2026-08-04
Anomalous magnetic moment of the electron
Geometric derivation
This study extends the geometric model of the electron in which its internal state is represented by two nearby branches of a single closed wave path. The focus here is on frequencies rather than the classical calculation of the magnetic moment from current and loop area. This approach is closer to the experiment: in a Penning trap, the electron magnetic moment is determined from a ratio of measured frequencies.
The main question can be stated simply. If a wave must successively traverse both internal branches and two connecting segments before fully returning to its initial state, its path becomes slightly longer. Can the corresponding decrease in frequency reproduce the leading term of the electron's anomalous magnetic moment?
We will show that the additional length of the two connecting segments increases the effective radius of the complete cycle by \(r_e/(2\pi)\). Preserving the “frequency × radius” invariant then gives the relative frequency lag
\[\tag{1} \boxed{ \frac{\Omega_0-\Omega_\Gamma}{\Omega_\Gamma} = \frac{\alpha_{\mathrm{fs}}}{2\pi} }. \] The right-hand side coincides with the leading Schwinger term in the quantum-electrodynamic expansion of the electron anomaly. Two levels of this result must be distinguished from the outset. The geometric ratio (1) follows from the adopted two-branch model. Its mapping to experimental frequencies requires a separate hypothesis about the action of the external magnetic field. This distinction will be maintained throughout. The two-sheet contour is treated as the geometric basis of the leading approximation; subsequent corrections are hypothesized to involve additional splitting levels. Their coefficients and signs are not derived here.
Introduction
In the Dirac equation, the magnetic g-factor of a point electron is \(g=2\). It is convenient to separate this baseline value from a small correction by defining the magnetic moment anomaly:
\[\tag{2} a_e=\frac{g-2}{2}. \] If \(g=2\) held exactly, \(a_e\) would vanish. Experiment, however, shows a small positive deviation. Its leading contribution is
\[\tag{3} a_e^{(1)}=\frac{\alpha_{\mathrm{fs}}}{2\pi}. \] The factor \(\alpha_{\mathrm{fs}}\) characterizes the electromagnetic coupling. In the proposed geometric interpretation, the divisor \(2\pi\) is associated with a complete circular traversal. In standard quantum electrodynamics, Eq. (3) arises as a one-loop radiative contribution. The present model addresses a different question: whether the same dimensionless quantity can emerge as a geometric ratio of two nearby frequencies of an internal contour.
Such agreement does not, by itself, replace the quantum-electrodynamic calculation. It may, however, suggest an internal geometry underlying the leading anomaly term and explain why the divisor \(2\pi\) appears.
1. What the Experiment Measures
The electron magnetic moment cannot be observed as a charge moving around a visible internal circle. In precision experiments, a single electron is confined in a Penning trap. The external magnetic field establishes the cyclotron frequency \(\omega_c\), while the spin state is characterized by the frequency \(\omega_s\).
For an idealized free electron, these frequencies satisfy
\[\tag{4} \omega_s=\frac{g}{2}\,\omega_c. \] Their difference is called the anomaly frequency:
\[\tag{5} \omega_a=\omega_s-\omega_c. \] The anomaly is therefore expressed through a frequency ratio rather than the absolute magnitude of an internal current:
\[\tag{6} \boxed{ a_e=\frac{\omega_a}{\omega_c} =\frac{\omega_s-\omega_c}{\omega_c} }. \] Equation (6) is central to the construction that follows. It sets a requirement for a frequency-based model: the internal frequencies must be related to the measured spin and cyclotron frequencies so that their relative difference equals \(\alpha_{\mathrm{fs}}/(2\pi)\) to leading order. The common magnetic-field scale cancels in the ratio. A frequency-based approach therefore does not require assigning a classical current or an assumed magnetic area to the internal contour in advance.
2. The Standard Explanation: From the Dirac Equation to Quantum Electrodynamics
Before turning to the geometric model, it is useful to distinguish two results of standard theory. The Dirac equation describes the electron as a relativistic spin-\(1/2\) particle and already gives the magnetic g-factor \(g=2\) at leading order. The small deviation from two does not follow from the Dirac equation alone. It appears in quantum electrodynamics after the electromagnetic field is quantized.
In a popular account, the additional mechanism is pictured as follows. An electron emits a virtual photon, continues interacting with the external electromagnetic field, and then absorbs that same virtual photon. The magnetic vertex being measured is thus dressed by an internal photon loop. This is not a literal observable photon traveling along a definite trajectory, but a diagrammatic representation of an intermediate quantum state that cannot be registered separately.
This process is sometimes described as the electron interacting with its own electromagnetic field. That phrase is a useful visual analogy, but more precisely it is electron self-interaction through the quantized electromagnetic field. The same electron emits and absorbs the virtual photon, while the external field used to measure the magnetic moment enters at a separate vertex. Even at the leading, one-loop order, the calculation has the schematic form
\[\tag{6a} \begin{aligned} \Lambda^\mu(p',p) &\;\propto\; e^2\!\int\!\frac{d^4k}{(2\pi)^4} \;\gamma^\alpha \frac{\not p'-\not k+m_e}{(p'-k)^2-m_e^2+i0} \;\gamma^\mu \\[4pt] &\qquad\times \frac{\not p-\not k+m_e}{(p-k)^2-m_e^2+i0} \;\gamma_\alpha \frac{1}{k^2+i0} \quad\Longrightarrow\quad a_e=F_2(0)=\frac{\alpha_{\mathrm{fs}}}{2\pi}+\ldots \end{aligned} \] Here \(p\) and \(p'\) are the electron momenta before and after interaction with the external field, \(k\) is the virtual photon's four-momentum, and the integral runs over all possible values of \(k\). The two electron denominators describe intermediate electron states; the factor \(1/(k^2+i0)\) represents the virtual photon; and the central matrix \(\gamma^\mu\) represents interaction with the external electromagnetic field. The proportionality sign indicates that gauge, regularization, and renormalization details of the full calculation have been omitted.
Extracting the magnetic form factor \(F_2(0)\) from this vertex, Schwinger's one-loop calculation gives the leading term \(\alpha_{\mathrm{fs}}/(2\pi)\). Higher powers of \(\alpha_{\mathrm{fs}}\) arise from more complex diagrams involving multiple virtual photons, electron–positron loops, and other quantum corrections. In the standard explanation, the divisor \(2\pi\) thus results from evaluating a four-dimensional loop integral, rather than being specified in advance as a geometric circumference factor.
The model proposed below reproduces only this leading coefficient by a different route: \(\alpha_{\mathrm{fs}}\) appears as a ratio of two internal scales, and \(1/(2\pi)\) arises from the additional length of two connecting segments in the complete two-branch cycle. The numerical agreement is significant, but the physical content of the two results remains different. Quantum electrodynamics derives the coupling to the magnetic field and the subsequent expansion terms; in the geometric model, the magnetic projection \(K(B)\) and higher-order corrections remain to be constructed.
3. Two Electron Scales
The fundamental difference between the two approaches is especially clear here. In quantum electrodynamics, the leading anomaly term results from a one-loop calculation of the electron's interaction with a virtual photon and the external electromagnetic field. In the proposed model, the same quantity appears more simply through a geometric relation between two electron scales: the internal radius \(r_e\) and the radius \(R\) of the closed two-branch path. We therefore examine the ratio of these scales and its connection to the additional path length. Identifying the resulting ratio with the magnetic anomaly will require a separate physical mapping hypothesis.
The previously developed model distinguishes the internal electromagnetic scale \(r_e\) from the electron's reduced Compton wavelength. To simplify the notation, we denote the latter by \(R\):
\[\tag{7} R=\overline\lambda_C=\frac{\hbar}{m_ec}. \] The scale \(r_e\) is defined by equating the electromagnetic energy scale with the rest energy:
\[\tag{8} r_e=\frac{e^2}{4\pi\varepsilon_0m_ec^2}. \] These two lengths arise from different physical conditions. Their ratio therefore provides a natural dimensionless coefficient. Substituting (7) and (8) shows that it is exactly the fine-structure constant:
\[\tag{9} \boxed{ \frac{r_e}{R}=\alpha_{\mathrm{fs}} }. \] A detailed construction of this relation is presented in “The Geometric Origin of the Square of the Fine-Structure Constant and the Parameters of the Bohr Atom.” Here Eq. (9) will be used as an established relation.
Two frequency scales must be distinguished. The internal electromagnetic scale corresponds to a frequency of order \(c/r_e\), whereas the length \(R\) corresponds to the Compton angular frequency \(c/R\). This article concerns the second scale, for which the ratio \(r_e/R\) produces a small correction of order \(\alpha_{\mathrm{fs}}\).
4. The Radially Split Internal Contour
Suppose the electron's internal contour has two nearby branches of radii \(R_+\) and \(R_-\). They are symmetrically placed about the mean radius \(R\), and their separation equals the internal electromagnetic scale:
\[\tag{10} R_\pm=R\pm\frac{r_e}{2}, \qquad R_+-R_-=r_e. \] If these branches are treated as two independent circular motions and their radii are simply averaged, the linear correction vanishes: the mean is again \(R\). The existence of two radii alone therefore does not produce a term of first order in \(r_e/R\).
A new possibility arises if the branches belong to one continuous path rather than two independent waves. The wave traverses the outer branch, crosses to the inner branch, traverses it, and then returns to the original branch. The complete state closes only after both circles and both connecting segments have been traversed.
In the simplest geometry, the connecting segments are radial. Each has length \(r_e\). The complete path therefore consists of four parts: two circles and two connecting line segments.
\[\tag{11} L_\Gamma=2\pi R_++2\pi R_-+2r_e. \] The radii sum to \(2R\), so the expression immediately simplifies:
\[\tag{12} \boxed{ L_\Gamma=4\pi R+2r_e }. \] The first term represents two complete circular traversals. The second arises solely from the need to cross between the branches twice. This small addition produces the frequency lag of first order in \(\alpha_{\mathrm{fs}}\).
Radial connecting segments are the simplest approximation. If the connection follows a curved trajectory, its length may differ from \(r_e\), changing the numerical coefficient of the correction. Obtaining precisely \(1/(2\pi)\) therefore relies on a specific geometric assumption: the total added length per complete cycle is \(2r_e\).
5. The Effective Radius of the Complete Cycle
The complete two-branch traversal comprises two circular revolutions and corresponds to a geometric phase of \(4\pi\). It can conveniently be replaced by an equivalent contour of effective radius \(R_{\mathrm{eff}}\). We define this radius through the total path length alone, rather than through area:
\[\tag{13} L_\Gamma=4\pi R_{\mathrm{eff}}. \] Comparing Eqs. (12) and (13) gives
\[\tag{14} \boxed{ R_{\mathrm{eff}}=R+\frac{r_e}{2\pi} }. \] Using the ratio (9), the same formula can be written in dimensionless form:
\[\tag{15} R_{\mathrm{eff}} =R\left(1+\frac{\alpha_{\mathrm{fs}}}{2\pi}\right). \] Here \(R_{\mathrm{eff}}\) is not a third internal electron orbit. Nor is it introduced as the radius of an area enclosed by a classical current. It is only the radius of a uniform \(4\pi\) cycle with the same length and traversal time as the original two-branch path.
This definition matters for the sign of the result. The additional connecting segments increase the path length, so \(R_{\mathrm{eff}}>R\). At an unchanged speed, the wave takes longer to complete the cycle, and the corresponding frequency must decrease.
6. The Frequency of Two-Branch Closure
To relate spatial scale and frequency, we use a wave-motion invariant. It has the same form for the original radius and the complete contour:
\[\tag{16} \Omega_0R=c, \qquad \Omega_\Gamma R_{\mathrm{eff}}=c. \] The notation \(\Omega\) is deliberate. The quantities \(\Omega_0\) and \(\Omega_\Gamma\) are internal geometric frequencies of the model; at this stage, they are not identified with the experimental frequencies \(\omega_s\) and \(\omega_c\).
Equation (16) shows that an increased effective radius lowers the frequency of complete closure:
\[\tag{17} \Omega_\Gamma =\frac{\Omega_0} {1+\alpha_{\mathrm{fs}}/(2\pi)} <\Omega_0. \] The meaning of this frequency requires clarification. Without the connecting segments, the two circular traversals have a total length \(4\pi R\) and take a time \(4\pi R/c\). Since the full geometric phase of this cycle is \(4\pi\), its phase rate is \(c/R=\Omega_0\). Adding the connecting segments increases the total time to \(4\pi R_{\mathrm{eff}}/c\), reducing the phase rate to \(c/R_{\mathrm{eff}}=\Omega_\Gamma\).
We now calculate the relative lag of the new frequency. Because the same invariant applies to both cases, the expression simplifies without an approximate expansion:
\[\tag{18} \frac{\Omega_0-\Omega_\Gamma}{\Omega_\Gamma} =\frac{R_{\mathrm{eff}}-R}{R}. \] The radius difference is already known from (14). We therefore obtain
\[\tag{19} \boxed{ \frac{\Omega_0-\Omega_\Gamma}{\Omega_\Gamma} =\frac{r_e}{2\pi R} =\frac{\alpha_{\mathrm{fs}}}{2\pi} }. \] Equation (19) is the central geometric result of this article. It is exact under the adopted assumptions: the branch separation is \(r_e\), the full path contains two connecting segments of length \(r_e\), and the same frequency–radius invariant holds for the original and effective contours.
The positive sign of the relative frequency lag follows from the increased path length: the full traversal frequency is lower than the baseline frequency, so \(\Omega_0-\Omega_\Gamma\) is positive. This is the sign of the geometric frequency ratio. Its correspondence to a positive magnetic-moment deviation requires the hypothesis introduced in the next section.
7. From Internal Frequencies to Experiment
A separate physical assumption is required at this stage. The internal frequencies \(\Omega_0\) and \(\Omega_\Gamma\) are extremely high and exist in the model even without an external magnetic field. The experimental frequencies \(\omega_s\) and \(\omega_c\), by contrast, depend on the field \(B\). Direct numerical equality between these pairs is therefore impossible.
As a leading approximation, suppose the magnetic field projects both internal frequencies into the observed dynamics through the same coefficient \(K(B)\):
\[\tag{20} \omega_s=K(B)\Omega_0, \qquad \omega_c=K(B)\Omega_\Gamma. \] The common coefficient may involve the magnetic field, charge, mass, and geometry of external motion. Its specific form is unnecessary for the ratio under consideration, because the identical factor cancels.
\[\tag{21} \frac{\omega_s-\omega_c}{\omega_c} = \frac{\Omega_0-\Omega_\Gamma}{\Omega_\Gamma}. \] According to the experimental definition (6), the left-hand side is \(a_e\); the right-hand side has already been obtained in (19). Under this frequency projection, the two-branch geometry therefore reproduces the leading anomaly term:
\[\tag{22} \boxed{ a_e^{(1)}=\frac{\alpha_{\mathrm{fs}}}{2\pi} }. \] The corresponding approximation for the magnetic g-factor is
\[\tag{23} g^{(1)}=2\left(1+\frac{\alpha_{\mathrm{fs}}}{2\pi}\right). \] The common coefficient \(K(B)\) has not yet been derived from the state operator. Equation (20) is therefore a hypothesis of identical magnetic projection, rather than an established consequence of split geometry. Cancellation of the magnetic field in the experimental ratio is consistent with this description, but does not by itself establish that the measured frequencies are projections of these particular internal frequencies. Testing the hypothesis requires independently deriving their coupling to the external field.
7.1. The Leading Approximation and Further Splittings
This study retains the minimal two-sheet geometry: one wave successively traverses two nearby circular branches and two radial connecting segments. Within the adopted contour, it gives an exact relative path-length increase and the corresponding frequency ratio (19). Identifying this ratio with the positive leading term of the magnetic anomaly relies on the magnetic projection hypothesis (20).
The two-sheet geometry reproduces the coefficient of the leading anomalous correction. Its correspondence to the positive magnetic anomaly is adopted as a physical mapping hypothesis. Corrections of higher orders are hypothesized to involve additional splitting levels of the internal state.
Let us separate the leading contribution from the as-yet undetermined remainder:
\[\tag{23a} \boxed{ a_e=\frac{\alpha_{\mathrm{fs}}}{2\pi}+\Delta a_{\mathrm{split}}, \qquad \Delta a_{\mathrm{split}}=O\!\left(\alpha_{\mathrm{fs}}^2\right). } \] The notation \(\Delta a_{\mathrm{split}}\) expresses the adopted hypothesis about the origin of higher-order corrections. It does not imply that their mechanism has already been constructed. If the remainder admits a power expansion, it can be written as
\[\tag{23b} \Delta a_{\mathrm{split}} =C_2\left(\frac{\alpha_{\mathrm{fs}}}{\pi}\right)^2 +C_3\left(\frac{\alpha_{\mathrm{fs}}}{\pi}\right)^3+\ldots. \] The coefficients \(C_2,C_3,\ldots\), their signs, and the physical rules coupling the levels are not determined in this article. One additional splitting level is not automatically identified with the next power of \(\alpha_{\mathrm{fs}}\): that correspondence must follow from its physical mapping. The coefficients cannot be obtained merely by counting branches, or imported from quantum electrodynamics and presented as geometrically derived.
The exactness of the geometric equality (19) is compatible with the approximate status of the magnetic result (22). The former concerns the length and frequencies of the chosen contour; the latter concerns their physical interpretation. A complete magnetic description may refine the mapping of internal frequencies and include additional channels. A contribution must not be included in the baseline contour and then counted again as a splitting correction.
A general framework for additional levels, interchannel couplings, and inverse synthesis is presented in “The Splitting Principle: Hierarchy, Horizontal Couplings, and Experimental Tests”. It offers a possible formalism for further development, but does not yet calculate the coefficients in (23b).
8. Expressing the Internal Frequency through the Operator \(J\)
In split geometry, the particle's complete state is specified by the operator
\[\tag{24} J(a,b)=\j^a(-\j)^b =\ep{}e^{i\pi b} +\em{}e^{i\pi a}, \qquad |J|=1. \] The parameter \(a\) describes the internal state, while \(b\) describes the particle's external motion. To examine the intrinsic internal geometry, the electron's center may be taken to be at rest. Then \(b=0\), and the operator reduces to
\[\tag{25} J(t)=\jmath^{a(t)}, \qquad a(t)=\frac{\Omega t}{\pi}. \] For the baseline contour, (25) uses \(\Omega=\Omega_0\); for complete two-branch closure, it uses \(\Omega=\Omega_\Gamma\). Reducing the frequency does not violate the normalization \(|J|=1\): the rate of internal phase evolution changes, while the magnitude of the complete state remains unchanged.
A value of \(J(t)\) alone, however, does not distinguish the two geometric branches. After the first revolution, the operator's phase may repeat even though the wave is now on the other branch. The complete internal state must therefore also contain a discrete branch label, \(\eta=+1\) or \(\eta=-1\):
\[\tag{26} \mathcal S(t)=\bigl(J(t),\eta(t)\bigr). \] After one revolution, the branch changes; after the second, both the phase and label return to their initial values. In this restricted sense, the two-branch contour has a period of \(4\pi\). Equation (26) is not yet a complete spinor derivation of an electron rotation through \(720^\circ\); it identifies the additional information that must accompany the operator \(J\) to describe successive traversal of the two branches.
The next task is to obtain the label \(\eta\) as an intrinsic state of an extended operator rather than an external designation. At the same time, it is necessary to derive from \(J(a,b)\) how the external magnetic field produces the common projection coefficient \(K(B)\).
9. Why a Frequency-Based Approach Is Preferable to an Area Calculation
For a classical closed current, the magnetic moment is the product of current and oriented area. In the two-branch model, however, an ambiguity immediately arises: radial connecting segments increase the traversal time but contribute no oriented area, because the position vector and path element are parallel along such a segment.
With charge and transport speed unchanged, the increased period reduces the current. For the adopted contour with radial connecting segments, a calculation using current and oriented area gives a negative linear correction, opposite to the positive magnetic anomaly. Declaring \(R_{\mathrm{eff}}\) to be the magnetic-area radius in advance would produce a positive answer, but would require additional geometric justification.
The frequency-based approach derives the period ratio separately, but does not by itself eliminate the need for consistency with the magnetic response. In this approach, \(R_{\mathrm{eff}}\) is defined exclusively through the length and time of complete closure. Frequencies are then compared—the quantities whose ratio is used experimentally. The derivation of (19) is therefore independent of any assumed area assigned to the connecting segments.
This does not make the distribution of current and magnetic area irrelevant. It remains important for a complete construction of the magnetic moment. The frequency interpretation of the leading anomaly term retains the explicitly stated hypothesis (20). Reconciling it with the charge and current distribution remains an open task; hypothesizing higher-order additions does not, by itself, resolve the sign discrepancy in the simple current calculation.
10. Scope of the Result
Within the adopted model, the following statements have been obtained in sequence:
1. The internal branches are separated by \(r_e\), and the ratio of this scale to the reduced Compton wavelength is \(\alpha_{\mathrm{fs}}\).
2. One wave successively traverses both branches and two radial connecting segments. The total path length is therefore \(4\pi R+2r_e\).
3. The equivalent radius associated with the period is \(R+r_e/(2\pi)\).
4. The invariant \(\Omega R=c\) converts the additional length into an exact relative frequency lag of \(\alpha_{\mathrm{fs}}/(2\pi)\).
5. If the external magnetic field projects the two internal frequencies identically, this ratio coincides with the leading term of the electron anomaly.
This article nevertheless does not provide a complete derivation of the magnetic moment. The interaction of the operator \(J\) with the magnetic field must be obtained separately, the common coefficient \(K(B)\) justified, the description of the internal branch label reconciled with an extended operator, and the hypothesized origin of higher powers of \(\alpha_{\mathrm{fs}}\) tested. The proposed geometry must also be checked for consistency with all previously constructed electron properties: mass, charge, spin, and Coulomb interaction.
The result is therefore restricted to the leading approximation. Further splittings are proposed as a possible source of the remainder \(\Delta a_{\mathrm{split}}=O(\alpha_{\mathrm{fs}}^2)\). Until their physical mapping is constructed, Eq. (23b) specifies a direction for further work rather than a calculated sequence of corrections.
Conclusion
Experimentally, the electron's anomalous magnetic moment manifests as a small difference between two frequencies. This suggests seeking its geometric origin through a difference in closure times of two internal wave states, rather than through an assumed rotation of a charged body.
In the radially split model, the full path of one wave contains two circles and two connecting segments. The additional length \(2r_e\) increases the effective radius of the \(4\pi\) cycle by \(r_e/(2\pi)\). Since the product of the internal frequency and the corresponding radius remains \(c\), the complete two-branch traversal frequency is slightly lower than the baseline frequency.
The relative lag between these frequencies is
\[\tag{27} \boxed{ \frac{\Omega_0-\Omega_\Gamma}{\Omega_\Gamma} =\frac{\alpha_{\mathrm{fs}}}{2\pi} }. \] The factor \(1/(2\pi)\) thus arises from converting the total added length of two connecting segments into an effective radius of the full \(4\pi\) cycle, while the fine-structure constant arises from the ratio of the internal scales \(r_e/R\).
If the external magnetic field projects the baseline and full-closure frequencies identically, Eq. (27) directly yields the leading anomalous magnetic moment term. The geometric result follows from the specified radii and connecting-segment geometry. Its physical correspondence to the positive anomaly remains a hypothesis and requires deriving the magnetic projection mechanism directly from the operator \(J(a,b)\).
As a working hypothesis of the model, the leading term is associated with two-sheet closure, while corrections of higher orders are tentatively associated with additional splittings. The minimal geometry is retained; the higher-order coefficients and their consistent relation to the magnetic moment remain subjects for further research.
Materials used
- X. Fan, T. G. Myers, B. A. D. Sukra, G. Gabrielse. Measurement of the Electron Magnetic Moment. Physical Review Letters, 130, 071801 (2023).
- J. Schwinger. On Quantum-Electrodynamics and the Magnetic Moment of the Electron. Physical Review, 73, 416–417 (1948).
- P. A. M. Dirac. The Quantum Theory of the Electron. Proceedings of the Royal Society A, 117, 610–624 (1928).
- T. Aoyama, M. Hayakawa, T. Kinoshita, M. Nio. Tenth-Order QED Contribution to the Electron g−2 and an Improved Value of the Fine Structure Constant. Physical Review Letters, 109, 111807 (2012).