No need for dark-matter, dark-energy or inflation, once ordinary matter is properly represented?

In a recent Foundations of Physics paper [5] by the current author it was shown that, when the self-force problem of classical electrodynamics is properly solved, it becomes a plausible ontology underlying the statistical description of quantum mechanics. In the current paper we extend this result, showing that ordinary matter, thus represented, possibly suffices in explaining the outstanding observations currently requiring for this task the contrived notions of dark-matter, dark-energy and inflation. The single mandatory `fix' to classical electrodynamics, demystifying both very small and very large scale physics, should be contrasted with other ad hoc solutions to either problems. Instrumental to our cosmological model is scale covariance (and `spontaneous breaking' thereof), a formal symmetry of classical electrodynamics treated on equal footing with its Poincare covariance, which is incompatible with the (absolute) metrical attributes of the GR metric tensor.


Introduction
At the turn of the twentieth century there was an absolute ruler to theoretical physics: classical electrodynamics (CE). It covered all physical phenomena but gravitation, was compactly formulated, fairly accurate, and even compatible with the newly discovered theory of relativity. Yet, everyone knew that the king was naked-that CE was not really a theory due to the so called classical self force problem. The force exerted by a point charge on itself is ill defined, and the electromagnetic energy density is non integrable at its location. Bridging between the two conflicting sides of the monarch were, and still are 1 , ad hoc 'cheats', enabling physicists to extract results from a non theory, but, as we explain in this paper, no single method resulted in a theory consistent with the full range of experiments to which CE is successfully applied.
The lack of a single consistent theory of CE does not seem to have greatly bothered physicists, so long as their methods of circumventing the self force problem led to reasonable agreement with experiments. However, new experimental results were rapidly accumulating, which could no longer be explained using those methods. The photoelectric and Compton's effects appeared to be completely at odd with the notion of a smooth EM field, while the electrostatic repulsion of protons in the nucleus of an atom seemed necessitating an additional balancing 'strong' force, the inclusion of which would still leave unexplained the very existence of stable solid matter. It seemed at the time that the naked monarch must accede his throne-his lack of garments being the least of reasons for that.
In the feverish 2 quest for a heir that ensued, physicists naturally looked for a close relative of their once omnipotent provider, but hampered by the undeveloped relevant mathematical machinery of their time 3 , the search radius was rather small. This resulted in a successor-QM-which while formally looked just like CE (only with operators replacing C-numbers, and commutators instead of Poisson brackets) was a completely different creature-a statistical theory! The unexpected transition ignited unrest in the kingdom, persisting to this very day and, as we show in section 4, deservedly so. QM only describes certain statistical aspects of the theory they should have crowned instead of CE and was never supposed to inherit the latter. It is an additional fundamental law of nature, complementing that heir rather than rivaling it.
In section 3 we return, in a sense, to those decisive days of search for a successor to CE, but not before clarifying what is the essence of CE which made it into a king in the first place. Armed with the vast mathematical arsenal which was not yet available to the founders of QM, we then construct a well defined theory of localized currents capturing that essence, and point to the fact that all previous attempts to hide the king's nakedness failed to do so. This theory, dubbed extended charge dynamics (ECD), is shown to be not merely a ghost behind QM, but rather an autonomous theory of testable predictions, which also eliminates the closely related self-force problem of general relativity.
At the turn of the twentieth century CE was the absolute ruler of theoretical physics. It is possible that all that was missing for its hegemony to last were proper garments, and a queen by his side-a statistical theory-to complement him.

Manifestly scale covariant classical electrodynamics
The following is a brief review of classical electrodynamics of interacting point charges. Except for the case of massless charges, it is equivalent to the presentation appearing in any standard book on the matter, but contains a few twists which should later ease the transition to ECD.
A note about dimensions in this paper. The custom of attaching a 'dimension' (in the usual sense of mass, length, mass/length, etc.) to constants and variables appearing in the equations of physics, not only does it lead to awkward combinations (e.g. elements in some abstract algebra expressed in kilos...) but, in fact, it is unnecessary. Any physically meaningful statement involves only pure real numbers, expressing the ratio between two quantities of the same 'dimensionality'. Accordingly, throughout this paper functions de-fined on Minkowski's space-time, M, have their values in the relevant abstract mathematical space, viz. no 'dimension' is attached to those objects, and points in space-time are indexed by four labels-just real numbers. The labeling convention of the (affine) space-time grid is chosen so that any vector x ∈ M retains its "norm squared" x µ g µν x ν under any Lorentz transformation. As the Lorentz group parametrically depends on some real parameter, c, the set of permissible labeling conventions is also c dependent, and contains elements which are related to each other by some Poincaré transformation and an arbitrary scaling operation x → λx, for some λ > 0.
Classical electrodynamics of N interacting charges in Minkowaki's space M is given by the set of world-lines k γ s ≡ k γ(s) : R → M, k = 1 . . . N , parametrized by the Lorentz scalar s, and by an EM potential A for which the following action is extremal Above, F µν = ∂ µ A ν − ∂ ν A µ is the antisymmetric Faraday tensor, q some coupling constant, and F 2 ≡ F µν F µν . Variation of (1) with respect to any γ yields the Lorentz force equation, governing the motion of a charge in a fixed EM fieldγ Multiplying both sides of (2) byγ µ and using the antisymmetry of F , we get that d dsγ 2 = 0, henceγ 2 is conserved by the s-evolution. This is a direct consequence of the s-independence of the Lorentz force, and can also be expressed as the conservation of a 'mass-squared current' Defining m = γ 2 ≡ dτ ds with τ = s (dγ) 2 the proper-time, equation (2) takes the familiar form with x(τ ) = γ (s(τ )) above standing for the same world-line parametrized by proper-time. We see that the (conserved) effective mass m emerges as a constant of motion associated with a particular solution rather than entering the equations as a fixed parameter. Equation (2), however, is more general than (4), and supports solutions conserving a negativeγ 2 (tachyons -irrespective of their questionable reality) as well as a vanishingγ 2 . 4 The second ingredient of classical electrodynamics, obtained by variation of (1) with respect to A, is Maxwell's inhomogeneous equations, prescribing an EM potential given the world-lines of all charges with k j(x) = q ∞ −∞ ds δ (4) x − k γ s kγ s (6) the electric current associated with charge k, which is conserved, The current on the r.h.s. of (5) obviously defines F only up to a solution to the homogeneous Maxwell's equation ∂ ν F νµ = 0.

Scale covariance
The above unorthodox formulation of classical electrodynamics highlights its scale covariance, a much ignored symmetry of CE which, nevertheless, is just as appealing a symmetry as translational covariance (Poincaré covariance in general). Any privileged scale appearing in the description of nature, just like any privileged position, should better be an attribute of a specific solution and not of the equations themselves which ought to support all properly scaled versions of a solution. As there seems to be some confusion regarding scale covariance, we try to clarify its exact meaning next. The Poincaré group plays a fundamental role in any theory, whether covariant or not. In particular, this means that a. If some coordinate system is suitable for describing the theory then so is any other system related to the first by a Poincaré transformation. b. Under the above change in coordinate systems, the parameters of the theory must transform under some representation of the Poincaré group. Poincaré covariant theories are those distinguished theories containing only Poincaré invariant parameters. c. The physical content of the theory is identified with invariants of the Poincaré group, viz., attributes transforming under its trivial representation which are therefore independent of the coordinate system.
Elevating the one-parameter group of scale transformations to the status of the Poincaré group amounts to extending the latter with a a dilation operation, x → λx for any λ > 0. By b above, we should also assign a scaling dimension, D Ω to each object, Ω, dictating the latter's transformation under scaling of space-time, Ω → λ −D Ω Ω, and by c, only dimensionless quantities have physical meanings (The custom of attaching 'dimensional units' to measurable quantities, such as a kilo or a meter, guarantees that in addition to the scale dependent measurement, another scale dependent gauge is specified, yielding a scale independent ratio.) Note, however, that the assignment of scaling dimensions to objects of a theory is not unique unless the theory is scale covariant, viz., contains parameters of scaling dimension zero only (even in this latter case one can distinguish between theories leaving an action invariant thereby facilitating the derivation of a conserved current associated with scaling symmetry, and those theories only preserving the equations. CE falls into the first category).
Back to the case of classical electrodynamics, we can see that the scaled variables also solve (2) and (5) The simplicity in which scale covariance emerges in classical electrodynamics is due to the representation of a charge by a mathematical point, obviously invariant under scaling of space-time. As we shall see, achieving scale covariance with extended charges is a lot more difficult, as no dimensionful parameter may be introduced into the theory from which the charge can inherit its typical scale.

The constitutive relations of CE
Associated with each charge is a 'matter' energy-momentum (e-m) tensor, formally satisfying ∂ ν k m νµ = F µν k j ν , Likewise, associated with the EM potential is a unique gauge invariant and symmetric 5 EM e-m tensor formally satisfying Poynting's theorem where only use of (5) and the identity has been made in establishing (12). Summing (10) over k and adding to (12) we get a symmetric conserved e-m tensor of the combined matter-radiation system, the conservation of which can also be established form the invariance of the action (1) under translations. Note that the obvious coupling between matter and radiation notwithstanding, the conserved e-m tensor in (14) splits into two pure contributions. Equation (10) and Maxwell's equations (5), together with electric charge conservation and the form (11) of the canonical EM tensor are dubbed in this paper the constitutive relations of CE, and in the sequel shall assume a status of axioms rather than of derived relations. For non intersecting world lines, it is easily shown that (14) ⇔ (10) and either one can be used as an axiom.
Finally, for future reference, we note that associated with the scaling symmetry (8) is an interesting conserved 'dilatation current' However, the conserved dilatation charge, d 3 x ξ 0 , depends on the choice of origin for both space-time, and the n parameterizations of k γ, and is therefore difficult to interpret.

The classical self-force problem
The self-force problem of CE refers to the fact that the EM potential, A, generated by (5) is non differentiable everywhere on the world lineγ ≡ ∪ s γ s , traced by γ, rendering ill defined the Lorentz force-the r.h.s. of (2)-as well as the r.h.s. of the constitutive relation (10) (even in the distributional sense). A reminder of this appears in the form of non integrable singularities on theγ's of the EM energy density Θ 00 , making the energy of a system of particles likewise ill defined. Fixing the self-force problem amounts to turning a non-theory into a (mathematically well defined) theory, and there is no obvious 'right way' of doing so. The simplest way, which often leads to good agreement with experiment, is to eliminate the self generated field from F when computing the Lorenz force acting on a particle. For this to be possible one needs to be able to uniquely define the contribution of each charge to the total field F , and the prevailing method is to take the retarded Lienard-Wiechert potential of the charge as that field. The r.h.s. of (2) is rendered well defined this way, but the constitutive relations no longer hold true even in a formal way-their validity follows from the existence of an action, (1), not discriminating between the contributions of the different charges to F . In his celebrated work on the self force problem, [6], Dirac further develops the above method by replacing the Lorentz force equation (2) with a more complicated equationthe so called Abraham-Lorentz-Dirac equation. His motivation, which we also adopt in the current paper, is to view the constitutive relations as the defining axioms of CE, from which well defined equations are to be derived, respecting these relations. 6 The rational behind this approach is clear: The infinitely detailed dynamics of point charges or the singular EM field generated by them are never the actual subject of observations in experiments to which CE is successfully applied, but rather the constitutive relations in their integral forms. For example, the thin tracks left by charges in bubble chambers do not resolve the dynamics on scales beneath the width of the track and can be deduced from a hypothetical pair {j, m}, localized about a common world line, satisfying the constitutive relations (10) and (7) (see appendix D). Likewise, the phenomenon of radiation resistance can be explained on the basis of Poynting's theorem (12) and e-m conservation (14)-provided both are well defined-and requires no mention of a damping self-force resisting the motion of the charges. In both cases, as in virtually all other cases in the (vaguely defined) scope of CE, one can circumvent the ill-definedness of the system (2) and (5) by resorting to the constitutive relations.
Despite Dirac's acknowledgment of the central role played by the constitutive relations, his proposal for solving the self force problem turns out to be inconsistent with the very constitutive relations on which it was founded. In fact, in the general case of a set of point charges interacting according to Dirac, it is not even known if any expression exists which can be interpreted as e-m conservation. This contradiction, which can be traced to some dubious mathematics employed in his derivation, is not very surprising, as the equations for point charges which are consistent with the constitutive relations-at least in a formal sense-are just the original ones, (2) and (5) from which the constitutive relations were derived in section 2.2.
In another classic work [11][12], Wheeler and Feynman gave a surprising new look at Dirac's electrodynamics. Elaborating the formalism of action-at-a-distance electrodynamics, they found a locally conserved and integrable e-m tensor for a set of point charges interacting through their half advanced plus half retarded Lienard-Wiechert potentials, without self interaction. Under certain assumptions, a subset of charges surrounded by sufficiently many other charges, behaves in accordance with Dirac's theory. Note, however that the converse is not true, namely, not every set of charges interacting according to Dirac is also such a sufficiently surrounded subset of charges interacting according to Wheeler and Feynman, so the latter's e-m tensor is generally not conserved in the Dirac case. Moreover, the form of the canonical EM tensor in that expression radically differs from (11), admitting both negative values for its energy density component as well as nonzero values at places where the EM field due to all charges vanishes. Wheeler and Feynman's theory, therefore, can hardly be claimed to be consistent with the full range of experiments to which CE is successfully applied. Instead, it is some well defined theory of interacting point charges admitting an integrable and conserved e-m tensor, and its formal affinity to (ill defined) CE is just an inevitable consequence of their common symmetry group. What is most interesting, however, is that inclusion of the advanced Lienard-Wiechert potential in that theory is mandatory in order to get a conserved e-m tensor. This feature, we shall see, caries to ECD.

Extended currents
It appears from the above examples that a mathematically well defined realization of the constitutive relations is incompatible with the notion of line currents. In a second class of solutions to the self-force problem, one therefore substitutes for the distributions (6) and (9) regular currents both localized aboutγ. The regularity of the electric current implies a smooth potential onγ, rendering the Lorentz force (2) well defined. Various proposals can be found in the literature, all utilizing a 'rigid constriction' is the sense that the extended currents are uniquely determined by γ. This is not only the simplest way to eliminate the singularity of A onγ but also the only one allowing to retain the Lorentz force equation (2). Below, we shall employ a novel rigid construction which will take us one step towards ECD. Unlike the alternatives, generally restricted to sufficiently small accelerations or a fixed mass shell constraint, this one it applicable to an arbitrary γ, including those reversing direction in time.
The idea is to substitute for δ (4) in (6) a finite approximation of a delta function, respecting the symmetries of the theory. In Euclidean four dimensional space this is straightforward: δ (4) (x) → a −4 f (x/a) for any normalized spherically symmetric f and some small a. In Minkowski's space this is more tricky. So first we note that the current is conserved and significantly differs from the -independent current only up to a distance from γ s of the order of √ (in the rest frame of γ s for some constant α). Taking the derivative of (17) with respect to we therefore get a conserved current which is significant only inside a ball of radius ∼ √ in the rest frame of γ, reducing to the line current (6) in the limit → 0. Pushing the derivative into the integral, the regular function appears as a finite approximation of the invariant δ (4) (x) entering (6). This can indeed be directly verified. Note, however, that even for a compactly supported f , (20) is non vanishing in some neighborhood of the light-cone x 2 = 0 for an arbitrarily large (light like) x. 7 Consequently, the current (19) is never compactly supported and can be shown to have an (integrable) algebraically decaying 'halo'. We see that the obvious way of covariantly generalizing Lorentz's construction of a finite-size electron, leads to weakly localized currents. There are, nevertheless, three major difficulties with the above extended current approach to the self-force problem. First, it introduces an arbitrary function-an infinite set of parameters-into single-parameter CE. Second, the dimensionful parameter spoils the scale-covariance of CE. Finally, the constitutive relations are still not satisfied, the problem being with the constitutive relation (10). To show this, we regularize the e-m tensor (9) for some regular function g, and notice that the value of the l.h.s. of (10) at any x depends only on the value of F onγ, whereas the r.h.s. depends also on the local value F (x). Taking the limit → 0 apparently solves this problem by restricting the support of both sides of (10) toγ, but in that limit, in addition to the expected Abraham-Lorentz-Dirac radiation reaction force, an additional force of the form −Cγ appears, with C → ∞ in that limit. This means that, indeed, the constitutive relation (10) is satisfied in the limit → 0, but only because the dynamics of the charges trivialize to uniform motion due to their infinite mass. No scaling of the mass or the coupling q with can restore non trivial dynamics-the only way to do so is to arbitrarily set C = 0 (or equivalently, 'absorb' this infinite term into the mass of the particle) which reproduces Dirac's theory. Summarizing, CE of point charges cannot satisfy the constitutive relations while CE of rigid extended charges further spoils scale covariance and introduces infinitely many new parameters.

Extended Charge Dynamics
Our starting point in the construction of currents satisfying the constitutive relations is the electric current (19) and expression (21) for the e-m tensor m. We saw above that the 'rigidity' of the covariant integrands in both currents leads to violation of the constitutive relations, while their nonsingular nature further spoils scale covariance. To fix both problems we substitute for them more 'vibrant' integrands which do depend on the local field F , and whose characteristic scale surfaces naturally without introducing extra dimensionfull parameters. To this end, let us look at the proper-time Schrödinger equation (also known as a five dimensional Schrödinger equation, or Stueckelberg's equation), with the gauge covariant derivative andh some real dimensionless 'quantum parameter', not to be confused with . It can be shown by standard means that solutions of (22) satisfy a continuity equation and four relations The common implications of the non relativistic counterparts of (24) and (25) are probability conservation and Ehrenfest's theorem respectively, and readily carry to the relativistic case by integrating each over space-time. Localized wave-packets can then be shown to trace classical paths when the EM field varies slowly over their extent. Yet, another implication of (24) and (25) which has no direct nonrelativistic counterpart is obtained by integrating the two equations over s rather than space-time. The s-independent current is conserved and the constitutive relation (10) is satisfied by j and Associating a unique φ with each particle and taking the sum of the corresponding currents, j, as the source of Maxwell's equations (5), the constitutive relations are fully satisfied, and the full symmetry group-scale covariance in particular-is retained. The above realization of the constitutive relations, nevertheless, is apparently inconsistent with the condition of localized j and m. The dispersion inherent in the Schrödinger evolution (22) implies that a localized wave-packet gradually spreads even in a potential free spacetime. In collisions with an external potential the situation is even worse, and may result in a rapid loss of localization. This means that the wave-packet could maintain its localization under the s-evolution (22) only if somehow the EM potential generated by its associated current j, creates a binding trap, but the prospects of such a solution are dim as the self generated Coulomb potential is repulsive rather than attractive. It is further unlikely that such a self-trapping solution, even if it exists in some otherwise potential free region of space-time, would retain its localization following violent (realistic) interactions with EM potential generated by other charges. Finally, as we shall show in section 4.4, equation (22) and its associated currents admit a much more natural interpretation in terms of ensembles of particle, making the single particle interpretation seem rather contrived.
It appears inevitable that for (22) to be useful in the realization of the constitutive relations by means of localized currents, an additional localization mechanism for the wave packet must be introduced into the formalism. In [9], this mechanism takes the form of a (point) 'delta function potential', δ (4) (x − γ s ), moving along someγ in Minkowski's space, which is added to the Hamiltonian in (22), preventing the wave function from spreading by the binding action of the potential. The resulting formalism, dubbed in [9] extended charge dynamics (ECD), leads to j and m which are both localized aboutγ and, by the scale invariance of the point potential, scale covariance is not breached while the constitutive relations are still satisfied in M/ ∪ k kγ .

The central ECD system
It would appear that one must first specify γ in order to solve for the wave function φ in the presence of the delta function potential, but a second equation specifying how φ must 'guide' 8 γ turns the relationship between φ and γ into a symbiotic one whose purpose is explained below. Associated with each particle, then, is a pair { k φ, k γ}, k = 1 . . . N , satisfying two coupled equations dubbed the central ECD system, which could therefore be looked at as a replacement for the Lorentz force equation (2). The first makes explicit the delta function potential (see [9] for a formal derivation) and reads (omitting the particle index on φ and γ) Note that, indeed, (28) defines a vector space for φ. The second is the 'guiding equation', stating that γ s must follow a local extremum of the modulus squared of φ, Above, G(x, x ; s) is the propagator of a proper-time Schrödinger equation, viz., solution of (22) satisfying the initial condition (in the distributional sense), and is some parameter of dimension 2, ultimately taken to zero, which is needed for the construction of the scale-invariant delta function potential.
It turns out that solutions, φ(x, s), of (22) in the presence of a delta function potential δ (4) (x − γ s ) contain a distribution on the light cone of γ s . As both J and M are bilinears in φ and its adjoint, a meaningless product of two distributions is formed as a result of naively taking the → 0 limit of j and m. A similar product of distributions is the source of much of the troubles in QFT and is overcome by two steps: covariant regularization, rendering all results finite, followed by 'renormalization', viz., making sense of the limit when the regulator is removed. Likewise, in ECD a covariant regulator, , is built into the formalism, and the counterpart of the renormalization step takes the form of a simple covariant prescription and yields smooth currents, satisfying the constitutive relations at ∀x / ∈γ in a mathematically well defined way. In particular, the EM energy density, Θ 00 , is integrable leading to a finite self EM energy. Note thatγ is excluded from the domain of all ECD currents (line currents supported onγ are formed in the limit → 0 by a mechanism similar to that presented in section 2.3.1). The central ECD system in the form (28) (29) involves a delicate → 0 limit which was not fully defined in [9]. The precise definition, given in appendix A, combined with the above regularization scheme has a remarkably simple justification: Both ECD currents (31) and (32) have an integrable singularity on the world lineγ. The central ECD system is nothing but the condition that no electric charge nor energy-momentum leak into those 'world-sinks' onγ. Without such a condition, the local constitutive relations are basically useless as those cannot be used to derive integral conservation laws. This conclusion appeared implicitly in [9] but is rigorously demonstrated only in the current paper.
Spin. The above procedure of realizing the constitutive relations involves scalar k φ's, but a similar method also exists in which each k φ transforms under an arbitrary representation of the Lorentz group. The spin of a particle therefore merely labels the particular method used to construct the ordinary currents j and m associated with the particle, and adds nothing conceptually new to the formalism. An example of spin-1 2 ECD is discussed in appendix E.

The nature of particles in ECD
The simplest possible problem in ECD is that of single stationary particle in an otherwise void universe. That is, the very existence of a particle is due to a nontrivial localized solution, viz. A = 0, for the coupled ECD equations. In a naive approach, this amounts to guessing a potential A, then solving the central ECD system (28),(29) for a pair {φ, γ}, from which the electric current (31) is computed, and 'hoping' that this current, along with the initial guess A, indeed solves Maxwell's equation (5).
Using a small-h approximation of the propagator, we show in appendix B that such solutions must indeed be particle-like, represented by integrable currents which are localized about their centerγ, and this conclusion is not an artifact of the small-h analysis but rather a direct consequence of equation (28).
It is speculated that different such stationary (more generally, non radiating...) solutions represent different elementary particles whose attributes, such as effective mass and electric charge, can be computed using the expression derived in appendix C.2. This is a direct test of ECD's validity, but as the set of equations coupling A and {φ, γ} is very difficult to solve even in this case, necessitating an extensive use of non standard numerical calculations, no success can be reported as of yet. However, the stage is completely set for such detailed analysis, and it is the hope of the author that this paper will motivate others to take part in this endeavor.
An elementary particle solution (or any other solution for that matter) must come with an 'antiparticle' solution to the ECD equations. This is a consequence of the symmetry of ECD under a 'CPT' transformation In fact, scalar ECD enjoys an even larger symmetry group, C: . However, spin-1 2 ECD, presented in appendix E, enjoys the CPT symmetry only. This symmetry implies that our naive notion of timereversal-'running the movies backward'-is not a symmetry of micro-physics and will be further mentioned in the context of the observed arrow of time.

The necessity for advanced solutions of Maxwell's equations
In a universe in which no particles imply no EM field, a solution of Maxwell's equations is uniquely determined by the conserved current, j, on their r.h.s. due to all particles. The most general such dependence which is both Lorentz and gauge covariant takes the form for some space-time dependent functional, α, of the current j, where Kret adv are the advanced and retarded Green's function of (5), defined by 9 In ill defined CE of section 2, α ≡ 0 is taken as a definition. Modulo the self force problem, the fact that CE admits a formulation in terms of an initial value problem means that indeed, solutions of CE may be found containing only retarded fields. In contrast, the ECD current j on the r.h.s. of Maxwell's equations also depends on A both explicitly through the gauge covariant derivative D, and implicitly via φ's dependence on A. An α ≡ 0 proviso, as in CE, is inconsistent with the ECD equations, not admitting an initial value formulation, and α would generally vary across space-time.
That advanced solutions of Maxwell's equations are on equal footing with retarded ones is outraging from the perspective of the (almost) consensual paradigm which accepts only retarded solutions as physically meaningful. One can think of two major reasons for this outrage. The first is the parallelism which which is often drawn with 'contrived' advanced solutions of other physical wave equations (e.g. surface waves in a pond converging on a point and ejecting a pebble). This parallelism, however, is a blatant repetition of the historical mistake which led to the invention of the aether. The formal mathematical similarity between the d'Alembertian-the only linear, Lorentz invariant second-order differential operatorand other (suitably scaled) wave operators, is no more than a misfortunate coincidence. Has this coincidence had some real substance to it, then application of the Lorentz transformation to the wave equation describing the propagation of sound, for example, would have yielded a meaningful result. It is quit remarkable that over a century after the existence of the aether was refuted, and the geometrization revolution of Minkowski in mind, terms such as 'wave' and 'propagation' are still as widely used in the context of electromagnetic phenomena as in the nineteenth century.
The second, stronger case for rejecting advanced solutions is observational. While as a general rule, we shall challenge this assertion in section 4, it is indeed true that, on a macroscopic scale there are no obvious signs of advanced radiation, e.g., no macroscopic object is observed anywhere spontaneously increasing its energy content by the convergence of advanced radiation on it. This so called radiation arrow of time, previously built into CE by the exclusion of advanced fields, can be explained by decomposing the global EM potential, (34), into a retarded piece, solution of Maxwell's equations (5) plus a 'vacuum' piece, solution of the (sourceless) homogeneous Maxwell's equations In ill defined CE an α ≡ 0 postulate resolves our dilemma but, as previously pointed, such a proviso is inconsistent with the ECD equations. However, we can consistently assume that α 'statistically vanishes', namely, that α, hence also A vac , is a rapidly fluctuating function of space-time such that the integrated Poynting flux associated with it across any macroscopic time-like surface is small, and that this small value statistically fluctuates around a zero mean. As the change in the e-m content of any three-volume can be read from the integrated Poynting flux across it's surface, the above assumptions are sufficient to explain why only the Poynting flux associated with retarded fields should be considered in macroscopic e-m balance. On the scale of individual particles, in contrast, the contribution of the vacuum filed is indispensable, as we shall see in section 4.2.
By redefining α → 1 − α, the 'ret' and 'adv' labels in (34) are swapped, and we get an oppositely pointing radiation arrow of time. The above analysis is therefore not a 'derivation' of the observed direction of the arrow of time within ECD, but rather a demonstration of the consistency of the ECD formalism with the existence of such an arrow, while the anthropic principle 10 explains the observed direction of the arrow. 4 The compatibility of ECD with QM

The block universe
In its greatest generality, ECD provides a rule for filling empty space-time with energy and momentum. A typical such e-m distribution is concentrated around world lines associated with particles, and in the vicinity of light cones with apexes on those world lines, corresponding to radiative process. This rule permits a very restrictive yet infinite set of such e-m distributions, one of which allegedly describes our universe. It is crucial to note that, while some features are common to all e-m distributions permitted by ECD, others are unique to the specific one filling our universe and, therefore, ECD alone is an incomplete description of the universe. The result of any conceivable experiment requires knowledge of that specific e-m distribution-that space-time structure-which includes not only a description of the experiment but also of the experimenter. 11 This view of the universe, as a fixed four dimensional 'block' filled with e-m (as oppose to a three dimensional universe evolving with time) goes by the name "the block universe". In fact, every relativistic theory, CE in particular, can be seen as a covariant way of generating such block universes, the dynamical equations of the theory being just the means of doing so. Tragically, though, CE admits also a Lorentz covariant initial value formulation. This feature, not shared by ECD, nor by time symmetric action-at-a-distance electrodynamics, facilitated a convenient but deceiving cut between the observer/experimenter, specifying the initial conditions on some space-like surface, and nature, propagating them forward in time. In contrast, the role of the experimenter in the block universe is more subtle: A well prepared one should be equipped with a 'photo album' containing closeups of the global structure or of generic parts shared by all admissible structures, and focus on that 'photo' best describing his/her experiment-preparation and measurement stages in a single 'photo'.
Then came the quantum crisis. Convinced by the triumph of CE that nature is deterministic, experimenters repeated their experiments with identical initial conditions set to their systems, but nature, so they reasoned to their embarrassment, chose this time to propagate them to different final states. The possibility that the different outcomes of apparently identical experimental settings are due to some variables which are hidden from the experi- Figure 1: The space-time substructure involved in correlation experiments menter but participate in the dynamics was later excluded by Bell [3], but this, too, did not serve as a warning sign that the initial value formulation of physical theories, dating back to pre-relativistic times and carried to relativistic physics by an ill defined theory, should be abandoned. Instead, the blame was put on the experimenter: Identically prepared systems do follow identical evolutions, and it is the metaphysical intervention of the experimenter in the act of measurement which is responsible for the discordant results.
The block universe approach, not constrained by an initial value formulation, offers a simple explanation to nature's 'indeterminism'. Experiments are never repeated. Instead, different parts of the space-time structure, supported on different times, correspond to different repetitions of an experiment. One can then 'chop off' two segments from the structure corresponding to two repetitions of the experiment, bring them to a common origin, and discover that they may coincide on their 'preparation part' yet differ on their 'measurement part'-just like two buildings may have identical basements but different penthouses. Collecting many such segments agreeing on their preparation stages, the probability of obtaining each observational outcome may be calculated. Quantum Mechanics is the statistical description of ensembles of such 'segments', cut from the space-time structure.
The block universe offers at once a simple explanation to observed violations of Bell's inequalities. Figure 1 depicts a typical space time substructure involved in a correlation experiment: Two nucleons escape a nucleus in a radioactive decay, each arriving at a polarimeter set to some orientation. Assuming, with Bell, that the measurement of each polarimeter is determined solely by its orientation and by regions of the 'nucleus tree' lying in the measurement's past light-cone, one can bound the degree to which the readings of the two polarimeters can be correlated. Nevertheless, Bell's assumptions are clearly at odd with the concept of the block universe. Rather than only reasoning that the details of the nucleus are manifested in the readings of each polarimeter, it is equally legitimate to expect the opposite, viz., that the readings are manifested in the nucleus. This so called 'retro-causal' influence, [1], explains how two remote particles appear to be exchanging mutual knowledge: Just like in a physical tree, only rarely can the motion of two distant branches be read from the motion of the trunk (e.g., an acoustic wave generated by the impact of an ax on the trunk, then propagating upwards to the branches). In generic cases, that motion is a global attribute of the the entire tree, with waves running in both the up and down directions, forming standing waves.
Most importantly, as the statistical aspects of ensembles of substructures of the global space-time structure describing our universe, viz. QM, are not fully encoded in the ECD equations, QM must be seen as an additional fundamental law of nature complementing ECD on statistical matters rather than rivaling it.
One cannot prove the above conjecture regarding the relation between QM and ECD based on purely theoretical arguments. As noted, QM allegedly encodes information about the particular ECD structure describing our universe which obviously contains information beyond the ECD equations proper. However, disproving that conjecture may be simple. It is enough to show, for example, that ECD particles cannot diffract or tunnel. Bellow, we focus on such outstanding predictions of QM which hitherto were seen as demonstrating the incompatibility of CE with it, and show that they all receive clear explanations within ECD.

Particle aspects of the EM field
Perhaps the first phenomenon which comes to mind in the above context is the photon which seems completely at odd with the notion of a smooth EM field. A key role in explaining photon related phenomena is played by the rapidly fluctuating vacuum field, (38), which we next turn our attention to.
Let us begin with a few general observations about the vacuum field: It is due to all particles in the universe, contains both advanced and retarded components and its form around a point, x ∈ M is determined by all currents in the neighborhood of the light cone of x. Since the intensity of radiation fields drops as one over the distance squared from the source, the influence of remote currents intersecting the light cone of x affects A vac (x) less than closer ones, but as the average number of particles in a spherical shell around x increases as the radius squared in a statistically homogeneous universe, the vacuum field a genuine attribute of the entire universe. However, local inhomogeneities in the distribution of charge in the universe are also manifested in the vacuum field in the form of statistical deviations from that 'universal part' of it.
We have argued in section 3.3 that the vacuum field plays no role in macroscopic radiation phenomena. More specifically, we show in appendix D that for any space-time volume, C, bounded by a time-like surface, T , and two space-like surfaces, Σ 1 , Σ 2 , the difference between the energy-momentum (e-m) content of those latter two can be read from the integrated Poynting flux across T . The insignificance of the vacuum field in macroscopic radiation phenomena entails that the part of this Poynting flux which is computed from bilinears in the vacuum field is negligible when either e-m contents of the Σ's becomes sufficiently large-the scale being the e-m content associated with a single particle. We further assume that cross terms in the vacuum field and the retarded field superpose incoherently, leaving only the Poynting flux computed from F ret . Nevertheless, when the Σ's enclose a single particle only, or a small number of them, the Poynting flux across T associated with A vac may be comparable with their e-m content and must not be neglected. Such is the case in the photoelectric effect: An advanced field associated with a particle-which is part of the vacuum field-converges on the particle, significantly increasing its e-m content. In Compton's effect a similar situation occurs but the jolting of the charge is accompanied by the generation of a strong retarded field which must also be added to e-m balance (see figure 2).

The 'conspiracy' leading to the invention of the photon
The insignificance of the vacuum field in macroscopic e-m balance on the one hand, and the existence of violent local fluctuations in it delivering e-m to particles on the other hand, imply that on average, the rate of e-m gained by particles in, say, a gas chamber, is proportional to the Poynting flux associated with retarded fields impinging on the chamber. This, of course, is verified in experiments, but the prevailing explanation given to this results is that the Poynting vector only describes the average e-m density associated with light corpusclesphotons-which eventually collide with gas particles in the chamber, delivering to them their e-m in a sequence of sudden acts.
The two explanations of the photoelectric effect can apparently be confronted if we now Figure 3: Space time structures involved in photo-detection place two gas chambers, or 'photodetectors', instead of one. A source emitting a single 'photon' implies a single 'photon detection' at most, whereas in the ECD model, two independently operating photodetectors (which are prevented from cross talking by partitions) can at times both fire. And indeed, such 'single-photon sources' (e.g. a molecule excited by a femtoseconds laser pulse, and then allowed to spontaneously decay) can be made, and the observed anticorrelation between the readings of the two detectors rules in favor of the photon model. Nevertheless, the block universe model leads to a simple explanation to the above observed anticorrelation also within the ECD framework. Figure 3 shows two substructures cut from the space-time structure, one corresponding to a single detection and one to double detection, with only the relevant part of the EM energy highlighted. The two substructures generally differ on all their parts, the 'emission vertex' part (marked with a circle) in particular, as the two solutions of the emitting particles involve different EM fields due to the absence of advanced fields from the left particle in (a). In a more metaphorical language, the advanced field jolting the particle in the right detector in (a), 'tunes' the source so as to prevent it from triggering the left detector.
Although both (a) and (b) are valid substructures, the frequency of their occurrence in the global structure needs not be similar. A 'single photon source' is defined as a source for which structure (a) is significantly more frequent than (b) (perhaps even, segments such as (b) appear with zero probability or are absent all together). Note that the nature of the source, being part of the substructure, strongly influences the relative frequencies of the latter. For strongly attenuated laser light, for example, it is found that both appear with equal frequencies, whereas for a light source of thermal origin, substructure (b) is more frequent. The branch of QM dealing with such statistical questions is quantum optics.
In actual experiments, e.g. [10], the retarded field of the source is relayed to the detecting charges by other charges, comprising mirrors, beam-splitters, fiber-optics etc. The crucial point is that, whatever optical path exists between the source and the detector, by means of retarded fields, there must necessarily exist a reverse path leading from the detector to the source via advanced fields. 12

Wave aspects of particles
In the previous section we saw how certain statistical properties of the space-time structure may explain particle aspects of the EM field. In the current section we show how other properties can explain apparently wave-like behavior on the part of particles. Here, again, the vacuum field plays a crucial part, but unlike in the the photoelectric and Compton's effects, its role in the current case is only to slightly perturb the path of the particle in the retarded field (due to all particles-self retarded field included.) This perturbation, nonetheless, incorporates global information about the scatterer into the path of a particle, as explained in section 4.2, implying that a particle passing through one slit in a double slit experiment is sensitive to the status of the other slit.
The smallness of the perturbation necessitates special experimental settings, facilitating the amplification of a feeble effect to a detectable level. There are exactly two such distinct amplification techniques. The first, used in scattering experiments, is geometric, relying on the huge distance between the scatterer and the detection screen which translates minute deflection angles into visible fringes. That the role of the vacuum field in this case is limited to a slight perturbation to the otherwise classical path of a particle in the external field, is supported by the fact that upon 'low passing' the observed scattering cross-section one reproduces the classical cross-section, similarly low-passed. This low-passed cross-section can be obtained either analytically, e.g. by convolving it with a kernel whose width is larger than the width of the fringes, or physically, by illuminating the particle with a weak source of light which also destroys the fringes. In this latter case, any reflection from the particle (sometimes referred to as 'measurement') entails a change in the form of the vacuum field at the location of the particle which varies between different scattered particles, hence the delicate statistical signature left by the scatterer in the vacuum field is destroyed, leaving only the classical cross section. Finally, the proportionality of the particle's deflection angle to its inverse momentum, implied by de Broglie's relations, surfaces naturally: The (small) deflection angle, α, of a particle in a scattering experiment is proportional to the transverse velocity acquired by it in passing near the scatterer, and is inversely proportional to its incident velocity. For a given perturbation, the former is inversely proportional to the particle's mass, hence α ∝ momentum −1 .
The second amplification technique, implemented in interferometers, relies on the ability of chaotic systems to amplify small perturbations. In a Mach-Zehnder configuration (a) used in neutron interferometers, for example, the beam-splitters (BS) and mirrors are crystals of macroscopic thickness, forming a huge lattice of scatterers in which a particle undergoes multiple scatterings before exiting. Even at the classical level, the dynamics in such a maze is highly chaotic, meaning, in particular, that the classical cross section obtained by averaging over the impact parameter, is utterly meaningless 13 . Small local deviations of a particle from its classical path induced by the vacuum field can therefore drastically impact the final angle at which the particle exists the interferometer. However, a neutron interferometer is a macroscopic device which can measure one meter across. Interference effects in a beam of neutrons taking place on such large scales (many 13 This method of obtaining the scattering cross section is consistent only for potentials for which the dynamics of the scattered particle is integrable. When applied to so called 'chaotic targets', the cross section becomes a fractal set defined on the unit sphere. An arbitrarily small perturbation to the potential representing the scatterer, completely modifies this set, including its course grained properties. But since an arbitrarily small perturbation always exists, the modeling of the scattering experiment using classical point dynamics is an insufficient abstraction. Any meaningful modeling of a physical experiment must incorporate the perturbing effect of the 'rest of the universe' in such a way that it can either be neglected below a certain threshold, or else incorporated into the model. Classical point dynamics-classical electrodynamics to be precise-fails to meet this criterion (and this has nothing to do with chaoticity in the usual sense of exponential sensitivity to initial conditions, but rather with the infinite time a particle gets trapped in chaotic targets).
The above situation drastically changes when modeling the experiment using quantum mechanics. The quantum mechanical differential cross section is always a smooth function, converging to a smooth distribution on the unit sphere as any perturbation to the potential representing the scatterer, or any coupling to the environment, are removed. While practically, it may not always be a problem-free tool for predicting the cross section (e.g. when the wavelength of the particle is much smaller than the scale of a classically chaotic scatterer) the above consistency criterion is always met. orders of magnitude larger than in scattering experiments on micron scale targets) must be due to similar scale interference effects in the vacuum field. That interference of EM radiation is supported by the very same interferometer -at least in a certain frequency band-is demonstrated by the use of neutron interferometers also for X-ray interferometry. The vacuum field, which does not carry e-m, is essentially a standing wave, so the beam splitters and reflectors in the interferometer set boundary conditions for this standing wave. A possible test of ECD could therefore be a neutron interferometer made of a crystal which does not scatter EM waves. As the mechanisms of scattering neutrons and EM radiation are different, this is not an entirely unlikely possibility.
The chaoticity of the underlaying classical dynamics is crucial for the operation of the interferometer. Suppose we remove BS2 from the apparatus (b). The influence of the vacuum field on the dynamics of a particle passing in region R is now marginal, and the particle continues its straight classical path, almost unperturbed, as follows from momentum conservation. This should be contrasted with (c), 'surrealistic' trajectories predicted by Bohmian mechanics, taking the other direction [2]. Without a reasonable explanation to such a breach of momentum conservation, Bohmian trajectories cannot be taken seriously as representing physical reality.

The meaning of the wave function
The role played by the vacuum field in the previous section, as the source of apparently non-classical behavior on the part of particles, closely resembles the role played by the 'zero point field' in stochastic electrodynamics (SED; see e.g. [5] and references therein). That theory is essentially Dirac's electrodynamics (see section 2.3) in a fluctuating EM background field, and as such does not satisfy the constitutive relations. These are not only necessary in order for a theory to be compatible with the predictions of CE but, as explained below, also to establish the compatibility of a theory with QM, raising doubt as to whether SED can really be the 'beable' underlying QM statistical predictions. Nevertheless, SED has had some impressive quantitative success in reproducing certain quantum mechanical results based on the concept of ensemble average, and it is therefore tempting to apply similar methods to ECD. However, the ECD counterparts of those methods are not only infinitely more complicated due to the extended structure of an ECD particle, but they also expose the 'deception' inherent in any alleged derivation of a statistical theory from a single system theory: one must postulate an ensemble over which the statistics is to be computed. When the single system equations are sufficiently simple, the postulated ensemble can be compactly defined, camouflaging the fact that critical information besides the single-system equations has been added to the computation. The definition of an ensemble of ECD structures requires an infinity of such postulates, making manifest the status of QM as a fundamental law of nature, on equal footings with the underlying single-system theory-allegedly ECD-and further explains why QM could have predated ECD (or whatever underlying theory).
The autonomous status of QM notwithstanding, it is constrained by the fact that it describes statistical properties of ECD substructures, each respecting the constitutive relations.
To check whether single-body QM is compatible with those, let us look at a collection of time slices of the global structure, corresponding to repetitions of an experiment. Each such substructure may involve a different distinguished particle, as in a scattering experiment, or the same particle-say, a radiating electron in a trap. If we now bring all time slices to approximately a common support in time, and add them together, we get for our distinguished particle an electric ensemble current. The reader can verify that the scattering cross section as well as any other measurable statistical expression produced by single-body QM, such as the spectrum of atoms, can be read from the ensemble current-an ordinary, conserved four-current.
Consider, next, an ECD substructure in the ensemble, indexed by e, and let k denote the distinguished particle in the substructure, e.g. the scattered particle. Using (34) we decompose the global EM potential into an external retarded field which is assumed constant throughout the ensemble, and a self field which varies across the ensemble incorporating also the vacuum field (38). Thus to each substructure, e, in the ensemble there correspond distinguished electric current e j and e-m tensor e m (note that the particle index k is omitted for economical reasons), and an EM potential e A sel , satisfying the constitutive relation (10) ∂ ν e m νµ = (F ext µν + e F sel µν ) e j ν .
Summing 14 (41) over the ensemble, we get for our ensemble quantities Next, we convolve (42) with a normalized Lorentz invariant kernel of the form (20). Assuming that this convolution eliminates the rapidly fluctuating f ens -an assumption to which we return below-we are left with an identical equation for the low-passed quantities without an f ens term. As shown in section 3, a systematic way of obtaining a conserved current j ens which, along with m ens satisfies the f ens -free (42), is via the five-dimensional Schrödinger's equation (22), for any choice ofh and q. Steady state solutions (in s) turn it into a Klein-Gordon equation for an ensemble of particles of a particular mass. The Klein-Gordon current is therefore consistent with its statistical interpretation as an electric current associated with an ensemble of ECD particles, and so is any linear combination of such currents, corresponding to so called 'statistical mixtures of wave-functions'. Note that the only feature of ECD used in establishing the above necessary conditions for its complementary statistical theory was the constitutive relations, suggesting that they play a central role in QM as well. As in SED, the spread of the wave-function is due to the vacuum field and the self force, having variable impact on individual members of the ensemble.
Returning to the elimination of the f ens term in (42) via a convolution with a kernel, this can be justified provided that the correlation length of f ens , is much smaller than the extent, √ , of the kernel (20). Now, the reader can verify that if the width of j ens 0 is on the order of the width of the individual e j 0 , then the correlation length of f ens must also be on that same order. It follows that the KG wave function cannot be consistently localized on scales beneath that correlation length, in line with the known result that relativistic wave equations run into interpretational difficulties when localized on scales smaller than the Compton length of the particle (see e.g. chapter 2 of [7]). Finally, we can appreciate why relativistic wave equations are successfully applied to the photoelectric effect but not the closely related Compton's effect. In the latter, the direction of the self force jolting individual particles is strongly correlated with the direction of the incident wave, hence the correlation length of f ens is on the order of the (macroscopic) external field, whereas in the former, the mass of the nucleus trapping the electron implies an essentially isotropic ionization direction, consistent with a small correlation length of f ens .

High energy physics
The immensely rich interaction of elementary ECD charges at close range opens up a possibility for a complete reformulation of physics at small scale. As all ECD currents, and in particular the individual electric currents associates with each charge, depend on the global EM field, at a sufficiently close rage the particles become so strongly entangled that it becomes almost meaningless to even speak of separate interacting particles. Instead, one gets a 'condensate' whose only reference to the number of its constituent particles is the number of world lines on which all ECD currents become (integrable) singular. It therefore seems redundant at this stage to try to extend ECD beyond its current structure in order to account for the strong force, which could be just a close-rage manifestation of the EM force.
Yet another feature of ECD which may be relevant at the nucleus scale, involves the ECD counterpart of the classical mass-squared current (3)-expression (98) derived in appendix C. Unlike (3), that current is only 'almost conserved', potentially loosing some of the masssquared charge of a particle to a world sink onγ (or, equivalently, gaining some charge therefrom). By 'almost' we refer to the fact that the said leakage is proportional to the third power of the small quantum parameterh. Now, a particle cannot just alter its masssquared charge without, in effect, becoming another particle. The new particle can be just a scaled version of the original (recalling that the mass-squared charge has dimension −2) if, say, the leakage is minimal, or it can be an entirely different particle, not related to the original by a scaling transformation, albeit with the same electric charge and spin. Unless the two particles have identical rest masses (which is never the case if the two are related by a scaling transformation, as energy has dimension −1) such a leakage must be accompanied by the release/absorption of energy. At the atomic scale there is no indication of that ever happening, but in the extreme EM fields existing at subatomic scales, the leakage may intensify, resulting in such particle conversions (Could a positron jolted from a nucleus be just a lean incarnation of a proton?) Assuming ECD is indeed the theory describing our universe, the relevance of the vacuum field may vary among different applications. For example, the binding of nuclei in a molecule may very well be due to the frenetic motion of the electrons in between them, perpetuated by energy exchange with the fluctuating vacuum field, in which case the Schrödinger wave function, encoding time-averaged quantities, is the best description of the system one can hope for from a practical stand point (as in the SED picture). At the scale of a single nucleus, in contrast, the vacuum field may have a diminished role. This possibility is supported by our previous observation in the context of scattering experiments, that heavier particles are less prone to the capricious nature of the vacuum field and moreover, that at sufficiently small scales the inter-particle forces greatly exceed those related to interaction with the vacuum field. This latter case offers the possibility of describing small scale physical processes as isolated ECD structures, giving a clear view of what is going on 'behind the hood' of those processes. This is clearly not just a philosophical gain, but may also have practical value. For example, a schematic view of two ECD structures describing the fusion of two light nuclei is depicted in figure 4. A full zoom of those structures may reveal the reasons behind the smaller velocities of the colliding nuclei, involved in structure (b) (e.g. specific orientation of the colliding nuclei, applied external field etc.) facilitating less demanding conditions for achieving controlled fusion. Such detailed considerations lie beyond the descriptive power of current alternatives to this description, such as QCD, which (assuming ECD is a valid theory of course) describes at most statistical aspects of ECD in the relevant domain.

Particle creation and annihilation
Nothing in the constitutive relations prevents a particle from 'reversing its direction in time'. In CE this scenario is of course prohibited by the mass-sell conditionγ 2 = const, constraininġ γ to lie inside the light-cone of γ, but this constraint does not carry to the γ part of an ECD particle, offering a simple geometrical picture of pair creation/annihilation (see figure 5). As a particle and its antiparticle have opposite signs for both their electric charges, and their mass-squared charges, such annihilation/creation scenarios respect the conservation laws of the two. The energy of a particle, however, equals that of its antiparticle. In annihilation processes, either EM radiation must be released or else a different pair (pairs) must be created, in order to respect energy conservation. The topology ofγ is further not constrained to just a line. Small loops, representing creation followed by annihilation may also exist which, combined with the vacuum field, respect local e-m conservation. Such 'twin abortions' may play a fundamental role in the statistical character of the vacuum field.

Gravitation
At the heart of any physical theory is a labeling scheme for events in space-time, viz., a coordinate system. Much of the development of theoretical physics over the years can be seen as a gradual increase in the flexibility of choosing a coordinate system for space-time, naturally accompanied by increasingly severer constraints on candidate physical theories, consistent with that greater flexibility. The freedom of choosing an arbitrary scale for a coordinate system, endorsed in the current paper, increases the (already large) set of permissible coordinate systems related to each other by a Poincaré transformation, but at the same time necessitated a very unusual mathematical construction in order for ECD to comply with scale covariance. The ultimate step in that direction is, of course, general covariance-the freedom to choose an arbitrary coordinate system to label space-time. This is not only an esthetically appealing symmetry, but it also avoids the circularity involved in the definition of inertial frames (to corroborate the laws of physics defined in inertial frames, one first needs to construct an inertial frame, which is only defined by the condition that the laws of physics hold in it). This ultimate freedom, as explained, does not come without a heavy price. At least one tensorial field-a 'metric'-must be involved (there are no generally covariant theories of lower rank tensors), the affine structure of Minkowski's space is lost, and only generally covariant quantities are meaningful.
General relativity (GR) is essentially the simplest generally covariant theory. Although more complicated, it shares with CE the same mathematical skeleton: Einstein's field equations in the presence of a source (with some gravitational constantḠ) the counterpart of Maxwell's equations (5), specifying a metric field given an e-m source, p µν , and the geodesic equationγ the analogue of the Lorentz force equation (2)-governing the motion of a point particle in a given metric field (hidden in the connection Γ). As in CE, to conform with scale covariance (being a subgroup of general covariance) the source of Einstein's field equations associated with a particle is supported on the latter's world line, and just like in CE, the metric generated by such a source is singular on the world line, rendering the geodesic equation of a particle ill defined. This is (misleadingly) referred to as the gravitational self force problem, making GR also a successful non-theory. At this point we should follow the rational of section 2.3, and seek similar constitutive relations capturing the essence of GR. However, (44) involves matter as a source, and matter is allegedly described by ECD, so our constitutive relations for GR must be compatible with a generally covariant extension of ECD.
The task of extending ECD to curved space-time is straightforward, resulting in a generally covariant version of the constitutive relations, involving covariant derivative instead of simple ones. The ill defined Lorentz force and geodesic equations of point particles, are thereby merged into a single well defined generally covariant central ECD system in a fixed background metric, guaranteeing that no electric charge nor e-m leak into world sinks onγ.
Next, we complement generally covariant ECD in a background metric with Einstein's field equations (44), where p = k k m + Θ is now covariantly conserved (the covariant counterpart of (14)), thereby elevating the metric to a status of a dynamical variable. Note that, as in the case of the ECD electric current (31)-the source of Maxwell's equation-which depends on the EM field, the source, p, of Einstein's equations depends also on the metric and its first derivatives. Note also that the geodesic equation can be recovered from the generally covariant ECD system using the same techniques employed in appendix B.

Possible applications of generally covariant ECD
GR has been tested to a high degree of accuracy only in a very limited range of gravitational curvatures. By (ill defined) GR it is usually meant: the geodesic equation in an external metric. Note that the implied assumption that the test particle does not perturb the external metric does not constitute an approximation, which can only be regarded as such if an exact analysis is well defined (because of the nonlinearity of Einstein's field equation, one cannot even decompose the metric into partial contributions, each generated by an individual particle, as in (linear) Maxwell's equations (5)).
In extreme gravitational curvatures-either small or large-GR, thus resolved, can hardly be said to be a well tested theory. In extremely small curvatures, such as on the outskirts of galaxies, GR fails colossally unless a very specific distribution of undetectable 'dark matter' is assumed to fill space (a rather peculiar conjecture given that its sole motivation is to salvage a non-theory.) A quantitative analysis of generally covariant ECD may reveal an essential deviation from simple geodesics rather in weak gravitational fields, explaining the MOND phenomenology. It should be noted in this regard that current alternatives to GR which reproduce the MOND phenomenology, such as STVG and TeVeS, are as non-theories as GR is, being based on point particles.
Black holes. The name "black hole" refers to a singular solution of the homogeneous Einstein's field equation, viz., p ≡ 0, and therefore does not explicitly involve matter. The degree to which such a solution, especially around the singularity, represents physical reality is questionable, with most experts on the matter maintaining that quantum gravity must take over in this regime. Regardless of the fact that such a theory has not yet been formulated, nor of the unclear meaning of such a theory (a statistical theory of generally covariant ECD?), generally covariant ECD is as well-defined at the center of a black hole as anywhere else. Instead of a nonphysical singularity one should find there a dense ECD condensate-possibly of unique character-but still respecting the constitutive relations, hence no charge nor e-m can disappear from the center of a black hole. Moreover, it may even turn out that no physical black hole can even exist once matter is properly incorporated into the model.
Gravitational waves. As of today, gravitational waves remain elusive, notwithstanding increasingly more sensitive detection techniques. And yet, in the controversy surrounding their existence, lasting for almost a century, the supporters of that possibility currently have the upper hand (at least in the number of academic positions they hold). This is largely due to the phenomenal success of Einstein's quadrupole formula (QF) for the generation of gravitational waves, in describing the orbital decay of the Hulse-Taylor binary pulsar. Much of the controversy in the latter years therefore focused on the question of whether QF can be considered a prediction of GR. Now, being a non theory, GR has no predictions. To evaluate the validity of QF or, in general, of the feasibility of gravitational waves generation/detection, one must first turn it into a theory, as generally covariant ECD does.

Cosmology
In section 3.2 it was speculated that different stationary and isolated solutions of the ECD equations represent different elementary particles, but by the scaling symmetry of ECD, to each such isolated solution there corresponds an infinite family of scaled versions, sharing the same electric charge and spin. This raises the question as to the source of the apparent 'spontaneous scaling symmetry breaking', viz., the absence of an observed continuum of masses associated with each elementary particle. Of clear relevance to this question is the ECD mass-squared current (98), which is not identically conserved. As noted in section 5.1, a change in the associated charge must be accompanied by a change in the self energy of the particle. The common self energy of all electrons could therefore be due to their coupling/entanglement by the vacuum field. For example, if we assume that the statistical tendency of heavier-than-average electrons is to loose weight, and that of the lighter-thanaverage electrons is to gain, the equilibrium mass distribution should have a narrow width at most. More generally, as all elementary particles are coupled by the vacuum field, it is equally natural to assume that they all share the same mass-squared charge, which fixes their relative self energies.
Once in such an equilibrium state, nothing fixes that common mass-squared charge of elementary particles to its observed value. In principle, this value can slowly shift, resulting in a corresponding shift in the 'expansion charge' of the entire universe-expression (100), the ECD counterpart of the classical dilatation charge (15)-which, like the ECD masssquared charge, is only almost conserved. This offers an alternative explanation for the source of galactic redshifts: A collective increase in the mass of elementary particles leads to a Hubble-like relation, as light collected from remote galaxies is emitted at an epoch of lower mass (hence longer wavelength) which is proportional to the distance between the emitter and the observer, for any (static) observer. Note that such a scenario may also be interpreted as an expansion of the universe, as more 'standard length gauges' (such as an electron) can fit between two galaxies as time passes. Indeed, the (generally covariant) expansion charge has both a geometrical piece and a compensating matter piece, setting the scale for the metric.
Finally, as in the case of black-holes, a singularity cannot appear in any cosmological model, such as the 'big bang' model, based on generally covariant ECD.

Conclusion
Any candidate for a novel fundamental physical theory should meet a minimal set of requirements. First, it must be a theory-a well defined piece of mathematics. Secondly, it must be compatible with existing well tested experimental results. Thirdly, the theory must have novel testable predictions. As demonstrated in this paper, requirements one and three are fully met by ECD. The second requirement is, of course, the most demanding, given the enormous body of knowledge under consideration. Rather than trying to cover all fields of physics, the emphasis in the paper was on the 'naturality' of ECD's explanations to some of the more puzzling observations. For example, unlike in SED, the existence of a fluctuating vacuum field is an inevitable consequence of the mathematical structure of ECD, incorporating the EM field into the source of Maxwell's equations. Quantum non-locality is likewise inevitable in the same sense that two branches of a tree cannot always be treated as separate entities. Finally, when all the mathematical dust settles following the construction of the ECD equations, we are left with simple charge and e-m conservation-probably the two most well tested phenomena-and with an extended symmetry group. The few physicists who do not appreciate the esthetics in scale-covariance, are forced to accept it, being a subgroup of general covariance. Any attempt to solve the gravitational self-force problem by means of extended particles must therefore reduce to a scale covariant theory in Minkowski's space.
A lot of work must still be invested in order for ECD to qualify as a successful theory. The author has already made some progress in solving relatively simple problems, but as no ready-made numerical tools, let alone analytic tools, come close to solving the ECD equations, the verdict of ECD awaits further advance on this highly technical front.

Appendices A The 'fine tuned' central ECD system
As all ECD currents are computed in the limit → 0, the central ECD system (28) and (29) is given bellow an operational definition for small only. To this end, we would need the small-s form of the propagator G. Plugging the ansatz into (22), with the free propagator computed for A ≡ 0, and expanding Φ (not necessarily real) in powers of s, Φ(x, x , s) = Φ 0 (x, x ) + Φ 1 (x, x )s + . . ., higher orders of Φ k can recursively be computed with Φ 0 alone incorporating the initial condition (30) in the form Φ 0 (x , x ) = 0 (note the manifest gauge covariance of this scheme to any order k). For our purpose, Φ 0 is enough. A simple calculation gives the gauge covariant phase where the integral is taken along the straight path connecting x with x.
Focusing first on (28), we see that, for fixed γ and G, it is in fact an equation for a function f R (s) ≡ φ(γ s , s). Indeed, plugging an ansatz for f R into the r.h.s. of (28), one can compute φ(x, s) ∀s, x, and in particular for x = γ s , which we call f L (s). The linear map f R → f L (which, using G(x , x; s) = G * (x, x ; −s), can be shown to be formally selfadjoint) must therefore send f R to itself, for (28) to have a solution. Now, the universal, viz. A-independent, i/(2πhs) 2 divergence of G(y, y, s) for s → 0 and any y, implies f R → f R + O( ), so the nontrivial content of (28) is in this O( ) term, which we write as f r . In [9], lim →0 f r = 0 was implied as the content of (28). While this may turn out to be true for some specific solutions (a freely moving particle, for example), equation (28) should take a more relaxed form Im lim where, as usual, 'Im' is the imaginary part of the entire product to its right. Moving next to the second ECD equation, (29), conveniently rewritten as a similar isolation of the nontrivial content exists. For further use, however, we first want to isolate the contribution of the small s divergence of G to φ(x, s), for a general x other than γ s . Substituting (46) into (28), and expanding the integrand around s to first order in s − s: Using and (50) where the above equality follows from (53), φ r (γ s , s) = f r (s) and (49). The fine-tuned definition of (29) is therefore Using the above definitions, (49) can also be written as More insight into this fine tuned central ECD system is given in the sequel. For the time being, let us just note that it is invariant under the original symmetry group of ECD. In particular, the system is invariant under under a gauge transformation and under scaling of space-time directly following from the transformation of the propagator under scaling Regarding this last symmetry, two points should be noted. First, for a finite it relates between solutions of different theories, indexed by different values of . It is only because is ultimately eliminated from all results, via an → 0 limit, that scaling can be considered a symmetry of ECD. The second point concerns the scaling dimension, 0, of φ s and φ r . By the symmetry (57), that dimension can be arbitrarily chosen. However, to comply with scale covariance j must have dimension −3, hence this special choice.

A.1 ECD currents
All ECD currents have the common form where B is some bilinear in φ and φ * . Using the decomposition (52) we write By the same arguments as in section 2.3.1, j ss (x) can be shown to reduce to the line current which is not necessarily conserved as f R (s) 2 may be s-dependent, and is discarded of in ECD. Likewise, the {s,s} contribution of all ECD currents is a distribution supported on γ albeit generally containing more complex distributions, involving also derivatives of line distributions.
Moving to the {r, r} term, this piece gives a nonsingular contribution which is well localized aroundγ in a region referred to as the core. The localization mechanism of the core is explained within the semiclassical approximation in appendix B. Finally, the integrand of the {r, s} term in the s integral, behaves in the limit → 0 as a regular, well localized piece, coming from the r part, multiplying a δ (ξ 2 ) coming from the s part. This piece generates a singularity onγ to which no charge leaks by virtue of (28). It also decays at large distances fromγ much more slowly than the core, and is therefore dubbed the 'halo' of the current.

B A semiclassical analysis of ECD
In this section we analyze the ECD system using a smallh approximation of the propagator known as the semiclassical propagator, Above, is the action of the classical path β such that β 0 = x and β s = x, and F -the so called Van-Vleck determinant -is the gauge-invariant classical quantity, given by the determinant We shall assume that, for a given s, there exists a unique path connecting x with x. In fact, it is only under this premise that the unitarity of G sc (x, x ; s) can be established. Moreover, a plurality of paths, completely arbitrarily, supports interference effects but excludes diffraction, viz., no classical path can lead to the classical shadow of a scatterer. The semiclassical propagator becomes exact for small s, so the singular-regular decomposition (52) of φ is consistent with the approximation, the latter affecting only the accuracy of φ r . 15 Let us next show that to leading order inh and some fixed potential A, the fine-tuned central ECD system is solved by any classical γ, and by a corresponding ansatz of the form with C ∈ C an arbitrary constant.
Substituting in (28), G → G sc , x → γ s and x → γ s , we first note that γ is the classical path in A, connecting γ s with γ s . Using I (γ s , γ s , s − s ) + I (γ s , γ 0 , s ) = I (γ s , γ 0 , s) we get with some real functional of the EM field and its first derivative (its local neighborhood in an exact analysis) onγ, such that lim →0 [R(s, ) − 2/ ] is finite, implying that (56) is satisfied. Moving next to the second refined ECD equation, (55), and pushing ∂ into the integral in (28), Theh∂F term in (71) can be neglected for smallh. Using a relativistic variant of the Hamilton-Jacobi theory (see appendix B in [9]), we can write which is independent of s . Together with (69) we therefore get which vanishes by (56), hence (55) is satisfied. The phase of the integrand is independent of s only for x = γ s , as manifested in (68). Otherwise, the family of paths connecting γ s with x, and that connecting γ s with γ 0 , traverse different parts of the potential and do not even lie on the same mass-shell. The phase is therefore a rapidly oscillating function of s for smallh and/or x lying far from γ s . In the terminology of section A.1, using the accuracy of the semiclassical propagator for small s, it can readily be shown that the {r, r} piece of a current is well focused aroundγ. The {r, s} term can further be shown to have an integrable r −1 singularity which is not a mere artifact of the semiclassical approximation, as the latter affects only the regular piece φ r . In the case of the electric current (31), this singularity leads to a constant (integrable) EM energy density onγ, as oppose to a non integrable singularity in the classical case, but it also implies a discontinuous EM field atγ (a non differentiable A there). This means that for x and x both lying onγ, the semiclassical propagator (64) is ill defined. In order to utilize this useful approximation one therefore must work with a finite which removes the discontinuity of A onγ, subtract from the current the {s, s} piece, and take the limit → 0 as a final step.

C The constitutive relations
To prove the conservation of the ECD electric current (31), we first need the following lemma, whose proof is obtained by direct computation.
Lemma. Let f (x, s) and g(x, s) be any (not necessarily square integrable) two solutions of the homogeneous Schrödinger equation (22), then This lemma is just a differential manifestation of unitarity of the Schrödinger evolutionhence the divergence. Turning now to equation (28), and its complex conjugate, we get by direct differentiation Focusing on the first term on the r.h.s. of (76), we note that, as G is a homogeneous solution of Schrödinger's equation, we can apply our lemma to that term, which therefore reads Integrating (76) with respect to s, the left-hand side vanishes (we can safely assume it goes to zero for all x, s , s as |s| → ∞), and the derivative ∂ µ can be pulled out of the triple integral in the first term. The reader can verify that this triple integral, after application of lim →0 ∂ −1 , is just ∂ µ j µ , with j given by (31) and φ, φ * are explicated using (74), (75) respectively. The ECD electric current is therefore conserved, provided the s integral over the second term in (76), after application of lim →0 ∂ −1 to it, vanishes in the distributional sense.
Let us then show that this is indeed the case. Integrating the second term with respect to s, and using ∂ s U( ; s − s ) = δ(s − s − ) + δ(s − s + ), that term reads

Using (74) and (75), this becomes
Writing φ = φ s + φ r above, and using the short-s propagator (46) plus the explicit form, (51), of φ s , one can show that application of lim →0 ∂ −1 to (79) results in a distribution supported onγ-a 'line sink'-which is composed of two pieces: one coming from φ s and one-from φ r . The s piece is just the (not necessarily vanishing) divergence of the line current (63) and is therefore of no concern to us. The second piece reads and represents a 'line sink in Minkowski's space' associated with the singularity of j onγ. By virtue of (56), no leakage of charge occurs at those sinks, as one can establish the timeindependence of the charge by integrating ∂ · j = 0 over a volume in Minkowski's space, and apply Stoke's theorem, to get a conserved quantity. A more explicit way of demonstrating the conservation of charge, avoiding the use of distributions, is shown next.

C.1 Line sinks in Minkowski's space
To gain a more explicit geometrical insight into the meaning of a 'line sink in Minkowski's space', consider a small space-like three-tube, T , surroundingγ, the construction of which proceeds as follows. Let β(τ ) = γ (s(τ )) be the world lineγ, parametrized by proper time τ = s (dγ) 2 , and let x → τ r be the retarded light-cone map defined by the relations Let the 'retarded radius' of x be r = η ·β τr .
The (retarded) three-tube of radius ρ is defined as the space-like three surface It can be shown in a standard way that the directed surface element normal to x ∈ T ρ is where dΩ is the surface element on the two-sphere. Let Σ 1 and Σ 2 be two time-like surfaces, intersecting T ρ and T R . Applying Stoke's theorem to the interior of the three surface composed of T ρ , T R , Σ 1 and Σ 2 , and using ∂ · j = 0 there, we get Realistically assuming that the second term on the r.h.s. of (85) vanishes for R → ∞, we get that the 'leakage' of the charge, Σ 2 dΣ 2 · j − Σ 1 dΣ 1 · j, equals to − lim ρ→0 Tρ dT ρ · j.
As dT ρ = O(ρ 2 ), the leakage only involves the piece of j diverging as r −2 . This piece, reads where s r = s (τ r )), and γ sa is the corresponding advanced point onγ, defined by Focusing first on the contribution of s r , and using a technique similar to that leading to (83), we get where m = dτ /ds needs not be constant. In the limit ρ → 0, using ∂ 1 ξ·γs r · ∂r| r=ρ → m −1 , the contribution of s r to the flux across T ρ is most easily computed Tρ dT ρ · j = qh 2 π dΩ dτ r m −1 Im φ r * (β τr , τ r )f R (τ r ) = 4qh 2 π 2 ds r Im φ r * (β sr , s r )f R (s r ) .
The contribution of s a to the flux of j is more easily computed across a different, (advanced) T ρ , and gives the same result in the limit ρ → 0. The fact that ρ can be taken arbitrarily small, in conjunction with the conservation of j(x) for x / ∈γ, implies that the flux of j across any three-tube, T = ∂C, with C a three-cylinder containingγ, equals twice the value in (87), when C is shrunk toγ. Changing the dummy variable s r → s in (87), the formal content of (80) receives a clear meaning using Stoke's theorem which vanishes by virtue of (56).

C.2 Energy-momentum conservation
The conservation of the ECD energy momentum tensor can be established by the same technique used in the previous section. To explore yet another technique, as well as to illustrate the role played by symmetries of ECD in the context of conservation laws, consider the following functional and let φ(x, s) be given by (74) for some fixed A(x) and γ s . Using directly following from the definition of φ, we calculate A [φ + δφ] and, after some integrations by parts, we get for the first variation (90) Choosing δφ = ∂φ · a, corresponding to φ(x, s) → φ(x + a, s), with infinitesimal a(x), vanishing sufficiently fast for large |x| so as to render δA well defined, we get in a standard way with j and m given by (31) and (32) respectively without the lim →0 ∂ −1 operation . Applying now lim →0 ∂ −1 to (91), the r.h.s. can be analyzed using the same technique used in the computation of (79). This gives which vanishes by virtue of (55) for any a. The arbitrariness of a implies the constitutive relation in the distributional sense. Just like the electric current j, the matter e-m tensor m can easily be shown to be a smooth function of x, implying pointwise equality in (93). Equation (55) in the central ECD system, by which (92) vanishes, appears therefore as the condition that no mechanical energy or momentum leak into a sink onγ. Not surprisingly, m is not conserved, due to broken translation covariance induced by A(x). To compensate for this, using Noether's theorem, we construct an 'equally non conserved' radiation e-m tensor, and subtract the two. Consider, then, the following functional of A(x), for fixed k j, (k labels the different particles) By the Euler Lagrange equations, we get Maxwell's equations, (5), with k k j as a source. As before, infinitesimally shifting the argument of an extremal A, viz. A(x) → A(x + a) ⇒ δA µ = ∂ ν A µ a ν , and following a standard symmetrization procedure of the resultant e-m tensor (adding a conserved chargeless piece ∂ λ F νλ A µ ) leads to the canonical (viz. symmetric and traceless) 'radiation e-m tensor' (11). Summing (93) over the different particles, k, and adding to (95), we get a conserved, symmetric e-m tensor, ∂ ν p νµ = 0 , with p = Θ + k k m . (97)

C.3 Charges leaking into world sinks
Both methods used above, can be applied to prove the conservation of the mass-squared current -the counterpart of (3) b(x) = lim In the first method, used to establish the conservation of j, the counterpart of (73) is ∂ s (g * Hf ) = ∂ · Reh∂ s g * Df , corresponding to the invariance of the Hamiltonian (in the Heisenberg picture) under the Schrödinger evolution. In the variational approach, the conservation follows from the (formal) invariance of (88) φ(x, s) → φ(x, s + s 0 ). However, the leakage to the sink onγ, between γ s 1 and γ s 2 , is given by 8π 2h3 is not guaranteed to vanish. Note that this leakage (whether positive or negative) is a 'highly quantum' phenomenon -proportional toh 2 (the term ∂ s φ r generally diverges ash −1 ). Similarly, associated with the formal invariance of (88) under is a locally conserved dilatation current, the counterpart of the classical current (15), The leakage to the sinks on kγ is due to the second term, involving the mass-squared of the particles. A leakage of mass, therefore, also modifies the scale-charge of a solution. their spin which merely labels different realizations of the constitutive relations-relations among ordinary tensors-by means of the ECD construction, and adds no genuinely new ingredients to the formalism. To translate (103) into an equation forγ-roughly speaking the center of the currentwe first 'low-pass' (103), viz., convolve it with a normalized kernel, w(s), to remove possible fluctuations in γ which are due to internal vibrations in the particle. It is easy to see then that for a sufficiently wide w, the r.h.s. of (103) becomes independent of the details of the foliation hence also the l.h.s. of (103). Next, we make the reasonable assumptions that the low-passed p is locally (s-wise) proportional to the low-passedγ, with an s-independent proportionality constant G. This latter assumption is nothing but the condition that the same particle is being investigated at different s's, namely, that the average momentum of the particle can be deduced from its average velocity. Under this assumption, using the same notation for the low-passed γ, (103) becomes withw(s, x) defined by x ∈ Σ s ⇒w(s, x) = w(s − s ). For a sufficiently isolated particle, expression (34) for A provides a convenient decomposition of F in (104) into a self field, F sel generated by the isolated particle, and an external field F ext generated by the rest of the particles. For a slowly varying F ext on the scale set by w the r.h.s. of (104) can be written QF ext µν (γ s )γ ν + F sel µν j ν γs , with Q = Σs dΣ·j the s-independent electric charge. Neglecting the self force, (105) becomes just the Lorentz force and the constant G in (104) is identified as p 2 /γ 2 , where p 2 is the Lorentz invariant rest-energy of the charge. The above analysis demonstrates that when the effect of the (well defined) self force can be neglected, e.g. when the current is approximately spherically symmetric, the Lorentz force equation is reproduced on scales larger than the extent of the particle. This explains the partial success of simply ignoring the self force as a solution to the self force problem. In some cases, in contrast, the self force dominates the dynamics leading to such a colossal failure of this approximation that physicist mistakenly reasoned that CE must be abandoned altogether.
Equation (102), somewhat artificially divide the change in the momentum of a particle into a work of the Lorentz force, plus a 'radiative' contribution, T dT ν m νµ , of the associated e-m density m. A more symmetric treatment of 'matter' and the EM field is provided by the conservation of the total e-m, p in (14). Applying Stoke's theorem to ∂p = 0, and using the same construction as in figure 1, we get p µ (s + δ) − p µ (s) = − T dT ν p νµ , with the total four-momentum content of Σ(s)∩C, including the EM part coming from Θ which is finite in ECD. If we assume, as previously, that the flux of p across T is purely of EM origin, we arrive at the conclusion that, for a sufficiently isolated particle (or a bound aggregate of particles), the change in momentum can be read from the flux of the Poynting vector across a time-like surface surrounding it. Note that no approximation whatsoever is involved this time.

E Spin-1 2 ECD
In a spin-1 2 version of ECD, the following modifications are made. The wave-function φ is a bispinor (C 4 -valued), transforming in a Lorentz transformation according to ρ (e ω ) φ ≡ e −i/4 σµν ω µν φ , for e ω ∈ SO(3, 1) , where σ µν = i 2 [γ µ , γ ν ], with γ µ Dirac matrices (not to be confused with γ the trajectory). The propagator is now a complex, 4 × 4 matrix, transforming under the adjoint representation, satisfying with the initial condition (30) at s → 0 reading δ (4) (x − x )δ αβ , where δ αβ is the identity operator in spinor-space, and g is some dimensionless 'gyromagnetic' constant of the theory. The transition to spin-1 2 ECD is rendered easy by the observation that all expressions in scalar ECD are sums of bilinears of the form a * b, which can be seen as a Lorentz invariant scalar product in C 1 . Defining an inner product in spinor space (instead of C 1 ) with γ 0 the Dirac matrix diag(1, 1, −1, −1) (again, not to be confused with γ the trajectory) and substituting a * b → (a, b) in all bilinears, all the results of scalar ECD are retained. The Lorentz invariance of (110) follows from the Hermiticity of σ µν with respect to that inner product, viz. (σ µν ) † = γ 0 σ µν γ 0 , and from (γ 0 ) 2 = 1.
Let us illustrate this procedure for important cases. By a direct calculation of the shorts propagator of (109), as in section A, the spin can be show to affect the O(s) terms in the expansion of Φ, leading to an equally simple φ s , the counterpart of (51), from which the regular part of all ECD currents can be obtained. The action, (88), from which all conservation laws can be derived, gets an extra spin term while the counterpart of the electric current, (31), derived from φ, is now a sum of an 'orbital current' and a 'spin current' j µ (x) ≡ j orbµ + j spnµ = lim →0 ∂ −1 ds qIm (φ, D µ φ) − g∂ ν (φ, σ νµ φ) , for x / ∈γ . (112) Each of the terms composing j is individually conserved and gauge invariant. The conservation of the orbital current follows from the U (1) invariance of (111), while conservation of the spin current follows directly from the antisymmetry of σ. This current has an interesting property that its monopole vanishes identically. Calculating in an arbitrary frame, using the antisymmetry of σ, and assuming j spn i (x) → 0 for |x| → ∞ The counterpart of (93) becomes ∂ ν k m orb νµ + g νµ k l = F µ ν k j orb ν + lim →0 ∂ −1 g 2 ds k φ, σ λρ k φ ∂ µ F λρ , for x / ∈γ , (114) with m orb the same as (32) with a * b → (a, b) in all bilinears, and l(x) = lim →0 ∂ −1 g 2 ds φ, F λρ σ λρ φ .
We shall see below that the 'spin force' density appearing on r.h.s. of (114) is an artifact of misidentifying the expression in brackets on the l.h.s. as the e-m tensor. Using (13) and the antisymmetry of σ and F , (114) can be rewritten as with k m spnνµ = g νµ k l + g ds k φ, σ ν λ F λµ k φ , x / ∈ ∪ k kγ As k k j generates A, we clearly have Summing (115) over k, and adding to (116), we get the locally conserved e-m tensor