Exact classical approach to the electron's self-energy and anomalous g-factor

Consider an isolated electron, with rest frame of reference R, rest mass m_e and electric charge ${\bar {e}}=-e$, where e denotes the elementary charge. Let $r_{ce}=\alpha \bar {\lambda }_{ce}$ denote the electron's electromagnetic radius at rest, where α is the fine structure constant and $\bar {\lambda }_{ce}$ is the corresponding reduced Compton wavelength. Let ρ_e and $\rho _{m_{e}}$ be the electron's charge and rest mass densities defined within the spatial domain $\Omega _{ce}\in \mathbb {R}^{3}$, with boundary $\partial \Omega _{ce}$ and volume V_ce . The boundary $\partial \Omega _{ce}$ is considered as smooth and spherically symmetric with radius r_ce . In accordance with the experimental observations [47,48] the distribution of charge and mass in Ω_ce is considered as isotropic, with $\rho _{e}\rho ^{-1}_{m_{e}}=em^{-1}_{e}$.

Let the electron be characterized by an effective rest mass

$$\begin{equation} M_e=\int_{\Omega_{ce}}\rho_{M_e}\textrm{d}v, \end{equation} \tag{ 1 }$$

where the effective mass density $\rho _{M_{e}}=\rho _{M_{e}}(r)$ is a smooth function over $r\in (0,+\infty )$. Here, r is time-independent radial parameter.

Let $\mathbf {u}_{e}$, with unit vector $\boldsymbol {\kappa }$ and magnitude $|\mathbf {u}_{e}|=u_{e}$, be the electron's relative velocity with respect to the observational rest frame of reference O and $\mathbf {p}_{e}=\gamma _{e}m_{e}\mathbf {u}_{e}$ be its momentum, where γ_e is the corresponding Lorentz factor. Furthermore, let $\mathbf {r}_{e}$, with $|\mathbf {r}_{e}|=r_{e}$, be an intrinsic field vector oscillating about the origin of R, with angular velocity $\boldsymbol {\omega }_{e}$ and areal velocity

$$\begin{equation} \mathbf{f}_e=\frac12(\mathbf{r}_e\times \tilde{\mathbf{u}}_e), \end{equation} \tag{ 2 }$$

where $\tilde {\mathbf {u}}_{e}$ is the corresponding tangential velocity, with $|\tilde {\mathbf {u}}_{e}|=\tilde {u}_{e}, \tilde {\mathbf {u}}_{e}=\boldsymbol {\omega }_{e}\times \mathbf {r}_{e}$ and $\tilde {\mathbf {u}}_{e}\cdot \mathbf {u}_{e}=0$. Here, r_e and $\tilde {u}_{e}$ are time-independent conjugate quantities and $\bar {\lambda }_{ce}$ is not an intrinsic wavelength characterizing the electron at rest. It represents the lower bound in the range of the electron's observable radius r_e . Since the areal velocity is intrinsic, we have the constraint

$$\begin{equation} r_e\tilde{u}_e=\bar{\lambda}_{ce}c, \end{equation} \tag{ 3 }$$

where c is the light speed in vacuum. We would like to point out, furthermore, that the magnitude of relative velocity $\mathbf {u}_{e}$ is intrinsic and not an average quantity. Therefore, in the Lorentz boost it remains invariant.

From eq. (2) there follows that the dynamics of $\mathbf {r}_{e}$ underlie the occurrence of angular magnetic moment

$$\begin{equation} \boldsymbol{\mu}_{e}=-\frac12r_e\int_{\Omega_{ce}}G_e \mathbf{j}_{e}\textrm{d}v, \end{equation} \tag{ 4 }$$

where $\mathbf {j}_{e}=\rho _{e}r_{e}\boldsymbol {\omega }_{e}$ is a pseudovector associated to the charge density current $\mathbf {J}_{e}=\gamma _{e}\rho _{e}\mathbf {u}_{e}$. Here, G_e is the integrand g-factor, where the latter reads

$$\begin{equation} g_e=\frac{2}{V_{ce}}\int_{\Omega_{ce}}G_e\textrm{d}v, \qquad G_e=\frac{e\rho_{M_e}}{m_e\rho_{e}}. \end{equation} \tag{ 5 }$$

Since $g_{e}=2(1+a_{e})$, we further have $M_{e}=m_{e}(1+a_{e})$, where a_e is the electron's anomalous g-factor.

The pseudodensity current $\mathbf {j}_{e}$ occurs not only in O but also in R and satisfies $\gamma _{e}\mathbf {j}_{e}\cdot \mathbf {J}_{e}=|\mathbf {J}_{e}|^{2}$. The free electron cannot be at rest with respect to any observer and $u_{e}=\tilde {u}_{e}$. Since $\dot {\mathbf {u}}_{e}$ and $\dot {\boldsymbol {\omega }}_{e}$ are zero vectors the scalar $\varphi _{e}=\varphi _{e}(r)$ and vector $\mathbf {A}_{e}=\mathbf {A}_{e}(r)$ potentials of the electromagnetic field in O do not depend explicitly on time and according to the Lorentz transformations, we have

$$\begin{equation} \varphi_{e}(r)=\gamma_e\eta_e\psi_{e}(r),\qquad \mathbf{A}_{e}(r)=2\,\gamma_e\frac{\mathbf{u}_e}{c^2}\psi_{e}(r), \end{equation} \tag{ 6 }$$

where $\eta _{e}=1+\beta ^{2}_{e}$, with $\beta _{e}=u_{e}c^{-1}$. The function $\psi _{e}(r)$ is regularized to the origin of R with respect to O and reads

$$\begin{equation} \psi_{e}(r)=\frac{\bar{e}}{4\,\pi\varepsilon_or}\bigg(1-e^{-\frac{\gamma_er}{(1+a_e)\lambda_{ce}}}\bigg), \end{equation} \tag{ 7 }$$

where ε_o is the vacuum permittivity. The function given in eq. (7) satisfies the classical field equation

$$\begin{equation} \Delta_r\psi_{e}(r)-\chi^2_e\phi_{e}(r)=0, \end{equation} \tag{ 8 }$$

where Δ_r is the radial part of the Laplace operator in spherical coordinates and $\chi _{e}=\gamma _{e}((1+a_{e})\lambda _{ce})^{-1}$ is a scaling constant. For $u_{e}<c$, both functions in eq. (8) satisfy the boundary conditions

$$\begin{equation} \psi_{e}(r),\phi_{e}(r)= \left\{\begin{array}{@{}l@{\quad}l@{}} 0, & r\to\infty, \\ \dfrac{\gamma_e\bar{e}}{4\,\pi\varepsilon_o(1+a_e)\lambda_{ce}}, & r\to0. \end{array} \right. \end{equation} \tag{ 9 }$$

The scale constant gives the lower bound of the electromagnetic field spatial frequency with respect to an observer in O. The function $\phi _{e}(r)$ is associated to a Yukawa potential contributing to the electron's self-energy, with scaling constant $\gamma _{e}c((1+a_{e})h)^{-1}$, where h is Planck's constant. Therefore, we have $\chi _{e}=\gamma _{e}m_{e}c((1+a_{e})h)^{-1}$ showing that within the quantum theory the self-interaction would be governed by an off shell photon with effective mass $m_{e}(1+a_{e})^{-1}$. Comparison between the Coulomb and regularized potentials is shown in fig. 1.

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For an observer in O the dependence on r of the electromagnetic field potentials given in eq. (6) is confined within the plane perpendicular to the electron's relative velocity. Accordingly, the Coulomb and Lorenz gauge are satisfied trivially and the electromagnetic field potentials do not obey the Liénard-Wiechert representation [49–52]. The electromagnetic field does not propagate at speed c independently of the electron and it does not classify as an on shell coupling between the electric and magnetic fields. The system will exhibit neither spontaneous emission nor absorption of photons. The electromagnetic field is confined to the electron with Umov-Poynting vector $\mu ^{-1}_{o}(\mathbf {E}_{e}\times \mathbf {B}_{e})=\varepsilon _{o}E^{2}_{e}\mathbf {u}_{e}$, where $\mathbf {E}_{e}$ is the corresponding electric field, $\mathbf {B}_{e}$ is the magnetic one, μ_o is the vacuum magnetic permeability and $\varepsilon _{o}E^{2}_{e}$ is the corresponding energy density. Integrating the latter over $\mathbb {R}^{3}$ by accounting for the representation $E_{e}=|\nabla _{r}\varphi _{e}(r)|$, we obtain the energy of the electromagnetic field. We have

$$\begin{align} W_e(r) & = \alpha c^2\,\gamma^2_e\eta^2_em_e\frac{\bar{\lambda}_{ce}}{2r} \bigg(2-4e^{\frac{\gamma_er}{(1 + a_e)\lambda_{ce}}} + 2e^{2\,\frac{\gamma_er}{(1 + a_e)\lambda_{ce}}} \\ &\quad + \frac{\gamma_er}{(1 + a_e)\lambda_{ce}}\bigg)e^{-2\,\frac{\gamma_er}{(1 + a_e)\lambda_{ce}}}, \end{align}$$

where

$$\begin{equation} \lim_{r\to0}W_e(r)=\frac{\alpha c^2\,\gamma^3_e\eta^2_em_e}{4\,\pi(1+a_e)}. \end{equation} \tag{ 10 }$$

The dependence on r is depicted in the inset of fig. 1.

The limits given in eqs. (9) and (10) underline that for $u_{e}<c$ the radial dependence of the electromagnetic field has a removable singularity ensuring the electron's self-energy takes finite value. Moreover, at high momentum scale the theory remains consistent, since the limit $u_{e}\to c$ corresponds to an open system.

Since the system is closed, with the action of no additional fields, the electron exhibits no exchange of energy and hence momentum. Consequently, neither external nor net self-forces [52–54] are acting on the electron. The system's energy remains purely kinetic, with Hamiltonian not depending explicitly on time. However, the considered system definitely exhibits a type of self-interaction, with energy that is not directionally specific. The corresponding Hamiltonian reads

$$\begin{equation} H_e=\gamma_em_ec^2 + \Sigma_e, \end{equation} \tag{ 11 }$$

where Σ_e is the electromagnetic self-energy. In particular, Σ_e equals the spatial average over the domain Ω_ce of the interaction energy $\bar {e}\varphi _{e}$ describing the electron's electromagnetic self-interaction in O. Moreover, the electron's self-energy is an effective kinetic-energy generated by the corresponding self-interaction yielding larger mass density $\rho _{M_{e}}>\rho _{m_{e}}$, with $\rho _{M_{e}}$ satisfying eq. (1). In the absence of self-interaction $\rho _{M_{e}}=\rho _{m_{e}}$ and the system is electrically neutral. Thus, we have

$$\begin{equation} \Sigma_e=\gamma_e c^2\,\int_{\Omega_{ce}}\rho_{M_e}-\rho_{m_e}\textrm{d}v, \end{equation} \tag{ 12 }$$

where

$$\begin{align} \rho_{M_e}& = \rho_{m_e}\left(1+\eta_e\frac{r_{ce}}{r}\left(1-e^{-\frac{\gamma_er}{(1+a_e)\lambda_{ce}}}\right)\right) \nonumber\\ & = \rho_{m_e}\left (1 + \frac{\eta_e\bar{e}}{m_ec^2}\psi_e\right). \end{align} \tag{ 13 }$$

The self-energy term given in eq. (12) is intrinsic and not a potential energy of a gradient field. Represented within the mathematical framework of quantum theory, it will remain invariant with respect to the electron'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 [46,55,56]. Moreover, having an exact value of a scaling constant regularizing the electrons interactions in solids, like χ_e from the cutoff term $\phi _{e}(r)$ represented explicitly in eq. (7) regularizing the electromagnetic field potentials in eq. (6), is a key point in facilitating the successful application of time-dependent density functional theory in studying electric polarization-switching mechanism in ferroelectric systems [57,58].

The Hamiltonian given in eq. (11) has a corresponding density related to the dynamics of the field vector $\mathbf {r}_{e}$. According to eqs. (3) and (13), for $r=r_{e}$, we have

$$\begin{equation} \mathcal{H}_e=c^2\,\rho_{m_e} \left(\gamma_e+\frac{\alpha}{m_ec}\mathcal{P}_e\right), \end{equation} \tag{ 14 }$$

where

$$\begin{equation} \mathcal{P}_e = m_e\gamma_e\eta_e\tilde{u}_e\left(1-e^{-\frac{\gamma_ec}{2\,\pi(1 + a_e)\tilde{u}_e}}\right) \end{equation}$$

is the generalized momentum. From eq. (14) we obtain the equations of motion as

$$\begin{equation} \tilde{u}_e=\int_{\Omega_{ce}}\frac{\partial\mathcal{H}_e}{\partial\mathcal{P}_e}\textrm{d}v, \qquad \dot{\mathcal{P}}_e=0, \end{equation} \tag{ 15 }$$

describing the electron's intrinsic dynamics.

The magnetic moment presented in eq. (4) is intrinsic. It is a classical representation of the electron's spin magnetic moment with included anomalous component. The anomalous component occurs as a result of the regularized self-interaction (see eq. (12)). In particular, substituting the explicit representation of G_e from eq. (5) in eq. (4), we get

$$\begin{equation} \boldsymbol{\mu}_{e}=-\frac{1}{2}g_e\mu_B\boldsymbol{\kappa}, \end{equation} \tag{ 18 }$$

where μ_B is the Bohr magneton. Here, stepping closer to the frontier of quantum theory, we take into account the equality $\bar {\lambda }_{ce}m_{e}c=\hbar$.

The intrinsic magnetic moment is related to neither rotation nor spinning of the electron's electric charge or rest mass, with R remaining inertial. It occurs since the electron's intrinsic field vector $\mathbf {r}_{e}$ is oscillating with angular frequency $\omega _{e}=\tilde {u}_{e}r^{-1}_{e}$, see eq. (2). Therefore, as the origin of R is moving relatively to an observer, with velocity $\mathbf {u}_{e}$, the electron appears as a circularly polarized traveling wave of amplitude $A_{e}=r_{e}$ and carrying rest mass M_e , see fig. 3. Accordingly, the electron's dynamics in O is associated to the vector field $\boldsymbol {\Phi }_{e}(\mathrm {x})=A_{e}\varPhi _{e}(\mathrm {x})\mathbf {n}_{e}$, where $\mathrm {x}\in \mathbb {R}^{1,3}$ is the position four-vector of the origin of electron's rest frame of reference and $\mathbf {n}_{e}\in \mathbb {C}^{3}$ is the field's unit vector. Here, at a certain point in spacetime the phase factor $\varPhi _{e}(\mathrm {x})$ of the propagator satisfies the Klein-Gordon equation

$$\begin{equation} \left(\square+\frac{M^2_ec^2}{\hbar^2}\right)\varPhi_e(\mathrm{x})=0. \end{equation} \tag{ 19 }$$

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The electron's intrinsic field vector is represented by the vector field in O according to the relation $\mathbf {r}_{e}=\sqrt {2}\mathrm {Re}\{\boldsymbol {\Phi }_{e}(\mathrm {x}=ct)\}$, where $\mathbf {n}_{e}=(\frac {1}{\sqrt {2}},\pm \frac {i}{\sqrt {2}},0)$.

Taking into account eqs. (11) and (12), we obtain the effective mass-energy relation

$$\begin{equation} \mathcal{E}_e=\gamma_{e}M_ec^2, \end{equation} \tag{ 20 }$$

where for the electromagnetic self-energy we get

$$\begin{equation} \Sigma_e = a_e\gamma_{e}m_ec^2. \end{equation}$$

Here we take into account that $M_{e}=m_{e}(1+a_{e})$. From eqs. (19) and (20) there follows that the electron is characterized by the effective momentum $P_{e}=\gamma _{\alpha }M_{e}u_{e}$ satisfying the energy-momentum relation

$$\begin{equation} \mathcal{E}_e=\sqrt{M^2_ec^4+P^2_ec^2}. \end{equation} \tag{ 21 }$$

Moreover, the effective rest mass, energy and momentum related by eq. (21) are physical characteristics of the spinor field $\Psi _{e}(x_\mu )$ that effectively accounts for the discussed self-interaction and satisfies the Dirac equation

$$\begin{equation} (i\hbar\gamma^\mu\partial_\mu-M_ec)\Psi_e(x_\mu)=0. \end{equation} \tag{ 22 }$$

In contrast to the Dirac spinor in standard quantum theory the spinor field in eq. (22) accounts for the anomalous component in the electron's spin magnetic moment.

From eqs. (3) and (16), we calculate the free electron field vector magnitude. We have $r_{e}=r_{B}$, where r_B is the Bohr radius. Moreover, expanding in power series the Lorentz factor in eq. (20), we get

$$\begin{equation} M_ec^2 + \frac{\alpha^2}{2}M_ec^2 + \frac{3\,\alpha^4}{8}M_ec^2 + \cdots, \end{equation}$$

where the second term is the non-relativistic oscillation energy of the free electron field vector. Omitting the anomalous component it equals the absolute value of the ground-state energy of the hydrogen atom. Therefore, the ground state of a hydrogen atom preserves the amplitude of the free electron field vector. Consequently, the corresponding total angular momentum equals the spin one. As a result of the proton-electron interaction the electron's net relative velocity equals zero and the dynamics of $\mathbf {r}_{e}$ appears as not directionally specific. At any instant of time, instead of a helix (see fig. 3), the oscillation of the electron field vector $\mathbf {r}_{e}$ indicates a point on a sphere representing the first hydrogen atomic orbital, see fig. 4. When excited, the magnitude and dynamics of the electron field vector change obtaining different orbital shape and hence electron cloud.

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According to the current picture, the electron cloud does not represent a moving around the nucleus point-like charge and mass with motion exclusively stabilized by the uncertainty of corresponding position and momentum. The electron's charge and mass only seemingly orbit around the atomic nucleus. In particular, the atomic shells and subshells appear as different modes of oscillation of $\mathbf {r}_{e}=\mathbf {r}_{e}(\theta ,\varphi )$ related to the stationary wave function $\varPhi _{e}(\varrho ,\theta ,\varphi )=R_{e}(\varrho )Y_{e}(\theta ,\varphi )$, where $\varrho , \theta$ and φ are the radial coordinate, polar and azimuthal angles, respectively. The associated vector field is $\boldsymbol {\Phi }_{e}(\theta ,\varphi )=A_{e}Y_{e}(\theta ,\varphi )\mathbf {n}_{e}$, where $\mathbf {n}_{e}$ is the radial unit vector. The stationary wave function satisfies the Helmholtz equation

$$\begin{equation} \left(\Delta+\zeta^2(\varrho)\right)\varPhi_{e}(\varrho,\theta,\varphi)=0, \end{equation} \tag{ 23 }$$

where $\zeta ^{2}(\varrho )=2m_{e}\hbar ^{-2}(E-U(\varrho )), E$ is the total energy of the oscillator and $U(\varrho )=-m_{e}c^{2}r_{ce}\varrho ^{-1}$ is its non-regularized potential energy. Note that as in eq. (19), the delta operator in eq. (23) is associated only to the kinetics of $\mathbf {r}_{e}$ and not to the motion of electron's charge and mass. Therefore, in the corresponding Schrödinger equation the momentum operator of the electron will be associated only to the dynamics of its field vector. Moreover, the hydrogen wave functions will be related to the probability of finding the electron's field vector pointing at a certain direction in space and having a particular magnitude. The position and momentum associated to the field vector will be naturally related via the Heisenberg uncertainty relations.

The solutions of eq. (23) are orthogonal polynomials associated to the eigenstates of the hydrogen atom. We have $\varPhi _{e}(\varrho ,\theta ,\varphi )\to \varPhi _{nlm}(\varrho ,\theta ,\varphi )\equiv \Lambda _{lm}R_{nl}(\varrho )Y^{m}_{l}(\theta ,\varphi )$, where $R_{nl}(\varrho )$ are the radial wave functions and $Y^{m}_{l}(\theta ,\varphi )$ the spherical harmonics. Moreover, n, l and m are the prime, orbital and magnetic quantum numbers, respectively. The oscillator's energy is

$$\begin{equation} E_n = -\frac{m_ec^2r_{ce}}{2n^2r_B}. \end{equation}$$

For all n and l, the field's amplitude $A_{e}\to A_{nl}$ is the peak amplitude equal to the value of ϱ at the global maximum of the function $|\varrho R_{nl}(\varrho )|$. Furthermore, $Y_{e}(\theta ,\varphi )\to \Lambda _{lm}Y^{m}_{l}(\theta ,\varphi )$, where for all l and m the normalization constant Λ_lm preserves the peak amplitude. In particular, for $m,\theta =\{0,0\,\},\{\pm l,0.5\,\pi \}$, we have the constraint $\Lambda _{lm}|Y^{m}_{l}(\theta ,0)|=1$. Thus, for the electron field vector we have $\mathbf {r}_{e}\to \mathbf {r}_{nlm}=A_{nl} \mathrm {Re}\{\Lambda _{lm}Y^{m}_{l}(\theta ,\varphi )\}\mathbf {n}_{e}$, where the corresponding field reads $\boldsymbol {\Phi }_{e}(\theta ,\varphi )\to \boldsymbol {\Phi }_{nlm}(\theta ,\varphi )=A_{nl} \Lambda _{lm}Y^{m}_{l}(\theta ,\varphi )\mathbf {n}_{e}$.

The angular dynamics related to three different atomic orbitals, with $A_{10}=r_{B}, A_{21}=4r_{B}$ and $A_{32}=9r_{B}$, is depicted in fig. 4.