Self-Interactions, Self-Energy and the Electromagnetic Contribution to the Anomalous g-Factor

1. Introduction

The anomalous magnetic moment and the intrinsic dynamics of non-composite particles have been considered as unique features of the quantum field theory since the beginning of its elaboration [1,2,3,4,5]. The electron’s anomalous magnetic moment and its fundamental properties are the first to be studied and realized. With the aid of quantum electrodynamics the value of corresponding g-factor was predicted with a stunning accuracy [6,7,8,9,10,11], leaving no space for mistrusting its effectiveness. Following, were the properties of the two remaining leptons carrying elementary charge, the muon and tau. Determining with high accuracy the dynamics of both massive leptons is an active field of research. The muon’s anomalous magnetic moment is still puzzling the community aiming to reduce the gap between theory and experiment [12,13,14,15,16,17,18,19], recently known to be of about $0.58$%. On the other hand, having a very short lifetime and being the massive among all leptons, measuring and predicting tau anomalous magnetic moment is a challenging task requiring great efforts [20,21,22,23,24,25,26]. Although, there is a serious discrepancy between theory and experiment, such efforts may have the potential to shed more light on the contribution of high order hadronic terms thus aiding in resolving the inconsistency in the muon data.

The microscopic electrodynamics underlying the occurrence of anomalous magnetic moment is indispensably related to the nature of self-interactions [1,27,28]. The latter are believed to be uniquely addressable by the regularization and renormalization methods of quantum theory [29,30,31,32,33], with no classical analog. Yet, even within the standard methods of quantum mechanics, the evaluation of self-Coulomb energy in multi-electron systems still poses a challenge [34,35].

The quantum and classical theories are believed to have no interconnection pertaining to the occurrence of anomalous magnetic moment and self-energy divergence problem. Therefore, a classical method with the potential to quantify the anomalous magnetic moment and the self-interactions in the absence of singularities and electromagnetic radiation may contribute significantly in establishing a better interrelationship between both theories.

The present paper propose an exact approach quantifying the electromagnetic contribution to the anomalous magnetic moment and related dynamics of isolated non-composite particle carrying elementary electric charge. The approach build on the methods of classical electrodynamics in studying the microscopic dynamics of self-symmetric systems, like the considered one, by removing all singular points. It incorporates a particular spatial averaging procedure and regularization of the electromagnetic field potentials confining the field to the particle itself. Accordingly, the effect of self-repulsion is accounted for without singularity, or violation of the conservation laws, allowing a complete microscopic description of the system’s energy state. Two types of self-interactions are predicted, the self-Coulomb and self-Zeeman ones. Essentially, the considered approach results in a system of transcendental equations of motion satisfied by the particle’s velocity and the anomalous g-factor of electromagnetic origin. The derived system of equations ensure fast and accurate computational results and predicts the value of anomalous g-factor with the same success the perturbative approach of quantum electrodynamics does. In the present work, the computations are carried out with accuracy matching one prat in a billion.

The rest of the paper is organized as follows. The mathematical notation of all fundamental physical quantities characterizing the system under consideration along with essential interrelationships between the introduced observables are presented in Section 2. Furthermore, the section discusses the occurrence of self-interactions and their explicit representation. Lagrangian and Hamiltonian density representations are also given. The main results are outlined in Section 3, with computations carried out on Wolfram Mathematica. Section 4 summarizes the used approach and obtained results.

2. Theoretical Background

2.1. General Considerations

In the present study all representations are restricted to the mathematical framework of the classical relativistic mechanics and electrodynamics, overlooking all relevant quantum mechanical representations except the ones of immense significance. For the sake of clarity, all physical quantities and equations of motion are represented within the standard three-dimensional vector formalism. The four-vector convention is omitted, since the representation in Minkowski space with the relevant Lorentz group is straightforward [36,37,38].

Consider an isolated system composed of single non-composite particle of type q, with rest frame of reference $\mathbf{R}$, rest mass $m_q$ and electric charge $e_q^{\pm }=\pm e$$\forall q$, where e denotes the elementary charge. Let $r_{cq}=\alpha {\overline{\lambda }}_{cq}$ be the particle’s electromagnetic radius at rest, where $\alpha$ and ${\overline{\lambda }}_{cq}$ are the fine structure constant and reduced Compton wavelength, respectively. Let ${\mathbf{u}}_q$, with $\mathit{\kappa }·{\mathbf{u}}_q=u_q$ and ${\dot{u}}_q=0$, be the velocity associated to the considered particle and ${\mathbf{p}}_q={\gamma }_q{m}_q{\mathbf{u}}_q$ the corresponding momentum in the observer’s rest frame $\mathbf{O}$, where ${\gamma }_q$ is the corresponding Lorentz factor. As we will see later in the discussion, for the considered system $u_q=0\iff \nexists e$. Furthermore, let $r_q={\overline{\lambda }}_q$, with ${\lambda }_q={\lambda }_{cq}$$\sqrt{{\beta }_q^{-2}-1}$, be the particle’s electromagnetic radius in $\mathbf{O}$, where ${\lambda }_q$ is the particle’s intrinsic wavelength, ${\beta }_q=u_q{c}^{-1}$ and c denotes the light speed in vacuum. Since the particle alone represents the only center of symmetry in the system, $r_q$ and $p_q$ are conjugate intrinsic variables, satisfying $r_q{p}_q={\overline{\lambda }}_{cq}{m}_{q}c$.

Let ${\rho }_{|e_q^{\pm }|}$ and ${\rho }_{m_q}$, with

$${\rho }_{\mathrm{s}}=\frac{2}{V_{cq}}{\int }_0^{\infty }\mathrm{s}\delta (\overline{\lambda }-\alpha {\overline{\lambda }}_{cq})\mathrm{d}\overline{\lambda },\mathrm{s}=\left\{\right|e_q^{\pm }|,m_q\},$$

be the charge and rest mass densities, respectively, such that

$$V_{cq}=8\pi {\int }_0^{\infty }{\int }_0^{\overline{\lambda }}\delta (\overline{\lambda }-\alpha {\overline{\lambda }}_{cq})r^2\mathrm{d}r\mathrm{d}\overline{\lambda },$$

(1)

and

$$e_q^{\pm }=\pm {\int }_{{\Omega }_{cq}}{\rho }_{|e_q^{\pm }|}\mathrm{d}\mathrm{v},m_q={\int }_{{\Omega }_{cq}}{\rho }_{m_q}\mathrm{d}\mathrm{v},$$

(2)

where the spatial domain ${\Omega }_{cq}\in {\mathbb{R}}^3$, with a boundary $\partial {\Omega }_{cq}$, is such that $V_{cq}={\int }_{{\Omega }_{cq}}\mathrm{d}\mathrm{v}$.

Let for all q the system is characterized by an effective rest mass given by

$$M_q={\int }_{{\Omega }_{cq}}{\rho }_{M_q}\mathrm{d}\mathrm{v},$$

(3)

where the corresponding density ${\rho }_{M_q}={\rho }_{M_q}\left(r\right)$ is a smooth function over $r\in (0,+\infty )$.

The system posses intrinsic angular momentum ${\mathbf{f}}_q$ and a corresponding magnetic one ${\mathit{\mu }}_q$, with magnitudes ${\mathrm{f}}_q$ and ${\mu }_q$, respectively. In accordance to (3), we have

$${\mathbf{f}}_q={\int }_{{\Omega }_{cq}}{\mathbf{r}}_q\times {\mathbf{j}}_{M_q}\mathrm{d}\mathrm{v},{\mathit{\mu }}_q=\frac{1}{2}{\int }_{{\Omega }_{cq}}{\mathbf{r}}_q\times G_q{\mathbf{j}}_{e_q^{\pm }}\mathrm{d}\mathrm{v},$$

(4)

where ${\mathbf{j}}_{M_q}={\gamma }_q{\rho }_{M_q}{\mathbf{u}}_q$ and ${\mathbf{j}}_{e_q^{\pm }}=\pm {\gamma }_q{\rho }_{|e_q^{\pm }|}{\mathbf{u}}_q$ are the system’s mass and charge density currents, respectively. Here, $G_q$ is the integrand g-factor, where the latter reads

$$g_e=\frac{1}{V_{cq}}{\int }_{{\Omega }_{cq}}{G}_q\mathrm{d}\mathrm{v},G_q=2\frac{|e_q^{\pm }|{\rho }_{M_q}}{m_q{\rho }_{|e_q^{\pm }|}}.$$

(5)

We further have $g_e=2(1+a_e)$, where $a_e$ is the electromagnetic contribution to the anomalous g-factor.

Let us point out that according to (5), the fraction $M_q{m}_q^{-1}$ is a constant for all q. We would like to stress, furthermore, that ${\mathbf{f}}_q$ is not an orbital angular momentum, since the system does not posses a center of symmetry independent from the particle. In the vector diagram sketching the dynamics of the considered system, the position vector ${\mathbf{r}}_q$ associated to the density currents will remain conjugate to the particle’s momentum for an infinite time. In other words, the dynamics related to both physical quantities given in Equation (4) is not time and space independent event for an observer in $\mathbf{O}$ and hence it cannot be associated to a free spinning sphere of radius $r_{cq}$. The particle’s momentum and corresponding currents are intrinsic, not relative to an observer and not a consequence of independent external action.

All of the above considered relations and definitions hold for ${\mathbf{u}}_q={\mathbf{v}}_q+{\tilde{\mathbf{v}}}_q$, where ${\mathbf{v}}_q$ is the velocity component characterizing a node moving relative to the origin of $\mathbf{O}$ and ${\tilde{\mathbf{v}}}_q$ is the velocity component associated to an oscillation occurring with respect to that node. In this regard, the particle has a standing wave representation that is naturally related to some uncertainty. The corresponding oscillation dynamics can be further studied within the classical wave theory, or with the more rigorous methods of the quantum theory. Such studies lie beyond the goals of the present research and may be discussed elsewhere.

2.2. Field Observables

Since the system is isolated, or $u_q\ne u_q\left(t\right)$, and the particle posses elementary electric charge confined within the time independent spatial domain ${\Omega }_{cq}$, the resulting electric current and its density are time and spatially independent. Accordingly, the electric ${\mathbf{E}}_q$ and magnetic ${\mathbf{B}}_q$ components of the corresponding electromagnetic field in $\mathbf{O}$ are time independent and spatially conjugated to the particle’s momentum. Thus, we have the relations ${\mathbf{E}}_q=-\nabla {\phi }_q$ and ${\mathbf{B}}_q=\nabla \times {\mathbf{A}}_q$, where ${\phi }_q={\phi }_q\left(r\right)$ and ${\mathbf{A}}_q={\mathbf{A}}_q\left(r\right)$ are the scalar and vector potentials of the electromagnetic field in $\mathbf{O}$. In particular, we have

$${\phi }_q={\gamma }_q{\varphi }_q,{\mathbf{A}}_q={\gamma }_q\frac{{\mathbf{u}}_q}{c^2}{\varphi }_q,{\mathbf{B}}_q=\frac{1}{c^2}({\mathbf{u}}_q\times {\mathbf{E}}_q),$$

(6)

where the function ${\varphi }_q={\varphi }_q\left(r\right)$, given with respect to the observer’s rest frame, is regularized to the origin of $\mathbf{R}$ and reads

$${\varphi }_q\left(r\right)=\frac{e_q^{\pm }}{4\pi {\epsilon }_{o}r}\left(1-e^{-\frac{{\gamma }_{q}r}{(1+a_e){\lambda }_{cq}}}\right),$$

(7)

with ${\epsilon }_o$ denoting the vacuum permittivity. The function given in Equation (7) is solution to the superposition of Poisson and Helmholtz equations, with constant of integration equal to zero at the limit $r\to \infty$ and minimal wavelength of the electromagnetic field equal to $(1+a_e){\lambda }_{cq}$$\sqrt{1-{\beta }_q^2}$ at $r\to 0$. Graphical representation of the difference between the Coulomb potential and the one given in Equation (7) is shown in Figure 1.

We would like to note that in Cartesian coordinates, the components of the vector potential given in (6) read $A_q^{\tau }={\gamma }_q{c}^{-2}{u}_q^{\tau }{\varphi }_q(\beta ,\nu )$, where $\tau \ne \beta \ne \nu =\{x,y,z\}$ and $u_q^{\tau }$ are the components of ${\mathbf{u}}_q$. The potentials given in Equation (6) satisfy trivially both choices of gauge the Coulomb and Lorenz one.

Owing to Equation (7), we obtain the analytical expression of the electromagnetic field’s energy, with density ${\epsilon }_o{E}_q^2$, for all q. Thus, integrating over ${\mathbb{R}}^3$, we get

$$W_q\left(r\right)=\alpha c^2{\gamma }_q^2{m}_q\frac{{\overline{\lambda }}_{cq}}{2r}\left(2-4e^{\frac{{\gamma }_{q}r}{(1+a_e){\lambda }_{cq}}}+2e^{2\frac{{\gamma }_{q}r}{(1+a_e){\lambda }_{cq}}}+\frac{{\gamma }_{q}r}{(1+a_e){\lambda }_{cq}}\right)e^{-2\frac{{\gamma }_{q}r}{(1+a_e){\lambda }_{cq}}},$$

(8)

where the constant of integration is fixed at $r\to \infty$. Graphical representation is depicted in the inset in Figure 1.

From Equations (7) and (8), we obtain two important limits. We have,

$$\underset{r\to 0}{lim}{\phi }_q\left(r\right)=\frac{{\gamma }_q^2{e}_q^{\pm }}{4\pi {\epsilon }_o(1+a_e){\lambda }_{cq}}\mathrm{and}\underset{r\to 0}{lim}{W}_q\left(r\right)=\frac{\alpha c^2{\gamma }_q^3{m}_q}{4\pi (1+a_e)},$$

respectively. These results point out that the energy of generated electromagnetic field does not vanish at $r\to 0$ and the potential energy is a constant ensuring that the self-interactions do not vanish at the given limit and do not associate to singularity points. Hence, on any gradient surface, including all points on the boundary $\partial {\Omega }_{cq}$, the presence of electromagnetic field appears as a steady electromagnetic veil making the particle to appear as a very round object [39]. In addition, for the corresponding Umov-Poynting vector, we have ${\mu }_o^{-1}({\mathbf{E}}_q\times {\mathbf{B}}_q)={\epsilon }_o{E}_q^2{\mathbf{u}}_q$, where ${\mu }_o$ is the vacuum magnetic permeability.

Note that, the discussed electromagnetic field is consistently confined to the particle. It depends exclusively on the particle’s intrinsic characteristics and does not obey the Liénard-Wiechert representation [38,40,41,42]. Hence, the electromagnetic field does not propagate at speed c independently from the particle and does not classify as an on shell coupling between the electric and magnetic field components. We neither observe spontaneous emission nor absorption of photons. In this regard, the steady presence of electromagnetic field around the particle can be considered as a classical analog of the foam of virtual photons presented in quantum electrodynamics.

2.3. Electromagnetic Self-Energy and Self-Interactions

Since the system is closed, with the action of no additional fields, the particle exhibits no exchange of energy and hence momentum. Accordingly, neither external nor net self-forces [38,43,44] are acting on the particle and the energy of the system remains purely kinetic. However, the considered system definitely exhibits two types of self-interactions of electromagnetic nature, with average energy depending only on $u_q$.

The Hamiltonian describing the total energy of electromagnetically self-interacting particle read

$$H_q={\gamma }_q{m}_q{c}^2+{\Sigma }_q+Z_q,$$

(9)

where the energy terms ${\Sigma }_q$ and $Z_q$ are associated to the self-Coulomb and self-Zeeman interactions, respectively.

The self-Coulomb term is the spatial average over the domain ${\Omega }_{cq}$ of the interaction energy between the particle’s charge and its own electromagnetic field. It reads

$${\Sigma }_q={\gamma }_q{c}^2{\int }_{{\Omega }_{cq}}{\rho }_{M_q}-{\rho }_{m_q}\mathrm{d}\mathrm{v},$$

(10)

where

$${\rho }_{M_q}={\rho }_{m_q}\left(1+{\eta }_q\frac{r_{cq}}{r}\left(1-e^{-\frac{{\gamma }_{q}r}{(1+a_e){\lambda }_{cq}}}\right)\right),$$

(11)

${\eta }_q=1+{\beta }_q^2$. Here, we have taken into account the relation ${\rho }_{m_q}{c}^2=\left|e_q^{\pm }\right|{\rho }_{|e_q^{\pm }|}{\left(4\pi {\epsilon }_o{r}_{cq}\right)}^{-1}$.

The self-Zeeman term represents the spatial average over the domain ${\Omega }_{cq}$ of the interaction energy between the system’s intrinsic magnetic moment presented in (4) and the induced magnetic field given in (6). Since the self-Zeeman interaction is time independent, the corresponding integrand satisfies $-{\mathit{\mu }}_q·{\mathbf{B}}_q=-{\mu }_q{B}_q$, where in accordance to (6), we have $B_q=c^{-2}{\gamma }_q{u}_q|\nabla {\varphi }_q|$. Thus, taking into account relations (4), according to which

$${\mu }_q=e{\overline{\lambda }}_{cq}c\frac{M_q}{m_q}\mathrm{and}{\mathrm{f}}_q={\overline{\lambda }}_{cq}{M}_{q}c,$$

we obtain

$$Z_q=-\frac{{\mathrm{f}}_q{p}_q}{m_q^2{\eta }_q}{\int }_{{\Omega }_{cq}}|\nabla {\rho }_{M_q}|\mathrm{d}\mathrm{v}.$$

(12)

The self-energy terms given in Equations (10) and (12) are intrinsic to the considered particle and hence invariant to an external influence. Therefore, represented in terms of quantum theory they will remain invariant with respect to the particle’s orbital state in many-body systems. That may be of benefit to the researchers studying multi-electron systems with the aid of computational methods that fail to account for the self-interactions without generating errors, see for example the case of Kohn-Sham density functional theory [35,45,46].

2.4. The Lagrangian and Hamiltonian Density

In Section 2.3 we discuss the two allowed by the conservation laws types of electromagnetic self-interactions that take place only within the domain ${\Omega }_{cq}$ and have particular energy densities.

In general, the Hamiltonian (9) has a corresponding density and Lagrangian. The latter reads

$${\mathcal{L}}_q={\mathbf{p}}_q·{\tilde{\mathbf{u}}}_q-{\gamma }_q{c}^2{\rho }_{M_q}+\frac{M_q{p}_q^3}{m_q^3{\eta }_q}\mathit{\kappa }·{\nabla }_p{\rho }_{M_q},$$

(13)

where

$${\tilde{\mathbf{u}}}_q=\mathit{\kappa }\frac{2}{V_{cq}}{\int }_0^{\infty }{u}_q\left(\overline{\lambda }\right)\delta (\overline{\lambda }-{\overline{\lambda }}_q)\mathrm{d}\overline{\lambda }$$

(14)

is the generalized velocity and

$${\rho }_{M_q}={\rho }_{m_q}\left(1+\frac{{\eta }_q}{m_q}\frac{\alpha }{c}{p}_q\left(1-e^{-\frac{{\gamma }_q{m}_{q}c}{2\pi (1+a_e)p_q}}\right)\right)$$

(15)

is the momentum representation of the effective mass density. Here, we take into account that $\mathit{\kappa }·{\nabla }_p{\rho }_{M_q}\in {\mathbb{R}}_{+}$ for all $u_q\in (0,c)$, such that $|\nabla {\rho }_{M_q}\left(r_q\right)|\to p_q^2{\left({\lambda }_{cq}c{m}_q\right)}^{-1}\mathit{\kappa }·{\nabla }_p{\rho }_{M_q}\left(p_q\right)$.

For the corresponding Hamiltonian density, we have

$${\mathcal{H}}_q={\gamma }_q{c}^2{\rho }_{M_q}-\frac{M_q{p}_q^3}{m_q^3{\eta }_q}\mathit{\kappa }·{\nabla }_p{\rho }_{M_q},$$

(16)

where the Hamiltonian’s equations read

$${\tilde{\mathbf{u}}}_q={\nabla }_p{\mathcal{H}}_q,\mathrm{and}{\dot{p}}_q=0.$$

(17)

We would like to point out that Equations (13) to (17) are derived by bearing in mind that the components of ${\mathbf{u}}_q$ are constrained over time. We have ${\mathbf{u}}_q\left(t\right)·{\mathbf{u}}_q\left(t\right)=u_q^2$, with $u_q\ne u_q\left(t\right)$.

3. Results

3.1. Equations of Motion

Following the Hamiltonian’s equations (17), the generalized velocity representation (14) and (5), we obtain the system’s equations of motion.

Working with the magnitude of the particle’s velocity for convenience, we have

$$u_q={\int }_{{\Omega }_{cq}}\mathit{\kappa }·{\nabla }_p{\mathcal{H}}_q\mathrm{d}\mathrm{v}.$$

(18)

Integrating over the domain ${\Omega }_{cq}$, we obtain one of the equations. It is a transcendental equation and reads

$$\alpha c{\gamma }_q{\tilde{\eta }}_q\left(1-\left(1+\frac{c}{2\pi (1+a_e)u_q}\right)e^{-\frac{c}{2\pi (1+a_e)u_q}}\right)-\frac{\alpha c{\gamma }_q^2}{4{\pi }^2(1+a_e)}{e}^{-\frac{c}{2\pi (1+a_e)u_q}}-u_q=0,$$

(19)

where ${\tilde{\eta }}_q={\eta }_q+3(1+a_e){\gamma }_q{\beta }_q^2$.

On the other hand, taking into account the explicit representation of the effective mass density (11), from (5) we obtain a second equation. For all q, we have

$$3{\eta }_q\left(\frac{1}{2}-\left(\frac{1-e^{-\frac{{\gamma }_q\alpha }{2\pi (1+a_e)}}\left(1+\frac{{\gamma }_q\alpha }{2\pi (1+a_e)}\right)}{{\left(\frac{{\gamma }_q\alpha }{2\pi (1+a_e)}\right)}^2}\right)\right)-a_e=0.$$

(20)

Equations (19) and (20) represent system of transcendental equations of motion, with $u_q=u_e$ for all q and $m_q$, where $u_e=const.$, for $t\in [0,+\infty )$. The solutions yield to the exact value of the particle’s energy represented by the Hamiltonian (9) and essentially to that of the electromagnetic contribution to the anomalous g-factor. It is worth noting that from Equation (19) follows that ${\beta }_e\approx \alpha$.

Within the considered theoretical approach $u_e$ is invariant for all observers, with relative quantities being the velocities ${\mathbf{v}}_q$ and ${\tilde{\mathbf{v}}}_q$, see Section 2.1. Accordingly, although the system’s total energy associated to the Hamiltonian given in Equation (9) remains invariant, the particle’s behavior with respect to $\mathbf{O}$ remains relative. Thus, for different observers the particle will have different behavior. In the case $|{\tilde{\mathbf{v}}}_q|⋙|{\mathbf{v}}_q|$ the particle will appear as more localized and associated to a high frequency oscillation in time. On the opposite, for $|{\mathbf{v}}_q|⋙|{\tilde{\mathbf{v}}}_q|$ the particle will be associated to a traveling wave of lower frequency. In both cases, however, because of it’s electrical charge yielding $u_e\ne 0$ for an infinite time the particle cannot be at rest in any of the observers’ rest frame of references. Moreover, since the magnitudes of ${\mathbf{v}}_q$ and ${\tilde{\mathbf{v}}}_q$ are time independent it follows that the oscillation dynamics will be underpinned by a plane of isotropy, with ${\mathbf{v}}_q·{\tilde{\mathbf{v}}}_q=0$.

3.2. Effective Mass-Energy Equivalence

Taking into account the contribution only of the electric part of the self-interactions in (9), we obtain the effective mass-energy relation. Accounting for the first two terms on the right hand side after the corresponding equality symbol, we get

$${\mathcal{E}}_q={\gamma }_e{M}_q{c}^2,$$

(21)

where for the self-Coulomb energy term we get ${\Sigma }_q=a_e{\gamma }_e{m}_q{c}^2$ and for the effective rest mass we have $M_q=m_q(1+a_e)$. Here, ${\gamma }_e$ is a constant for all q, obtained by solving Equations (19) and (20), see Section 3.1. We would like to point out once more that within the considered mathematical framework if $\nexists e$, then $\nexists a_e$ and the particle’s speed equals zero. Thus, the energy of a free electrically charged non-composite particle of rest mass $m_q$ is always higher than the energy of a free electrically neutral non-composite particle of the same rest mass. Furthermore, according to Equation (21) the former will always be characterized by an effective momentum given by $P_q={\gamma }_e{M}_q{u}_e$ and yielding to the effective energy-momentum representation

$${\mathcal{E}}_q=\sqrt{M_q^2{c}^4+P_q^2{c}^2}.$$

(22)

It is important to note that the effective rest mass, energy and momentum related by Equation (22) are physical characteristics of the spinor field ${\Psi }_q\left(x_{\mu }\right)$ that effectively account for the self-Coulomb energy of the q-th particle and satisfies the Dirac equation

$$(i\hslash {\gamma }^{\mu }{\partial }_{\mu }-M_{q}c){\Psi }_q\left(x_{\mu }\right)=0.$$

(23)

With the aid of Equation (23) the free-particle representation of the corresponding field holds ensuing that the quantum system under consideration remains isolated and hence the conservation laws are not violated. Moreover, Equation (23) accounts for the electromagnetic contribution to the particle’s anomalous g-factor, predicting $g_e=2(1+a_e)$ for all q.

3.3. The Anomalous g-Factor

The computations of the electromagnetic component of the particle’s anomalous g-factor and the corresponding speed are carried out for $\alpha ={137.035999084}^{-1}$ [47]. Solving the system of transcendental Equations (19) and (20) for the particle’s speed we obtain $u_e=2.18821606262\times {10}^6$ ms${}^{-1}$, with corresponding value of $a_e$ given in the second row of Table 1. Compared to the experimentally obtained [11] and calculated with the aid of quantum electrodynamics [7,8] electron’s anomalous g-factor, the numerical value obtained by solving Equations (19) and (20), with the given value of the fine structure constant, is accurate to a one part in a billion. To the best of our knowledge, the result given in Table 1 is the most accurate classical evaluation of the electromagnetic contribution to the electron’s anomalous g-factor reported to the present days. We would like to point out that the obtained accuracy holds further for $\alpha ={137.035999206}^{-1}$ [48], with $u_e=2.18821616430\times {10}^6$ ms${}^{-1}$. The accuracy may be improved with additional refinement of the fine structure constant after the ninth digit behind the decimal point.

4. Summary

With the aid of essential regularization (Section 2.2) of the electromagnetic field potentials and spatial averaging procedure (Section 2.3), elaborated within the formalism of classical electrodynamics, the present paper reports an exact approach quantifying on a microscopic level the electrodynamics of isolated system comprised of non-composite particle of arbitrary rest mass and possessing elementary electric charge. The proposed approach overcomes all singularities arising in the conventional methods of classical electrodynamics thus uncovering in details the physical nature of electromagnetic self-interactions and the occurrence of anomalous magnetic moment in the system. It quantifies the particle’s dynamics exactly through a system of transcendental equations of motion obtained in the absence of any approximations and at the classical limit, see Section 3.1. Essentially, the solutions of these equations give the exact values of the particle’s intrinsic velocity and the electromagnetic contribution to the anomalous g-factor (see Section 3.3). The derived transcendental equations ensure fast computations and provide highly accurate results competitive to the perturbative method of quantum electrodynamics, see the comparison in Table 1.

In general, the proposed approach may be build on and integrated into the mathematical framework of the classical and quantum field theories. In that regard, more contributions to the particle’s magnetic moment may be calculated exactly. With the appropriate gauge fields and regularization one may introduce additional self-interactions and generalize the system of transcendental equations discussed in Section 3.1.

In conclusion, it appears that the present study supports a thesis pointing out that the anomalous magnetic moment may not be a unique feature to the quantum theory and its occurrence may have a solid classical description. Moreover, it may be the case that the perturbative method of quantum field theory is not unique to that feature and in addition there exist an exact method for its calculation. On that account, pushing the boundaries of classical or quantum field theories further to test the possibility for the exact calculation of even higher contributions in the anomalous magnetic moment of non-composite particles is worth a shot. The proposed regularization to the scalar potential and the related self-Coulomb and self-Zeeman interactions, furthermore, may significantly facilitate the optimization of some computational methods in solid state physics making them self-consistent. Especially these methods studying multi-electron systems that fail to address the self-interactions without the consideration of additional corrections.