Depending on the experiment, the most emblematic subatomic particle, the electron, has been found to interact as a point-like corpuscle in scattering experiments [1], or to behave as an extensible wave [2]. Elaborating on Bohr's interpretation of Quantum Mechanics [3], Heisenberg concluded that particles could neither be represented nor even apprehended by the human mind, and that only their abstract mathematical description existed [4]. For de Broglie however, “abstract presentations have no physical reality. Only the movement of elements localized in space, in the course of time, has physical reality” [5]. Hence, modern physics has identified with unprecedented precision the interactions and their underlying principles, has successfully described its environment, spacetime, but still lacks a characterization of the nature of its “objects,” the particles themselves.

Consequently, several kinds of electron models have been proposed: extended models [6], point-like models, and mixed models in which a point-like corpuscle follows an extended trajectory [7]. Early attempts included the spherical models of Abraham [8] and Lorentz [9], which led to theories of electromagnetic mass [10–13]. Spherical models soon evolved into the so-called ring models of Parson [14], Webster [15], Allen [16], and Compton [17], constituted of rotating infinitesimal charges and verifying the properties of classical magnetic moment and Compton scattering. Essential constraints however, such as electron cohesion and stability, could not be satisfied: new putative forces, denoted Poincaré stresses [18], were suggested to maintain the cohesion of the negatively charged electron. The abstract descriptions of quantum mechanical theories [19, 20] then successfully accounted for the wave-like behavior of the electron and probabilistically predicted [21] the values of most observables by considering a point-like particle, yet failed at interpreting fundamental constants or explaining how a point-like corpuscle could have spin or a finite energy density. Paradoxically, quantum mechanics revived geometric models when Schrödinger noticed within the Dirac equation itself a rapid oscillatory trembling motion, which he called Zitterbewegung (zbw) [22], exhibiting microcurrents arising at light velocity c. Surprisingly, the electron seemed to follow a helical trajectory of radius ƛ_c, the reduced Compton wavelength, surrounding the average travel direction (Figure 1A). Several such zbw models, identifying spin with orbital angular momentum, were interpreted classically [27–29]. Subsequent electrodynamical or hydrodynamical models involved fluids with spin [30], current loops of a certain thickness [31], Dirac-like Equations [32, 33], moving charged membranes [34], plasmoid fibers [35], or toroidal geometry [35, 36]. Wondering whether zbw could be a real phenomenon, Hestenes emphasized the need to investigate the electron substructure, suggested zbw could originate in the electron self-interaction [37], and showed zbw was compatible with the ring models [38].

With the development of realist models of the electron emerged theories of electromagnetic mass. At first, the spherical models of Abraham [8], and Lorentz [9] seemed to fail to recover Einstein's relation E = mc² due to the appearance of a factor 4/3, but later proved to be compatible, once relativistic corrections were accounted for [12]. Stability of the sphere however still relied on Poincaré stresses or unknown surface tension [34], and electron mass could not be predicted from an objective criterion, but depended on the value taken by an arbitrary parameter, whose value is unconstrained, i.e., the radius of the sphere. Of note, the mass of subatomic particles is not predicted by quantum theories, and their values need to be inserted in calculations [12]. Most ring models [14–17] are prior to the discoveries of the spin, anomalous magnetic moment, and quantum mechanics. The ring model of Bergman and Wesley [31] exhibited cohesion and stability, but the expression for mass still involved an arbitrary parameter (i.e., width of current loop), and the substance constituting the electron remained indeterminate. More recently, Consa proposed a point-like electron following a toroidal trajectory [36], recovered mass independently of any arbitrary parameter, but did not specify how the trajectory developed nor demonstrated its stability. To our knowledge, the Virial theorem, which should be satisfied since the electron is a bound system, has not been considered in electron models. Potential energy is often equated to +mc², although cohesion potential energy should be negative for a bound system [as it is for the atom for instance [19]]. Kinetic energy is not usually accounted for, even though Lorentz [9], Hestenes [38] and others [e.g., [32]] noted the existence of a rotating motion and wondered whether kinetic energy did not contribute to rest mass. For Barut and Bracken, rest mass energy of the particle is the energy of the internal motion in the rest frame [29].

Hence, several issues remain to be addressed regarding the electron: for instance, which forces could cause the puzzling helical trajectory? What could be the nature of the substance constituting the electron? Could an electrodynamical description account for electron cohesion and stability? And could Lorentz' hypothesis advocating the electromagnetic origin of mass be simultaneously implemented from an objective criterion, instead of an arbitrary parameter?

In this study, we present a relativistic electrodynamical model of the electron at rest, in which charged subparticles follow definite trajectories. The model is based on two main hypotheses: (i) the existence of charged colorless subparticles called triolets, (ii) the assumption that triolets revolve at light velocity on coplanar circular orbits, constituting an envelope and nucleus, depending on their electromagnetic charges. As the electron is coherent, it is assumed that the model satisfies the Virial theorem. Constraints capturing the measured values of several observables (classical and anomalous magnetic moments, spin, Compton wavelength, kinetic energy) are formulated. Using Liénard-Wichert potentials, we then determine the specific kinds and numbers of triolets satisfying envelope and nucleus stability. Remarkably, we find that these kinds and numbers are precisely those that allow predicting electron mass and muon mass electromagnetically directly from the substructure, thus implementing Lorentz' hypothesis. Electron mass is effectively derived from an expression of substructure stability, which constitutes an objective criterion in our view. Our system also illustrates the possibility of constructing causal, local, objective, and realist models of particles beneath the Compton scale. Finally, we discuss novel perspectives suggested by the model, relative to the understanding of wave-corpuscle duality and to its relation to quantum theory.

In a previous study, we proposed that just six kinds of indestructible elementary subparticles denoted sparks, bearing electric charge ±e/6 and a specific strong interaction color charge, are necessary and sufficient to reconstruct all subatomic particles, so that sparks are conserved and reorganized across particle decays and annihilations [Avner, Boillot, Richard, submitted]. Since sparks are subject to both the strong and electromagnetic interactions, with the former dominating at short distances [20], groups of three sparks could presumably assemble beforehand to form composite colorless particles, thereafter called triolets, bearing charge +e/6, –e/6, +e/2, or –e/2 (Figure 1B). Henceforth, we shall suppose that the electron is exclusively composed of triolets, which travel at light velocity [7], exhibit some intrinsic angular momentum L_trlt, and being colorless, are submitted to electromagnetic and centrifugal forces only (hypothesis A).

Following de Broglie's proposition, we aim at constructing a plausible electrodynamical model of the electron at rest, in which positive and negative triolets form an electromagnetically bound system, exhibit the zbw microcurrents, and account for all experimentally measured observables. The electron is considered here as a particle of a certain extension, composed of revolving charged subparticles, the triolets, thereby exhibiting magnetic moment and intrinsic angular momentum (its spin) sensed by other particles. We know that the measured value of the electron magnetic moment is the sum of Bohr magneton μ_B = –eℏ/2m, predicted by both classical physics and quantum mechanics, where ℏ is reduced Planck constant, m the electron mass, and e the elementary charge, and an anomalous magnetic moment [39], which accounts for a small fraction a_anml ≃ 0.001159 of the previous and is only predicted by quantum electrodynamics [20]. Remarkably, the value of the classical magnetic moment of the electron can be derived by considering a charge (–e) revolving on a circular orbit of radius ƛ_c [19]. Hence, we reckoned the classical and anomalous magnetic moments could, respectively, be produced by two different components of the electron, namely a negatively charged envelope and a neutrally charged nucleus, also possibly responsible for the electron's wavelike and corpuscular behaviors, respectively. The peculiar helical trajectory of the electron predicted by zbw could then be naturally apprehended by considering that zbw describes the dynamics of envelope triolets, which are attracted and bound to the nucleus (Figure 1C). Electron spin could correspond to the sum of angular momenta of envelope triolets. Moreover, we shall regard electron mass as being a manifestation of the total electromagnetic cohesion energy E of the particle, as Lorentz hypothesized [9], through Einstein's formula m = E/c². The latter interpretation of the mass is naturally suggested by the observation that the muon possesses a mass ~206.77 times bigger than that of the electron, while its Compton wavelength is ~206.77 times smaller, as would be the case for a mass of electromagnetic origin, presenting a potential proportional to inverse distance.

The net electromagnetic forces acting on any particular envelope triolet should mostly depend on its surrounding triolets. The envelope could be organized into a complex structure, with triolets irregularly distributed along the orbits, or revolving at various radii, or experiencing fluctuations. To facilitate calculations however, we chose to make approximations and consider triolets at radial equilibrium rotating in the same direction on four coplanar circular orbits of different radii depending on their four different electromagnetic charges (hypothesis B, Figure 2). In our model, positive and negative nucleus triolets are intertwined to maintain their cohesion and could rotate along two close yet separate orbits due to the charged envelope. This could cause in turn a similar arrangement in the envelope, which would exhibit predominantly intertwined triolets, in spite of the excess of negative triolets. We are aware our model is only an approximation, even if we reckon that a collection of fluctuating ±e/6 and ±e/2 triolets traveling at light velocity could possibly converge toward such a configuration. Because of their stronger charges, ±e/2 triolets could be more tightly bound and form a condensed nucleus, while ±e/6 triolets would be bound more loosely and constitute the envelope.

In addition, as the electron is a bound system whose inner potentials allegedly depend on position coordinates only and not velocities (justification is given below), the Virial theorem should be verified [40]: for inverse square law electromagnetic interactions, one typically has E = U/2 and E = –T, where T is the total internal kinetic energy and U the internal potential energy. Therefore, T and U should, respectively, amount to +mc² and −2mc², resulting in total internal energy E = T + U = –mc² corresponding to electron mass, the minus sign being indicative of a bound system. Finally, we shall admit that, for the electron at rest, envelope triolets approximately follow a circular trajectory of radius ƛ_c = ℏ/mc, as suggested by the classical derivation of Bohr's magneton, and by zbw-like models. Interpretations of fundamental constants associated to the electron, such as reduced Planck constant ℏ and fine-structure constant α = e²/4πε₀ℏc (where ε₀ designates vacuum permittivity), should also emerge from the model.

Our system captures the measured values of charge, magnetic moments, spin and kinetic energy, and will be validated by showing that cohesion and stability can be satisfied, and potential energy (and thus electron mass) can be recovered. Let us here mathematically formulate the constraints: (i) a charge –e carried by N_env = N_env+ + N_env− triolets of charge ±e/n_env at the envelope; (ii) a classical magnetic moment μ_B generated by envelope triolets rotating at radii ρ_env+ = η_env+ƛ_c, ρ_env− = η_env−ƛ_c, and producing currents I_env+, I_env−; (iii) an anomalous magnetic moment a_anml·μ_B generated by N_nuc nucleus triolets (N_nuc+ = N_nuc−) of charge ±e/n_nuc rotating in the same direction as envelope triolets at radii ρ_nuc+ = η_nuc+ƛ_c, ρ_nuc− = η_nuc−ƛ_c, with momentum p_nuc+ ≃ p_nuc− = p_nuc and producing currents I_nuc+, I_nuc−; (iv) an internal kinetic energy T = ∑_ip_ic = +mc²; (v) a spin S_env = +ℏ/2 generated by envelope triolets of momentum p_env+, p_env−:

The fact that the muon has same spin as the electron, despite possessing a smaller Compton wavelength and the same number of triolets according to our chemical theory [Avner, Boillot, Richard, submitted], suggests that the angular momentum of envelope triolets could be a constant ρ_env+p_env+ ≃ ρ_env−p_env− ≡ L_trlt,env, yielding from (5):

L_trlt,env is possibly determined by the triangular substructure of envelope triolets made of three strongly interacting sparks, and could be at the basis of Planck's constant. Further constraints are also deduced (see Values of Observables) from Equations (1–5):

where b_env and b_nuc are dimensionless numbers. Assuming that η_env+ ≃ η_env− ≡ η_env ≃ 1, we deduce (in Values of Observables) from Equations (10–11) that the kinetic energies of the envelope and nucleus are approximately equal T_env+ ≃ T_nuc ≃ +mc²/2, leading to relation:

Furthermore, the electromagnetic force acting on a nucleus triolet due to the envelope charge and current and the electromagnetic force exerted on an envelope triolet due to the net nucleus magnetic moment were derived but found to be negligible when compared to intra-component interactions. This suggests that each component is only loosely bound to the other, almost constituting an independent system, and thus verifies the Virial theorem independently (see Values of Observables), yielding for potential energies U_env ≃ U_nuc ≃ –mc² and total energies E_env ≃ E_nuc ≃ –mc²/2. The system of Equations (9–11) further allows to determine η_nuc+ (Values of Observables) for each value of N_nuc+:

while η_nuc− is then given by Equation (9).

System cohesion and stability can be formulated by ensuring triolets are at radial equilibrium. As triolets are electrically charged and travel at light velocity, we use Liénard-Wichert potentials from relativistic electrodynamics [41] to express the radial components of electric field E_ij⊥ and magnetic field B_ij emitted by triolet T_j of charge q_j at retarded time t', radius ρ_j and retarded angle θ'_j, and sensed at distance R_ij—electromagnetic fields traveling at light velocity in vacuum—by triolet T_i arriving at the vertical (angle 0), radius ρ_i, at time t (Figure 3A). From known electrodynamical expressions [41] for these fields, using cylindrical unit vectors and coordinates, and Figure 3B, we deduce (Forces and Potentials):

where R_ij and γ_j are defined by:

Note that these fields depend on position coordinates only, not velocities, thereby justifying the use of the Virial theorem. We then derive expressions (Forces and Potentials) for the net radial Lorentz force F_ij⊥ due to triolet T_j exerted on triolet T_i belonging to the same component, and for the centrifugal force F_ctfg,i experienced by triolet T_i:

where b_i stands for b_env (respectively, b_nuc) when T_i belongs to the envelope (resp. the nucleus). In the electron at rest, assuming triolets remain at radial equilibrium, the net radial component of the Lorentz force exerted by other triolets should compensate the centrifugal force. Neglecting the small contribution of the envelope onto the nucleus and vice-versa, and expressing equilibrium for triolet T_i along the radial direction and rearranging (Triolets at Radial Equilibrium), we obtain for the envelope and nucleus:

where sgn(i·j) is the sign of the product of the charges of triolets T_i and T_j, and α is the fine-structure constant, which is found to be related to the ratio between the centrifugal force and the net radial electromagnetic force experienced by any single triolet inside the electron. We assume positive and negative triolets are intertwined and uniformly distributed along the orbits except—as negative triolets are more numerous at the envelope—consecutive negative envelope triolets, which presumably repel to produce stretches of alternatively charged triolets separated by empty space (Figure 2). Let d_env designate the distance (using the number of missing triolets as units) between the stretches. The expressions under the sums in Equations (20–21) can be calculated by first considering the non-retarded angular positions θ_j of triolets distributed along the circular orbit, then by determining the corresponding retarded angles θ'_j, as illustrated in Retarted Angles, using Newton's recursion method for instance onto transcendental equation:

and then deriving γ_j from Equation (17). Equations (20–21) will help us derive adequate values for N_env, N_nuc, n_env, n_nuc, b_env, b_nuc.

The potential energy due to interactions between the nucleus and envelope being negligible, the total potential energy of our system is approximately U_tot ≃ U_env + U_nuc, where U_env, and U_nuc are, respectively, the envelope and nucleus potential energies, which are evaluated in Potential Energy:

where H_ij = R_ij/ƛ_c. Assuming η_env+ ≃ η_env− ≃ 1, we demonstrate (Potential Energy) from b_env=2N_env (11) and Equation (20), which expresses the radial stability of every envelope triolet, that Equation (23) yields U_env≃ –mc². Likewise, assuming η_nuc+ ≃ η_nuc−, we demonstrate (Potential Energy) from b_nucη_nuc = 2N_nuc (12) and Equation (21), which expresses the radial stability of every nucleus triolet, that Equation (24) yields U_nuc ≃ –mc². Hence, we find: U_tot ≃ U_env + U_nuc ≃ −2mc², as expected from the Virial theorem, and recover electron mass. As substructure stability implies radial equilibrium for all envelope and nucleus triolets (20–21), it allows predicting electron mass. We find it remarkable that the same number of triolets allows to recover both substructure stability and electron mass.

The problem then reduces to determining triolet configurations, i.e., sets of values for {n_env, n_nuc, N_env+, N_env−, N_nuc, b_env, b_nuc, η_env+, η_env−, η_nuc+, η_nuc−, d_env}, that verify radial equilibrium for every triolet and correctly predict the total energy. We shall estimate the stability and total energy in three different models of the envelope successively, each lying at a different level of approximation. The three models are: the one-orbit model, where all envelope triolets rotate on the same orbit η_env+≃η_env−≡η_env; the two-orbits model, where positively-charged envelope triolets revolve on orbit of radius η_env+ and negative triolets at radius η_env−; the n-orbits model where every envelope triolet i rotates on a circular orbit of specific but fixed radius η_i.

We shall first estimate the number of triolets N_env present in the envelope by considering the one-orbit model. Assuming η_env+ ≃ η_env− and η_nuc+ ≃ η_nuc−, we have R_ij ≃ 2ρ_icosγ_j (Figure 3C) both at the envelope and nucleus, and Equations (20–21) can be approximated to:

Recalling that b_env is related to N_env via b_env = 2N_env (11), and setting values for input parameters {n_env, d_env}, the iteration over N_env values in Equation (25) enabled us to determine values for b_env and N_env approximately verifying Equations (25) and (11) simultaneously. Due to the asymmetry in the arrangement of envelope triolets, we found these Equations were satisfied for different values of N_env depending on the triolet T_i under consideration. In the case n_env = 6, d_env = 0 for instance, we found positive triolets approximatively satisfied these conditions for N_env ≃ 108, while negative triolets did so for N_env ≃ 144, thus justifying the necessity of considering two distinct orbits in the envelope. Although these figures should be regarded as merely indicative, cases d_env = 1 and d_env = 2 also pointed at average value N_env = 126, corresponding to N_env+ = 60 and N_env−= 66, and we shall be considering only this case in the remainder of our analysis. For the nucleus, in the absence of a constraint like Equation (11), values for b_nuc and η_nuc satisfying Equations (26) and (12) simultaneously were determined for every iterated value of N_nuc. However, when accounting for the correction due to envelope current (first two terms, Triolets at Radial Equilibrium):

the best estimate appeared to be N_nuc = 18 (Table 1). Note that input values other than n_env = 6, n_nuc = 2 did not yield any possible solutions.

Now, considering n_env = 6, n_nuc = 2, d_env = 2, η_env+ ≃ η_env− ≃ 1 (one-orbit model), and putting in the value obtained above for N_env, we evaluated potential energies U_env, U_nuc using Equations (13, 23–24) and found U_env = −0.997·mc² (Table 2), U_nuc ≃ −1.000·mc² (Table 1). The total potential energy therefore amounts to U_tot ≃ −1.997·mc², close to our expected result. Hence, recalling that kinetic energies satisfy T_env ≃ T_nuc ≃ +mc²/2, then T_tot ≃ T_env + T_nuc ≃ +mc², E_tot ≃ T_tot+ U_tot ≃ –mc², and the mass of the electron is deduced directly from our model substructure. Likewise, since the muon is seen as an excited state of the electron [6] according to our chemical theory [Avner, Boillot, Richard, submitted], presumably displaying a similar arrangement of triolets albeit on a smaller scale, muon mass can also be successfully calculated by replacing m by muon mass m_μ in expressions (23–24), or equivalently ƛ_c by the reduced muonic Compton Wavelength ƛ_muon.

We next evaluated the cohesion and stability of individual triolets. For the symmetric nucleus, we computed the right-hand side G_nuc of Equation (21) for every triolet; for n_nuc = 2, N_nuc = 18 for instance, accounting for the correction due to the envelope current, we obtained G_nuc(η_nuc+) ≃ 136.83, G_nuc(η_nuc−) ≃ 137.24 (Table 1). For the asymmetric envelope, which can be divided into six identical stretches of 21 triolets in the case N_env = 126, we computed the right-hand sides G_env of Equation (20) for every triolet belonging to the first stretch and compared the results with the left-hand side 1/α ≃ 137.036, which they should yield if triolets were truly at radial equilibrium. For the one-orbit model, setting n_env = 6, values of G_env disagreed with the expected value for all values of d_env (the case d_env = 2 is given in Table 3). Clearly, in these conditions at least, the centrifugal and net electromagnetic forces fail to compensate and to ensure radial equilibrium, one dominating over the other, and triolets would be moving radially as well as azimuthally. Therefore, we considered the two-orbits model with η_env+≃ 0.977, η_env− ≃ 1.023, for which we obtained an acceptable energy value (Table 2). Once again, we found that radial equilibrium was not verified for many envelope triolets, especially for consecutive negative triolets or those adjacent to them (Table 3). Hence, we decided to complicate our model again and considered envelope triolets orbiting at various but fixed radii ρ_i (n-orbits model) instead of the probably too general ρ_env+ and ρ_env−. We heuristically determined fixed radii exhibiting reasonable stability for all envelope triolets, then used an optimization algorithm, described in Optimization Algorithm, to make every triolet tend toward radial equilibrium, minimizing criterion K, the average absolute deviation from 1/α per triolet:

which effectively constitutes a measure of global stability of envelope triolets. Our algorithm converged toward a solution yielding acceptable energy and global stability (Table 2). The stability values G_env(η_i) of individual envelope triolets belonging to the first stretch in the n-orbits model are shown in Table 3: most values appeared to be close to 1/α. We found that our optimization algorithm nicely converged toward stable solutions. However, the latter were highly dependent on initial conditions, and a thorough optimization study is needed to ensure local minima are avoided.

In this study, we presented a relativistic electrodynamical model of the electron based on natural interpretations of its associated observables. Our electron model is composed of triolets that revolve along coplanar circular orbits constituting an envelope and nucleus, which could be responsible for its wavelike and corpuscular behaviors, respectively. These two components would thus constitute a natural solution to wave-corpuscle duality. Capturing the values of charge, spin, magnetic moments, Compton wavelength and kinetic energy, we created a triolet-based configuration that verified cohesion and stability without invoking Poincaré stresses, and predicted electron and muon mass, defined as electromagnetic cohesion energy, directly from substructure stability. Importantly, our model accounts for kinetic energy and presents a negative cohesion potential energy, in agreement with the Virial theorem. In our model, the numbers of triolets in the envelope and nucleus are the adjusting parameters, and the same numbers are found to account both for substructure stability and electron mass. Notably, electron mass can be derived directly from an expression of substructure stability. Our study therefore implements Lorentz' hypothesis, which advocates the electromagnetic origin of mass, from an objective criterion, even if satisfaction of the criterion itself relies on two parameters, i.e., the numbers of triolets in the envelope and nucleus. Noteworthy, these parameters are not arbitrary, but instead are strongly constrained by several relations (11, 12, 20, 21, 27) that fix their values in our model. Altogether, we believe our study establishes that deterministic electrodynamical models of subatomic particles can be constructed beneath the Compton scale, in agreement with an objectively realist conception of physics.

Envelope triolets could also fluctuate radially or otherwise in time, possibly constituting a periodic wave that revolves at light velocity. This system has not been investigated here, but is of interest because this periodic wave could correspond to the wave associated to the electron, first imagined by de Broglie and later represented by wavefunction |ψ> in Schrödinger's wave mechanics or Dirac's quantum mechanics. It is conceivable that a wave made of envelope triolets, if it exists, attracts and drives the nucleus in the manner of the de Broglie-Bohm guiding wave [23, 24], sensing the electromagnetic fields generated by the envelopes belonging to other particles. Hence, envelope triolets could undulate and incarnate wavefunction |ψ>, whose concrete existence has recently been reconsidered [42]. Note further that nucleus triolets could also form a wave, reminiscent of the second wave described in de Broglie's double solution theory [23]. Specifically, triolets could propagate in a highly dynamical manner and experience irregular fluctuations, as in the hydrodynamical model of Bohm and Vigier [25]. Importantly, it has been suggested that solutions of this type could account both for quantum phenomena [26] and for quantum principles [37]. Bell also wrote that such solutions were compatible with the predictions of quantum mechanics [43]. Further, it is conceivable that such a complex envelope can exhibit several stable states, much like modes for a vibrating rope. These could correspond to the eigenstates of quantum mechanics. In the general case, the envelope would be in an unstable state, but could converge toward one of its eigenstates upon measurement, which could be conceived as the sum of interactions between system subparticles and apparatus subparticles. Such propositions constitute an interpretation of von Neumann's reduction of the wave packet [44], and would provide a possible solution to the measurement problem of quantum mechanics [45].

These considerations suggest that quantum theories, which encompass all subatomic phenomena and whose standard interpretation states that everything is intrinsically probabilistic, could eventually emerge [46, 47] from a relativistic electrodynamical description in agreement with the deterministic paradigm, which supports the causality principle, objective reality, and governs macroscopic physics. In this perspective, Schrödinger and Dirac Equations would constitute high-level descriptions of the dynamics of envelope triolets. Our study therefore provides new insight regarding the unification of the two apparently irreconcilable paradigms in physics: the deterministic and quantum paradigms.

Now, how exactly does the electron appear to be point-like in corpuscular interactions? How does our model relate to the observation that the electron seems spherical [48], or that its spin, charge and orbital components seem to be separable [49–51]? How would the moving electron, which exhibits a wave satisfying de Broglie relation p=h/λ, be described? Could our description be regarded as an attempt to create a corpuscular counterpart to wave mechanics? Could analogous electrodynamical models be similarly constructed for other subatomic particles [52]? Could our extended model of the electron bring insight to the nature of molecular bonding, or to the arrangement of electrons inside atoms? And finally, what would be the implications for the interpretation of quantum mechanics [45]? How would quantum properties, such as the existence of eigenstates, the measurement problem or entanglement, and quantum phenomena, such as the two-slits experiment or the one-dimensional potential well, be understood in the light of our model? We believe the aforementioned questions should stimulate discussion and foster novel investigations.