Neoclassical models of charged particles

Classical electrodynamics (CED) has achieved great success in its domain of application, but despite this success, it has remained a theory that lacks complete self-consistency. It is worthwhile trying to make CED a self-consistent theory, because many important phenomena lie within its scope, and because modern field theories have been modelled on it. Alternative approaches to CED might help finding a definite formulation, and they might also lead to the prediction of new phenomena. Here we report two main results. The first one derives from standard CED. It is shown that the motion of a charged particle is ruled not only by the Lorentz equation, but also by equations that are formally identical to Maxwell equations. The latter hold for a velocity field and follow as a strict logical consequence of Hamilton’s action principle for a single particle. We construct a tensor with the velocity field in the same way as the electromagnetic tensor is constructed with the four potential. The two tensors are shown to be proportional to one another. As a consequence, and without leaving the realm of standard CED, one can envision new phenomena for a charged particle, which parallel those involving electromagnetic fields. The second result refers to a field-free approach to CED. This approach confirms the simultaneous validity of Maxwell-like and Lorentz equations as rulers of charged particle motion.


Introduction
The problems we can address with the tools of classical electrodynamics (CED) can be divided in two classes. In one class, charge distributions and currents are given and one seeks the electromagnetic field they produce. This field is found by solving Maxwell equations. In the second class, the electromagnetic fields are specified and one seeks to determine the motion of charged particles that are acted upon by these fields. This motion is found by solving the Lorentz equation of motion. The complete problem, in which both the electromagnetic fields and the motion of charged particles are self-consistently obtained, is an open problem (see, e.g. [1], Chap. 16). Only approximate solutions for special cases are known. That is, CED was never completed to a self-consistent theory. It was also not superseded by quantum electrodynamics (QED). The latter does not answer fundamental questions of the classical formulation. For example, how the emitted radiation of an accelerated charge affects its motion. Heuristic modifications of the Lorentz equation have physically inconsistent consequences, such as selfacceleration and infinite self-energies [1][2][3]. QED does not address particle motion and, on the other hand, it must deal with its own infinities. To be sure, renormalization procedures of quantum field theory are widely seen as an acceptable tool to deal with infinities; but these infinities can also be seen as a sign that the theory lacks selfconsistency. QED does not contain CED as a limiting case. Rather, QED, the archetype of modern field theories, is modelled on CED as its quantum version. There is, however, no systematic quantization procedure. What one essentially did to 'quantize' classical formulations was to proceed by trial and error. For instance, the Gupta-Bleuler formalism for covariant quantization required the ad-hoc separation of a Hilbert space into physical and unphysical states [4]. Non-covariant quantization did not require such a separation. Another example of internal inconsistencies is given by the handling of 'virtual' processes, which are deprived of physical reality and yet assumed to be the very cause of observable, physical effects such as the Lamb shift and the Casimir force. These inconsistencies were accepted because QED proved to be highly successful.
We should strive for a better understanding of CED, not only in the pursue of having a complete, selfconsistent theory that applies to many phenomena, but also because new phenomena might be predicted by following alternative approaches [5][6][7][8]. Here we report two such alternative approaches. The first one is formulated fully within the standard framework of CED, but taking into account a mathematical feature that has been largely neglected in physics. It relates to Hamilton's action principle, from which the Lorentz equation derives. As we will see, the Lorentz equation follows from a field equation that rules the dynamics of the so-called Mayer field of extremals. Using this formulation, we can show that the motion of a charged particle is ruled not only by the Lorentz equation, but by Maxwell-like equations as well. Such a result, which is a strict logical consequence of the standard approach, might have manifold consequences that lead to the prediction of new phenomena. The second, alternative approach to CED is a field-free one. This type of approach has been put forward in the past [9] and is sometimes referred to as 'action-at-a-distance electrodynamics' [2]. Our model, however, does not require action at a distance. It is based on the same causal principle that connects a source, which is at one place, with the field it produces at other places.
We are used to see fields as physical entities, even though we admit that they can be observed only indirectly, through their action on material particles. In actuality, the electric field is defined as force per unit charge, and the magnetic field is defined as force per unit current [5]. Forces cannot exist independently of the bodies on which they act. There is an obvious inconsistency in referring to forces that propagate without acting on anything. This inconsistency, though, has not been an obstacle for developing a successful theory. In fact, success has been the ultimate reason for giving fields the same status as material particles. However, it could be that fields are just an indirect way of describing the properties of an entity, about which we know very little: the electromagnetic vacuum. It is clear that vacuum is not the same as emptiness. Vacuum is a physical object on its own. Indeed, vacuum has two well known physical properties: electric permittivity (ò 0 ) and magnetic permeability (μ 0 ). The origin of these properties is, however, unknown. While quantum vacuum fluctuations could give us some clue as to the origin of ò 0 and μ 0 , more experimental input is certainly necessary to construct a model of vacuum. It is regrettable that almost all we know about the electromagnetic properties of vacuum stems from experiments that were already performed on the nineteenth-century. The fortuitous, historical development of physics has certainly determined that the study of vacuum was largely neglected. The historical development has also contributed to establish the idea that the study of vacuum can be conducted only by resorting to quantum concepts [10][11][12][13][14][15][16]. We cannot exclude beforehand that a better understanding of a 'classical' vacuum is precisely what we need to characterize the quantum-classical border. It is our hope that the results reported in this work help drawing interest to the study of vacuum, no matter the tools employed, classical or quantal.
The rest of the paper is organized as follows. In section 2, we summarize the standard formulation of CED. In section 3, we give a short account of Carathédory's approach to the calculus of variations, which is required to derive the Maxwell-like equations that we present in section 4. Next, in section 5 we present an alternative, fieldfree formulation of CED. In section 6 we make some closing remarks.

The Maxwell-Lorentz equations
For future reference, we recall here the basic equations of CED. First, we have Maxwell equations, written in covariant form (Greek indices take values from 0 to 3 and a sum over repeated indices is understood): The electromagnetic tensor F μ ν is defined in terms of the four-potential A μ , by F μ ν = ∂ μ A ν − ∂ ν A μ . Given the source current j μ and appropriate boundary conditions, F μ ν is determined by the above equations. It should be noted that equation (2) is an identity that holds irrespective of the physical meaning of A μ . Equations (1) and (2) are, respectively, the inhomogeneous and homogeneous Maxwell equations. Using three-vector notation, they read Second, we have the Lorentz equation that rules the motion of a particle with charge e and mass m, subjected to the action of a given field F μ ν : In three-vector notation and considering derivatives with respect to time (t = x 0 /c) instead of derivatives with respect to s, equation (6) reads, for ν = 1, 2, 3, It is worth noting that equation (8) is not an independent equation. It follows from equation (7), as can be readily shown.
Both Maxwell equations and the Lorentz equation can be shown to follow from variational principles. In the case of Maxwell equations, the variational principle involves a Lagrangian density. In the case of the Lorentz equation, the variational principle involves a Lagrangian. Modern field theories are formulated in terms of variational principles that involve Lagrangian densities. As we shall see next, the variational principle that leads to the Lorentz equation has also a field related to it. That is, one can derive field equations from an action principle that entails a Lagrangian, not a Lagrangian density. This is clearly shown in Carathéodory's approach to the calculus of variations [17]. We present next a very short account of this approach.

Carathéodory's formulation
Hamilton's action principle establishes that particles move along curves  that render extremal an action integral I, i.e, . In physics, it is seldom mentioned that  can exist only as a member of a whole field of extremals, known as a Mayer field [17][18][19]. An extremal curve is a solution of the Euler-Lagrange equations for the Lagrangian L x x , ( )  . The variational problem, which consists in finding an extremal curve, in fact requires finding a whole set of extremals. The sought-after extremal curve is a particular member of a field of extremals. This is the so-called imbedding theorem in the calculus of variations [17][18][19], basically a consequence of continuity assumptions. The left panel of figure 1 illustrates the usual approach in classical physics: one focuses on a single extremal, the one that describes the motion of a single particle. The right panel shows instead the whole picture: the sought-after extremal cannot be an isolated one. It can exist only as a member of a field of extremals, a so-called Mayer field [19]. The domain under consideration is of utmost importance. One and the same Lagrangian can lead to different families of extremals, depending on the domain under consideration and its associated boundary conditions. The existence of a Mayer field requires that some integrability conditions are satisfied. Carathéodory's formulation [17] makes clear how these conditions relate to the Euler-Lagrange equations.
are the same as those satisfying the so-called 'equivalent variational problem' . In Carathéodory's approach, instead of seeking for a curve  that makes I extremal, we seek for local extremal values of some quantities, as shown below. In this setting, L is considered to be a Lorentz-invariant function of x and the velocity-field v(x), i.e. L = L(x, v(x)). Moreover, L must be homogeneous of first order in the velocities: L(x, λv(x)) = λL(x, v(x)), for λ > 0. This is required, for the variational problem to be invariant under changes of the curve parameter. The extremal curve x s s s s : , , is a curve in space-time. The parameter s, used for describing , has no physical meaning. The only thing that matters is how the position ( In order to find v(x), we impose the following condition: and require that the expression on the left-hand side has zero as a stationary value with respect to variations of v.
Equations (10) and (11) are Carathéodory's 'fundamental equations'. On applying the integrability conditions ∂ 2 S/∂x μ ∂x ν = ∂ 2 S/∂x ν ∂x μ to equation (11), we get Equation (12) leads to the equations of motion. To show this, we first derive equation (10) with respect to x μ , getting Because of equation (11), the above equation reduces to On using ∂ 2 S/∂x μ ∂x ν = ∂ 2 S/∂x ν ∂x μ first and then (11), we get On evaluating this last relation along a single extremal, dx μ /ds = v μ (x(s)), we readily see that the right-hand side is the derivative of ∂L(x, v)/∂v μ with respect to s, i.e. d(∂L/∂v μ )/ds. We thus obtain the Euler-Lagrange equations: One usually focuses on a single extremal curve, even though this curve must be embedded in a whole family of extremals. Right panel: the complete picture. C is just one member of a whole family of curves, whose collective behavior is ruled by field equations.
Let E μ (L) stand for the left-hand side of equation (17). It can be shown that the E μ are components of a covariant vector, called Euler vector. This vector satisfies the identity E x 0 º m m  [18]. Due to this identity, only three of the four Euler-Lagrange equations, E μ (L) = 0, are independent. One can then replace any of these four equations by another one, e.g. x x 1 1 2 ( ) = n n   , which 'fixes the parameter' s. Carathéodory's approach brings to the fore that any particular curve, which satisfies the Euler-Lagrange equations, is embedded in a whole field of extremal curves. The region under consideration is assumed to be simple-connected and without focal points, i.e. any point within the region belongs to a single extremal curve. Particle motion can occur along any of these extremal curves. By considering the whole Mayer field of extremals, we can study particle dynamics in terms of field dynamics. We do this next, for the particular case of a charged particle that is subjected to the action of an electromagnetic field.

Maxwell-like equations ruling the motion of charged particles
Let us consider a particle of charge e and mass m, which is subjected to the action of an electromagnetic field. The field is given by the four-potential A μ . The standard Lagrangian for this case reads is Minkowski's metric tensor. On substituting this L in equation (12), we get That is, w μ is a normalized velocity field: w μ w μ = 1. By defining the antisymmetric tensor This shows that M μν and F μν differ from one another only by a constant. Let us stress that equation (22) follows from the standard formulation of charged-particle motion, which is based on Hamilton's action principle. We have only carried this formulation to its logical end. Hence, the remarkable relationship between M μν and F μν , given by equation (22), is a logical consequence of Hamilton's action principle. The tensors M μν and F μν have two quite different physical contents. M μν refers to particle motion (see equation (21)) and F μν represents the (given) electromagnetic field that causes this motion. We usually assume that F μν is a physical entity on its own, even though we can take notice of its existence only indirectly, by observing how the field acts on test particles. All the information we have access to about F μν is encoded in the motion of test particles, a motion that is ruled by the Lorentz equation. Now, F μν satisfies Maxwell's equation (1): ∂ μ F μ ν = (4π/c)j ν , where j ν is the field's source. Hence, equation (22) (1) and (2). While the pair (1), (2) rules the dynamics of the electromagnetic field, the pair (23), (24) refers to particle dynamics. Maxwell equations have been amply studied. The physical consequences of equations (23) and (24) are open to study. Let us stress that the latter do not refer to a stream of noninteracting charges or to a charged fluid, but to the Mayer field that is associated to a single particle. In old-fashioned quantum parlance, this Mayer field could be referred to as a 'pilot field'. The streamlines of this field are potential, i.e. virtual trajectories. Only one of these trajectories is actually realized, the one that goes through the point where the particle is located at some 'initial' time. All features of an actual, single trajectory are inextricably tied to the ones that characterize the field. However, there may be collective features that cannot be exhibited by a single trajectory, e.g. those which derive from possible interferences between different subsets of the whole field. Interference effects can be induced by imposing appropriate boundary conditions, for instance by means of a two-slit configuration. What happens to F μν in this case, must also happen to M μν .
When addressing a single extremal curve of the Mayer field, our results reduce to the standard ones. To show this, we multiply (19) with w μ and sum over μ. We can then use that w μ ∂ ν w μ = 0, which follows from w μ w μ = 1, thereby obtaining On evaluating equation (25) along a single extremal, whose differential equation is dx μ /ds = w μ (x(s)), we get the Lorentz equation (see equation (6) Let us stress that w μ is a normalized velocity field (w μ w μ = 1) by definition: Note that we seek for the integral curves of w μ , those satisfying dx μ /ds = w μ (x(s)). The integral curves of w μ are thus given by functions x μ (s), with s corresponding to the arc-length: . This is not the case for the integral curves of, say, v μ , which satisfy In summary, the Lorentz equation (26) for a single extremal follows from the more general equation (25), which holds for a field of extremals. It is worth mentioning that back in 1952, Dirac proposed equation (25) as a possible, heuristic generalization of (26) [20]. However, Dirac addressed a 'stream of electrons', the density of which was assumed to be so small, that repulsive forces could be neglected. Dirac's goal was to develop 'a new classical theory of electrons' [20]. We stress that the Mayer field w μ we have addressed should not be confused with a field describing a fluid of weakly interacting charges. The Mayer field w μ belongs to a single charge.
It is also worth mentioning that on writing equation (23), we eliminated the electromagnetic field. This is an old goal that arose from viewing fields as auxiliary quantities, deprived from physical reality. As already mentioned, fields can indeed be observed only indirectly, through their action on test charges. A theoretical formulation of physical phenomena, whose goal is to contain only observable quantities, has no place for fields. Schwarzschild, Tetrode, Fokker, Feynman and Wheeler were among those who proposed new classical electrodynamic theories without fields [9]. These theories were based on the concept of action at a distance. Only at this price fields could be eliminated from the description. In the present formulation, the field was eliminated without invoking action at a distance. However, Maxwell equations were invoked to arrive at equation (23). There is an alternative approach, which does not require the electromagnetic field. We present this approach next.

Field-free formulation of classical electrodynamics
In order to motivate a causal, field-free formulation of CED, let us see first how Maxwell equations can be derived from two basic principles.

Standard formulation: Maxwell equations derived from two principles
Maxwell equations can be derived from two basic principles: charge conservation and the causal connection between a source-current and the electromagnetic field that this source generates. The first principle, charge conservation, leads to the continuity equation, which written in covariant form reads The second principle establishes the causal connection between the current j x ( ) ¢ m at space-time point x¢ and the field A μ (x) it generates at any other space-time point x. This field is given by where G x x ( ) -¢ is a Green function or propagator that is assumed to satisfy As a consequence of the above two principles, it follows, first, that , because j μ vanishes at infinity. Hence, equation (27) implies that the field A μ defined by equation (28) is in the Lorentz gauge; see equation (30).
We see that, on account of its definition, A μ satisfies the equations We can next define the (gauge invariant) antisymmetric tensor F μ ν = ∂ μ A ν − ∂ ν A μ . On taking the derivative ∂ μ F μ ν , it follows at once, from equations (31) and (32), that Moreover, from the definition of F μ ν , it follows that In this way, one describes the interaction between source and probe, an interaction that is mediated by the field. It is however possible to dispose of the field, so that one makes reference to the two interacting currents only, as we show next.

Field-free formulation of CED
From the two principles we have assumed, we can obviously keep the first one, charge conservation, when dealing with both source and probe currents. The second principle causally connects the source-current with the field A μ it generates. This principle essentially stipulates how the presence of the current affects its surroundings and how this effect propagates to different places. Mathematically, this information is encoded in the Green function or propagator. This function is implicitly given by the equation it satisfies, or explicitly in some specific cases. Anyhow, whenever the propagator depends on c , the electromagnetic vacuum is playing a role. This holds true, irrespective of the system of units being used. In some systems, μ 0 is an assigned quantity, not a measured quantity. What matters is that, assigned or measured, μ 0 represents a physical, quantifiable property that we ascribe to vacuum. We can thus assume that the presence of a source-current affects vacuum in a way that is essentially described by the propagator. Instead of connecting the source-current with an intermediary field A μ , we can connect source and probe via the vacuum. That is, the causal, linear connection that was assumed in equation (28) can also be assumed for the two involved currents, the source and the probe. Following this line of thinking, we take now as a second principle that source current j s m and probe current j p m are causally related by the propagator G x x ( ) -¢ , which we correspondingly assume to be a retarded Green function that satisfies equation (29). Hence, where κ is a constant that makes equation (35) dimensionally correct. While equations (28) and (35) are mathematically the same, they have two different physical contents. Equation (28) is interpreted as giving the field A μ (x) at all space-time points x, a field that is produced by a well-localized source-current j s m . In the standard formulation of CED, the current j s m is assumed to produce a field A μ (x) at all space-time points. On the other hand, equation (35) connects two well-localized currents with one another. We may interpret the probe current j p m entering equation (35) as we interpreted the Mayer field w μ in the standard formulation of section 4.
That is, if a probe charge is at x, then its associated current is given by j x p ( ) m of equation ( couples to the probe charge, whose motion is then ruled by equation (46). In the field-free formulation (see section 5.2), the action of the source current on the probe current is given by equation (35). It is assumed that the connection between j s m and j p m follows from the electromagnetic properties of vacuum, characterized by ò 0 and μ 0 . The electromagnetic properties of vacuum are effectively encoded in a propagator, or Green function . As we said before, A μ and j p m are mathematically the same, but not so physically. Equations (28) and (35) imply that j c A 4 p ( ) k p = m m , a relationship that parallels equation (22).
Moreover, the relationship between equations (44) and (46) has a counterpart in the relationship between equation (42) and the Hamilton-Jacobi equation that involves A μ , as we show next. Equation (11), when applied to the Lagrangian (18), leads to w μ = (1/mc)(∂ μ S − (e/c)A μ ). From η μ ν w μ w ν = 1, we get the Hamilton-Jacobi equation example, if F μν is predicted to have wave behavior, so will also M μν . In such a case, wave-particle duality acquires a concrete meaning, with wave features referring to w μ (x), and particle features referring to x μ (s). Generally, optical phenomena-such as optical cloaks and optical beams carrying orbital angular momentum-should have a counterpart in the case of M μν . One must, however, keep in mind that M μν is related to a set of virtual trajectories, which are the integral curves of w μ (x). Only one of these trajectories is actually realized by a single charge.
As for the field-free approach to CED that we discussed in this work, it also led us to conclude that the motion of charged particles fulfil both Maxwell-like equations and a Lorentz-like equation. In this approach, the causal connection between a source current j s m and a probe current j p m was not mediated by an electromagnetic field A μ . The standard relation between j s m and A μ was replaced by a similar relation between j s m and j p m . In both cases, there is a response at x to the presence of a current j x s ( ) ¢ m at x¢. This response is taken to be A μ (x) in standard CED and j x p ( ) m in the field-free formulation. In the two cases, the mediator is what we call 'vacuum', a substrate that is endowed with physical reality whenever we ascribe to it electromagnetic properties, which are quantified by ò 0 and μ 0 . Experimentally, these quantities can be accurately measured using basic tools such as capacitors and current-carrying wires, respectively. Theoretically, vacuum is addressed using quantum tools. The quantum vacuum is supposed to be the result of virtual processes that involve all possible sorts of particles and interactions. It should be worthwhile to explore how far we can go with a theoretical description of vacuum that uses only classical tools, such as those employed in linear response theory [24][25][26][27][28][29]. Describing vacuum as a physical object would give 'self-action' a consistent physical meaning. Indeed, self-action could be understood to stand for the back-action of vacuum on any charge. Charges would then be considered and treated as open systems, i.e. systems that interact with their surrounding, in this case with vacuum.
In summary, it seems to be firmly established that the motion of a charged particle obeys not only the Lorentz equation, but also Maxwell-like equations. The physical consequences that derive from this result are presently unknown and pose a challenge for future research.

Data availability statement
No new data were created or analysed in this study.