2026-07-12
Internal motion of the electron and the origin of its magnetic moment
In the previous two parts, we introduced the phase basis of two independent complex planes and demonstrated how the action of the modulation operator changes the internal state and external motion of a particle. This framework will now be applied to an electron at rest. The main goal of this article is to relate the internal phase periodicity to the closed two-sheet geometry of the electron and directly derive the fundamental scale of its magnetic moment.
The key result will be the geometric derivation of the Bohr magneton. Unlike the purely formal substitution of a ready-made gyromagnetic factor, here we first find the internal round-trip period, the equivalent current, and the total oriented area of the two sheets. Only then will the result be compared with the standard notation for the electron's magnetic moment.
This article examines the fundamental, unperturbed magnetic moment of the electron. Its small anomalous component is associated with the final splitting of the two branches and is detailed in a separate paper, "The Anomalous Magnetic Moment of the Electron".
1. Connection with Previous Sections
The complete final phase state is written by the operator
\[\tag{1} J(a,b)=\j^a(-\j)^b. \] In an idempotent basis, it takes the form
\[\tag{2} J(a,b) =\ep e^{i\pi b} +\em e^{i\pi a}. \] Parameter a describes the internal phase, and 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
\[\tag{3} J_{\mathrm{int}}(t) =J(a(t),0) =\ep+\em e^{i\pi a(t)}. \] If the intrinsic angular frequency is \(\omega\), then
\[\tag{4} a(t)=\frac{\omega t}{\pi}, \qquad J_{\mathrm{int}}(t) =\ep+\em e^{i\omega t}. \] The elements \(\ep,i\ep,\em,i\em\) are coordinates of the finite phase space. They are not identified with the laboratory coordinates \(ct,x,y,z\). A full discussion of this distinction is given in parts one and part two.
2. Phase Dynamics of a Constant Norm
Multiplication of the operator by the speed of light introduces the full phase-geometric dynamics:
\[\tag{5} V_J(t)=cJ_{\mathrm{int}}(t). \] Since the two idempotent planes are orthogonal, the finite operator preserves the unit norm:
\[\tag{6} J_{\mathrm{int}}(t) \overline{J}_{\mathrm{int}}(t) =1. \] The quantity \(V_J=cJ\) is not the laboratory velocity of the electron center. It describes the passage through an internal phase state with a constant norm. The corresponding phase curve can be obtained by integration:
\[\tag{7} \mathbf R_J(t) =\int V_J(t)\,dt. \] For a rotating component with a frequency \(\omega\), a characteristic phase radius arises.
\[\tag{8} r(\omega)=\frac{c}{\omega}. \] This formula expresses the general principle of the relationship between period and scale. However, the phase circle itself is not yet a literal trajectory of a point charge. For a physical description of the electron, it is necessary to separately define a mapping of the phase periodicity onto a closed internal wave contour.
3. Two Internal Scales of the Electron
In the geometry of the electron, two radii must be distinguished. The first is the reduced Compton length:
\[\tag{9} \boxed{ R=\overline{\lambda}_C =\frac{\hbar}{m_ec} }. \] The second is the classical radius of the electron:
\[\tag{10} \boxed{ r_e =\frac{e^2}{4\pi\varepsilon_0m_ec^2} }. \] These scales are related by the fine-structure constant:
\[\tag{11} \boxed{ r_e=\alpha_{\mathrm{fs}}R, \qquad \frac{r_e}{R}=\alpha_{\mathrm{fs}} }. \] The radius \(R\) defines the average scale of a closed magnetic circuit. The shorter length \(r_e\) characterizes the internal electromagnetic scale and, in the split model, determines the distance between two close branches. Therefore, these radii refer to different levels of the same geometry and should not replace each other.
4. Two Related Frequencies
Each internal scale corresponds to a natural frequency. For the electromagnetic scale \(r_e\), we introduce the frequency \(omega_{\mathrm{int}}\), and for the Compton scale \(R\), we introduce the frequency \(\Omega_C\):
\[\tag{12} \omega_{\mathrm{int}}r_e=c, \qquad \Omega_CR=c. \] 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:
\[\tag{14} \Omega_C =\frac{m_ec^2}{\hbar}. \] Therefore, the internal mass projection is written as
\[\tag{15} \boxed{ m_ec^2 =\hbar\Omega_C =\alpha_{\mathrm{fs}} \hbar\omega_{\mathrm{int}} }. \] The fine structure constant here relates the two internal frequencies and the two geometric scales. It is not interpreted as the inverse Lorentz factor. The Lorentz factor relates to the external motion of an already formed particle, while \(\alpha_{\mathrm{fs}}\) participates in the internal formation of rest energy.
5. From the Phase Circle to the Physical Contour
The phase periodicity of the operator must be mapped onto the physical structure of the electron. We denote this mapping symbolically:
\[\tag{16} \mathbf R_J(t) \longrightarrow \Gamma_e. \] Here \(\Gamma_e\) is not the trajectory of the electron's center, but the closed path of the internal charge wave. In a deeply split geometry, this path has two closely spaced branches:
\[\tag{17} \boxed{ R_+=R+\frac{r_e}{2}, \qquad R_-=R-\frac{r_e}{2} }. \] The same wave sequentially traverses both branches. The average radius \(R\) determines the fundamental magnetic scale, and the distance
\[\tag{18} R_+-R_-=r_e \] describes a deeper splitting of the inner contour.
The transition from a phase operator to a two-sheet charge contour is a physical hypothesis of the model. The operator algebra allows for independent splitting, but the specific spatial mapping of the branches requires a separate physical interpretation.
6. Two-sheeted closure
First, let us consider the fundamental magnetic moment in the zeroth approximation in the small ratio \(r_e/R=\alpha_{\mathrm{fs}}\). In this approximation, both branches have the same radius projection \(R\), but remain distinct sheets of the internal state.
The length of each circular path is
\[\tag{19} L_1=L_2=2\pi R. \] The complete internal state returns to its original configuration only after successively traversing both sheets:
\[\tag{20} \boxed{ L_0=L_1+L_2=4\pi R }. \] At the internal wave propagation speed of \(c\), the total period is
\[\tag{21} \boxed{ T_0=\frac{L_0}{c} =\frac{4\pi R}{c} }. \] After the first traverse, the phase may have the same observed direction, but the wave is on a different sheet. Only the second traverse restores both the phase and the sheet of the internal state. Therefore, a complete closure corresponds to a period of \(4\pi\).
7. Equivalent Internal Current
A closed passage of a charge wave is equivalent to an electric current. In time \(T_0\), the charge passes through the entire two-sheet cycle, so the equivalent current is
\[\tag{22} I_0=\frac{e}{T_0}. \] Substituting the period (21), we obtain
\[\tag{23} \boxed{ I_0=\frac{ec}{4\pi R} }. \] We are not talking about the rotation of a rigid charged body. The current \(I_0\) is the effective characteristic of a single continuous charge wave closed on two consecutive sheets.
8. Total Oriented Area
Both sheets are traversed in the same circular orientation. Therefore, their contributions to the magnetic area are additive. In the basic approximation, each sheet covers an area of \(\pi R^2\), and their sum is
\[\tag{24} \boxed{ A_{\Sigma} =\pi R^2+\pi R^2 =2\pi R^2 }. \] The two terms in (24) do not imply the presence of two independent charges. These are two consecutive parts of a single complete circuit. If the orientations of the circuits were opposite, their magnetic contributions would cancel each other out. For the electron state, the coordinated passage of both branches in the same orientation is considered.
9. Geometric Derivation of the Bohr Magneton
The magnetic moment of a closed circuit is determined by the product of the equivalent current and the total oriented area:
\[\tag{25} \mu_e^{(0)}=I_0A_{\Sigma}. \] Using (23) and (24), we find
\[\tag{26} \mu_e^{(0)} =\frac{ec}{4\pi R} \,2\pi R^2 =\frac{ecR}{2}. \] Now let's substitute the geometric radius of the main contour:
\[\tag{27} R=\frac{\hbar}{m_ec}. \] Then
\[\tag{28} \boxed{ \mu_e^{(0)} =\frac{ec}{2} \frac{\hbar}{m_ec} =\frac{e\hbar}{2m_e} =\mu_B }. \] The fundamental magnetic moment arises from three geometric properties: the internal wave velocity \(c\), the Compton radius \(R=\overline{\lambda}_C\), and the two-sheet period \(4\pi\). No additional factorThe \(1/\alpha_{\mathrm{fs}}\) term is not introduced in this derivation.
10. Direction of the Magnetic Moment
The oriented contour defines the normal \(\widehat{\mathbf n}_s\). For a positive charge, the direction of the magnetic moment is determined by the right-hand screw rule. The electron's charge is negative, so its magnetic moment is directed opposite to the chosen normal of the internal state:
\[\tag{29} \boxed{ \boldsymbol{\mu}_e^{(0)} =-\mu_B\widehat{\mathbf n}_s }. \] The model must distinguish several independent characteristics. The choice of one of two orthogonal phase planes determines the orientation of the spin state. The direction of the closed loop determines the orientation of the internal current. The sign of the charge refers to a separate property of the total particle operator, and the parameter \(b\) describes the external motion of the center. These characteristics can be related by a physical mapping, but should not be identified a priori.
11. The geometric meaning of the factor g = 2
In standard gyromagnetic notation, the magnetic moment is related to the spin by the relation
\[\tag{30} \boldsymbol{\mu} =g\frac{q}{2m_e}\mathbf S. \] The two-sheet periodicity of the internal state corresponds to the spin
\[\tag{31} |\mathbf S|=\frac{\hbar}{2}. \] Comparing this result with the value already obtained from geometry
\[\tag{32} |\boldsymbol{\mu}_e^{(0)}| =\frac{e\hbar}{2m_e}, \] We find the fundamental gyromagnetic factor:
\[\tag{33} \boxed{g_0=2}. \] Thus, the value \(g_0=2\) is not used as the initial substitution when calculating the magnetic moment. It appears after comparing the geometrically determined moment with the half-spin of the two-sheet state. A more detailed discussion of the origin of the period \(4\pi\) and spin is given in the article "Geometric Model of Electron Spin".
12. Anomalous component of the magnetic moment
The main result (28) was obtained in the zeroth approximation, when two sheets are projected onto a single average radius \(R\). Taking into account the finite splitting, the branches have radii \(R_+\) and \(R_-\), and two transitions of length on the order of \(r_e\) appear between them.
The total length of the split path becomes equal to
\[\tag{34} \boxed{ L_{\Gamma} =2\pi R_+ +2\pi R_- +2r_e =4\pi R+2r_e }. \] The relative addition to the unperturbed length \(L_0=4\pi R\) is
\[\tag{35} \boxed{ \frac{L_{\Gamma}-L_0}{L_0} =\frac{2r_e}{4\pi R} =\frac{r_e}{2\pi R} =\frac{\alpha_{\mathrm{fs}}}{2\pi} }. \] The same dimensionless quantity forms the first term of the magnetic moment anomaly:
\[\tag{36} a_e^{(1)} =\frac{\alpha_{\mathrm{fs}}}{2\pi}. \] As a result, the measured magnetic moment modulus is written as
\[\tag{37} |\boldsymbol{\mu}_e| =\mu_B(1+a_e). \] Formula (35) shows the geometric scale of the correction, but the transition from the added length to the observed anomalous frequency requires a separate magnetic mapping. This step, its limitations, and its relationship to the experimental determination of \(a_e\) are discussed in detail in the article "Anomalous Magnetic Moment of the Electron. Geometric Derivation".
13. Comparison with Zitterbewegung
The resulting construction bears a qualitative similarity to the motion known in Dirac theory as zitterbewegung. In both cases, a rapid internal periodicity related to the speed of light and the Compton scale is assumed, while the motion of the particle's center is considered separately.
However, this similarity does not imply the identity of the two models. In the Dirac equation, zitterbewegung arises from the interference of the positive and negative energy components of the spinor. In the present model, internal periodicity is introduced through the phase operator and its mapping onto a two-sheeted closed contour. Therefore, the comparison should be understood as an indication of the common scale and common type of internal dynamics, rather than as an established mathematical equivalence.
14. What follows from the geometry and what remains a hypothesis?
After accepting the two-sheet contour, the following follows directly from its geometry:
1. total unperturbed path \(L_0=4\pi R\);
2. period \(T_0=4\pi R/c\);
3. equivalent current \(I_0=ec/(4\pi R)\);
4. total oriented area \(A_{\Sigma}=2\pi R^2\);
5. main magnetic moment \(\mu_e^{(0)}=\mu_B\);
6. after comparison with spin \(\hbar/2\) — value \(g_0=2\).
2. period \(T_0=4\pi R/c\);
3. equivalent current \(I_0=ec/(4\pi R)\);
4. total oriented area \(A_{\Sigma}=2\pi R^2\);
5. main magnetic moment \(\mu_e^{(0)}=\mu_B\);
6. after comparison with spin \(\hbar/2\) — value \(g_0=2\).
PhysicalThe model's hypotheses remain:
1. Mapping the phase dynamics of \(J_{\mathrm{int}}(t)\) onto a two-sheet charge contour;
2. Physical origin of spatial splitting \(R_+-R_-=r_e\);
3. Relationship between the direction of the deep cycle and the sign of the charge;
4. Mechanism of interaction of the internal contour with the external magnetic field;
5. Frequency mapping of the geometric additive to the observed anomaly;
6. Origin of the subsequent terms of the expansion of the anomalous magnetic moment.
2. Physical origin of spatial splitting \(R_+-R_-=r_e\);
3. Relationship between the direction of the deep cycle and the sign of the charge;
4. Mechanism of interaction of the internal contour with the external magnetic field;
5. Frequency mapping of the geometric additive to the observed anomaly;
6. Origin of the subsequent terms of the expansion of the anomalous magnetic moment.
This separation is necessary: the algebra of the phase operator, the geometry of the adopted contour, and its physical mapping belong to different levels of model construction.
Conclusion
The new phase basis allows us to describe the internal periodicity of the electron without identifying the phase coordinates with laboratory space-time. For a physical electron, this periodicity maps onto a closed two-sheet contour, the average radius of which is equal to the reduced Compton length.
Complete state recovery requires two round-trips and corresponds to a length of \(4\pi R\). During this time, one charge wave successively covers two identically oriented areas. Therefore, the geometric sequence has the form
\[\tag{38} \boxed{ \begin{gathered} J_{\mathrm{int}}(t) \longrightarrow \text{two-sheet contour}, \qquad R=\overline{\lambda}_C, \qquad L_0=4\pi R, \\[4pt] I_0=\frac{ec}{4\pi R}, \qquad A_{\Sigma}=2\pi R^2, \\[4pt] \mu_e^{(0)} =I_0A_{\Sigma} =\frac{e\hbar}{2m_e} =\mu_B. \end{gathered} }. \] Thus, the fundamental magnetic moment of the electron is determined by the internal wave velocity, the Compton scale, and the two-sheeted geometry of the closure. The classical radius \(r_e\) defines a deeper splitting of this contour and thereby links the fundamental geometry of the electron with the small anomalous component of its magnetic moment.
Materials used
- Wikipedia. Fine-structure constant.
- Wikipedia. Electron magnetic moment.