconducting field is multiply connected , we must allow flux quanta to be outside the superconducting region but enclosed by the path    @xmath260    here the phase field corresponds to the momentum field of cooper pairs ( [ eq-4ps ] ) in a type ii superconductor . in the case of a superconducting line we extrapolate all results to the limit @xmath261 and @xmath262 such that the line radius @xmath61 ( [ eq - lambdal ] ) remains finite , at its characteristic constant value . the phase field must also satisfy the general relationship between @xmath99 and @xmath14 implied by maxwell s equations with sources    @xmath263    in terms of the four - vector potential we obtain    @xmath264    this is the usual equation relating the electromagnetic potential to its source current .    in a superconducting field the london equations ( [ eq - london ] ) give    @xmath265    or , with the use of    @xmath266    we obtain    @xmath267    implying that the supercurrents respond to the electromagnetic field at the same point in spacetime . using the equation of continuity ( which follows from maxwell s equations )    @xmath268    we obtain    @xmath269 \ + \   { { 4\pi } \over c}\ j^\nu\ = 0\ ] ]    so that    @xmath270    this equation depends only on the london equations and maxwell equations , so it is _ always _ true for a supercurrent obeying ( [ eq-4supercurrent ] ) . the general solution of ( [ eq-4potto4current ] ) in a superconducting field is simply    @xmath271    where we have used ( [ eq-4ps ] ) and ( [ eq-4supercurrent ] ) . using ( [ eq - lorentzgauge ] ) , ( [ eq - london ] ) , ( [ eq - continuity ] ) and ( [ eq - daljej ] ) we find @xmath272    so we indeed have the solution .    this general solution ( [ eq - generalsol ] ) is useful because @xmath273 can often be determined . in a simply connected superconducting field far from other objects    @xmath274    as is well known for the 3-vector @xmath89 from static situations@xcite , so that    @xmath275    barut and bornzin@xcite claim that the relation between the four - vector potential and the source current density uses the standard retarded green function . this is _ not _ the case in a superconducting field because the electromagnetic field generates supercurrents ( [ eq - emgensupercurrent ] ) , and this is not taken into account in the standard formalism . thus in a superconducting field the electromagnetic field does _ not _ propagate outward from the source supercurrents like a wave . the 4-vector potential is proportional to the supercurrent density at the _ same _ position in spacetime , with the possible addition of a field with zero curl  which can not affect the electromagnetic field . the principle of causality holds that no information can travel faster than the speed of light . this principle is _ not _ required by special relativity since it is violated by an object that moves with a finite size  that does not propagate as a wave . we have seen that in a superconducting line the electromagnetic field does not propagate like a wave . motion as an object of finite size satisfies ( [ eq-4potto4current ] ) and ( [ eq - generalsol ] ) because @xmath276 , @xmath277 and @xmath278 move together . so we assert that the entire cross section of a line moves together as a unit , continuing to satisfy the quantization condition ( [ eq - multisup ] ) for a multiply connected region . this is a violation of the principle of causality on the microscopic scale .    using the phase field we can illustrate this violation directly . for a path which includes the centerline of a superconducting line , we obtain from ( [ eq - multisup ] ) in the transverse rest frame of an element of the line centered at the point @xmath279    @xmath280    if the line contains one quantum of flux ( taking @xmath281 for convenience ) . here the @xmath282 refers to the transverse rest frame at @xmath279 , moving at @xmath283 ( see ( [ eq - beta ] ) ) relative to the lab ( default rest frame , see section  [ sec - introduction ] ) . because of equation ( [ eq - intpseh ] ) we can no longer have @xmath284 in the presence of the centerline . but the london equations ( [ eq - london ] ) must hold outside of the centerline , so within the line    @xmath285    we presume that @xmath286 , since @xmath287 would be equal to @xmath64 if not for the flux quantum at the core of the line . we now consider motion in the lab but restrict consideration to nonrelativistic motion transverse to the average direction of the line ( @xmath288 ) see ( [ eq - beta ] ) . we then use the fact that an appropriate vector field can be specified by its divergence and curl@xcite to find@xcite that the phase field is completely specified by its behavior on the centerline    @xmath289    evaluating ( [ eq - pstx ] ) we simply obtain ( [ eq - psx ] ) , the static result . but this is valid for the moving line as long as relativistic effects of the transverse motion can be neglected . this indicates that the whole transverse cross section of the line moves together  as an object with finite size . this type of motion is required by the quantization condition ( [ eq - intpseh ] ) and by the nature of the superconducting line . this type of motion is theoretically convenient because the lagrangian density for the filamentary line with interactions ( [ eq - lagrangiandwint ] ) can be used to derive the equations of motion for the superconducting line with only minor modifications . there is no need to consider added degrees of freedom for the transverse size of the line . the state of the entire line can still be completely specified as a function of only two parameters , which can still be taken as @xmath200 and @xmath199 . now @xmath290 refers to the position in time and space of a point on the centerline of the line . because this model is _ acausal _ , it is also _ nonlocal _ ; the response at a point is affected by a field over an extended region of spacetime . ( see section  [ ssec - selfintline ] . ) the nonlocality of this model makes even self interactions finite , and indeed self interactions are an important part of the model .      in order to consider the superconducting line , we generalize the filamentary model of ( [ eq - filemdual]-[eq - fil4current ] ) . for instance the electromagnetic field ( [ eq - filem ] ) is a rank 2 tensor in 4 dimensions , because @xmath291 ( and @xmath292 ) are position 4-vectors , the @xmath293-function is lorentz invariant , and the integral formalism shown is independent of the type of parameterization as long as @xmath199 covers the full length of the line , and @xmath200 covers the full time under consideration . we must modify the @xmath293-function to give the line a finite transverse size . prescriptions for doing this are frame dependent , so we must start in the transverse rest frame of an element of the line . then we can obtain more general results via lorentz transformation to the lab .    we evaluate ( [ eq - filem ] ) in the transverse rest frame of an element of the superconducting line by making the replacement    @xmath294    where    @xmath295    approaches a dirac @xmath293-function in the dimensions transverse to the line as @xmath296 . here @xmath297 is the perpendicular vector from the centerline to point @xmath298 and this perpendicular intersects the centerline at @xmath299 . ( if there is more than one such point , we must add the contributions in order to find the total field . ) in the transverse rest frame    @xmath300    @xmath301    where both of these quantities are 4-vectors because @xmath302 and @xmath303 represent the proper time and a length element transverse to @xmath304 in the rest frame of an element of the line . hence the electromagnetic field in the transverse rest frame of this element of the string is given by    @xmath305    @xmath306    @xmath307    @xmath308    where the matrix contains components of the vector @xmath309 giving the direction of the centerline in the transverse rest frame of the line at the point @xmath310 . so the electromagnetic field in this frame is given by    @xmath311 @xmath312    this looks like the point at which we started ( [ eq - bline ] ) , but now the unit vector @xmath313 does not necessarily point along the @xmath243 axis and its direction can change as a function of @xmath200 and @xmath199 .    by performing a tranverse lorentz transformation from the rest frame of the above element of the line to the lab we find the electromagnetic field of the line ( for the line ( [ eq - yzline ] ) moving in the @xmath314 plane ) in the lab    @xmath315    @xmath316    where @xmath317 is the result of lorentz transformation of @xmath318 , selecting events simultaneous in the lab to describe the moving field . the extra @xmath319 factors arising from the transformation of the field enable @xmath317 to take the direction of the field when expressed in lab coordinates . in fact the net result of the lorentz transformation , expressed in lab coordinates is the change @xmath320 . since @xmath318 and @xmath317 are unit vectors , they are , in fact , equal . the directions of the line in its rest frame and the lab are the same . here we use    @xmath321    @xmath322    and    @xmath323    is the pancaking cylindrical radius of the line expressed in lab coordinates . by our use of a lorentz transformation we are ignoring the fact that the rest frame of the element of the line is an accelerating reference frame . according to meissner , thorne , and wheeler@xcite this approximation , using a comoving inertial frame , is permissible as long as inertial forces are small and typical accelerations are much less than @xmath324 where @xmath325 is the amplitude of a typical oscillation . we will assume that these conditions hold true except near the ends of a line . we wish to use maxwell s equations to find the supercurrent in the lab . maxwell s equations act at the _ field _ point @xmath41 and therefore we do not consider the explicit space and time dependence of the string coordinate @xmath38 , which acts effectively as a source point . neither do we consider the explicit space and time dependence of @xmath319 and @xmath326 which is determined by @xmath38 . however , the field at * x * can change with time due to motion of the string . since the electromagnetic field does not change ( is _ quantized _ ) in the rest frame of the string , all time dependence is due to motion . so at a fixed field point @xmath41 , the change of the magnetic induction field with time is given by    @xmath327    @xmath328    so for the line ( [ eq - yzline ] ) moving in the @xmath314 plane    @xmath329    here @xmath304 is the velocity of the line at the field point @xmath41 which is the same as the velocity of the point on the centerline @xmath38 associated with @xmath41 , since the cross section of the line moves together . for similar reasons the change of the electric field with time is given by    @xmath330    and if the line satisfies ( [ eq - yzline ] )    @xmath331    using maxwell s equations we find    @xmath332    where @xmath176 is the instantaneous radius of curvature at @xmath333 . this current satisfies the equation of continuity ( [ eq - continuity ] ) exactly . details of these calculations are included in appendix  [ ssec - instradius ] , where the derivatives of @xmath40 are explicitly evaluated . it is also shown in appendix  [ ssec - calcemfield ] that if we apply equation ( [ eq - generalsol ] ) and attempt to derive the electromagnetic field ( [ eq - efieldline ] ) and ( [ eq - bfieldline ] ) from ( [ eq - supercurrentyz ] ) we do not get precise agreement . we get the expected result plus remainder terms proportional to @xmath334 and @xmath335 which are small for large radii of curvature . we take this to imply that our simple procedure , which involves giving the delta function a finite size and using a lorentz transfomation from a comoving inertial frame , does not provide an exact solution for the electromagnetic field . it will be sufficient , however , for our present purposes .    performing an inverse lorentz transformation we find the supercurrent in the rest frame of an element of the line :    @xmath336 the parameters @xmath319 and @xmath326 enter this expression because we have defined @xmath176 in the lab . ( see appendix  [ ssec - instradius ] ) the charge density is zero in the rest frame . it is shown in appendix  [ ssec - energyofbending ] that the energy of the finite size line is independent of the instantaneous radius of curvature @xmath176 at a given point . this fact is important because we do not include any term in the lagrangian density for the energy of bending . so in the absence of interactions , general wave solutions of the type ( [ eq - yzline ] ) and ( [ eq - circularline ] ) traveling in the positive @xmath106-direction could occur with the superconducting line . replacing the argument @xmath337 with @xmath338 produces solutions traveling in the negative @xmath106-direction . following the sign conventions of bjorken and drell@xcite ( which are consistent with jackson ) the action ( except for weak and hadronic modes ) is    @xmath339    where no mass term appears in the lagrangian because we assert that mass is not fundamental . here @xmath340 is the mechanical string lagrangian density ( [ eq - lagrangiandensity ] ) for line @xmath341 , @xmath342 is the electromagnetic current due to electric charges , and @xmath343 is the electromagnetic current due to superconducting lines . applying the principle of least action to the superconducting line with finite size , using ( [ eq - deltatrans ] ) in ( [ eq - fil4current ] ) , it can be shown that the equation of motion for a given line , instead of ( [ eq - filmotionwint ] ) , is given in the case of interactions by    @xmath344    where    @xmath345    is the average current over the cross section of the line weighted by the shape of the line field . this averaging must be performed in the transverse rest frame of the line . but the resulting current is a 4-vector , and can be transformed to any frame ( within the approximations which have been discussed@xcite ) . indeed , since the transverse rest frame varies with the line element in question for solutions of the type ( [ eq - yzline ] ) , it is most convenient for these solutions to think of the quantities @xmath346 , @xmath347 , and @xmath348 transformed from the rest frame to the lab . then if we wish to make a change of parameterization to the particular parameterization of section  [ ssec - neglecting ] , we can do so . ( it is only important that the parameterization be the same on both sides all equations . )    for a given line we work in the large radius of curvature approximation , @xmath349 , where @xmath350 is the radius of curvature of the centerline so that all points which make a significant contribution to the average are perpendicular to the line at a single unique point ( always true for points closer to the line than @xmath350 ) . to evaluate the interaction of the line with itself , we use equation ( [ eq - supercurrentyz * ] ) in the interaction term and obtain ( with @xmath351 )    @xmath352    @xmath353    where we have used the fact that the radius of curvature of the centerline can be related to the shape of the trajectory ( see appendix  [ ssec - radiusandshape ] ) according to the equation    @xmath354    in order to learn how to transform the radius of curvature from the lab to the line s rest frame    @xmath355    because of azimuthal symmetry the first term in ( [ eq - selfintline ] ) gives a result of zero . unless the radius of curvature is infinite , the second term in ( [ eq - selfintline ] ) is easily seen to be finite but not equal to zero . in fact if @xmath356 , we obtain    @xmath357    hence we have evaluated the self interaction of the superconducting line . it is finite and calculable in spite of the fact that the field diverges along the centerline . the averaging required by the principle of least action leads to a finite result .    because the self - interaction term in equation ( [ eq - eqofmotionwint ] ) is not zero , and traveling waves are an exact solution of this equation without interactions , there are _ no _ traveling wave solutions propagating at the velocity of light for the motion of a single isolated superconducting line once self - interactions are included . wave motion of a single superconducting line may still be possible when the photon quantization condition is satisfied but then the line is not isolated , and we are currently unable to describe the motion . now we are ready to define what we mean by the wave function of the electron in this model which plays such a prominent role in the dirac quantization condition ( [ eq - singv ] ) . we define the electron wave function to be the change in the ether caused by the bare electron , i.e. by the end of the appropriate line . this change will be the response of the ether to the bare electron s electromagnetic field . ( see section  [ ssec - masselectron ] . ) since the bare electron s electromagnetic field has infinite extent , the wave function of the electron also has infinite extent  as in standard quantum theory , and consistent with our reasoning in section  [ ssec - fq ] .    on the microscopic scale , this wave function is not continuous ( differing from the standard model ) . it is a mesh of line pairs . each line pair is continuous . we should realize that the cross sections of our line pairs are small on the atomic scale and mostly fit into the holes in matter . the outer radius of a line pair ( section  [ ssec - static ] ) must be much smaller than the bohr radius of the hydrogen atom : @xmath358 . hence there are many holes in the wave function which constitute a larger version of dirac s  nodal lines " . if three of the line pairs form a complete circuit around such a hole , and if magnetic field links the circuit , the circuit will superconduct in response to the vector potential ( figure  [ fig - fluxlhalf ] ) , causing quantization of magnetic flux ( [ eq - fq ] ) through the circuit . if the applied flux through the complete circuit is less than half a flux quantum , the net flux will be brought to zero . if the applied flux through the circuit is greater than half a flux quantum , experimental evidence@xcite indicates that the supercurrent in the complete circuit will _ increase _ the flux as necessary to match the nearest flux quantum ( figure  [ fig - fluxghalf ] ) . we will see that the photon can be considered as an excitation of the wave function of the electron .    to 2.6 in     to 2.6 in we seek the types of waves which can propagate at the speed of light on the two lines of an isolated pair . we attempt to find them using the known solutions ( [ eq - yzline ] ) and ( [ eq - circularline ] ) without the interaction terms . from the principle of least action for the twoline system we find    @xmath359    @xmath360 \delta { y^\mu_1}\ ] ]    @xmath361 \delta { y^\mu_2 } \biggr \ }   \ \ \ .\ ] ]    due to the interaction terms the variations in @xmath362 and @xmath363 are coupled so we can not set the coefficients of @xmath364 and @xmath365 separately to zero , without first knowing the solutions to the equations of motion . to handle this situation , we define the 4-vectors    @xmath366    as the average and half the difference between the wave solutions of line 1 and line 2 . then we find    @xmath367    we express the mechanical string lagrangian densities ( [ eq - lagrangiandensity ] ) for line 1 and line 2 in terms of @xmath368 and @xmath325 :    @xmath369    @xmath370      to find the common mode solutions we set    @xmath371    where @xmath372 is a constant 4-vector . note that @xmath325 has no time component and no @xmath106-component by construction . thus    @xmath373    and the twoline system exhibits no excitation . it moves together as one system . since the common mode solutions exhibit no excitations , and hence no net electromagnetic field at a given @xmath374 , we interpret them as gravitational waves . ( see section  [ sec - generalrel ] . )    since @xmath325 is constant , @xmath375 and @xmath376 , hence the mechanical lagrangian density of the system becomes    @xmath377    just twice the lagrangian density ( [ eq - lagrangiandensity ] ) . since for common mode solutions @xmath378 the variation of the action becomes    @xmath359    @xmath379\ ] ]    @xmath380    and the equation of motion becomes    @xmath381    @xmath382    where @xmath383 and @xmath368 must satisfy equations ( [ eq - yd ] ) and ( [ eq - ydif ] ) , and the left hand side of the equation refers to @xmath368 . we label the 8 interaction terms @xmath384 where @xmath385 or 2 corresponding to the line where the average is performed , @xmath386 or 2 corresponding to the line field which is averaged , and @xmath387 or 0 corresponding to the term in the current ( containing either a @xmath66 or a @xmath63 , see ( [ eq - supercurrentyz * ] ) ) . for the common mode solutions , line 1 and line 2 have the _ same _ instantaneous rest frame , which simplifies the averaging calculation .    as in section  [ ssec - selfintline ] , we find that @xmath388 . in order to obtain wave solutions with the remaining interaction terms we specify the constant 4-vector to be in the @xmath389-direction    @xmath390    ( labeling the constant length also @xmath372 ) if the motion is in the @xmath314 plane . note that @xmath391 since the fields are in opposite directions . ( see figure  [ fig - commonmodegeom ] . ) using the large radius of curvature approximation and equations ( [ eq - k0fact ] ) , ( [ eq - wronskian ] ) and the derivative of ( [ eq - wronskian ] ) we find    @xmath392    where the normalized separation of the two lines in the @xmath389 direction is given by    @xmath393    ( we quote the covariant components @xmath394 , so the interaction is _ attractive_. ) as expected , this interaction term is zero if @xmath395 and attractive for all @xmath396 greater than 0 . from rotational symmetry we see from figure  [ fig - commonmodegeom ] that    @xmath397    to 1.8 in   plane.,title="fig : " ]    after taking advantage of the fact that for this common mode geometry the instantaneous rest frames and instantaneous radii of curvature of the two lines are the same if the separation is perpendicular to the motion , we find that    @xmath398    if @xmath399 . since @xmath400 for common mode solutions , we find that    @xmath401    and the sum of all interaction terms for this common mode geometry is zero . hence , before considering the quantization conditions , common mode solutions  gravitational waves  of types ( [ eq - yzline ] ) can propagate freely at the speed of light if @xmath402 , the direction of the 3-vector associated with @xmath372 , is perpendicular to the motion . it is shown in appendix  [ sec - generalmotionincommonmode ] that the orientation of @xmath402 does not affect common mode motion . so common mode solutions of type ( [ eq - circularline ] ) can also propagate at the speed of light . we will see in section  [ ssec - generalityquant ] that the photon quantization conditions must still be satisfied , even in this case . but the quantization currents have no role beyond their role in keeping the separation @xmath372 constant . ( see section  [ ssec - tensionpair ] . )    since the orientation of @xmath402 does not matter in the common mode , and the sum of the interaction terms is zero in common mode , electromagnetic waves with @xmath402 spinning ( helical polarization ) can propagate at the speed of light on top of gravitational waves  as long as the photon quantization conditions are satisfied .      to find differential mode solutions we set    @xmath403    without variation , and therefore    @xmath404    we consider the case where @xmath325 has only transverse components ( unlike appendix  [ sec - generalmotionincommonmode ] ) so @xmath325 , @xmath405 and @xmath406 are all orthogonal to @xmath368 , @xmath407 and @xmath408 . the total mechanical lagrangian density becomes    @xmath409    if we simply define    @xmath410    the mechanical lagrangian density becomes    @xmath411    with the standard solutions of the form ( [ eq - yzline ] ) and ( [ eq - circularline ] ) without interaction . we refer to this differential solution as @xmath412 ( rather than @xmath362 ) to indicate generality . both @xmath362 and @xmath363 are solutions to this mechanical lagrangian without interaction . so for the differential mode    @xmath413    we require    @xmath414    and therefore    @xmath415    the variation of the action becomes    @xmath359    @xmath416   ( + \delta d^\mu)\ ] ]    @xmath417(- \delta d^\mu )    \biggr \}\ ] ]    @xmath418\ ] ]    @xmath419    and we see that the differential mode solutions involve the _ difference _ between the interaction terms of the two lines . the equation of motion for differential mode solutions becomes :    @xmath381    @xmath420    where @xmath421 and @xmath422 must satisfy ( [ eq - y1dy2d ] ) and the left hand side of the equation refers to @xmath412 ( hence either @xmath421 or @xmath422 ) .    _ a. radial excitation  linear polarization _ [ ssssec - radialexcitation ]    now we let @xmath423 in order to look for differential solutions of the type ( [ eq - yzline ] ) . we consider the moment in space and time shown in figure  [ fig - radialexcitation ] when the two lines reach maximum separation . at this moment the two lines will be at rest in the lab , hence they have common mode geometry in their joint rest frame . accordingly we can use ( [ eq - i121 ] ) and ( [ eq - i110pi120 ] ) to calculate the interaction terms for differential mode waves with radial excitation    @xmath424    @xmath425    @xmath426    @xmath427    at the distance @xmath428 of maximum separation ( with @xmath429 ) . to 2.6 in     we keep the sign conventions of the common mode interaction terms for clarity . since we must take the _ difference _ of these terms , it is very easy to see that the differential mode interaction does not vanish for the case of radial excitation . although we have only considered the point of maximum separation , clearly the nonvanishing of the interaction terms is general throughout the motion with radial excitation . the interaction terms ( [ eq - i121m211 ] ) and ( [ eq - i110p120m210m220 ] ) point in the @xmath430 direction , but this is no longer true away from maximum separation since @xmath431 and @xmath432 are _ not _ in opposite directions . since motion with radial excitation is in the same direction as nonzero interaction terms , we can not model differential mode waves with radial excitation propagating at the speed of light . if the photon quantization conditions are satisfied , such motion could still occur , but a model would require knowledge of the quantization currents which we currently do not understand . _ b. model of the photon helical polarization _ [ ssssec - helicalphoton ]    to model the photon we use solutions of the type ( [ eq - circularline ] ) with sinusoidal functions @xmath248 and @xmath229 , i.e. one fourier component , where @xmath433 is assumed constant . @xmath434    @xmath435    here @xmath436 is the direction in the rotating frame ( which is the rest frame of _ both _ lines along their entire length ) , which is initially vertical ( @xmath437 direction ) in the lab , @xmath438 and the wave has a frequency @xmath439 . with this choice we can achieve lack of radial excitation since    @xmath440    ( see figure  [ fig - helicalphoton ] ) . the whole system rotates with positive helicity . the velocities are given by    to 2.6 in , @xmath441.,title="fig : " ]    @xmath442    @xmath443    and the magnetic field directions are given by the following expressions :    @xmath444    @xmath445    since both lines are at rest in the rotating frame , we calculate the currents in this frame directly . we use ( [ eq - eline*]-[eq - bline * ] ) for the electromagnetic fields of both lines . then from appendix  [ ssec - generalcurl ] and [ ssec - helicalmodegeom ] we find the following expressions :    @xmath446    @xmath447    @xmath448    where @xmath449 is the instantaneous radius of curvature of the centerline , which is constant for both lines . hence we find :    @xmath450\ ] ]    @xmath451\ ] ]    these currents satisfy the equation of continuity ( [ eq - continuity ] ) exactly , but we still find a small inconsistency if we try to use ( [ eq - generalsol ] ) to recalculate the electromagnetic field ( see appendix  [ ssec - calcemfield ] ) . we find the interaction terms in helical mode    @xmath452\ ] ]    where we take into account that the radii of curvature for line 1 and line 2 point in opposite directions . ( hence @xmath453 and @xmath454 are different functions of position though the magnitude of @xmath455 is the same on both centerlines . ) we make a correction for the fact that @xmath456 and @xmath457 have different axes by changing @xmath458 into @xmath459 where @xmath460 is the cylindrical radius perpendicular to the @xmath243-axis , and symmetric for both bessel functions . we obtain    @xmath461    where    @xmath462    and in a similar fashion    @xmath463    so the sum of the interaction terms ( including the negative signs for differential mode )    @xmath464    @xmath465    is in the radial @xmath430 direction and attractive for the helical differential mode of the photon .    since the motion of the helical photon is orthogonal to the radial direction , it can propagate freely at the velocity of light . the quantization currents must play a particularly simple role . the left hand side of ( [ eq - eqofmdiff ] ) is zero in the radial direction , but the right hand side of ( [ eq - eqofmdiff ] ) , given by ( [ eq - inttermsdiff ] ) , is not zero in the radial direction . the quantization currents ( which have not been considered so far ) must freeze the radial separation of the two lines of the photon at @xmath466 . this allows us to isolate the effects of the quantization currents , which we do not fully understand , and model the helical photon .      in figure  [ fig - wfelectron ] we consider the wave function of an electron with 4-momentum @xmath467 . the wave function is a mesh of line pairs , each of which ( from quantum mechanics , as we will see in section  [ sec - quantmech ] ) oscillates at frequency @xmath468 and with a wavelength @xmath469 if @xmath470 points in the direction of the line pair and @xmath471 in the more general case , where @xmath472 is the direction of the line pair . we use the absolute value sign to indicate that no positive sense for @xmath472 is defined . hence @xmath473 is the phase field of this wave function .    to 1.5 in we have pictured the mesh as real and drawn it with regular spacing for clarity . in general , of couse , a wave function is complex . in reality neither the spacing nor the directions will be regular . closed superconducting circuits will be formed with many shapes ( triangles , rectangles , pentagons , ... ) and sizes , and in many different orientations .    in each circuit @xmath474    since @xmath475 is the phase field for this ( single - valued ) wave function . ( we are using a plane - wave electron for simplicity . the fourier composition of wave functions will be considered in section  [ sec - quantmech ] . ) we see that the phase field of an electron wave is more than just the electron s 4-momentum . it represents the entire wave function of a plane - wave electron . ( it is probably useful to think of it as a differential form@xcite . )    but now consider figure  [ fig - photonfixedt ] . a photon is propagating on one of the line pairs , hence on only one leg of one or more closed circuits .    to 1.5 in     for _ this _ line pair , the number of oscillations per unit length has increased and so the phase field is    @xmath476    the sum of the 4-momenta of the electron and the photon . hence , since this photon is propagating along only one of the line pairs , we must have ( from ( [ eq - multisup ] ) )    @xmath477    so that the photon s momentum and energy are quantized . for the electron s wave function to be single valued in the presence of the photon , ( [ eq - multisup ] ) must hold . thus ( [ eq - multisup ] ) must hold for _ all _ quanta , not just flux quanta . consider a circuit at _ fixed time _ ( figure  [ fig - photonfixedt ] ) . with @xmath478 ( and hence the direction of the relevant line pair ) defined to be in the @xmath243-direction , equation ( [ eq - pepg ] ) becomes    @xmath479    so that    @xmath480    we presume that the length of one photon is the wavelength @xmath481 , since it is the smallest length satisfying periodic boundary conditions  bringing the two lines back to their quiescent condition . consider a circuit at _ fixed z _ ( figure  [ fig - photonfixedz ] , with @xmath478 still in the @xmath243-direction )    to 1.5 in .,title="fig : " ]    @xmath482    where we presume the photon passes in a time interval of one period @xmath483 ( the smallest time duration satisfying periodic boundary conditions ) . so    @xmath484    and the photon must be _ quantized _ if it is to propagate on a line pair through the superconducting ether .    to consider whether or not a photon can exist we must integrate over the relevant line pair for at least a wavelength @xmath481 or a period of time @xmath72 and close the integral on nearby line pairs without excitation . if the quantization condition ( [ eq - pepg ] ) is satisfied , apparently the constant radial separation of the two lines of the helical photon is maintained at @xmath466 . this explains how a sine wave can start and stop instantly . the two lines of the photon are in general _ not isolated _ if the quantization condition ( [ eq - pepg ] ) is satisfied . the situation is similar to figure  [ fig - fluxghalf ] , where currents in the ether maintain a flux quantum . we observe from equations ( [ eq - pepg ] ) and ( [ eq - multisup ] ) that the quantization of the photon really does not depend on the photon being an excitation of any particular electron s wave function . it is a property of the mesh of superconducting lines that comprises the ether  and forms the wave functions of all electrons . even though we will show that the transverse motion of the helical photon is nonrelativistic , we attempt to make no approximations ( beyond the large radius of curvature approximation , which is well satisfied ) , because we will obtain precise results . we perform separate local lorentz transforms ( using ( [ eq - beta1 ] ) or ( [ eq - beta2 ] ) as appropriate ) on the electromagnetic field ( [ eq - eline*]-[eq - bline * ] ) in the rotating frame to find the electromagnetic fields of line 1 and line 2 in the lab :    @xmath485    @xmath486    @xmath487    @xmath488    @xmath489    the velocity of the entire cross section of each line is the velocity of its centerline in accord with section  [ ssec - causality ] . we note that the transverse components of * e * and * b * are equal for both lines and the transverse fields of the two lines add  but the net field in the @xmath243-direction is zero . the poynting vector of the helical photon , expressing the flow of energy per unit area per unit time , is given by    @xmath490    @xmath491 - { \hat x}^\ast \bigl [ b^{\ast}(1)^2 - b^{\ast}(2)^2 \bigr ] \biggr \}\ ] ]    which shows that some energy is directed along the @xmath243-axis and some energy circulates about the @xmath243-axis . at fixed @xmath106 , the energy flow per unit time is given by    @xmath492    with    @xmath493    and is totally in the @xmath243-direction . ( as for ( [ eq - tiltbessel ] ) we take into account the tilt of both bessel functions with respect to the @xmath243-axis . ) we wish to find the energy in one photon , @xmath494 . it should be given by the energy passing a fixed @xmath106 in one period . @xmath495    @xmath496    if we want to compare with measured photon energies we must make a _ binding energy correction _ , because when the line pair transfers the photon energy to an atom , the photon must provide the energy necessary to separate the lines , because one line will go to an electron and the other to the nucleus . so we see that the overlap term containing @xmath497 clearly does not belong in the energy transferred to an atom . we assert that the corrected energy @xmath498 transferred to an atom is just ( [ eq - egwoverlap ] ) without the overlap term , because it gives the sum of the electromagnetic energies transferred by each line separately . we presume that in the process of the lines separating , angular momentum about the common mode centerline is conserved , because the force between the lines is central  between the centers . so we can check our method by calculating the angular momentum about the common mode center  i.e . the spin of the photon . this binding energy correction is approximately a factor of two decrease in the energy .    using similar reasoning we can also neglect the effects of the photon quantization currents on the energy of the traveling photon because this energy is not transferred to the absorbing atom . quantization current effects are also part of the photon binding energy .    using the above correction we find    @xmath499    so    @xmath500    we will show that when the large radius of curvature approximation is satisfied @xmath501 is very close to @xmath114 . _ hence the quantization of the helical photon implies that the amplitude of the photon wave is an absolute constant ! _    if we assume that @xmath502 near the earth ( as we will infer from section  [ ssec - parameters ] ) , we have    @xmath503    since the line radius @xmath61 would give a finite size to the electron , from section  [ sec - introduction ] we have @xmath504 . so for a green photon ( @xmath505 ) we have    @xmath506    @xmath507    in addition we find    @xmath508    so    @xmath509    for helical geometry the large radius of curvature approximation    @xmath510    is well justified , for a visible photon .    in figures  [ fig - currentsfortogether ] and [ fig - currentsforapart ] , current distributions are shown with which the quantization condition could be maintained . in figure [ fig - currentsfortogether ] currents running on the outside surfaces of the line pairs tend to decrease the radial separation of the central pair , and in figure  [ fig - currentsforapart ] currents are shown which tend to separate the central pair . the dominant contributions are likely to come from the currents running on the outer surfaces of the central pair . currents in the same direction attract . currents in the opposite direction repel . hence in their simplest configuration , the photon quantization currents could form a loop around the outer surfaces of the central line pair for the length of the photon ( one wavelength ) . the angular momentum about a given point on the common mode centerline of the pair is given by    @xmath511    so at fixed @xmath106 the angular momentum per unit length is    @xmath512    perhaps the easiest was to evaluate ( [ eq - dlemdz ] ) is to realize that in taking moments of momentum , the entire momentum of each line can be considered to act at its centerline ( center of momentum ) . so , for example , for a term involving @xmath513 we have    @xmath514    @xmath515   \times ( -{\hat x}^\ast ) \ ] ]    @xmath516 \times ( -{\hat x}^\ast ) \ ] ]    @xmath517 \times ( -{\hat x}^\ast ) \ ] ]    @xmath518    therefore the spin of the photon is given by the angular momentum in one wavelength :    @xmath519    @xmath520    assuming the binding energy correction is valid , this result is exact except for the the large radius of curvature approximation ( [ eq - largeroc ] ) . figures [ fig - wfelectron ] , [ fig - photonfixedt ] , and [ fig - photonfixedz ] are simplifications . we will see ( section  [ ssec - motionbpp ] ) that the typical spacing of line pairs is less than @xmath521 , whereas the wavelength of a green photon is @xmath522 . thus the quantization of the photon involves many closed circuits . from figures [ fig - currentsfortogether ] and [ fig - currentsforapart ] , the currents which maintain the quantization of the helical photon , i.e. the constancy of the photon amplitude , can be complex and , as with figure  [ fig - fluxghalf ] , we do not have a detailed understanding , beyond the quantization condition itself , of what makes the currents flow as they do . but apparently such flow is required for single - valued wave functions , and so it happens . we have made progress with our classical picture but are now meeting difficulty in understanding the quantization condition itself .    to 1.5 in     to 1.5 in     an atom generates each photon on the relevant line pair , which acts as a transmission line . once the photon exists and satisfies the quantization condition ( [ eq - pepg ] ) , apparently it is maintained by the quantization currents as it transits the universe until it is absorbed by another atom , eventually . apparently these photon quantization currents persist forever , like currents in a superconducting loop , which is apparently what they are . we see from the fact that ( [ eq - pepg ] ) and ( [ eq - multisup ] ) are identical , that this quantization condition is a completely general result applying to the phase field of any particle traveling in the superconducting ether . the ether is a multiply connected superconductor . in particular this quantization condition must hold for any excitation of a line pair . it is _ not _ restricted to particles traveling at the speed of light , and must hold for matter waves and gravitational waves , as well as electromagnetic waves . we can now identify a matter wave as a gravitational ( i.e. common mode ) wave which is stimulated by a particle , so that it does not travel at the velocity of light , but rather travels with the particle . it is the particle s wave function .      in the case that there are no excitations of the line pairs under consideration we have    @xmath523    for any closed loop at a fixed time . if there were a change in the radial separation of any line pair in the circuit , this change would necessarily be quantized , contrary to our assumption . hence the radial separation in quiescent common mode and the tension of a line pair    @xmath524\ + \ \tau_{qc } \label{eq - tensiontau0}\ ] ]    must be constant _ for the whole universe _ at a given time . here @xmath525 is the tension ( energy per longitudinal length ) associated with the quantization currents for this line pair . it is convenient to think of the quantization currents in this case as flowing in opposite directions on the outer surfaces of each pair . these currents are likely to be the dominant quantization currents . we assume that they also persist forever . the radial separation , the line - pair tension and the quantization currents , however , must change with time as the universe expands , reflecting the conservation of energy . apparently this line pair tension and radial separation is a measure of world time@xcite . we use dirac s pole current formalism@xcite to find the electromagnetic field of the poles at the ends of a line . with our line of finite size the dual field of the line ( [ eq - filemdual ] ) becomes    @xmath526    with from ( [ eq - deltatrans ] )    @xmath527    the pole current is defined by the extension of maxwell s equations if magnetic poles were to really exist :    @xmath528    @xmath529    @xmath530    after an integration by parts we obtain the pole current    @xmath531    if we presume that @xmath221 and @xmath222 are sufficiently distant that they are of no concern , the pole current becomes    @xmath532    @xmath533    we get our expectation for a pole current from each end of the line of finite size _ except _ for the signs of both poles . the reason is that we actually cut off the integral over the line too soon . the dual field actually extends to infinity beyond the ends of our line . so by cutting off the dual field at the mechanical end of the line , we are _ absorbing _ the dual field field at the end of the line . this is the reason for the incorrect signs . contrary to our view dirac , excluded the line from physical reality because he wanted the pole field to appear as a monopole , in the space _ external _ to the line . therefore we take our dual field as the _ negative _ of dirac s dual field .    in our model maxwell s equations are satisfied everywhere , and the pole currrent is actually @xmath64 . @xmath534    but we can use dirac s pole current method to find the effective pole current outside the ends of our lines just by reversing the signs of the pole currents :    @xmath535    the field which appears to be absorbed in ( [ eq - dualcurrent ] ) , because we cut off the integral too soon , actually exits the line and spreads out in all directions . we assume , without justification , that there is simply a sharp , clean break at the end of the line  the superconducting field and the supercurrent simply stop . the implications of this simple phenomenological assumption are rich and interesting .    using maxwell s equations and ( [ eq - generalsol ] ) we can show that in the superconducting field :    @xmath536    @xmath537    therefore the electromagnetic field at the break at the end of a line should act as a source for the electromagnetic field beyond the line , which satisfies the homogeneous wave equation . we will see that the bare positron - pole ( using positive charge for convenience ) bounces around ( _ zitterbeweging _ ) so rapidly that a proper treatment of the shape of the field of the bare positron - pole is unnecessary , as long as the total outward flux is correct . in the limit @xmath296 we simply find    @xmath538    the current of point poles . therefore we can use the standard lienard - wiechert potentials@xcite with the substitutions    @xmath539    to find the electromagnetic field of the bare positron - pole    @xmath540_{ret } + \ { g_0 } \ \biggl [ { { { \hat r } \times \bigl \ { ( { \hat r}-{\bm \beta } ) \times   { \dot { \bm \beta } } \bigr \ } } \over { \kappa^3 r } }   \biggr ] _ { ret } \label{eq - bppbfield}\ ] ]    @xmath541_{ret } \label{eq - bppefield}\ ] ]    with    @xmath542    and where the expressions in brackets are evaluated at the retarded time    @xmath543    and    @xmath544    @xmath545    ( the bare positron - pole will acquire it s electric charge in section  [ ssec - electroncharge ] and then be called the bare positron . )    the field of the bare electron - pole is similar , coming from the negative pole in ( [ eq - poles ] ) the line field is , of course , present , but it is quantized and shielded by a superconducting sheath , so we can often neglect its interaction ( section  [ ssec - intercommutation ] ) with the surrounding line pairs .    in section [ ssec - neglecting ] , we saw that without considering interactions , the end of a line moves at the speed of light transverse to the line . it is interesting to note that , if the bare positron - pole moves at the speed of light , it s electromagnetic field ( [ eq - bppbfield]-[eq - bppefield ] ) is well behaved . in fact , we can see that in the absence of acceleration ( @xmath546 ) , the electromagnetic field of the bare positron - pole is actually @xmath64 at the speed of light .    in this universe , however , electromagnetic fields are always present . we will see that in the presence of external fields the speed of the bare positron - pole is less than the speed of light  but , unless the bare positron - pole is in a _ very _ strong field , its speed will be close to the speed of light . so the natural state of the end of a line is to move near the speed of light transverse to the line . this is in great contrast to our concept of a point particle whose natural state is to remain at rest . evidently , this is because the end of a line carries no inertia .      to solve the equations of motion for the bare positron - pole , we need to find the average of the line s dual field ( [ eq - lineemdual ] ) over the cross section of the line at the end points . in the rest frame of the bare positron - pole we find    @xmath547    since @xmath548 is the proper time , the vector    @xmath549    is a 4-vector . after transformation to the lab , the transformed vector is    @xmath550    remembering our definition of @xmath200 ( see ( [ eq - taudef ] ) ) . hence in the lab we find    @xmath551    in order to deal with the end of a line we _ pick _ the directions defined by @xmath552 and @xmath199 to be transverse to the motion . ( so they are no longer necessarily in the @xmath243 direction . ) then using the fact that @xmath553 and @xmath554 are both 4-vectors , we find    @xmath555    and so    @xmath556      in the presence of interactions , the boundary conditions ( [ eq - bcendofline ] ) which must be satisfied at the ends of a line become    @xmath557    and since @xmath219 is in general not zero , we obtain    @xmath558    or at each end point    @xmath559    where @xmath560 is given by ( [ eq - pldymup ] ) . the self interaction term is simply twice the mechanical term , so the equation of motion for _ either _ end point becomes    @xmath561    where    @xmath562    is a very strong field ( 3/2 times the average of the self field ) . we define @xmath563 as the direction of the component of the electric field which is perpendicular to the line ( @xmath317 )    @xmath564    and we also define    @xmath565    we assume that the velocity of the bare positron - pole is perpendicular to @xmath563    @xmath566    so the equations of motion become    @xmath567    @xmath568    @xmath569    @xmath570    usually we can neglect the averaging requirement for the external electromagnetic field , if it does not vary too rapidly over the cross section of the line . we use a simplified notation without averaging , but , in principle , we always have to take the average in the rest frame of the bare positron - pole and transform the resulting field to the lab ( as we did in section  [ ssec - selfintbareelectronpole ] ) . we first consider the case where the electric field is zero . in this case we find from ( [ eq - bpplmotion ] )    @xmath571 so it is obvious in this case that @xmath572 must be negative , i.e. the line orients both of its ends so that the external magnetic field is opposite the magnetic field of the line , as we would expect from energy considerations .    from ( [ eq - bpptmotion ] ) and ( [ eq - bpppmotion ] ) we find that @xmath244 is proportional to @xmath573 . @xmath574    hence we can solve both ( [ eq - bpptmotion ] ) and ( [ eq - bpppmotion ] ) by presuming that the ends of the line both align @xmath317 opposite the magnetic field so that @xmath575 . next we consider the case where the electric field and the magnetic field are perpendicular as is often the case with electromagnetic waves and is true of the field of another moving pole ( [ eq - bppbfield]-[eq - bppefield ] ) . in this case we can solve the equations of motion with the following presumptions :    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ \1 ) at each end of the line , @xmath317 aligns opposite the external magnetic field . \2 ) the external electric field defines the direction of @xmath563 , perpendicular to @xmath317 ( @xmath576 ) . \3 ) then @xmath226 and the velocity of the bare positron - pole is in the direction @xmath577 ( so @xmath578 ) perpendicular to the line , and the value of @xmath579 can be found from ( [ eq - bpplmotion ] ) ,    \4 ) equation ( [ eq - bpp3motion ] ) is satisfied because @xmath580 and @xmath581 are both zero . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _    so we have the solution for the motion of the bare positron - pole in the case that the external electric and magnetic induction fields are perpendicular . the bare positron - pole carries no inertia so it responds to the field conditions instantaneously , as it moves and the external fields change . in the general case where @xmath582 , the end of the line must orient itself such that the @xmath563 component of the lorentz force on the moving pole is @xmath64 . this requires a nonzero value of @xmath580 . if @xmath583 , it is easy to find the appropriate orientation of @xmath317 if @xmath584 is small . the general case requires the solution of a transcendental equation . ( see appendix  [ ssec - orientation ] . )    in all cases we use ( [ eq - bpplmotion ] ) to find the velocity of the moving pole :    @xmath585    and    @xmath586    where the @xmath587 sign takes the same value in both equations . we see from ( [ eq-1ogamp ] ) that we must have the @xmath588 sign because @xmath572 and @xmath589 are independent , i.e. @xmath572 could be zero . hence the velocity of the bare positron - pole is given by    @xmath590    @xmath591    using the minimal value of the line - pair density of the ether given by ( [ eq - dldvnum ] ) we can estimate @xmath158 , the typical distance between line pairs    @xmath592    we find @xmath593    and hence @xmath594    is the typical time between collisions of the bare positron - pole with line pairs at the minimal density of ( [ eq - dldvnum ] ) . during each collision the external electromagnetic field changes radically in both magnitude and direction and the bare positron - pole travels near the velocity of light in a new direction . the main causes of the external field are    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ \(1 ) reflection of the pole field ( [ eq - bppbfield]-[eq - bppefield ] ) by the superconducting line pairs ( see ( [ eq - pealemac ] ) and ( [ eq - memalbmac ] ) ) , and    \(2 ) currents in closed superconducting loops , stimulated by the pole magnetic field , bringing the magnetic flux linking each loop to zero because of flux quantization _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _    thus the motion of the bare positron - pole is _ zitterbewegung _ , jittery random - walk motion with a very short time between collisions . even though the bare positron - pole is traveling near the velocity of light , for many purposes it can be considered at rest in a frame traveling at a much smaller velocity . contrary to the general rule ( section  [ ssec - generalityquant ] ) this zitterbewegung motion is _ not _ quantized , because the end of a line ( electron ) is not part of a closed loop .      at this point we are forced to make an assumption which does not follow from the principle of least action applied to the superconducting line . in order to agree with experiment we need to essentially put the appropriate electric charges at the ends of each line . we will try to do this so that the electron receives the correct charge and also the correct spin . we can do this while avoiding questions about how we attach the charge by assuming that in exiting the end of a line the electromagnetic field undergoes a phase transition . we are implying that the core of a line is a third phase , in addition to the superconducting field and normal space outside the line .    in the rest frame of the bare positron - pole ( still using posititve charge for convenience ) , this phase transition is given by    @xmath595    where , outside the line ,    @xmath596    is the expected magnetic induction field of a point pole . the magnetic induction @xmath56 is constant across the phase boundary , so that the maxwell equations are satisfied in this phase transition . the conservation of charge is not violated since an electron and positron are assumed to be pair produced when a line breaks . assumption ( [ eq - phasetrans ] ) is equivalent to the transformation on the charges    @xmath597    and so is our model for the origin of electric charge . we can take our new assumption about charges conveniently into account with the substitution    @xmath598    in equation ( [ eq - bppbfield ] ) . then if @xmath599 is equation ( [ eq - bppbfield ] ) and @xmath600 is equation ( [ eq - bppefield ] ) with the replacement @xmath601 , we can express the total electromagnetic field ( _ except _ the line field ) of the bare positron as a complex vector in the form    @xmath602    @xmath603    where @xmath604 is just ( [ eq - bppbfield ] ) with the substitution @xmath605 . it will be convenient to use this complex notation . the real field comes from the magnetic charge , and the imaginary field ( which , of course , is actually physical ) comes from the electric charge . thus we are saying that the bare positron is almost a _ dyon_@xcite , but with a _ spin - dependent _ magnetic charge .    in section  [ ssec - motionbpp ] the motion of the bare positron - pole was discussed entirely on the basis of the principle of least action applied to the superconducting line , before making our assumption on the origin of electric charge ( [ eq - origincharge ] ) . we note that the equations of motion ( [ eq - motionep ] ) or ( [ eq - motionepparts ] ) do _ not _ involve the bare positron - pole s pole field if we evaluate the self field just inside the end of the line . they do involve the pole field if we evaluate the self field just outside the line . our assumption ( [ eq - origincharge ] ) makes a difference only outside the line . with or without this assumption the dominant external electromagnetic field will be due to the reflection ( including the role of current loops ) of the electromagnetic field of the bare positron . we , therefore , presume that the character of the motion of the bare positron ( _ zitterbewegung _ ) is unchanged by our assumption on the origin of electric charge ( [ eq - origincharge ] ) . we prove that this is true in appendix  [ ssec - motionelectron ] . however , by adding an electric charge , we have added angular momentum ( spin ) to the end of each line . this will affect the orientation of each end of the line . so far we do _ not _ have a model of the magnetic moment of the electron or positron , so we expect our results for the _ orientation _ of the end of each line to be _ nevertheless we presume that the microscopic motion of the bare positron is still zitterbewegung .    from section  [ ssec - motionbpp ] we expect rapid changes in the external electromagnetic field felt by the bare positron on a time scale of about @xmath606 or less . ( with a higher line - pair density of the ether , this time would be shorter . ) no measurement device in human possession can measure this time dependence , so we consider the electromagnetic field averaged over a time interval of    @xmath607    during which there are at least @xmath608 collisions , enough to form a reliable measurement of the average field .    note that although the bare positron is rapidly accelerating , it _ does not radiate _ because the magnetic induction field is blocked by the closed loops maintaining flux quantization . hence we can see immediately that    @xmath609    the time - avearaged magnetic induction field is zero at macroscopic distances from the positron . the magnetic induction field is reflected back toward the bare positron with no energy loss ( by superconducting loops ) , where it is reflected again by other current loops , and again , and again until the sense of direction of the field is lost , because the current loops are in random orientations . hence , effectively , all the magnetic induction field from the bare positron is contained in a very small region near the moving positron , and so in this region , it is _ very _ intense . we work in the effective rest frame of the bare positron the frame in which it is at rest over @xmath610 and hence    @xmath611    if we average over the bare positron s spin and are far from any masses the only unique vector is in the direction    @xmath612    which we presume is reasonably constant over @xmath610 . hence in finding the average electromagnetic field of the positron we can see that all terms transverse to @xmath613 average to zero . hence    @xmath614    the average microscopic electric field near the bare positron is given by    @xmath615    the macroscopic electric field of the positron is    @xmath616    as expected . thus the measured charge of the positron is    @xmath617    when the dielectric constant of the ether is taken into account . hence the measured charge of the electron is not constant . it should vary with the dielectric constant of the ether as the universe expands . a portion of the cosmological red shift could be due to the change of this dielectric constant with time . the expected change with time is in the observed direction . ignore the positron s line field _ , and ignore , for the moment , magnetic screening due to flux quantization , we can find the effective magnetic charge on the positron ( working with positive charge ) using@xcite the modified maxwell equation    @xmath618    the magnetic field at macroscopic distances from the positron will be    @xmath619    but the magnetic induction field will be    @xmath620    determined by the strength of the effective free pole ( magnetic charge ) .    both of these fields will be blocked by screening currents due to flux quantization . we do not yet have a model for these screening currents , but we can at least take these screening currents into account through the use of a _ screening function _ @xmath621 which cuts off the magnetic field of the positron as it would be , on average , cut off by screening currents . @xmath622    the actual shape of the magnetic induction field of a positron will vary from this smooth model according to the actual positions of the closed circuit current loops relative to the particular positron in question . equation ( [ eq - bmacscreenedpole ] ) just represents the average shape of the magnetic induction field coming from the positron itself . reflected fields are not included . it is well known@xcite that the electromagnetic angular momentum of a point monopole @xmath623 in the presence of a point charge @xmath188 is    @xmath624    where the angular momentum vector points in the direction from the point charge to the point pole ( if both are positive ) . since our model of the positron is almost a dyon , we can calculate the spin of the positron if we assume that the line is filamentary , and that it points in the @xmath243-direction , where it can make no contribution to the angular momentum . we obtain    @xmath625    this calculation assumes that the line radius is small enough that the point charge and point pole approximations can be used . we also assume that the screening currents are sufficiently far away that they do not affect the calculation . the calculation is independent of the separation between the pole and the charge , but there must be some separation to determine the _ direction _ of the spin .    since there is no natural separation between the charge and the pole in our assumption ( [ eq - origincharge ] ) , we must look to experiment to gain insight . for a positive charge we assume that the spin is in the same direction as the magnetic moment of the _ line _ ( @xmath317 ) . ( we do not yet have a model of the magnetic moment of the positron . ) figure 14 shows schematically the small separation between pole and charge required to have the spin point in the assumed direction for a positron . figure  [ fig - electronspin ] also shows the situation for an electron . for both particles the assumed spin direction is _ opposite _ to the line .    to 1.8 in     it is interesting to note that the microscopic calculation of the spin ( [ eq - leg]-[eq - spinpositron ] ) gives the same result as the macroscopic calculation    @xmath626    the microscopic calculation is probably the most relevant since the dominant contributions to either integral ( [ eq - spinpositron ] ) or ( [ eq - legmac ] ) come from regions very near the pole and charge . the separation between the pole and the charge is the only length involved and defines the importance of contributions to the integrals . in the final integration over the @xmath106-direction , half of the integral is contributed by the region between the pole and charge . since this separation is apparently _ very _ small ( see equation  [ eq - rasmall ] ) , the approximation that the screening fields do not affect the integral is probably valid .      from sections [ ssec - motionbpp ] and [ ssec - electroncharge ] we see that the electromagnetic field of the bare positron ( again considering positive charge ) does not carry inertia , i. e. mass . in the average rest frame of the bare positron , however , the superconducting ether will certainly respond to the average electromagnetic field of the bare positron . this response implies a potential energy , which we interpret as mass . the wave function of the positron is the response of the ether to the electromagnetic field of the bare postron@xcite    @xmath627    where the sum is over all line pairs which respond to the bare positron . so the mass of the positron is just the potential energy of its wave function in its rest frame . this response is a discontinuous function of position ( as in section  [ ssec - wfelectron ] ) . it is only defined at equilibrium positions ( in absence of distortion ) of line pairs . the wave function is a vector field , but it always points toward ( or away from ) the average position of the bare positron , so we can take it as a scalar without loss of information . so the mass of the positron is given by    @xmath628 \label{eq - mesum}\ ] ]    where the integrals have the same end points in space , but differing paths . the points in space @xmath629 and @xmath630 are chosen to be on line pair @xmath341 but beyond the response of line pair @xmath341 to the bare positron . the first integral is over the actual path of the distorted line pair @xmath341 using the actual tension @xmath631 which could be different from @xmath632 in general , because this line pair is part of a matter wave , a quantized excitation of the ether . ( but see below ( [ eq - tensiontau0 ] ) and ( [ eq - metau0 ] ) where we will find , in fact , that @xmath633 ) . the second integral is over the straight line path ( in the @xmath243-direction ) which the line pair would have in absence of the positron . since in this case the line pair is assumed not to be in an excited state , the tension is @xmath632 ( [ eq - tensiontau0 ] ) . the mass of the positron ( or electron ) in this model is clearly calculable . at small distances there is no convergence problem since we sum over a finite number of line pairs . at large distances we can approximate the sum ( [ eq - mesum ] ) by an integral . all integrals clearly converge since we will see that the electromagnetic force causing the response of a line pair falls off like @xmath634 where @xmath635 is the magnitude of the separation from the bare positron s average position . the zitterbewegung motion of the bare positron is far too fast to affect the ether s response . with typical times of @xmath606 , very large energies would be required for such a response , because all responses are quantized . the ether responds to the _ average _ electromagnetic field of the bare positron , as it undergoes its zitterbewegung motion in its wave function . the quantized changes occur over long times and can therefore have low energy . we calculate the response of a line pair to the average electromagnetic field of the bare positron in its average rest frame , where the calculation is static . if @xmath173 is the electric dipole moment per unit length of the line pair in question , the electric force per unit length on the line pair is    @xmath636    but since the dipole moment per unit length is induced by the electric field of the positron itself we have    @xmath637    where @xmath638 is the polarizability per unit length of the line pair , which in general depends on @xmath639 , the angle between the electric field and the line pair@xcite . ( see figure  [ fig - lpstars12 ] . )    to 1.5 in     hence the electric force per unit length on the line pair    @xmath640    is proportional to the gradient of the square of the electric field independent of the sign of the charge of the electron or positron . the component of the opposite force along the line pair is the gravitational force on the electron or positron . it is independent of the sign of the charge because it is _ induced_. hence the electric force which the positron exerts on a line pair is given by    @xmath641    and is always _ attractive _ , as required for the force of gravity in our region of the universe . ( we assume that we can neglect the second term in this case because @xmath642 , but see section [ ssec-1or3 ] )    the effective magnetic charge of the positron will also cause a force on our line pair given by@xcite    @xmath643_i\ = \ \bigl [ ( { \bf m } \times \nabla)\ \times\ { \bf b}^{mac } \bigr ] _ i\ = \ \sum_k m_k \nabla_i b_k^{mac}\ ] ]    where @xmath172 is the magnetic moment per unit length of the line pair . the induced magnetic moment is _ opposite _ to the applied magnetic induction field    @xmath644    where the polarizability is the same as in the electric case . hence the magnetic force per unit length on a line pair is    @xmath645    again independent of the sign of the charge . using ( [ eq - bmacscreenedpole ] ) the magnetic force per unit length is    @xmath646    _ repulsive _ and independent of the dielectric constant of the ether . the screening function @xmath621 ( e.g. @xmath647 ) must be positive and strongly increasing as a function of @xmath635 . it can also vary slowly with the position of the positron ( e.g. @xmath648 above could vary ) due to extra line pairs from a passing star . because of this screening , we assume the electric force is dominant , except _ very _ close to the positron , and in the case that the dielectric constant of the ether is _ very _ large . thus the total force per unit length on a line pair is given by    @xmath649    @xmath650    in figure  [ fig - lpstars12 ] we consider a line pair stretching out from star 1 and passing through our positron s wave function on the way toward star 2 ( which could be _ very _ far away , e.g. over the horizon ) . in figure  [ fig - tensionbalance ] we see a closeup of how the force per unit length @xmath651 changes the tension , which is considered to point along the line pair in the direction from star 1 to star 2 ( except in figure  [ fig - tensionbalance](a ) ) .    to 1.5 in ( a ) considering the tension always to point from left to right.,title="fig : " ]    since the forces must balance ( in the static configuration )    @xmath652 so    @xmath653    and    @xmath654    where * x * is any point on the distorted line pair and @xmath655 is the initial tension vector at point @xmath629 . hence    @xmath656    the tension @xmath657 varies such that the effective tension , _ including _ the effects of the force due to the positron , is constant . the transverse components change the direction of the tension , while the longitudinal components can change the magnitude . it is convenient and reasonable to _ define _ the tension to include the longitudinal force of the positron . this force is effectively part of the tension of the responding line pair . then the magnitude of the tension is always @xmath632 in accord with ( [ eq - tensiontau0 ] ) , and so that the matter wave is a gravitational / common - mode wave . hence we replace @xmath631 by @xmath632 in ( [ eq - mesum ] ) and obtain for the mass of the positron    @xmath658    where @xmath659 is the length difference of line pair @xmath341 with the positron and without it . the response of a line pair extends far beyond the point of application of the force @xmath660 , just as the distortion of a bow string extends far beyond the point at which it is plucked . just how far the response extends should depend on the surrounding matter density because other charged particles will exert forces on this same line pair . these forces , which are exerted on the line pair under consideration by other charged particles , can also be considered using ( [ eq - taux ] ) . thus the calculation is a many body problem , by its nature , because everything is connected . appropriate assumptions must be made to find a numerical solution . this work has not yet been done . dependence on the surrounding matter density has been observed . the mass of the electron changes from it s  vacuum " value when the electron is in a solid@xcite . this change of mass in a solid is thought to be a collective effect . we are suggesting that mass may be a collective effect even in  vacuum " .    in order to gain insight into this mass calculation we construct a toy model in which we assume ( figure  [ fig - lpstars12toy ] ) that the line pair in question is fixed at a distance @xmath433 from the positron . we neglect the curvature of the line pair in evaluating the total force and consider the total force to act at the point of closest approach . we assume the electric force dominates so that ( taking only the first term ) the components of the force per unit length which the positron exerts on the line pair are given by    to 1.5 in     @xmath661    @xmath662    where we simply take @xmath663 @xcite . the total force perpendicular to the line is    @xmath664    where @xmath368 is the distance of closest approach of the line to the positron . the total longitudinal force ( along the line ) is zero but for future reference we calculate the force that the positron exerts on the line pair in the directions of star 1 and star 2 separately :    @xmath665    @xmath666    if @xmath667 is the displacement of the line pair at closest approach , from figure  [ fig - lpstars12toy ]    @xmath668    @xmath669    and since @xmath670 we have    @xmath671    since the line pair tension must balance @xmath672    @xmath673    we find    @xmath674    @xmath675    so the mass of the positron or electron in this toy model is given by    @xmath676    so , replacing the sum by an intrgral    @xmath677    we have ( since @xmath678 )    @xmath679    @xmath680    at this stage the only quantities for which we know a numerical value in this equation are @xmath188 and @xmath681 , but the formula illustrates what quantities are important in determining the mass of the electron in this model . we can cross check this formula with others which we will obtain . we expect order of magnitude accuracy if we were to evaluate @xmath682 and @xmath433 with care  using appropriate averaging , probably with a monte carlo . the cutoff @xmath682 should be evaluated so the integral in ( [ eq - sumtoint ] ) approximates the required sum . it depends , at least in principle , on @xmath145 and on the function being integrated ( here @xmath683 ) .    near the earth we will see ( section  [ ssec - parameters ] ) that the dielectric constant of the ether is estimated to be very close to @xmath114 . in this case we have @xmath684 and @xmath685 , the strength of the dirac monopole . hence as @xmath686 the force @xmath651 on a line pair is dominated by the pole force , and hence is _ repulsive_. so we expect the effective value of @xmath682 to be the size of a typical deflection with the full pole strength . outside @xmath682 ( or @xmath687 , reflecting radial symmetry of the configuration with many line pairs ) line pairs will be concentrated as the pole is rapidly screened and @xmath651 becomes attractive due to the electric force . since @xmath682 should depend only on the size of a typical deflection at full pole strength , we will simply take @xmath682 to be a constant ( at least in vacuum near the earth , relevant to the calculation of @xmath19 in this toy model ) . we discuss the response of our macroscopic electron in its mass shell ( wave function ) to an external electromagnetic field and the possible effects of its string in appendix  [ sec - response ] . we have seen in section  [ ssec - masselectron ] that the mass of the electron is proportional to @xmath145 the line pair density of the ether . we assumed that the line pair density is isotropic reflecting the large scale structure of the universe , for which ( from ( [ eq - dndadomega ] ) )    @xmath688    where the subscript references the _ background _ ether  line pairs for which star 1 and star 2 are _ very _ far away ( perhaps over the horizon ) . in contrast to the background ether , the line pair density from a star is certainly _ not _ isotropic . since we can approximately ( better than @xmath689 accuracy ) associate a line pair coming from the star for each neutron mass ( @xmath690 ) @xmath691    where the delta function in solid angle ensures that all line pairs point in the radial direction from the center of the star ( a simplification ) and where    @xmath692    where @xmath693 is the mass of the star and @xmath176 is the distance from the center of the star . in table [ table - lpdatsurface ] we list some line - pair densities at the surfaces of known astronomical objects , where sagittarius @xmath694 is a star at the center of our galaxy which is thought to be a supermassive black hole@xcite . also listed is the star at the center of the m87 galaxy which appears to be an even more massive black hole@xcite . for the two black holes we take    @xmath695    the radius at the surface to be the schwarzschild radius . we also quote @xmath158 values at the surface of the object using ( [ eq - ltyp ] ) . bject   = @xmath696   = @xmath697   = @xmath698   @xmath699 + earth @xmath700 @xmath701 @xmath702 @xmath703 + sun @xmath704 @xmath705 @xmath706 @xmath707 + sagittarius @xmath708 @xmath709 @xmath710 @xmath711 @xmath712 + m87 @xmath713 @xmath714 @xmath715 @xmath716 +    [ table - lpdatsurface ]    bject   = @xmath696   = @xmath717   = @xmath698   @xmath699 + sun @xmath704 @xmath718 @xmath719 @xmath720 + central bulge @xmath721 @xmath722 @xmath723 @xmath724 + background ether - minimum @xmath725 @xmath726 + background ether - measured @xmath727 @xmath728 +    [ table - lpdatearth ]    we note that the line - pair density at the surface of the earth , from the earth alone , is 11 orders of magnitude greater than our previously assumed minimum line - pair density of the background ether ( [ eq - dndanum ] ) ! assuming masses do not vary significantly with altitude above the surface of the earth@xcite , we infer that the line - pair density of the background ether must be 13 orders of magnitude or more higher than our minimum estimate ( [ eq - dndanum ] ) . this fact has significant implications for the discussion of inflation .    in table [ table - lpdatearth ] we list the line - pair densities at the earth due to our sun and the central bulge of our galaxy@xcite . the mass contained in this central bulge is only perhaps a factor of 2 less than the visible mass of our galaxy inside the orbit of the sun . we see that the contribution of the central bulge is much smaller than the contribution of the sun , which is much smaller than the contribution of the earth itself . we also list our new estimate for the minimum line - pair density of the background ether ( @xmath729 and our rough measurement ( [ eq - fulldensity ] ) .    we have painted a very static picture in discussing equation ( [ eq - dndastar ] ) . this static picture should be viewed as a statement of the average behavior line pairs emerging from a star . all changes from this picture are quantized , but changes will certainly occur . line pairs are attracted by charges of both signs . so if a star is moving through a field of line pairs , the line pairs will tend to stick to the charges , and stretch as the star continues to move . when the line pairs have stretched sufficiently that their tension force is strong enough to free them ( perhaps due to a changing angle ) , they will contract ( being perfectly elastic ) and move to the next row of charges . the line pairs will tend to form straight lines , however , because in that way they minimize their energy . most line pairs are part of the background ether so end , and hence are bound strongly , in stars which are _ very _ far away . assuming the universe is still expanding , star 1 and star 2 , at the ends of a typical line pair , are moving apart as the line - pair tension @xmath632 continues to decrease very slowly with time throughout the universe . two line pairs can bind to form a 4-line system , but the binding is weak compared to the binding for a pair .    in figures [ fig - lpstars12 ] and [ fig - lpstars12toy ] we identify the force of gravity from star 1 via our line pair @xmath341 on our standard positron to be ( from ( [ eq - fz1 ] ) )    @xmath730    in a similar way we identify    @xmath731    the force from star 2 via the same line pair on our positron . the situation is intrinsically 3-body and equations ( [ eq - fg1j ] ) and ( [ eq - fg2j ] ) represent the only self consistent way to isolate @xmath732 and @xmath733 respectively , as is required to describe the force of gravity . ( in principle we should include @xmath734 in the force from both stars , but this force adds to zero when we consider all line pairs near the positron . ) both stars exert attractive forces on the positron via this line pair @xmath341 . after all , line pair @xmath341 is _ part of both stars_. the force of gravity propagates instantaneously because both stars are already at the position of the positron . the quantum nature of these forces is illustrated by the fact that many line pairs from star 1 will come through the positron s wave function if it is sufficiently near star 1 . however , the nearest line pair coming from star 2 could be light years away if star 2 is over the horizon . thus star 2 will not exert significant force on the positron , similar to many other stars in random directions . thus the total force of gravity due to all line pairs from star 1 on the positron is    @xmath735    @xmath736    @xmath737    we observe that this force is not obviously proportional to the mass of the positron ( [ eq - me ] ) . we examine the possibility that this force obeys newton s law of universal gravitation , setting    @xmath738    and we find that    @xmath739    so    @xmath740    ( remember that we are only using a toy model here . ) the value of @xmath682 , like @xmath186 , must be constant within our solar sysem . the value of @xmath682 has an additional restriction . it must be the same whether the positron is attracted by the sun , or the earth , or a baseball . we note that the force of gravity on a single electron or positron has not been well measured@xcite . it is a very difficult measurement so we need a consistency check for this determination of @xmath19 . we consider a single electron which combines with a single proton to form a hydrogen atom  the smallest bit of neutral matter , for which newton s law of universal gravitation is well established in our solar system . we are trying to stay out of the nucleus so we simply assume that    @xmath741    the gravitational force on the proton from star 1 is a factor of @xmath183 times the gravitational force on the electron . so negelcting binding effects , the force on the hydrogen atom is just @xmath742 times the force on the electron . @xmath743    normal matter , neglecting binding effects , is just made up of proton - electron combinations , perhaps in the form of neutrons , so the total gravitational force on a mass @xmath744 ( _ assumed _ proportional to mass ) is just    @xmath745    so    @xmath746    and so we also obtain    @xmath747    if we compare our two values of @xmath19 we find    @xmath748    hence our two values of @xmath19 agree using a value of @xmath183 which indicates that the gravitational force exerted by star 1 is proportional to the mass of the object on which it acts . this proportionality is shown more generally by ( [ eq - proportionaltom ] ) . we have not discussed the gravitational force which our positron exerts on star 1 . it is zero unless the positron s line goes through the matter of star 1 . newton s third law is apparently violated by the quantization of the gravitational force , but it should be valid on average . ( the electromagnetic reaction forces to both gravitational forces , as we have defined them , do satisfy newton s third law , as we have shown . ) our calculation of @xmath19 only comes from a toy model , but it shows some important features . using the cgs values    @xmath749    @xmath750 we obtain    @xmath751    @xmath752    so apparently @xmath69 , the outer radius of a line pair , is very small . if @xmath682 is of order @xmath158 , @xmath753 !    since the gravitational field emerging from a star in this model is proportional to the number of line pairs emerging ( hence the number of nucleons in the star ) , the gravitational force in this model is _ not _ rigorously proportional to active mass ( the source of the field ) . we have simply _ assumed _ that the gravitational force is proportional to passive mass because we can not model the nucleus . for a rigorous treatment to an accuracy better than 1% and a discussion of experimental tests of our model of gravity , see appendix  [ sec - rigorousgravity ] .      in the presence of star 1 , the second term in ( [ eq - ftotal ] ) will lead to a gravitational force on our standard positron and we need to either consider it , or show that it is small . from ( [ eq - epsminus1 ] ) we find    @xmath755    where @xmath183 is the fractional concentration of superconducting fields , which implies that    @xmath756    near a star , @xmath183 will have two contrubution one from the background ether and the other from the star . for the star we have    @xmath757    where @xmath176 is the distance from the center of the star , and @xmath758 can not be greater than @xmath114 . since the superconducting fields can overlap , the fractional concentration _ free _ of superconductor is just the product of the respective concentrations without superconductor    @xmath759    implying that    @xmath760    hence , adding up the contributions of all relevant line pairs , the total force of gravity on the positron due to the gradient of @xmath186 is    @xmath761    also attractive . in calculating ( [ eq - fge ] ) we only included line pairs coming from star 1 , to calculate the total gravitational force from star 1 . the total force from the background ether was zero because it is isotropic . but here star 1 causes the dielectric constant of the ether to be _ anisotropic_. so all line pairs , even from the background ether , will exert a force on the positron in the direction of @xmath762 . the ratio of the magnitudes of the two forces of gravity is    @xmath763    for most circumstances    @xmath764    and we can neglect ( [ eq - fgeor3 ] ) with respect to ( [ eq - fge ] ) . however , in extreme circumstances where the fractional concentration of superconductor becomes very close to 1 , the @xmath765 term implies that @xmath766 could dominate . we rewrite ( [ eq - g ] ) to show the dependence of @xmath19 on @xmath186 , the dielectric constant of the ether :    @xmath767    actually @xmath20 ( [ eq - me ] ) and by assumption ( [ eq - fgp ] ) @xmath690 both should vary with @xmath186 . so we consider the quantity    @xmath768    proportional to the gravitational force on representative masses , to most effectively represent the weakening of the gravitational force as @xmath186 changes . if @xmath682 is constant , @xmath769 will weaken significantly in regions where @xmath186 is important , because @xmath186 becomes large near @xmath770 . thus the gravitational force should weaken compared to expectations when the density of line pairs becomes high , for instance near a black hole  where the gravitational force is expected to be strongest . since @xmath769 varies as @xmath771 , the effects of black holes could be considerably weaker than expected . this could contribute to the fact that ligo@xcite has not yet seen gravitational waves . on the other hand , the masses of black holes could be considerably larger than measured while assuming constant @xmath19 . as shown by ( [ eq - fgeepsofge ] ) in the limit @xmath770 , the @xmath754 gravitational force ( [ eq - fgeor3 ] ) could dominate ( [ eq - fge ] ) . but in this limit both electric terms in ( [ eq - ftotal ] ) should be dominated by the repulsive magnetic terms , which are screened but independent of @xmath186 . we will suggest in section  [ sec - rotations ] that the reversal of the gravitational force has indeed been seen , in a region which should have a _ very _ high density of line pairs . if this reversal has , indeed , been seen , a reversal should certainly have occurred in the high densities of the early universe . since we have so far been unable to derive newton s second law directly from the properties of the ether , we use einstein s principle of equivalence of gravitation and inertia@xcite ( extended slightly ) and consider a positron with acceleration @xmath772 . this acceleration is equivalent in all respects to a gravitational field which points _ opposite _ to @xmath772 . ( think of a scale in an elevator . ) @xmath773    the latter form applies if the gravitational field is due to star 1 . what we interpret as the gravitational force on the positron from star 1 , we have seen is actually a force of the line pairs from star 1 on the positron . @xmath774    hence in the equivalent situation of acceleration @xmath772 , line pairs exert a force on the positron given by    @xmath775    the acceleration must be due to some applied force , @xmath776 . the forces on the positron must balance :    @xmath777    @xmath778    hence the ether exerts a force equal to @xmath779 on an accelerating mass , leading to newton s second law of motion .    for a deeper understanding of newton s second law , including how it can be made consistent with the discussion of section  [ ssec - calcg ] , see appendix  [ sec - understandingnewton ] . as a result of the above discussion we see that the force exerted by the positron on the line pairs of the ether is given by    @xmath780    this force is exerted on the line pairs that go through the wave function of the positron . the line pairs that go through the positron s wave function respond by stretching and deforming ( not by accelerating like a massive particle ) . by our use of maxwell s equations we have assumed special relativity from the beginning of this paper . we point out how special relativity fits naturally into our construct . special relativity applies to the situation of a particle in the background ether , which is homogeneous and isotropic because it is far from any masses . if a photon passes this particle , its velocity is @xmath781 , because of the characteristics of the line pair ( without rest mass ) on which it is propagating independent of the motion of the source . this observation is sufficient@xcite to build the entire special theory of relativity . our concept of mass ( [ eq - metau0 ] ) fits very naturally into the overall construct , as potential energy of the ether .    for a discussion of the full meaning of lorentz invariance , see appendix  [ sec - understandingnewton ] . it should be possible to relate the quantized line - pair density of the ether , approximated by @xmath782 , to the metric tensor of general relativity in static situations . this work has not been done but has promise of uniting general relativity with quantum theory . the fact that @xmath19 is not constant must be taken into account . we have not yet done a serious job of studying the dynamics of the ether , e.g. using ( [ eq - fonpairs ] ) . such a study will be needed for a complete model . we note that there is an essential difference between line pairs in the ether and the structure of space - time in general relativity . line pairs are quantized objects which can _ move _ , and are not fixed in space - time . we also note here that the concept of gravitational waves used in this paper is quite different from the gravitational waves of the standard model@xcite . our concept of gravitational waves is appropriate to our quantized model . in this model , wave - particle duality is not a problem  it is natural . the particle is the bare electron . the wave is the response of the ether to the bare particle@xcite . since we started with dirac s quantization condition ( [ eq - singv ] ) it is not surprising that we are consistent with quantum mechanics with one exception . we construct a continuous wave function which models the displacement of each line pair in ( [ eq - psix ] ) in the average rest frame ( @xmath282 ) of the electron :    @xmath783    this function is a signed scalar . it interpolates between the displacements of the various line pairs to form a continuous function , as in standard quantum mechanics . but contrary to the expectations of standard quantum mechanics , this wave function does not spread out with increasing time . the electron at rest is at equilibrium with the ether  but subject to zitterbewegung which will manifest itself in slow motion after sufficiently long times . the whole wave function moves , but does not spread . this could be a measureable difference between our model and the standard model of quantum mechanics . thus we see that our electron s wave function is intrinsically localized in position space , but has a finite spread , and therefore must have some spread in momentum space . hence true plane - wave states do not exist . therefore we need to modify our discussion of the quantization of the photon ( equation ( [ eq - pepg ] ) and figures [ fig - wfelectron]-[fig - photonfixedz ] ) since we used a plane wave electron for simplicity . but there is really no essential difference from our previous discussion . the non - plane - wave electron state is a superposition of plane wave electrons with various values of @xmath784 and we have    @xmath785    for each @xmath784 in the superposition . this leads to the quantization of the photon ( [ eq - pepg ] ) as before . we can also obtain the quantization of matter waves by assuming that all matter waves are localized ( to any required accuracy consistent with the uncertainty principle ) in both position and momentum spaces . if we simply look on a larger scale at figure  [ fig - photonfixedt ] and replace the photon by a localized matter wave , then just as for the photon we will have    @xmath786    where we do not know in advance the precise value of @xmath784 , but presume it will correspond to a value of @xmath787 that is probable according to the momentum space wave function of our particle . in figure  [ fig - photonfixedt ] ( concentrating on the matter wave instead of the photon ) we pick a path of integration along the @xmath106-axis , defined to be parallel to @xmath787 , and integrate over one complete wavelength . from the fact that @xmath158 ( table  [ table - lpdatsurface ] ) is very small , we can do this to any required accuracy , and then turn to integrating over a @xmath368-leg , which is perpendicular to the momentum . we then use a @xmath106-leg which is far away , well outside the particle s wave function , where the contribution to the total integral is negligible . we then complete the integral with a @xmath368-leg as before . hence we obtain as in section  [ ssec - quantphoton ]    @xmath788    then using the standard formalism@xcite we find the momentum space wave function of our electron    @xmath789    in its rest frame . from our previous discussion ( section  [ ssec - masselectron ] ) both @xmath790 and @xmath791 are spherically symmetric , if we sum over spin states . hence the general wave function of the free electron can be written in the standard form , where    @xmath792    where @xmath793 is no longer restricted to the rest frame . the full wave function , including time dependence is given by    @xmath794    where @xmath784 is the total 4-momentum of the electron . in position space we have    @xmath795    just as in the normal formalism . this leads to schr " odinger s equation and the standard quantum mechanics of angular momentum . but until we have a model of the magnetic moment of the electron , we can not fully discuss spin . ( we could , of course , just take the value of the magnetic moment of the electron from experiment or the dirac equation , as standard quantum mechanics does . ) for instance for the hydrogen atom , an electron in the potential energy of a heavy proton , we obtain    @xmath796 \ \psi_{qm } \      = \ i\hbar\ { { \partial \psi_{qm } } \over { \partial t}}\ \ \    \label{sch}\ ] ]    the schr " odinger equation just like in standard quantum mechanics . the energy spectra for the two models are identical . in both models we have to add the effects of spin to get the measured spectra if we have succeeded in calculating @xmath19 , the entire concept of the planck mass is specious because @xmath19 is not fundamental . hence general relativity and the hot big bang model , in particular , need modification . but we can at least heuristically consider the topic of inflation in our model by using weinberg s result@xcite for evolution of the universe    @xmath797    where @xmath51 is the energy density of the universe , @xmath798 is the pressure , and @xmath176 is the cosmic scale factor ( sometimes called the radius of the universe ) . we suggest that the twoline potential energy ( [ eq - twolinepot ] ) was the energy source for inflation .    in the early universe , after the line pairs started to condense , we presume that they were radiation dominated , with many excitations traveling on them , so that both @xmath51 and @xmath798 were positive . the line pair density should have been extremely high so that the fractional concentration of superconductor approached one ( @xmath770 ) and the dielectric constant of the ether was very large ( @xmath799 ) . thus the effective charge of the electron approached zero and from ( [ eq - ftotal ] ) the gravitational force was initially _ repulsive_. therefore , heuristically , we can think of the value of @xmath19 in ( [ eq - evolutionr ] ) as negative and exponential expansion resulted . when the universe expanded to the point where the fractional concentration of superconductor became less than 1 , @xmath186 decreased , the force of gravity became attractive and the exponential expansion stopped , as long as    @xmath800    this should have been true through the eras of radiation dominance and matter dominance  until fairly recent times . using ( [ eq - fulldensity ] ) and ( [ eq - dndanum ] ) , we estimate the expansion factor during inflation to be @xmath22 , with a large uncertainty . dark energy consists of line pairs which stretch across the universe . the dark energy density is the energy density of the ether    @xmath801    note from ( [ eq - tensiontau0 ] ) that the energy / length due to the quantization currents has been included in @xmath632 . dark energy is completely dark . the only way it can be detected directly is by its gravitational effects , such as changes in a mass moving through it . the dark energy density decreases with time as the universe expands and the tension of the all the line pairs in the universe decreases slowly . the pressure due to the longitudinal tension is given by    @xmath802    so for the longitudinal tension alone we would have the ratio of pressure to energy density given by    @xmath803    if there were no transverse tension , we would take the average pressure over three directions and find    @xmath804      but there is tension in one of the transverse directions . we can easily calculate the transverse tension which the quantization currents need to balance in order to keep @xmath632 constant , for a length @xmath210 of a line pair . if @xmath805 locates the center of line 2 from line 1 ( section  [ sec - two ] ) , this tension is    @xmath806    the quantization currents could provide this transverse tension simply by flowing in opposite directions on the outer surfaces of the line pair  or they could communicate with other line pairs as shown in figure  [ fig - currentsforapart ] . in the latter case the transverse tension would be transmitted to other line pairs and    @xmath807    implying that @xmath808 for the ether and from ( [ eq - evolutionr ] ) the ether can contribute to the current acceleration of the universe . each line pair in this case is under tension in two directions . we use ( [ eq - me ] ) , ( [ eq - g ] ) , and ( [ eq - rholambda ] ) to estimate the line - pair density of the background ether in terms of the dark energy density , the mass of the electron , and newton s constant @xmath19    @xmath809    we have taken @xmath810 from the wmap collaboration@xcite    @xmath811    in order to obtain an estimate for the properties of the background ether , we guess that @xmath812 so that    @xmath813    we take @xmath433 to be the mean free path in air@xcite    @xmath814    we obtain    @xmath815    consistent with our minumum value from table  [ table - lpdatearth ] . we note that the line pair density of the background ether is of order 20 times the line - pair density of the sun at its surface . from ( [ eq - dldv ] ) we obtain @xmath816 .      using ( [ eq - rholambda ] ) and ( [ eq - fulldensity ] ) we find the line - pair tension of the universe at the current time :    @xmath817    then again using    @xmath818    we find from ( [ eq - rasmall ] )    @xmath819    we note that our guess for @xmath682 ( [ eq - yminguess ] ) and ( [ eq - ymintyp ] ) looks reasonable since @xmath682 must be @xmath820 , or it would affect the measured pointlike structure of the electron .    to estimate the fractional superconducting concentration of the background ether , we consider the fractional concentration in one volume    @xmath821    and find it to be negligible . so the dielectric constant of the ether is 1 near the earth and in all circumstances except the most extreme . since this model predicts the weakening of the gravitational force near strong gravitational sources because of the increasing dielectric constant of the ether , in principle it could account for dark matter . but the very low fractional concentration of superconductor in the background ether from ( [ eq - fb ] ) ( which should be similar in contribution to stars in our galaxy ) appears to rule this out . we need to discuss what happens to the line pairs , which begin in a star , when it rotates . if a line pair points along the axis of rotation , nothing happens . if it is off axis pointing parallel to the axis , it will twist about the axis of rotation . but this is no problem since line pairs can move through each other . if a line pair is off axis and pointing perpendicular to the axis , it will stretch and contract as the star rotates . this is no problem since the first line pair is matched with another line pair pointing in the opposite direction which stretches when the first contracts and vice versa . hence no net work is done by the star in rotating    however , as a star rotates , each bit of mass @xmath91 does exert a force on the line pairs in its wave function given by ( [ eq - fonpairs ] )    @xmath822    ( where @xmath823 is the cylindrical radius ) . thus in extreme cases , line pairs will be concentrated on the axis of a spinning mass . hence we achieve a natural qualitative understanding of agn ( active galactic nuclei ) jets in this model . the m87 galaxy ( in table  [ table - lpdatsurface ] ) has an agn / black - hole at it s center of a mass @xmath824 , which sends out about @xmath825 line pairs . many of these line pairs become concentrated on the axis of rotation , apparently sufficiently to reverse the force of gravity , producing jets . the velocities in these jets are high@xcite , because the particles involved have very low mass , according to ( [ eq - me ] ) because @xmath826 becomes small . ( actually if the force of gravity reverses , the dominant contributions to the mass ( [ eq - metau0 ] ) from ( [ eq - ftotal ] ) are repulsive , but we can still expect the mass to be small . ) we have tied a string onto every electron in the universe , therfore electrons are _ distinguishable_. this removes the theoretical underpinnings of the exclusion principle , which apparently pauli never believed@xcite anyway . but something must replace these underpinnings . we suggest that the pole force of the electron does part of the job . the pole has the strength of a dirac monopole ( if @xmath502 near the earth ) . screening currents tend to amplify the strength of the pole if you are in _ contact _ with the electron , i.e. inside the screening currents . so there should be _ very _ strong forces between electrons ( i.e , any objects ) in contact . we suggest the pole field of the electron is responsible for the impenetrability of matter . if you push down on a table , we suggest that the force which pushes back on you is the contact force of the pole fields of the electrons in the table . ( hence our lines sense pressure only at the ends . ) we postulate that the rest of the job is done by the line forces between electrons . the line force has infinite range , as does the spin glass spin - spin correlation length@xcite . we also suggest that the line force is responsible for the _ exchange _ force of ferromagnetism . elliot lieb has said@xcite that the reason the pauli exclusion principle is not a force is that you can not write it in hamiltonian form . so calling it a force is perhaps appropriate for this classical model . the line force between two electrons provides a strong attractive force between separated electrons of opposite spin . apparently any attractive force between electrons can lead to cooper pairs@xcite , so we expect the line forces between two electrons to contribute to the formation of cooper pairs , and hence that form of superconductivity . we note that line pairing should show an isotope effect since nuclei are involved . the isotope effect played a strong role in establishing the original cooper pair mechanism . the quantum of flux in a cooper pair system is @xmath827 , because the charge is due to two electrons . we have to explain why flux quanta of @xmath827 can exist in type ii superconductors , when only flux quanta of @xmath828 are allowed by the ether . we note that @xmath827 is a point of unstable equilibrium for line pairs in the ether . if the curl of the phase field is filamentary , the ether currents do nt know which way to change . i.e. whether to bring the flux to @xmath64 or @xmath828 we postulate that this position of unstable equilibrium with the ether currents is enough to let the cooper pair currents keep the flux steady at @xmath827 . it is interesting to note that if line forces between electrons contribute strongly to the formation of cooper pairs , then line pairs are pretty much the same thing as cooper pairs in a superconductor  and hence appropriate to the generation of mass@xcite according to some authors . the integer quantum hall effect can be taken@xcite as direct evidence for the quantization of magnetic flux in units of @xmath828 . in the fractional quantum hall effect , it is suggested@xcite that composite fermions are formed when one electron binds an even number of flux quanta ( @xmath828 ) . the fractional quantum hall effect is then explained when these composite fermions undergo the integer quantum hall effect , as if they were electrons . the currents needed to contain these flux quanta are not obviously available , so we suggest that the quantum hall effects present direct evidence for our unpaired lines of flux ( section  [ ssec - static ] ) . we suggest that the flux is contained by the superconducting fields which we postulate in section  [ ssec - superconfield ] . the force that binds the electron to the lines of flux is then given by ( [ eq - ftotal ] ) . we have seen that this same force on line pairs is the force of gravity . we obtain schr " odinger s equation for the hydrogen atom , and by implication for heavier atoms . we obtain newton s second law of motion for the electron , and by assumption for more massive objects . our model of gravity agrees with newtonian gravity within our solar system . therefore differences between the predictions of this model and the standard model should be hard to find . we focus here on experiments which could show a difference . we show in table  [ table - relevantexperiments ] relevant experimental results and indicate if the standard model and this model are consistent with these experiments .    1 . search for flux quantization outside cooper pair superconductors : + the most relevant experiment to this model is a search for flux quantization performed by tonomura _ et . they conclude that magnetic flux is not quantized in a torroidal ferromagnet . if this result is correct , it rules out this model , which is based on flux quantization . this experiment is one in a series of beautiful electron holography experiments that has studied the aharanov - bohm effect in fine detail . + however , we suggest that this group of experiments can not rule out this model . the reason is that this model changes the nature of the experimenter s probes . we attach a line of flux onto each of their electrons . examination of their methods@xcite shows that they did not pay attention to the spins of their probe electrons . consequently the lines of flux attached to their probe electrons may have spoiled their results . their geometry in dealing with normal samples was likely different from that which they used for measurements with superconductors . + the above argument applies to any search for flux quantization which uses electrons . + we consider the argument for flux quantization in section  [ ssec - fq ] to be very strong . we also note that flux quantization has apparently been observed outside of superconductors in the quantum hall effects g-2 of the electron : + we list in table  [ table - relevantexperiments ] the measurement of @xmath829 of the electron , for which the standard model has achieved very high precision agreement . so far , there is no prediction for this model , because we do not have a model of the magnetic moment of the electron . dark energy : + measurements of dark energy are pretty clearly not consistent with the standard model , but are consistent with this model of the ether ( section  [ sec - darkenergy ] ) . 4 .   inflation : + inflation in the standard model requires a scalar field , which has never been observed , plus reheating@xcite after the scalar field does its job . this model , at least heuristically ( section  [ sec - inflation ] ) , proceeds from the string era naturally , with no new requirements , since the strings are still here . agn jets : + this model , at least qualitatively ( section  [ sec - rotations ] ) , explains why the rotation axis of a black hole is special , why agn jets form , and why they have very high velocities . explanations using the standard model are doubtful . spin - spin correlation length + the spin - spin correlation length in spin glasses is infinite@xcite . every spin couples equally with every other spin regardless of how far they are apart . this is widely considered to be ridiculous , but it is not taken as a conflict with the standard model , because spin glasses are very complicated systems . in our model an infinite correlation length is natural . we suggest that these complicated systems which disagree with the standard model show a pattern . fractional quantum hall effect : + the fractional quantum hall effect is another case where the standard model is in apparent disagreement with experiment , but the disagreement is not taken very seriously because solids are complicated systems . our model has a natural interpretation of this effect ( section  [ sec - quantumhall ] ) . xperiment   standard model   this model   +   + flux quantization search@xcite yes no but see section  [ ssec - existing ] + g-2 of the electron yes no prediction + dark energy no yes + inflation requires scalar field , reheating yes + agn jets doubtful yes + spin - spin correlation length no , but complicated yes + fractional qh effect no , but complicated yes +    [ table - relevantexperiments ]      in addition we suggest the following experiments to measure a difference between this model and the standard model :    1 . find the line : + this model predicts that charged particles come with an attached line . in accelerators presumably the pair to the beam particle ( with which the beam particle forms a line pair ) , because of energy considerations , is among the image charge in the beam pipe . the stability of the beam should be strongly affected by the image charge in the beam pipe . apparently this is true . if a short section of ceramic beam pipe were to be used , the beam would probably be lost.@xcite it may be possible to directly observe the line in accelerators measure g accurately : + look for a material dependence ( of order a part in @xmath830 ) of the measured values of @xmath19 . see appendix  [ sec - rigorousgravity ] . 3 .   study the pomeron : + if this model is correct charged particles are not isolated and unitarity will be violated . in light of this possibility , the rising total cross section is interesting . this rise may be associated with the pomeron . 4 .   study the spreading of the electron wave function : + this model predicts that the wave function of an isolated electron will not spread . look for the pole of the electron : + from ( [ eq - ymintyp ] ) we see that measurements down to a scale of @xmath831 may prove sufficient . experiments able to make measurements on this scale may be able to measure the screening function @xmath832 in ( [ eq - ftotal ] ) . we collect here the remaining problems that we anticipate to be the most serious for this model :    1 . the disagreement with the flux quantization experiments of tonomura _ et al . _ is discussesd in section  [ ssec - existing ] . the violation of the theoretical underpinnings of the pauli exclusion principle is discussed in section  [ sec - pauli ] . 3 .   with the size of the outer radius of a line pair estimated by equation ( [ eq - ra ] ) the tension in an unpaired line ( [ eq - t ] ) is ridiculously high . this has observable consequences . other numbers appear to be fairly reasonable . if this model is correct , a way must be found to fix this problem . this problem seems similar in severity to a well known problem in the standard model the energy in the electric field of a point charge is infinite . the bare electron is modeled as the end of a superconducting line of magnetic flux . electric charge must be added by hand to this bare electron , but with the correct charge the electron also has the correct spin , @xmath18 . so the bare electron is almost a dyon , but with a spin - dependent magnetic charge . if the dielectric constant of the ether is 1 near the earth , the effective magnetic charge of the electron has the strength of a dirac monopole , but is strongly screened by the ether  superconducting line pairs from all the other electrons in the universe . the bare electron interacts with the ether and undergoes zitterbewegung , very rapid random walk motion , traveling near the speed of light in its average rest frame . the wave function of the electron is the response of the ether to the average electromagnetic field of the bare electron , explaining wave - particle duality in a natural way . quantum mechanics results from this wave nature . the mass of the electron is the potential energy of its wave function ( in its rest frame ) . the charge and mass of the electron are predicted to vary with the dielectric constant of the ether . this may affect the interpretation of the cosmological red shift . the photon is modeled as a helical wave , one wavelength long , traveling on a line pair . the spin of the helical photon is calculated to be @xmath17 . the ether is a mesh of superconducting line pairs , of a new type which do not conduct electrons or cooper pairs . these line pairs are under tension , so could represent dark energy . in general , quantization conditions arise because the ether is a multiply connected superconductor .    using a toy version of our model with measured values of newton s constant @xmath19 , the mass of the electron and the dark energy density of the universe , we estimate the line pair density near the earth to be @xmath833 . this is a factor of @xmath22 larger than would be expected without inflation . the gravitational force exerted by a star on a positron is due to the electromagnetic field of the positron interacting with the superconducting line pairs coming from the star . this force is independent of the sign of the positron s charge because it is _ induced_. the electric force is attractive but the magnetic force is repulsive . hence the gravitational force is predicted to weaken , then reverse , as the dielectric constant of the ether becomes large . the ether exerts a force equal to @xmath779 on an accelerating mass  which leads to newton s second law of motion . we have proven that both the principles of gauge invariance and microscopic causality are violated by the superconducting fields of this model . because this model is acausal , it is also nonlocal and even self interactions are finite . we suggest that the pauli exclusion principle is actually due to a very strong spin - dependent magnetic force : the pole force and line force between electrons . we suggest that the pole force , a contact force , is responsible for the impenetrability of matter . i thank my colleagues at stony brook for helpful discussions . this includes chris quigg who introduced me to the dirac string long ago . any mistakes , of course , are my own . this work was supported by the national science foundation . it is useful to define the angle @xmath834 so that @xmath835 gives the instantaneous slope of the string at any point for solution ( [ eq - yzline ] ) moving in the @xmath314 plane . @xmath836    here @xmath40 is the direction of the magnetic field ( [ eq - bfieldline ] ) at @xmath41 , the point in question . also @xmath40 is the direction of the centerline at the relevant point ( @xmath837 ) . ( for simplicity , we assume here that this point is unique as will be true in the case of sufficiently large radius of curvature . ) we follow the discussion of section  [ ssec - supercurrent ] in seeking the derivatives of @xmath838 with respect to position and time at the field point @xmath41 . this unit vector gives the direction of the magnetic field at @xmath41 and is assumed to be parallel transported from point @xmath38 on the centerline to @xmath41 for this purpose . note that the derivatives of @xmath317 will be _ different _ at point @xmath41 than on the centerline ( point * y * ) because the radius of curvature will in general be different for a curving line . if @xmath40 changes , the direction of its change must be perpendicular to itself :    @xmath839\ = \ d\theta[{\hat y } \ell_z-{\hat z } \ell_y ] \ = \   d\theta\ \hat{\theta}\ ] ]    we define the instantaneous radius of curvature @xmath176 by the equation    @xmath840    where we define the radius to be _ negative _ if the change in @xmath834 is positive , assuming positive @xmath841 . we make this choice of sign because a line with decreasing @xmath834 in @xmath314 motion matches the configuration of helical motion  and it is convenient to consider this radius of curvature positive . since @xmath210 changes only in the @xmath40 direction , we have    @xmath842    hence we can evaluate the partial derivatives of the components of @xmath40 , for solution ( [ eq - yzline ] ) motion in the @xmath314 plane :    @xmath843    @xmath844    @xmath845    @xmath846    since @xmath847 does not change in the rest frame of the string , we use section  [ ssec - supercurrent ] to find the partial derivatives of the components of @xmath317 with respect to time .    @xmath848    @xmath849    at a given point in time the instantaneous radius of curvature changes with the change in the position of the field point @xmath41 according to the equation    @xmath850    because at a given time the center of curvature is fixed , and therefore for solution ( [ eq - yzline ] ) the partial derivatives with respect to @xmath368 and @xmath106 are given by    @xmath851    @xmath852    and the partial derivative with respect to @xmath200 is    @xmath853    from ( [ eq - lhat ] ) we can find general expressions for the direction of the change with decreasing @xmath834 ( hence @xmath854 , defining positive @xmath176 ) and the expression for @xmath350 , the radius of curvature of the centerline . taking the differential of ( [ eq - lhat ] ) with respect to @xmath106 we find    @xmath855    @xmath856    @xmath857    and so noting    @xmath858    we find    @xmath859    for a displacement in the @xmath106-direction along the centerline . we note that the @xmath368-component of @xmath860 is always negative , but that the radius of curvature in @xmath314 motion can change dynamically between positive and negative values . if we consider the position of the centerline as a function of @xmath106 ( as an independent variable_not _ a field coordinate ) we can relate the radius of curvature on the centerline to the shape ( second derivative ) of the trajectory :    @xmath861    @xmath862    @xmath863    we use this equation to show how the radius of curvature is transformed in going from the lab to the line s rest frame :    @xmath864      we use a polar coordinate system with origin at the center of curvature to find @xmath865 . from the definition of the curl    @xmath866    where @xmath867 is the normal ( given by the right hand rule ) to the circulation of @xmath317 around the closed path @xmath868 , divided by the area enclosed ( see figure  [ fig - curl ] . ) we find    @xmath869 } \over { dr\ ( r+dr/2)\ d\phi } }      \ = \ { { { \hat n}\ dr\ d\phi } \over { dr\ r\ d\phi } } \ ] ]    @xmath870    where @xmath871 in the case of figure  [ fig - curl ] .    to 2.5 in .,title="fig : " ]      in common mode geometry appendix  [ ssec - instradius ] refers to @xmath872 which points generally in the @xmath873-direction . we need to discuss how @xmath874 differs , since @xmath875 . from ( [ eq - lhat ] ) we find    @xmath876    @xmath877    @xmath878    @xmath879    @xmath880    @xmath881    and all the spatial first derivatives are reversed ( when referenced to a fixed coordinate system , as must be true ) compared to the case of @xmath872 and expressed in terms of @xmath872 or @xmath874 as appropriate . but the radius of curvature on the centerline @xmath350 ( and by extension @xmath176 ) is the same for both lines .      for helical geometry from ( [ eq - lhat1 ] ) and ( [ eq - lhat2 ] ) we find , in the rest frame of _ both _ lines    @xmath882    @xmath883 so for line 1    @xmath884    the radius of curvature is a constant on the centerline . in the rotating rest frame we use the time of an observer at the center of rotation who is at rest in the lab . the derivatives of @xmath885 can be calculated from    @xmath886    but if we only need the curl , we can use ( [ eq - curl ] ) .    for line 2    @xmath887    @xmath888    and we see that the radius of curvature on the centerline is also a constant for line 2 , with the same constant value of @xmath889 as for line 1 . the derivatives of @xmath890 can be calculated from    @xmath891      in order to calculate the electromagnetic field from ( [ eq - supercurrentyz ] ) , we need to make use of the fact that @xmath667 and @xmath892 are related by the fact that @xmath823 is a cylindrical radius . selecting events which are simultaneous in the lab to define @xmath317 and @xmath823 , and requiring @xmath893 in the rest frame of the line , we find    @xmath894    using this relationship and ( [ eq - supercurrentyz ] ) in ( [ eq - generalsol ] ) we find for the electromagnetic field :    @xmath895    @xmath896    @xmath897    @xmath898 - \hat{z}\delta y\gamma^2[1 - 2\ell_y^2\beta^2 ] \bigr\}\ ] ]    @xmath899 \biggr\}\ ] ]    so the terms in @xmath334 and @xmath335 are not consistent with ( [ eq - efieldline ] ) and ( [ eq - bfieldline ] ) .    in helical geometry if we attempt to use ( [ eq - generalsol ] ) to try to calculate the electromagnetic field in the rest frame of line 1 , we obtain    @xmath900    @xmath901    which is consistent with ( [ eq - eline * ] ) but differs from ( [ eq - bline * ] ) by a term which is second order in @xmath902 and consequently very small . we consider the dependence of the energy of a curved but static line on the radius of curvature . we use a curvilinear coordinate system which bends with the line and has @xmath834 , @xmath51 and @xmath639 as independent variables with the elements of length @xmath903 , @xmath904 , and @xmath905 . the cylindrical radius of curvature @xmath176 at the point @xmath41 is given by    @xmath906    where @xmath350 is the radius of curvature of the centerline . we calculate the magnetic energy in a length @xmath907 of the line where    @xmath908    we find    @xmath909    @xmath910    so we obtain    @xmath911    as in ( [ eq - t0 ] ) . the energy of the line is independent of its curvature . one side of the centerline stretches , but the other side contracts , with no net change in energy . for common mode motion with the line separation perpendicular to the motion , we have shown that the interaction terms in ( [ eq - intcommode ] ) sum to zero and therefore solutions of type ( [ eq - yzline ] ) can propagate at the speed of light on the two lines of an isolated pair . now we would like to consider a line pair moving in the @xmath314 plane with the line separation also in the @xmath314 plane . in common mode the separation of the centerlines is fixed at @xmath912 perpendicular to both centerlines . this geometry has the complication that the corresponding points on the two lines have _ different _ values of @xmath913 though each is a solution of type ( [ eq - yzline ] ) . so    @xmath914    @xmath915    where @xmath372 is a constant and @xmath916 . using arguments similar to appendix  [ ssec - instradius ] and remembering that here @xmath106 is an independent variable  not a field coordinate , we find    @xmath917    @xmath918    where @xmath350 is the radius of curvature of the _ common mode centerline _ of both lines . so    @xmath919    @xmath920    we find that ( [ eq - lm1 ] ) and ( [ eq - lm2 ] ) become    @xmath921    @xmath922    so the total lagrangian becomes    @xmath923    identical to @xmath251 in ( [ eq - lm ] ) .    as in section  [ sssec - commonmodesol ] the common mode interaction terms add to zero when both lines are referenced to the common mode centerline . the two lines do not bend so that they have precisely the same shape . they have the same center of curvature when referenced to the common mode centerline , so that their individual centerlines have _ different _ radii of curvature but the radii of curvature of the fields are identical at any point    hence we get the same equations of common mode motion with the separation perpendicular to the plane of the motion as we get if the separation is in the plane of the motion . therefore we presume that the orientation of @xmath402 makes no difference to the common mode motion , and common mode solutions ( gravitational waves ) of type ( [ eq - circularline ] ) can also propagate at the speed of light on an isolated line pair . following the reasoning of section  [ ssec - motionbpp ] we have shown how the equations of motion of the the bare positron - pole ( [ eq - bpptmotion]-[eq - bpp3motion ] ) are solved in the case that @xmath924 ( where @xmath162 and @xmath56 refer to external fields ) . if @xmath317 is at an angle @xmath925 from the magnetic induction field @xmath56 and @xmath926 as shown in figure  [ fig - orientation ] , then from ( [ eq - bpplmotion ] ) and ( [ eq - bpp3motion ] ) we find    @xmath927    @xmath928 + b_0 \sqrt{b_0 ^ 2 + e^2 cos^2(\theta+\alpha ) - b^2 cos^2(\theta ) } }   \over { b_0 ^ 2 + e^2cos^2(\theta+\alpha ) } } \label{eq - transcendental}\ ] ]    which is the transcendental equation which must be solved for @xmath834 given @xmath248 , @xmath929 , and @xmath584 . if @xmath584 is small , we presume that @xmath834 will be small and @xmath579 is fixed , since all the cosine terms in ( [ eq - transcendental ] ) are equal to @xmath114 . then we have    @xmath930    @xmath931    showing that the orientation of @xmath838 is easily determined if @xmath584 is small .    to 3.0 in       the electric charge at the end of each line is easily incorporated into the principle of least action using the electric charge current in ( [ eq - actionnoh ] ) . the only unnatural requirement for this model is putting the charge and pole at ( nearly ) the same position at the end of each line , so that the end of a line becomes a almost a dyon , with a spin dependent magnetic charge . with both electric and magnetic charge , equation ( [ eq - motionepparts ] ) , the boundary condition to be satisfied at the end of each line , becomes :    @xmath932    we evaluate the boundary condition just outside the end of each line so that the electric charge is included . the pole self field is continuous so we can evaluate it just inside the line as we did in section  [ ssec - selfintbareelectronpole ] . the self field of the electric charge will be finite if we average over the cross section of the end of the line . then , since we switch to the point charge approximation to evaluate the motion of the bare positron , we can argue that the electric self energy is zero from symmetry . it is easy to see that the cross terms in the self energy are zero . evaluating them in the rest frame of the positron where @xmath933 , we see that the pole only interacts with a magnetic field , whereas the charge only interacts with an electric field . hence the self - energy terms are unchanged from the case of the positron - pole and the equation of motion becomes    @xmath934 \label{eq - eqofmotionwcharge}\ ] ]    where    @xmath935    is a small number if @xmath936 near the earth implying that the motion of the bare positron is dominated by the pole , which is much stronger than the charge . we can show this directly by combining terms in ( [ eq - eqofmotionwcharge ] ) to find    @xmath937    that the motion reduces to the case of the bare positron - pole . we simply make the substitutions    @xmath938    @xmath939    and the equation of motion of the bare positron reduces to the equation of motion of the bare positron - pole , in terms of these transformed fields . this model of the positron ( using positive charge for convenience ) has an electric charge of @xmath193 , a screened magnetic pole field of effective magnetic charge @xmath194 , and a line field that ensures that the total magnetic flux leaving the positron is zero . hence our positron ( assuming @xmath940 near the earth ) has the expected electric charge but no true magnetic charge . it has a magnetic dipole moment , which we are currently unable to model . therefore our model of the positron should behave like a physical positron if we assume that the positron s measured magnetic dipole moment is consistent with our model . the short answer to this question is no . the string tension is quantized and therefore can not transmit an increase in force to the positron . the string is perfectly elastic . if you pull on it in the direction of the string , you simply increase the string s length and do not affect the positron s motion . however , if you pull on the string perpendicularly to the string , you can affect the orientation of the string , and thereby change the positron s motion . so this fact brings out the importance of spin and developing a model of the magnetic moment that will be needed to make this model complete . following the line of reasoning which led to ( [ eq - dndastar ] ) and hence to ( [ eq - g ] ) , the line pair density coming from a star of arbitrary material is given by    @xmath941    where @xmath942 is one atomic mass , @xmath943 is the mass number of the material ( number of nucleons ) and @xmath944 is the relative atomic mass of the material . the brackets indicate the mass weighted mean and the subscript @xmath945 indicates that the ratio of the line pair density from a star to its mass depends on the material of the active mass ( source of the field ) . then the value of @xmath946 becomes    @xmath947    the size of the material dependence factor is typically @xmath948 . this factor approaches @xmath689 for hydrogen , but it is difficult to measure @xmath19 with a gas . there are wild fluctuations@xcite in existing measurements of @xmath19 and this model may be able to help@xcite bring understanding to these measurements .    this model can not assess whether or not the gravitational force is proportional to passive mass ( on which the field acts ) because we do not model the nucleus . the discussion of section  [ ssec - calcg ] indicates that when a line pair passes a charged particle , it will exert an attractive force on this particle , and this force will switch directions as the line pair passes . hence the force of a line pair on a particle _ will _ depend on the velocity of the line pair . if line pairs in our surroundings dominantly come from stars we can see , the  fixed " stars , we expect that the force of the ether on a charged particle _ will _ depend on the velocity of the particle , instead of only on the acceleration of the particle . however , we have seen that the average line pair ( section  [ ssec - estimatelpd ] ) is about a factor of @xmath22 larger than the size of our visible universe , probably because of inflation . this indicates that the total volume of our universe is about a factor of @xmath949 greater than the volume of our visible universe ! matter which went over our horizon early in the inflationary process will now be moving at velocities far beyond the velocity of light relative to us , but may have stars which are the source of line pairs near us . thus we postulate that the line pairs near us are moving isotropically with uniformly distributed velocities at least up to the velocity of light . this causes lorentz invariance . apparently these line pairs are not accelerating with respect to our inertial system@xcite .      c. bennet _ et al . _ , apjs * 148 * ( 2003 ) , p. 24 . m. k. gaillard , p. d. grannis , and f. j. sciulli , reviews of modern physics * 71 * , s96 ( march 1999 ) . b. abbott _ et al . _ , 0 collaboration , phys . letters * 82 * , 2457 ( 1999 ) . see , for example , donald h. perkins , _ introduction to high energy physics _ , addison - wesley ( 1987 ) , p. 21 and b. hatfield , _ quantum field theory of point particles and strings _ , addison - wesley ( 1992 ) , p. 479 . n. bohr , phys . rev . * 48 * , 696 ( 1935 ) . n. bohr ,  discussion with einstein on epistomological problems in atomic physics " , in _ albert einstein : philosopher - scientist _ , p. a. schilp , ed . , library of living philosophers , evanston , ill . ( 1949 ) , p. 232 . a. einstein , b. podolsky , and n. rosen , phys . rev . * 47 * , 777 , ( 1935 ) . albert einstein,reply to criticism " , in _ albert einstein : philosopher - scientist _ , p. a. schilp , ed . , library of living philosophers , evanston , ill . ( 1949 ) , p. 672 . see , for example , n. d. mermin , reviews of modern physics * 65 * , 803 ( 1993 ) . a. m. dirac , phys . rev . * 74 * , 817 ( 1948 ) . j. d. jackson , _ classical electrodynamics _ , john wiley & sons , inc . ( 1975 ) . ( we follow the conventions of the second edition rather than the third because the third edition does not use a consistent set of units . ) a. tonomura , _ the quantum world unveiled by electron waves _ , world scientific ( 1998 ) , p. 133 . d. j. fixsen _ _ , astrophys . j. * 420 * , 445 ( 1994 ) . michael r. douglas , jhep 05(2003)46 , p. 55 . roberto valandro , arxiv:0801.05842v2 , ph . d. thesis , p. 10 this is a readable introduction to the landscape . ernst mach , _ the science of mechanics _ , written in may , 1883 , translated by thomas j. mccormack , open court ( 1960 ) , p. 284 . p. a. m. dirac , proc . london * a133 * , 60 ( 1931 ) . the fact that a spinor changes into its negative under a @xmath950 rotation is irrelevant here . we are not rotating the wave function . f. london , _ superfluids _ , vol . i , john wiley & sons , inc . 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this is an attempt to construct a _ classical _ microscopic model of the electron which underlies quantum mechanics . 
 an electron is modeled , not as a point particle , but as the end of an electromagnetic string , a line of flux . 
 these lines stretch across cosmic distances , but are almost unobservable because they condense into pairs  which form the ether . 
 photons are modeled to propagate on these line pairs , which act effectively as wave guides . 
 these line pairs are also responsible for the force of gravity  which is electromagnetic in character .    using our classical interpretation of an electron as a quantized electromagnetic field 
 , we find that we achieve a natural interpretation of not only quantum mechanics , but also mass , spin , mach s principle , gravity , self interactions , newton s second law of motion , the quantization of the photon , wave - particle duality , inflation , dark energy , agn jets , the pauli exclusion principle , superconductivity , and the quantum hall effects . 
 mass generation and inflation are discussed without the use of scalar fields . 
 we attempt to calculate newton s gravitational constant g , the mass of the electron , the spin of the electron , and the spin of the photon  all in terms of planck s constant , the charge of the electron , the outer radius of a line , and the distribution of matter in the universe . 
 the predictions of this model and the standard model are both compared to experiment , emphasizing areas where the predictions differ . 
 this model s remaining problems are discussed . 
 this model is a classical field theory of interacting strings which explains a large number of diverse phenomena , even though it does not discuss the strong or weak interactions . 
 since the ether of this model is quite well determined , it might be useful in restricting the landscape of vacua in more complete string theories .