Maxwell–Lorentz Electrodynamics Revisited via the Lagrangian Formalism and Feynman Proper Time Paradigm

Classical electrodynamics is nowadays considered [1–4] as the most fundamental physical theory, largely owing to the depth of its theoretical foundations and wealth of experimental verifications. In this work we describe a new approach to the classical Maxwell theory, based on a vacuum field medium model, and reanalyze some of the modern classical electrodynamics problems related with the description of a charged point particle dynamics under an external electromagnetic field. We remark here that by “a charged point particle” we as usual understand an elementary material charged particle whose internal spatial structure is assumed to be unimportant and is not taken into account, if the contrary is not specified.

We shall use the least action principle to discuss, for various charged point particle dynamics the important physical principles characterizing the related electrodynamical vacuum field structure. In particular, the main classical relativistic relationships characterizing the charge point particle dynamics are obtained using the least action principle within Feynman’s approach to the Maxwell electromagnetic equations and the Lorentz type force derivation. Moreover, for each least action principle constructed, we describe the corresponding Hamiltonian pictures and present the related energy conservation laws. Making use of this modified least action approach, a classical hadronic string model is analyzed in detail.

The classical Lorentz force expression with respect to an arbitrary inertial reference frame has engendered many theoretical and experimental controversies, such as the relativistic potential energy impact on the charged point particle mass, the Aharonov–Bohm effect [5–7] and the Abraham–Lorentz–Dirac radiation force [1,2,8] expression. In an effort to explain this, R. Feynman [9] in his “Lectures on Physics” wrote:

“Now we would like to state the law that for quantum mechanics replaces the law F = qv × B. It will be the law that determines the behavior of quantum mechanical particles in an electromagnetic field. Since what happens is determined by amplitudes, the law must tell us how the magnetic influences affect the amplitudes; we are no longer dealing with the acceleration of the particle. The law is the following: the phase of the amplitude to arrive via any trajectory is changed by the presence of a magnetic field by an amount equal to the integral of the vector potential along the whole trajectory times the charge of the particle over Planck’s constant. That is,

These two expressions are correct not only for static fields, but together give the correct result for any electromagnetic field, static or dynamic. This is the law that replaces F = q(E + v × B).”

Consequently, the analysis of the Lorentz force subject to the assumed vacuum field medium is a very interesting and important problem, which was discussed by E. Fermi, G. Schott, R. Feynman, F. Dyson [9–14] and many other physicists. To describe the essence of the electrodynamic problems related to the description of charged point particle dynamics under external electromagnetic field, let us begin with the classical Lorentz force expression

Here ξ ∈ ℝ is a particle electric charge, u ∈ T (ℝ³) is its velocity [15,16] vector, expressed here in the light speed c units, is the associated external electric field and is the corresponding external magnetic field, acting on the charged particle, expressed in terms of suitable vector $A:M^4\to {\mathbb{E}}^3$ and scalar φ : M⁴ → ℝ potentials. Moreover, “∇” is the standard gradient operator with respect to the spatial variable $r\in {\mathbb{E}}^3$ and × is the usual vector product in three-dimensional Euclidean space ${\mathbb{E}}^3:=({ℝ}^3,<\cdot ,\cdot >)$, which is naturally endowed with the classical scalar product < ·, · >. These potentials are defined on the Minkowski space $M^4\simeq ℝ\times {\mathbb{E}}^3$, which models a chosen laboratory reference frame ${\mathcal{K}}_t$. Now, it is a well-known fact [2,3,9,17] that the force expression (1.1) does not take into account the dual influence of the charged particle on the electromagnetic field and should be considered valid only if the particle charge ξ → 0. This also means that expression (1.1) cannot be used for studying the interaction between two different moving charged point particles, as was pedagogically demonstrated in the classical manuals [2,9]. The classical Lorentz force expression (1.1) is a natural consequence of the interaction of a charged point particle with an ambient electromagnetic field, and its derivation based on the general principles of dynamics was analyzed in detail by R. Feynman and F. Dyson [9–11].

Taking this into account, it is natural to reanalyze this problem from the classical perspective, using only the Maxwell-Faraday wave theory aspect to specifying the corresponding vacuum field medium. Other questionable inferences from the classical electrodynamics theory, which strongly motivated the analysis in this work, are related both to an alternative interpretation of the well-known Lorenz condition, imposed on the four-vector of electromagnetic observable potentials (φ, A) : M⁴ → T^∗(M⁴) and the classical Lagrangian formulation [2] of charged particle dynamics under external electromagnetic field. The Lagrangian is strongly dependent on an important Einsteinian notion of the rest reference frame ${\mathcal{K}}_{\tau }$ and the related least action principle, so before explaining it in more detail, we first analyze the classical Maxwell electromagnetic theory from a strictly dynamical point of view.

Let us consider with respect to a laboratory reference frame ${\mathcal{K}}_t$ the additional Lorenz conditiona priori assuming the Lorentz invariant wave scalar field equation and the charge continuity equation where ρ : M⁴ → ℝ and $J:M^4\to {\mathbb{E}}^3$ are, respectively, the charge and current densities of the ambient matter. Then one can show [18,19] that the Lorentz invariant wave equation and the classical electromagnetic Maxwell field equations [1–3,9,17] hold for all (t, r) ∈ M⁴ with respect to the chosen laboratory reference frame ${\mathcal{K}}_t$.

Notice here that, inversely, Maxwell’s equations (1.8) do not directly reduce, via definitions (1.2) and (1.3), to the wave field Equations (1.5) and (1.7) without the Lorenz condition (1.4). This fact is very important and suggests that when it comes to a choice of governing equations, it may be reasonable to replace Maxwell’s equation (1.8) with the Lorenz condition (1.4) and the charge continuity Equation (1.6). To make the equivalence statement above more transparent, we formulate it as the following proposition.

Proposition 1.1. The Lorentz invariant wave equation (1.5) together with the Lorenz condition (1.4) for the observable potentials (φ, A) : M⁴ → T^∗(M⁴) and the charge continuity relationship (1.6) are completely equivalent to the Maxwell field equation (1.8).

Proof. Substituting (1.4), into (1.5), one easily obtains which implies the gradient expression

Taking into account the electric field definition (1.2), expression (1.10) reduces to which is the second of the first pair of Maxwell’s equations (1.8).

Now upon applying ∇× to definition (1.2), we find using the definition (1.3), that which is the first pair of the Maxwell equations (1.8). Upon differentiating Equation (1.5) with respect to the time t ∈ ℝ and taking into account the charge continuity Equation (1.6), one finds that

This is equivalent to the wave Equation (1.7) if one observes that the current vector $J:M^4\to {\mathbb{E}}^3$ is defined by means of the charge continuity equation (1.6) up to a vector function $\nabla \times S:M^4\to {\mathbb{E}}^3$. Now applying operation ∇× to the definition (1.3), it follows from the wave Equation (1.7) that which leads directly to which is the first of the second pair of the Maxwell equations (1.8). The final “no magnetic charge”equation in (1.8) follows directly from the elementary identity < ∇, ∇× >= 0, thereby completing the proof. □

This proposition allows to consider the observable potential functions (φ, A) : M⁴ → T^∗(M⁴) as fundamental ingredients of the ambient vacuum field medium, by means of which we can try to describe the related physical behavior of charged point particles imbedded in space-time M⁴. The following observation provides strong support for this approach:

Observation. The Lorenz condition (1.4) actually means the scalar potential field φ : M⁴ → ℝ continuity relationship, whose origin lies in some new field conservation law, characterizing the deep intrinsic structure of the vacuum field medium.

To make this observation more transparent and precise, let us recall the definition [2,3,9,17] of the electric current $J:M^4\to {\mathbb{E}}^3$ in the dynamical form where the vector u ∈ T (ℝ³) is the corresponding charge velocity. Thus, the following continuity relationship holds, which can easily be rewritten [20] as the integral conservation law for a charge inside any bounded domain ${\mathrm{\Omega }}_t\subset {\mathbb{E}}^3$, moving in the space-time M⁴ with respect to the natural evolution equation

The above reasoning leads to the following result.

Proposition 1.2. The Lorenz condition (1.4) is equivalent to the integral conservation lawwhere${\mathrm{\Omega }}_t\subset {\mathbb{E}}^3$ is any bounded domain, moving with respect to the charged point particle ξ evolution equationwhich represents the velocity vector of related local potential field changes propagating in the Minkowski space-time M⁴. Moreover, for a particle with the distributed charge density ρ : M⁴ → ℝ, the following Umov type local energy conservation relationshipholds for any t ∈ ℝ.

Proof. Consider first the corresponding solutions to potential field Equation (1.5), taking into account condition (1.15). Owing to the standard results from [2,9], one finds that which gives rise to the following form of the Lorenz condition (1.4):

This obviously can be rewritten [20] as the integral conservation law (1.19), so the expression (1.19) is verified.

To prove the local energy conservation relationship (1.21) it is necessary to combine the conditions (1.16), (1.23) and find that

Recall now that the infinitesimal volume transformation d³r = χ(t, r)d³r₀, where the Jacobian χ(t, r) := |∂r(t; r₀)/∂r₀| of the corresponding transformation $r:{\mathrm{\Omega }}_t{}_{{}_0}\to {\mathrm{\Omega }}_t$$,$ induced by the Cauchy problem for the differential relationship (1.20) for any t ∈ ℝ, satisfies the evolution equation easily following from (1.20). Then applying the operator ${\displaystyle {\int }_{{\mathrm{\Omega }}_{t_0}}(\dots ){\chi }^2{d}^3{r}_0}$ to the equality (1.24), one obtains

Here we denoted the conserved charge $\xi :={\displaystyle {\int }_{{\mathrm{\Omega }}_t}\rho (t,r)d^{3}r}$ and the local energy conservation quantity $ℰ(\xi ;{\mathrm{\Omega }}_t):={\displaystyle {\int }_{{\mathrm{\Omega }}_t}(\rho \phi \chi )d^{3}r}$. The latter quantity can be simplified, owing to the infinitesimal Lorentz invariance four-volume measure relationship d³r(t, r ) ∧ dt = d³r₀ ∧ dt₀. Here the variables (t, r) ∈ ℝ_t × Ω_t ⊂ M⁴ are, in the present context, taken with respect to the moving reference frame ${\mathcal{K}}_t$ related to the infinitesimal charge quantity dξ(t, r) := ρ(t, r)d³r. The variables $(t_0,r_0)\in {ℝ}_t{}_{{}_0}\times {\mathrm{\Omega }}_t{}_{{}_0}\subset M^4$ are taken with respect to the laboratory reference frame ${\mathcal{K}}_t{}_{{}_0}$, related to the infinitesimal charge quantity dξ(t₀, r₀) = ρ(t₀, r₀)d³r₀, satisfying the charge conservation invariance dξ(t, r) = dξ(t₀, r₀). The above mentioned infinitesimal Lorentz invariance relationships make it possible to calculate the local energy conservation quantity $ℰ$(ξ; Ω₀) as where we took into account that $dt=d{t}_0{\left(1-{\left|u\right|}^2\right)}^{1/2}$. Thus, owing to (1.26) and (1.27), the local energy conservation relationship (1.21) is satisfied, proving the proposition. □

The local energy conservation quantity (1.27) can be rewritten as where $dℰ(t,r)=d\xi (t,r)\phi (t,r){\left(1-{\left|u\right|}^2\right)}^{-}{}^{1/2}$ is the distributed in vacuum electromagnetic field energy density, related to the electric charge dξ(t, r) located at the point (t, r) ∈ M⁴.

The above proposition suggests a physically motivated interpretation of electrodynamic phenomena in terms of what should naturally be called the vacuum potential field, which determines the observable interactions between charged point particles. More precisely, we can a priori endow the ambient vacuum medium with a scalar potential energy field density function W := ξφ : M⁴ → ℝ, where ξ ∈ ℝ₊ is the value of an elementary charge quantity, satisfying the governing vacuum field equationstaking into account the external charged sources, which possess a virtual capability for disturbing the vacuum field medium. Moreover, this vacuum potential field function W : M⁴ → ℝ allows the natural potential energy interpretation, whose origin should be assigned not only to the charged interacting medium, but also to any other medium possessing interaction capabilities, including for instance, material particles, interacting due to gravity.

This leads naturally to the next important step, consisting in deriving the equation governing the corresponding potential field $\overline{W}:M^4\to ℝ$, assigned to a charged point particle moving in the vacuum field medium with velocity u ∈ T(ℝ³) and located at point $r(t)=R(t)\in {\mathbb{E}}^3$ at time t ∈ ℝ. As can be readily shown [18,19,21], the corresponding evolution equation governing the related potential field function $\overline{W}:M^4\to ℝ$, assigned to a moving in the space ${\mathbb{E}}^3$ charged particle ξ under the stationary distributed field sources, has the form

Where $\overline{W}:=W(t,r){|}_{r\to R}{}_{(t)}$, u(t) := dR(t)/dt at point particle location (t, R(t)) ∈ M⁴.

Similarly, if there are two interacting charged point particles, located at points r(t) = R(t) and $r_f(t)=R_f(t)\in {\mathbb{E}}^3$ at time t ∈ ℝ and moving, respectively, with velocities u := dR(t)/dt and u_f := dR_f(t)/dt, the corresponding potential field function ${\overline{W}}^{\prime }:M^4\to ℝ$, considered with respect to the reference frame ${\mathcal{K}}^{\prime }$ specified by Euclidean coordinates $(t\prime ,r-r_f)\in {\mathbb{E}}^4$ and moving with the velocity u_f ∈ T(ℝ³) subject to the laboratory reference frame ${\mathcal{K}}_t$, should satisfy [19,22] with respect to the reference frame ${\mathcal{K}}^{\prime }$ the dynamical equality where we have denoted the velocity vectors u′ := dr/dt′, u′_f := dr_f/dt′ ∈ T (ℝ³). The latter comes with respect to the laboratory reference frame ${\mathcal{K}}_t$ about the dynamical equality

The dynamical potential field Equations (1.30) and (1.31) appear to have important properties and can be used as means for representing classical electrodynamic phenomena. Consequently, we shall proceed to investigate their physical properties in more detail and compare them with classical results for Lorentz type forces arising in the electrodynamics of a moving charged point particles in an external electromagnetic field.

In this investigation, we were in part inspired by works [23–27] and studies [28,29] devoted to solving the classical problem of reconciling gravitational and electrodynamic charges in the Mach–Einstein ether paradigm. First, we shall revisit the classical Mach–Einstein relativistic electrodynamics of a moving charged point particle, and second, we study the resulting electrodynamic theories associated with our vacuum potential field dynamical Equations (1.30) and (1.31), making use of the fundamental Lagrangian and Hamiltonian formalisms which were devised in [18,30].

The classical relativistic electrodynamics of a freely moving charged point particle in the Minkowski space-time $M^4:=ℝ\times {\mathbb{E}}^3$ is based on the Lagrangian approach [2,3,9,17] with Lagrangian function where m₀ ∈ ℝ₊ is the so-called particle rest mass and u ∈ T (ℝ³) is its spatial velocity in the Euclidean space ${\mathbb{E}}^3$, expressed here and in the sequel in light speed units (with light speed c). The least action principle in the form for any fixed temporal interval [t₁, t₂] ⊂ ℝ gives rise to the well-known relativistic relationships for the mass of the particle the momentum of the particle and the energy of the particle

It follows from [2,3], that the origin of the Lagrangian (1.33) can be extracted from the action on the suitable temporal interval [τ_1,τ₂] ⊂ ℝ, where, by definition, and τ ∈ ℝ is the so-called, proper temporal parameter assigned to a freely moving particle with respect to the rest reference frame ${\mathcal{K}}_r$. The action (1.38) is rather questionable from the dynamical point of view, since it is physically defined with respect to the rest reference frame ${\mathcal{K}}_{\tau }$, giving rise to the constant action S = −m₀(τ₂ − τ₁), as the limits of integrations τ₁ < τ₂ ∈ ℝ were taken to be fixed from the very beginning. Moreover, considering this particle to have charge ξ ∈ ℝ and be moving in the Minkowski space-time M⁴ under action of an electromagnetic field $\left(\phi ,A\right)\in ℝ\times {\mathbb{E}}^3$, the corresponding classical (relativistic) action functional is chosen (see [2,3,9,17,18,30]) as follows: with respect to the rest reference system, parameterized by the Euclidean space-time variables $\left(\tau ,r\right)\in {\mathbb{E}}^4$, where we have denoted $\dot{r}:=dr/d\tau$ in contrast to the definition u := dr/dt. The action (1.40) can be rewritten with respect to the laboratory reference frame ${\mathcal{K}}_t$ the moving with velocity vector $u\in {\mathbb{E}}^3$ as on the suitable temporal interval [t₁, t₂] ⊂ ℝ, which gives rise to the following [2,3,9,17] dynamical expressions for the particle momentum and for the charged particle ξ energy, where, by definition, $P\in {\mathbb{E}}^3$ is the common momentum of the particle and the ambient electromagnetic field at a space-time point (t, r) ∈ M⁴.

The expression (1.43) for the particle energy $ℰ$₀ also is open to question, since the potential energy ξφ, entering additively, has no affect on the particle mass m = m₀(1 − |u|²)^−1/2. This was noticed by L. Brillouin [31], who remarked that the fact that the potential energy has no affect on the particle mass tells us that “… any possibility of existence of a particle mass related with an external potential energy, is completely excluded.” Moreover, it is necessary to stress here that the least action principle (1.41), formulated with respect to the laboratory reference frame ${\mathcal{K}}_t$ time parameter t ∈ ℝ, appears logically inadequate, for there is a strong physical inconsistency with other time parameters of the Lorentz equivalent reference frames. This was first mentioned by R. Feynman in [32], in his efforts to rewrite the Lorentz force expression with respect to the rest reference frame ${\mathcal{K}}_{\tau }$. This and other special relativity theory and electrodynamics problems stimulated many prominent physicists of the past [3,31–34] and present [21,24,26,35–44] to try to develop alternative relativity theories based on completely different space-time and matter structure principles.

There also is another controversial inference from the action expression (1.41). As one can easily show [2,3,9,17], the corresponding dynamical equation for the Lorentz force is given as

We have defined here, as before, for the corresponding electric field and for the related magnetic field, acting on the charged point particle ξ. The expression (1.44) means, in particular, that the Lorentz force F depends linearly on the particle velocity vector u ∈ T (ℝ³), and so there is a strong dependence on the reference frame with respect to which the charged particle ξ moves. Attempts to reconcile this and some related controversies [21,31,32,45] forced Einstein to devise his special relativity theory and proceed further to creating his general relativity theory trying to explain the gravity by means of geometrization of space-time and matter in the Universe. Here, we must mention that the classical Lagrangian function $ℒ$ in (1.41) is written in terms of a combination of terms expressed by means of both the Euclidean rest reference frame variables $\left(\tau ,r\right)\in {\mathbb{E}}^4$ and arbitrarily chosen Minkowski reference frame variables (t, r) ∈ M⁴.

These problems were recently analyzed using a completely different “no-geometry” approach [19,21,22], where new dynamical equations were derived, which were free of the controversial elements mentioned above. Moreover, this approach avoided the introduction of the well known Lorentz transformations of the space-time reference frames with respect to which the action functional (1.41) is invariant. From this point of view, there are interesting for discussion conclusions in [37,46–48], in which some electrodynamic models, possessing intrinsic Galilean and Poincaré–Lorentz symmetries, are reanalyzed from diverse geometrical points of view. Subject to a possible geometric space-type structure and the related vacuum field background, exerting the decisive influence on the particle dynamics, we need to mention the recent works [49,50] and the closely related classical articles [51,52]. Next, we shall revisit the results obtained in [18,19] from the classical Lagrangian and Hamiltonian formalisms [30] in order to shed new light on the physical underpinnings of the vacuum field theory approach to the study of combined electromagnetic and gravitational effects.

Ampère’s ingenious classical analysis of magnetically interacting to each other two electric currents in thin conductors, as is well known, was based [2,3,9,17] on the following experimental fact: the force between two electric currents depends on the distance between conductors, their mutual spatial orientation and the quantitative values of currents. Under the assumption of the infinitesimal superposition principle, A.M. Ampère derived a general analytical expression for the force between two infinitesimal elements of currents: where vectors r, r′ ∈ E³ point at infinitesimal currents dr = sdl, dr′ = s′dl′ with normalized orientation vectors $s,s\prime \in {\mathbb{E}}^3$ of two closed conductors l and l′ carrying currents I ∈ ℝ and I′ ∈ ℝ, respectively, and the unit vector n := (r − r′)/|r − r′|, the spatial orientations of these infinitesimal elements are fixed, and the function $\alpha :{\left({\mathbb{S}}^2\right)}^2\times {\mathbb{S}}^2\to ℝ$ is a real-valued smooth mapping. Then taking into account the mutual symmetry between the infinitesimal elements of currents dl and dl′, belonging respectively to these two electric conductors, the infinitesimal force (1.47) was assumed by Ampère to locally satisfy Newton’s third law: with the mapping where < ·, · > is the natural scalar product in ${\mathbb{E}}^3$ and k₁, k₂ ∈ ℝ are some still undetermined real and dimensionless parameters. The assumption (1.48) is apparently very restrictive and can be considered as reasonable only subject to a stationary system of conductors under regard, when the mutual action at a distance principle [2,9] can be applied. As J.C. Maxwell [53] observed “…we may draw the conclusions, first, that action and reaction are not always equal and opposite, and second, that apparatus may be constructed to generate any amount of work from its own resources. For let two oppositely electrified bodies A and B travel along the line joining them with equal velocities in the direction AB, then if either the potential or the attraction of the bodies at a given time is that due to their position at some former time (as these authors suppose), B, the foremost body, will attract A forwards more than B attracts A backwards. Now let A and B be kept apart by a rigid rod. The combined system, if set in motion in the direction AB, will pull in that direction with a force which may either continually augment the velocity, or may be used as an inexhaustible source of energy.”

Based on the fact that there is no possibility to measure the force between two infinitesimal current elements, A.M. Ampère took into account (1.48), (1.49) and calculated the corresponding force exerted by the whole conductor l′ on an infinitesimal current element of other conductor under regard: which can be equivalently transformed as owing to the integral identity which can be easily checked by means of integration by parts. Introducing the vector potential generated by the conductor l′ at point $r\in {\mathbb{E}}^3$, belonging to the infinitesimal element dl of the conductor l, the resulting infinitesimal force (1.50) gives rise to the following expression: where we have taken into account the standard magnetic field definition and the corresponding current density relationship

There are clearly many possible choices for the dimensionless parameters k₁, k₂ ∈ ℝ. In his analysis, Ampère chose the case when k₁ = 1, k₂ = −2 and obtained the now well-known magnetic force expression which easily reduces to the classical Lorentz expressionfor a force exerted by an external magnetic field on a point particle moving with a constant velocity u ∈ T (ℝ³) with an electric charge ξ ∈ ℝ.

If one takes an alternative choice and sets k₁ = 1, k₂ = −1, the expression (1.54) yields a modified magnetic Lorentz type force, exerted by an external magnetic field generated by a moving charged particle with a velocity u′ ∈ T (ℝ³) on a point particle, endowed with the electric charge ξ ∈ R and moving with a velocity u ∈ T (ℝ³) : which was briefly discussed in [21,54,55] and recently obtained and analyzed in detail from the Lagrangian point of view in [18,19,22,56] in the following equivalent to (1.32) infinitesimal form: where δA(r) ∈ T ^∗(R³) denotes the magnetic potential generated by an external charged point particle moving with velocity u_f ∈ T (ℝ³) and exerting the magnetic force δf_L(r) on the charged particle located at point r ∈ ℝ³ and moving with velocity u ∈ T (ℝ³) with respect to a common reference system ${\mathcal{K}}_t$. We also need to mention here that the modified Lorentz force expression (1.59) does not take into account the resulting pure electric force as the conductors l and l′ are considered to be electrically neutral. Simultaneously, we see that the magnetic potential has a physical significance in its own right [8,21,54,56] and has meaning in a way that extends beyond the calculation of force fields.

To obtain the Lorentz force (1.59) exerted by the external magnetic field generated by the whole conductor l′ on an infinitesimal current element dl of the conductor l, it is necessary to integrate the expression (1.60) along this conductor loop l′ : that is the equality where, by the electric field $E(r):=-\nabla \overline{W}-\partial A(r)/\partial t$. Now one can easily derive from (1.62) the desired Lorentz force expression (6.24), if one takes into account that the whole electric field E(r) ≃ 0 owing to the neutrality of the conductors. Concerning the latter it is worth mentioning the following remark of D. Kastler [57]:

“It is true that Ampere’s formula is no more admissible today, because it is based on the Newtonian idea of instantaneous action at a distance and it leads notably to the strange consequence that two consecutive elements of the same current should repel each other. Ampere presumed to have demonstrated experimentally this repulsion force, but on this point he was wrong. The modern method, the more rational in order to establish the existence of electrodynamics forces and to determine their value consists in starting from the electrostatic interaction law of Coulomb between two charges (two electrons), whose one of them is at rest in the adopted frame of reference and studying how the interaction forces transform when one goes, thanks to the Lorentz-Einstein relations, to a system of coordinates in which both charges are in motion. One sees the appearance of additional forces proportional to e²/c², e being the electrostatic charge and c the light velocity, hence one sees that not only the spin but also the magnetic moment of the electron are of relativistic origin - as Dirac has shown - but that the whole of electromagnetic forces has such an origin.”

Thereby, the above analysis of Ampère’s derivation of the magnetic force expression (1.54), as well as its consequences (1.59) and (1.60) make it possible to suppose that the missed modified Lorentz type force expression (1.61) could also be embedded into the classical relativistic Lagrangian and related Hamiltonian formalisms, eventually giving rise to new aspects and interpretations of many “strange” experimental phenomena observed during the past few centuries.

In the vacuum field theory approach to combining electromagnetism and gravity, devised in [18,19], the main vacuum potential field function $\overline{W}:M^4\to ℝ$, related to a charged point particle ξ under the external stationary distributed field sources, satisfies the dynamical Equation (1.29), namely in the case when the external charged particles are at rest, where, as above, u := dr/dt is the particle velocity with respect to some reference system.

To analyze the dynamical Equation (2.1) from the Lagrangian point of view, we write the corresponding action functional as expressed with respect to the rest reference frame ${\mathcal{K}}_{\tau }$. Fixing the proper temporal parameters τ₁ < τ₂ ∈ ℝ, one finds from the least action principle (δS = 0) that where, owing to (2.2), the corresponding Lagrangian function is

Recalling now the definition of the particle mass and the relationships from (2.3) we easily obtain exactly the dynamical Equation (2.1). Moreover, one now readily find that the dynamical mass, defined by means of expression (2.5), is given as which coincides with the Equation (1.35) of the preceding section. Now one can formulate the following proposition using the above results

Proposition 2.1. The alternative freely moving point particle electrodynamic model (2.1) allows the least action formulation (2.2) with respect to the “rest” reference frame variables, where the Lagrangian function is given by expression (2.4). Its electrodynamics is completely equivalent to that of a classical relativistic freely moving point particle, described in Subsection 1.2.

We proceed now to the case when our charged point particle ξ moves in the space-time with velocity vector u ∈ T (ℝ³) and interacts with another external charged point particle ξ_f, moving with velocity vector u_f ∈ T (ℝ³) with respect to a common reference frame ${\mathcal{K}}_t$. As was shown in [18,19], the respectively modified dynamical equation for the vacuum potential field function ${\overline{W}}^{\prime }:M^4\to ℝ$ subject to the moving reference frame ${\mathcal{K}}^{\prime }$ is given by equality (1.31), or where, as before, the velocity vectors u′ := dr/dt′, ${u^{\prime }}_f:=d{r}_f/dt\prime \in T({ℝ}^3)$. Since the external charged particle ξ_f moves in the space-time M⁴, it generates the related magnetic field B := ∇ × A, whose magnetic vector potentials $A:M^4\to {\mathbb{E}}^3$ and $A^{\prime }:M^4\to {\mathbb{E}}^3$ are defined, owing to the results of [18,19,21], as

Whence, taking into account that the field potential and the particle momentum $p\prime =-{\overline{W}}^{\prime }u\prime =-\overline{W}u$, equality (2.7) becomes equivalent to if considered with respect to the moving reference frame ${\mathcal{K}}^{\prime }$, or to the Lorentz type force equality if considered with respect to the laboratory reference frame ${\mathcal{K}}_t$, owing to the classical Lorentz invariance relationship (2.9), as the corresponding magnetic vector potential, generated by the external charged point test particle ξ_f with respect to the reference frame ${\mathcal{K}}^{\prime }$, is identically equal to zero. To imbed the dynamical Equation (2.11) into the classical Lagrangian formalism, we start from the following action functional, which naturally generalizes the functional (2.2):

Here, as before, ${\overline{W}}^{\prime }$ is the respectively calculated vacuum field potential $\overline{W}$ subject to the moving reference frame ${\mathcal{K}}^{\prime }$, $\dot{r}=u\prime dt\prime /d\tau$, ${\dot{r}}_f={u^{\prime }}_{f}dt\prime /d\tau$, $d\tau =dt\prime {\left(1-{\left|u\prime -{u^{\prime }}_f\right|}^2\right)}^{1/2}$, which take into account the relative velocity of the charged point particle ξ subject to the reference frame ${\mathcal{K}}^{\prime }$, specified by the Euclidean coordinates (t′, r − r_f) ∈ ℝ⁴, and moving simultaneously with velocity vector u_f ∈ T (ℝ³) with respect to the laboratory reference frame ${\mathcal{K}}_t$, specified by the Minkowski coordinates (t, r) ∈ M⁴ and related to those of the reference frame ${\mathcal{K}}^{\prime }$ and ${\mathcal{K}}_{\tau }$ by means of the following infinitesimal relationships:

So, it is clear in this case that our charged point particle ξ moves with the velocity vector $u\prime -{u^{\prime }}_f\in T({ℝ}^3)$ with respect to the reference frame ${\mathcal{K}}^{\prime }$ in which the external charged particle ξ_f is at rest. Thereby, we have reduced the problem of deriving the charged point particle ξ dynamical equation solved in Subsection 2.1.

Now we can compute the least action variational condition δS = 0, taking into account that, owing to (2.12), the corresponding Lagrangian function with respect to the rest reference frame ${\mathcal{K}}_{\tau }$ is given as

As a result of simple calculations, the generalized momentum of the charged particle ξ equals where, owing to (2.9) the vectors $p\prime :=-{\overline{W}}^{\prime }u\prime =-\overline{W}u=p\in {\mathbb{E}}^3,A\prime ={\overline{W}}^{\prime }{u^{\prime }}_f=\overline{W}{u}_f=A\in {\mathbb{E}}^3$, and giving rise to the dynamical equality with respect to the rest reference frame ${\mathcal{K}}_{\tau }$. As $dt\prime =d\tau {\left(1+{\left|\dot{r}-{\dot{r}}_f\right|}^2\right)}^{1/2}$ and ${\left(1+{\left|\dot{r}-{\dot{r}}_f\right|}^2\right)}^{1/2}={\left(1-{\left|u^{\prime }-{u^{\prime }}_f\right|}^2\right)}^{1/2}$, we obtain from (2.16) the equality exactly coinciding with equality (2.10) subject to the moving reference frame ${\mathcal{K}}^{\prime }$. Now, making use of expressions (2.13) and (2.9), one can rewrite (2.17) as that with respect to the laboratory reference frame ${\mathcal{K}}_t$: exactly coinciding with (2.11):

Remark 2.2. The Equation (2.19) allows to infer the following important and physically reasonable phenomenon: if the test charged point particle velocity u_f ∈ T (ℝ³) tends to the light velocity c = 1, the corresponding acceleration force $F_{ac}:=-\nabla \overline{W}\left(1-{\left|u_f\right|}^2\right)$ is vanishing. Thereby, the electromagnetic fields, generated by such rapidly moving charged point particles, have no influence on the dynamics of charged objects if observed with respect to an arbitrarily chosen laboratory reference frame ${\mathcal{K}}_t$.

The latter Equation (2.19) can be rewritten as or, using the well-known [2] identity where $a,b\in {\mathbb{E}}^3$ are arbitrary vector functions, in the standard Lorentz type form

The result (2.22), being before found and written down with respect to the moving reference frame ${\mathcal{K}}^{\prime }$ in [18,19,21] and in [58] yet with some inconsistency, makes it possible to formulate the next important proposition.

Proposition 2.3. The alternative classical relativistic electrodynamic model (2.10) allows the least action formulation based on the action functional (2.12) with respect to the rest reference frame${\mathcal{K}}_{\tau }$, where the Lagrangian function is given by expression (2.14). The resulting Lorentz type force expression equals (2.22) modified by the additional force component F_c := −∇ < ξA, u − u_f >, important for explanation [5–7] of the well known Aharonov–Bohm effect.

It is easy to see that the action functional (2.12) is written utilizes the classical Galilean transformations. If we now consider the action functional (2.2) for a charged point particle moving with respect to a reference frame ${\mathcal{K}}_{\tau }$, and take into account its interaction with an external magnetic field generated by the vector potential $A:M^4\to {\mathbb{E}}^3$, it can be naturally generalized as where $d\tau =dt{\left(1-{\left|u\right|}^2\right)}^{1/2}$.

Thus, the corresponding common particle-field momentum takes the form and satisfies where is the corresponding Lagrangian function. Since $d\tau =dt{\left(1-{\left|u\right|}^2\right)}^{1/2}$, one easily finds from (2.25) that

Upon substituting (2.24) into (2.27) and making use of the identity (2.21), we obtain the classical expression for the Lorentz force F acting on the moving charged point particle ξ : where, is its associated electric field and is the corresponding magnetic field. This result can be summarized as follows:

Proposition 2.4. The classical relativistic Lorentz force (2.28) allows the least action formulation (2.23) with respect to the rest reference frame variables, where the Lagrangian function is given by formula (2.26). Yet its electrodynamics, described by the Lorentz force (2.28), is not equivalent to the classical relativistic moving point particle electrodynamics, described by means of the Lorentz force (1.44), as the inertial mass expression$m=-\overline{W}$ does not coincide with that of (1.35).

Expressions (2.28) and (2.22) are equal to the gradient term F_c := −ξ∇ < A, u − u_f >, which reconciles the Lorentz forces acting on a charged moving particle ξ with respect to different reference frames. This fact is important for our vacuum field theory approach since it uses no special geometry and makes it possible to analyze both electromagnetic and gravitational fields simultaneously by employing the new definition of the dynamical mass by means of expression (2.5).

Any Lagrangian theory has an equivalent canonical Hamiltonian representation via the classical Legendre transformation [15,17,59–61]. As we have already formulated our vacuum field theory of a moving charged particle ξ in Lagrangian form, we proceed now to its Hamiltonian analysis making use of the action functionals (2.2), (2.14) and (2.23).

Take, first, the Lagrangian function (2.4) and the momentum expression (2.3) for defining the corresponding Hamiltonian function with respect to the moving reference frame ${\mathcal{K}}_{\tau }$ :

It is easy to show [15–17,59,61] that the Hamiltonian function (2.31) is a conservation law of the dynamical field Equation (2.1), that is for all τ, t ∈ ℝ which naturally leads to an energy interpretation of H. Thus, we can represent the particle energy as

Accordingly the Hamiltonian equivalent to the vacuum field Equation (2.1) can be written as and we have the following result.

Proposition 2.5. The alternative freely moving point particle electrodynamic model (2.1) allows the canonical Hamiltonian formulation (2.34) with respect to the “rest” reference frame variables, where the Hamiltonian function is given by expression (2.31). Its electrodynamics is completely equivalent to the classical relativistic freely moving point particle electrodynamics described in Subsection 2.1.

Analogously, one can now use the Lagrangian (2.14) to construct the Hamiltonian function for the dynamical field Equation (4.14), describing the motion of charged particle ξ in an external electromagnetic field in the canonical Hamiltonian form: where with respect to the laboratory reference frame ${\mathcal{K}}_t$. Here we took into account that, owing to definitions (2.8), (2.9) and (4.17), and, in particular, where A : M⁴→ ℝ³ is the related magnetic vector potential generated by the moving external charged particle ξ_f. Equations (2.35) can be rewritten with respect to the laboratory reference frame ${\mathcal{K}}_t$ in the form which coincides with the result (2.22).