2026-07-12
Internal motion of the electron and the origin of its magnetic moment
The previous two parts introduced a phase basis of two independent complex planes and considered the action of the modulation operator on the internal state and external motion of a particle. Here, this framework is applied to an electron whose center is at rest. The aim of this article is to specify a physical mapping of internal phase periodicity onto a closed charge-carrying contour and to examine which additional assumptions reproduce the principal magnetic-moment scale.
Once the contour radius, charge-transport speed, total-charge normalization, and oriented area are specified, the calculation yields the Bohr magneton. This is a conditional result of the adopted current mapping, rather than a derivation of all these premises from the phase operator. The comparison with spin is performed separately, using the standard spinor representation.
We consider the principal magnetic scale without the anomalous correction. A finite separation between the branches adds a geometric contribution to the path length, but its connection to a positive anomaly requires a separate dynamical justification. This issue is discussed in the article "The Anomalous Magnetic Moment of the Electron".
1. Connection to the previous parts
The finite-level part of the phase state is written using the operator
\[\tag{1} J(a,b)=\j^a(-\j)^b. \] In the idempotent basis, it takes the form
\[\tag{2} J(a,b) =\ep e^{i\pi b} +\em e^{i\pi a}. \] The parameter a describes the internal phase, while the parameter b describes the external state of motion. For an electron whose center is at rest in the chosen reference frame, we set b=0. Then
\[\tag{3} J_{\mathrm{int}}(t) =J(a(t),0) =\ep+\em e^{i\pi a(t)}. \] For a uniformly changing internal phase, let us introduce its angular frequency \(\omega\), defined with respect to time \(t\) in the rest frame of the center:
\[\tag{4} a(t)=\frac{\omega t}{\pi}, \qquad J_{\mathrm{int}}(t) =\ep+\em e^{i\omega t}. \] The elements \(\ep,i\ep,\em,i\em\) are basis elements of the finite-level phase space; the coordinates are their coefficients. This basis is not identified with the laboratory coordinates \(ct,x,y,z\). The distinction is discussed in the first and second parts.
2. Phase dynamics and algebraic unitarity
Let us define a formal phase quantity with the dimensions of velocity:
\[\tag{5} V_J(t)=cJ_{\mathrm{int}}(t). \] Under the adopted conjugation and idempotent relations, the operator satisfies the algebraic unitarity condition:
\[\tag{6} J_{\mathrm{int}}(t) \overline{J}_{\mathrm{int}}(t) =1. \] Equality (6) does not define a Hilbert-space norm, energy, or probability. Without a physical mapping, the quantity \(V_J=cJ\) also specifies neither the laboratory velocity of the center nor the charge-transport speed. Its formal integral defines a phase curve:
\[\tag{7} \mathbf R_J(t) =\int V_J(t)\,dt. \] For the rotating component with frequency \(\omega\), a characteristic phase radius arises:
\[\tag{8} r(\omega)=\frac{c}{\omega}. \] The radius in (8) pertains to the rotating component of the formal integral. The full integral (7) also contains the linear term \(ct\ep\) and is not a closed circle. Obtaining a closed physical contour requires a separate projection and a specification of periodicity. Below, the transport speed of the charge distribution is taken to be \(c\); this is an additional hypothesis, rather than a consequence of (6).
3. Two internal scales of the electron
Let us adopt the reduced Compton wavelength as the mean radius of the magnetic contour:
\[\tag{9} \boxed{ R=\overline{\lambda}_C =\frac{\hbar}{m_ec} }. \] The second scale is the classical electron radius, defined as a combination of constants:
\[\tag{10} \boxed{ r_e =\frac{e^2}{4\pi\varepsilon_0m_ec^2} }. \] These scales are related by the fine-structure constant:
\[\tag{11} \boxed{ r_e=\alpha_{\mathrm{fs}}R, \qquad \frac{r_e}{R}=\alpha_{\mathrm{fs}} }. \] In the model, \(R\) is adopted as the mean contour radius, and \(r_e\) as the separation between nearby branches. These are physical assignments of two known scales, rather than a proven derivation of the electron's spatial dimensions. In particular, the classical radius must not automatically be interpreted as a measured particle size.
4. Two related frequencies
Let us define two angular frequencies by the kinematic rule "speed divided by radius." For the scale \(r_e\), denote this frequency by \(\omega_{\mathrm{int}}\), and for \(R\) by \(\Omega_C\):
\[\tag{12} \omega_{\mathrm{int}}r_e=c, \qquad \Omega_CR=c. \] Using relation (11), we obtain
\[\tag{13} \boxed{ \Omega_C =\alpha_{\mathrm{fs}} \omega_{\mathrm{int}} }. \] The Compton frequency is directly related to the rest energy:
\[\tag{14} \Omega_C =\frac{m_ec^2}{\hbar}. \] Once \(R\) has been defined by (9), equality (14) is an identity. In this notation, the rest energy is written as
\[\tag{15} \boxed{ m_ec^2 =\hbar\Omega_C =\alpha_{\mathrm{fs}} \hbar\omega_{\mathrm{int}} }. \] Relations (13)–(15) follow from the definitions of the scales and frequencies. They do not by themselves derive mass from the internal field energy or establish a frequency-projection mechanism. Here, the constant \(\alpha_{\mathrm{fs}}\) is not an inverse Lorentz factor. The general phase frequency \(\omega\) in (4), the frequency \(\omega_{\mathrm{int}}=c/r_e\), and the frequency \(\Omega_C=c/R\) must be distinguished until a law relating them physically has been specified.
5. From the phase circle to the physical contour
A closed physical structure requires an additional mapping of the phase dynamics, including removal of the linear component of integral (7). Let us denote this mapping symbolically:
\[\tag{16} \mathbf R_J(t) \longrightarrow \Gamma_e. \] Here, \(\Gamma_e\) is the proposed closed transport path of the internal charge distribution, rather than the trajectory of the electron's center. For two nearby circular branches, let us adopt:
\[\tag{17} \boxed{ R_+=R+\frac{r_e}{2}, \qquad R_-=R-\frac{r_e}{2} }. \] The same wave traverses both branches sequentially. The mean radius \(R\) determines the principal magnetic scale, while the separation
\[\tag{18} R_+-R_-=r_e \] describes a deeper splitting of the internal contour.
The mapping onto a charge-carrying contour, its mean radius, and its splitting are physical hypotheses. The finite-level part \(J(a,b)\) does not by itself describe all the information in the two-sheeted extension: this requires an internal factor \(Q_\Gamma\) consistent with the article on spin. Charge is specified by a separate sector of the complete state; this article considers an electron with \(q=-e\), where \(e>0\).
6. Two-sheeted closure
First, consider the principal magnetic moment to zeroth order in the small ratio \(r_e/R=\alpha_{\mathrm{fs}}\). In this approximation, both branches project onto the same radius \(R\) but remain distinct sheets of the internal state.
The length of each circular traversal is
\[\tag{19} L_1=L_2=2\pi R. \] In the adopted two-sheeted extension, the complete internal state returns to its initial configuration after both sheets have been traversed sequentially. Neglecting the lengths of the connecting transitions:
\[\tag{20} \boxed{ L_0=L_1+L_2=4\pi R }. \] If the charge distribution is transported along the path at speed \(c\), the full period is
\[\tag{21} \boxed{ T_0=\frac{L_0}{c} =\frac{4\pi R}{c} }. \] One circular traversal corresponds to an internal geometric angle of \(2\pi\), and the full two-sheeted cycle to \(4\pi\). For the fundamental mode, which accumulates a phase of \(2\pi\) over the time \(T_0\), the temporal phase frequency is \(\omega_\Gamma=2\pi/T_0=c/(2R)=\Omega_C/2\). This is not the frequency \(\omega_{\mathrm{int}}\) in (12). The periodicity of the internal extension also does not prove an overall sign change of the complete spinor under a physical rotation: the law of physical rotations is introduced separately.
7. The equivalent internal current
To calculate the current, let us additionally adopt a stationary charge distribution confined to the entire thin sequential contour and normalized to a single electron charge. At constant speed \(c\), the uniform linear charge density is \(\lambda_q=q/L_0\), so the signed current is \(I_q=\lambda_qc=q/T_0\). Below, \(I_0=|I_q|\) denotes its positive magnitude:
\[\tag{22} I_0=\frac{e}{T_0}. \] Substituting the period (21), we obtain
\[\tag{23} \boxed{ I_0=\frac{ec}{4\pi R} }. \] This is an effective stationary current mapping, rather than the rotation of a rigid body or a point charge. Charge is normalized over the entire contour, rather than separately on each sheet. The physical realization must ensure charge conservation: \(\int\rho\,d^3r=q\) and \(\partial_t\rho+\nabla\cdot\mathbf j_q=0\). For a nonstationary charge wave, a separate justification is needed for why its time-averaged current yields (22); phase velocity alone does not determine charge transport.
8. The total oriented area
Let us additionally adopt the same orientation for both circular traversals. The vector area of the full path is defined as \(\mathbf A_\Gamma=\tfrac12\oint_\Gamma\mathbf r\times d\mathbf r\). To zeroth order, the connecting segments are neglected, and the two circular parts give \(\mathbf A_\Gamma=A_\Sigma\widehat{\mathbf n}_\Gamma\), where
\[\tag{24} \boxed{ A_{\Sigma} =\pi R^2+\pi R^2 =2\pi R^2 }. \] The two terms do not imply two independent charges: current normalization already accounts for the full path. For equal traversals with opposite orientations, their vector areas cancel. At finite separation, the contribution of the connecting segments to the area depends on their geometry and must be calculated separately. The same orientation of the branches is an assumption of the magnetic mapping, rather than a consequence of having two sheets.
9. Reproducing the Bohr magneton scale
For a localized stationary current, the magnetic moment in SI units is defined as \(\boldsymbol\mu=\tfrac12\int\mathbf r\times\mathbf j_q\,d^3r\). In the adopted thin-contour approximation, this gives \(\boldsymbol\mu=I_q\mathbf A_\Gamma\). Let \(\mu_e^{(0)}\) denote the positive magnetic-moment scale of the chosen contour:
\[\tag{25} \mu_e^{(0)}=I_0A_{\Sigma}. \] Using (23) and (24), we find
\[\tag{26} \mu_e^{(0)} =\frac{ec}{4\pi R} \,2\pi R^2 =\frac{ecR}{2}. \] Let us substitute the previously adopted contour radius:
\[\tag{27} R=\frac{\hbar}{m_ec}. \] Then
\[\tag{28} \boxed{ \mu_e^{(0)} =\frac{ec}{2} \frac{\hbar}{m_ec} =\frac{e\hbar}{2m_e} =\mu_B }. \] Result (28) follows from the combined adopted premises: \(R=\overline\lambda_C\), charge transport at speed \(c\), a total charge of \(-e\), a stationary current, and the same orientation of the branches. Doubling the number of traversals does not by itself increase the moment: for one circular traversal, \(T_1=2\pi R/c\), \(I_1=ec/(2\pi R)\), and \(A_1=\pi R^2\), so \(I_1A_1=ecR/2\) as well. More generally, for \(N\) traversals of a single normalized charge with the same orientation, the factors of \(N\) in the area and period cancel. Here, the two-sheeted structure specifies internal periodicity but does not independently explain the value of the magnetic scale.
10. Direction of the magnetic moment
The path orientation defines the geometric normal \(\widehat{\mathbf n}_\Gamma\). For \(q=-e\), the current-loop moment points opposite to this normal. To connect it to spin, let us additionally adopt the mapping \(\widehat{\mathbf n}_\Gamma\mapsto\widehat{\mathbf n}_s\) for a state with positive spin projection onto the chosen axis. The mean magnetic moment of this state is then
\[\tag{29} \boxed{\langle\widehat{\boldsymbol\mu}_e^{(0)}\rangle=-\mu_B\widehat{\mathbf n}_s}. \] The contour normal, spin state, current direction, charge sign, and external motion have different definitions. Interchanging the principal channels defines the realizations \(J_1(a,b)=J(a,b)\) and \(J_2(a,b)=J(b,a)\) but does not establish their orthogonality as spin states. This correspondence is introduced separately. The choice of realization does not change the charge \(q=-e\), while the parameter \(b\) retains its meaning as external motion. For the opposite spin projection, its connection to the opposite magnetic-moment orientation must also be part of the adopted mapping.
11. Comparison with spin and the factor g=2
Let us adopt the standard spin representation and the gyromagnetic operator relation. Here, \(g>0\), and the sign of the electron's magnetic moment is accounted for by its charge \(q=-e\):
\[\tag{30} \widehat{\boldsymbol\mu}=g\frac{q}{2m_e}\widehat{\mathbf S},\qquad q=-e,\quad e>0. \] Spin space is introduced separately from the contour geometry. For the electron, \(s=1/2\), the projection eigenvalues are \(\pm\hbar/2\), and the squared-spin operator satisfies \(\widehat{\mathbf S}^{\,2}=3\hbar^2I/4\). The quantum magnitude is therefore \(\sqrt3\hbar/2\). In a state with positive projection onto \(\widehat{\mathbf n}_s\), the mean spin vector takes the form
\[\tag{31} \langle\widehat{\mathbf S}\rangle=\frac{\hbar}{2}\widehat{\mathbf n}_s,\qquad \langle\widehat S_{\mathbf n_s}\rangle=\frac{\hbar}{2}. \] Thus, \(\hbar/2\) in (31) is the projection and the magnitude of the mean vector in a fully polarized state, rather than the quantum magnitude \(\sqrt{\langle\widehat{\mathbf S}^{\,2}\rangle}\). Let us compare the magnetic projection of this state with the positive contour scale:
\[\tag{32} |\langle\widehat\mu_{\mathbf n_s}^{(0)}\rangle|=\mu_e^{(0)}=\frac{e\hbar}{2m_e}. \] Equations (30)–(32) imply \(|\langle\widehat\mu_{\mathbf n_s}\rangle|=g_0e\hbar/(4m_e)=\mu_B\), hence
\[\tag{33} \boxed{g_0=2}. \] The value of \(g_0\) was not inserted into calculation (22)–(28). However, (33) is obtained only after adopting the charge-current normalization, radius, spin representation, and connection between contour orientation and spin. This reproduces \(g_0=2\) within the specified mapping, rather than independently deriving spin from the two-sheeted structure. The distinction between the internal cycle and physical rotations is discussed further in the article "The Two-Sheeted Cycle and the Spin Description of the Electron".
12. The anomalous component of the magnetic moment
The principal result (28) pertains to the zeroth-order approximation. At finite separation, the radii are \(R_+\) and \(R_-\). To estimate the length, let us additionally adopt two straight radial transitions, each of length \(r_e\); a different transition geometry will change this estimate.
With this choice of transitions, the full path length is
\[\tag{34} \boxed{ L_{\Gamma} =2\pi R_+ +2\pi R_- +2r_e =4\pi R+2r_e }. \] The relative addition to the unperturbed length \(L_0=4\pi R\) is
\[\tag{35} \boxed{ \frac{L_{\Gamma}-L_0}{L_0} =\frac{2r_e}{4\pi R} =\frac{r_e}{2\pi R} =\frac{\alpha_{\mathrm{fs}}}{2\pi} }. \] Numerically, this geometric coefficient coincides with the leading term of the anomaly in quantum electrodynamics. We give it as an external result for comparison, rather than as a consequence of (35):
\[\tag{36} a_e^{(1)} =\frac{\alpha_{\mathrm{fs}}}{2\pi}. \] In the standard spin description, \(a_e=(g-2)/2\). The positive magnitude of the magnetic projection in a fully polarized state is written as
\[\tag{37} |\langle\widehat\mu_{\mathbf n_s}\rangle|=\mu_B(1+a_e),\qquad g=2(1+a_e). \] Increasing the path length does not by itself produce a positive anomaly. At unchanged charge and speed, \(I=e c/L_\Gamma\) decreases. For two radial connections with zero contribution to the vector area, \(A_\Gamma=\pi(R_+^2+R_-^2)=2\pi R^2(1+\alpha_{\mathrm{fs}}^2/4)\). The same current calculation therefore gives \(\mu_\Gamma/\mu_B=(1+\alpha_{\mathrm{fs}}^2/4)/(1+\alpha_{\mathrm{fs}}/(2\pi))=1-\alpha_{\mathrm{fs}}/(2\pi)+O(\alpha_{\mathrm{fs}}^2)\). The linear correction has the opposite sign to (36). A positive anomaly requires additional dynamics or a different magnetic mapping, justified independently of the desired result. This issue is considered in the article "The Anomalous Magnetic Moment of the Electron: A Geometric Derivation".
13. Comparison with zitterbewegung
The construction has a qualitative resemblance to the internal periodicity discussed in connection with zitterbewegung in Dirac theory: it involves the speed-of-light scale and the Compton wavelength, while the motion of the center is considered separately. This is a comparison of scales, rather than proof of physical charge motion along the adopted contour.
In the standard description of a free Dirac particle, zitterbewegung is associated with interference between positive- and negative-energy components. The characteristic angular frequency in the rest frame is \(2m_ec^2/\hbar=2\Omega_C\); it does not coincide with the fundamental frequency of the two-sheeted cycle, \(\omega_\Gamma=\Omega_C/2\). In the present model, the phase operator and its spatial mapping are specified differently. Mathematical equivalence between the two constructions has therefore not been established.
14. What follows from the geometry and what remains hypothetical
Once the geometry, charge transport, and spin mapping are adopted, the following conditional results are obtained:
1. Neglecting the connecting segments, the full path is \(L_0=4\pi R\);
2. At transport speed \(c\), the period is \(T_0=4\pi R/c\);
3. For a stationary distribution of a single charge, the current magnitude is \(I_0=ec/(4\pi R)\);
4. For circular branches with the same orientation, the area is \(A_\Sigma=2\pi R^2\);
5. With \(R=\hbar/(m_ec)\), the positive magnetic scale is \(\mu_e^{(0)}=\mu_B\);
6. A separate comparison with the projection \(\hbar/2\) gives \(g_0=2\).
2. At transport speed \(c\), the period is \(T_0=4\pi R/c\);
3. For a stationary distribution of a single charge, the current magnitude is \(I_0=ec/(4\pi R)\);
4. For circular branches with the same orientation, the area is \(A_\Sigma=2\pi R^2\);
5. With \(R=\hbar/(m_ec)\), the positive magnetic scale is \(\mu_e^{(0)}=\mu_B\);
6. A separate comparison with the projection \(\hbar/2\) gives \(g_0=2\).
The following remain physical hypotheses of the model:
1. The mapping of phase dynamics and the internal factor \(Q_\Gamma\) onto a physical contour;
2. Justification of the radius \(R\), separation \(R_+-R_-=r_e\), speed, and charge-current normalization;
3. The correspondence of dynamical realizations to orthogonal spin states and contour orientation;
4. The mechanism of interaction with an external field, stability of the internal state, and absence of energy losses;
5. A dynamical mechanism for a positive anomaly that resolves the sign discrepancy of the simple current calculation;
6. The origin of higher-order correction terms. The charge sign is specified by an independent charge sector, rather than by the choice of sheet or spin direction.
2. Justification of the radius \(R\), separation \(R_+-R_-=r_e\), speed, and charge-current normalization;
3. The correspondence of dynamical realizations to orthogonal spin states and contour orientation;
4. The mechanism of interaction with an external field, stability of the internal state, and absence of energy losses;
5. A dynamical mechanism for a positive anomaly that resolves the sign discrepancy of the simple current calculation;
6. The origin of higher-order correction terms. The charge sign is specified by an independent charge sector, rather than by the choice of sheet or spin direction.
This distinction is necessary: the algebra of the phase operator, the geometry of the adopted contour, and its physical mapping belong to different levels of the model's construction.
Conclusion
The phase basis allows internal periodicity to be described without identifying phase space with laboratory spacetime. The mapping onto a two-sheeted charge-carrying contour, the Compton radius, the transport speed, and normalization to a single charge are introduced as separate physical premises.
In the adopted zeroth-order approximation, the full internal path comprises two traversals with a total length of \(4\pi R\). For stationary transport of a single charge and branches with the same orientation, the current and area yield the following conditional calculation:
\[\tag{38} \boxed{ \begin{gathered} J_{\mathrm{int}}(t) \longrightarrow \text{two-sheeted contour}, \qquad R=\overline{\lambda}_C, \qquad L_0=4\pi R, \\[4pt] I_0=\frac{ec}{4\pi R}, \qquad A_{\Sigma}=2\pi R^2, \\[4pt] \mu_e^{(0)} =I_0A_{\Sigma} =\frac{e\hbar}{2m_e} =\mu_B. \end{gathered} }. \] The calculation reproduces the magnetic scale \(\mu_B\) and, after adopting the standard spin representation and magnetic mapping, \(g_0=2\). The two-sheeted period alone derives neither spin nor an enhancement of the magnetic moment: the doublings of period and area cancel. The relative scale of the additional length, \(\alpha_{\mathrm{fs}}/(2\pi)\), also does not establish a positive anomaly. Connecting the internal geometry to charge current, spin, interaction with a field, and stable dynamics remains an essential part of the model's further development.
Materials used
- Wikipedia. Fine-structure constant.
- Wikipedia. Electron magnetic moment.