2026-08-04
Anomalous magnetic moment of the electron
Geometric derivation
This paper continues the geometric model of the electron, in which its internal state is represented by two closely spaced branches of a single closed wave path. The primary focus here will be on frequencies, rather than the classical calculation of the magnetic moment in terms of current and circuit area. This approach is closer to experiment: in a Penning trap, the electron's magnetic moment is determined from the ratio of the measured frequencies.
The main question of the paper can be formulated very simply. If a wave must sequentially traverse both internal branches and two transitions between them to fully return to its original state, its path becomes slightly longer. Could the corresponding decrease in frequency account for the first term of the electron's anomalous magnetic moment?
Below, it will be shown that the additional length of the two transitions increases the effective radius of the complete cycle by \(r_e/(2\pi)\). While maintaining the "frequency × radius" invariant, this leads to a 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 first, Schwinger term of the quantum-electrodynamic expansion of the electron anomaly. It is necessary to separate the two levels of the result beforehand. Geometric relation (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 explicitly maintained throughout the remainder of this paper.
Introduction
In the Dirac equation, the magnetic factor of a point electron is \(g=2\). It is convenient to separate this fundamental value from a small correction and introduce the magnetic moment anomaly:
\[\tag{2} a_e=\frac{g-2}{2}. \] If the exact equality \(g=2\) were satisfied, then \(a_e\) would vanish. However, experiment shows a small positive deviation. Its main contribution is given by
\[\tag{3} a_e^{(1)}=\frac{\alpha_{\mathrm{fs}}}{2\pi}. \] The factor \(\alpha_{\mathrm{fs}}\) characterizes the electromagnetic coupling, and the factor \(1/(2\pi)\) indicates the correlation of the correction with a full circular path. In standard quantum electrodynamics, formula (3) arises as a one-loop radiative contribution. The present model poses a different problem: to test whether the same dimensionless quantity can appear as the geometric ratio of two close frequencies of the inner circuit.
This coincidence in itself does not yet replace a quantum electrodynamic calculation. However, it can reveal what internal geometry may underlie the first term of the anomaly and why the divisor \(2\pi\) appears in it.
1. What is measured in the experiment?
The magnetic moment of an electron cannot be observed as the movement of a charge along a visible inner circle. In precise experiments, a single electron is confined in a Penning trap. The external magnetic field sets the cyclotron frequency \(\omega_c\), and the spin state is characterized by the frequency \(\omega_s\).
For an idealized free electron, these frequencies are related by the relation
\[\tag{4} \omega_s=\frac{g}{2}\,\omega_c. \] The difference between them is called the anomalous frequency:
\[\tag{5} \omega_a=\omega_s-\omega_c. \] Hence, the anomaly is expressed not through the absolute value of the internal current, but through the frequency ratio:
\[\tag{6} \boxed{ a_e=\frac{\omega_a}{\omega_c} =\frac{\omega_s-\omega_c}{\omega_c} }. \] Formula (6) is important for the further construction. It shows that to obtain the first term of the anomaly, it is sufficient to find two physically related frequencies whose relative difference is equal to \(\alpha_{\mathrm{fs}}/(2\pi)\). When calculating the ratio, the overall scale of the magnetic field is reduced. Therefore, the frequency approach does not require pre-assigning a classical current or some conventional magnetic area to the internal circuit.
2. Standard Explanation: From the Dirac Equation to Quantum Electrodynamics
Before moving on to the geometric model, it is useful to distinguish two results of the standard theory. The Dirac equation describes the electron as a relativistic particle with spin \(1/2\) and, even in the initial approximation, yields a magnetic factor of \(g=2\). However, the small deviation from 2 does not follow from the Dirac equation itself. It appears only in quantum electrodynamics, after quantization of the electromagnetic field.
In the popular presentation, this additional mechanism is depicted as follows. The electron emits a virtual photon, continues to interact with the external electromagnetic field, and then absorbs the same virtual photon. Thus, the measured magnetic peak turns out to beis surrounded by an internal photon loop. This is not the literal flight of the observed photon along a specific trajectory, but a graphical representation of an intermediate quantum state that cannot be detected separately.
This process is sometimes called the interaction of an electron with its own electromagnetic field. This phrase is convenient as a visual image, but strictly speaking, we are talking about the electron's self-interaction through a quantized electromagnetic field. A virtual photon is emitted and absorbed by the same electron, while the external field used to measure the magnetic moment enters a separate vertex. Even in the first, single-loop approximation, the corresponding calculation schematically looks like this:
\[\tag{6a} \begin{aligned} \Lambda^\mu(p',p) &\;\propto\; e^2\!\int\!\frac{d^4k}{(2\pi)^4} \;\gamma^\alpha \frac{\not p'-\not k+m_e}{(p'-k)^2-m_e^2+i0} \;\gamma^\mu \\[4pt] &\qquad\times \frac{\not p-\not k+m_e}{(p-k)^2-m_e^2+i0} \;\gamma_\alpha \frac{1}{k^2+i0} \quad\Longrightarrow\quad a_e=F_2(0)=\frac{\alpha_{\mathrm{fs}}}{2\pi}+\ldots \end{aligned} \] Here \(p\) and \(p'\) are the electron momenta before and after interaction with the external field, \(k\) is the four-momentum of the virtual photon, and integration is performed over all possible values of \(k\). The two electron denominators describe the intermediate states of the electron, the factor \(1/(k^2+i0)\) is the virtual photon, and the central matrix \(\gamma^\mu\) is the interaction with the external electromagnetic field. The proportionality sign reminds us that the gauge, regularization, and renormalization details of the full calculation are omitted here.
After extracting the magnetic form factor \(F_2(0)\) from this vertex, a one-loop Schwinger calculation yields the first term \(\alpha_{\mathrm{fs}}/(2\pi)\). Subsequent powers of \(\alpha_{\mathrm{fs}}\) arise from even more complex diagrams with multiple virtual photons, electron-positron loops, and other quantum corrections. Thus, in the standard explanation, the divisor \(2\pi\) appears as a result of calculating a four-dimensional loop integral, not as a predetermined geometric circumference.
The model proposed below reproduces only this first coefficient in a different way: \(\alpha_{\mathrm{fs}}\) arises as the ratio of two internal scales, and \(1/(2\pi)\) arises from the additional length of two transitions in a complete two-branch cycle. The numerical agreement between the two results is significant, but their physical content is still different. Quantum electrodynamics derives the connection with the magnetic field and subsequent terms of the expansion, whereas in the geometric model, the magnetic projection \(K(B)\) and higher-order corrections still need to be constructed.
3. Two Electron Scales
It is here that the fundamental difference between the two approaches becomes particularly apparent. In quantum electrodynamics, the first term of the anomaly arises as a result of a complex one-loop calculation of the electron's interaction with a virtual photon and an external electromagnetic field. In the proposed model, the same quantity arises much more simply from the geometric relationship between two electron scales: the internal radius \(r_e\) and the radius of the closed two-branch path \(R\). Therefore, we will not need to calculate quantum corrections further, but merely establish how these two scales are related.
The previously constructed model distinguishes between the internal electromagnetic scale \(r_e\) and the reduced Compton length of the electron. To avoid overloading the subsequent formulas, we denote the last one by \(R\):
\[\tag{7} R=\overline\lambda_C=\frac{\hbar}{m_ec}. \] The scale \(r_e\) is determined by the equality of the electromagnetic energy and the rest energy:
\[\tag{8} r_e=\frac{e^2}{4\pi\varepsilon_0m_ec^2}. \] These two lengths are obtained from different physical conditions. Therefore, their ratio forms a natural dimensionless coefficient. After substituting (7) and (8), it exactly coincides with the fine structure constant:
\[\tag{9} \boxed{ \frac{r_e}{R}=\alpha_{\mathrm{fs}} }. \] A detailed construction of this relation is given in the article "Geometric Origin of the Squared Fine Structure Constant and the Parameters of the Bohr Atom." Here, formula (9) will be used as an already obtained result.
It is important not to confuse the two frequency levels. The internal electromagnetic scale corresponds to a frequency of order \(c/r_e\), while the length \(R\) corresponds to the Compton angular frequency \(c/R\). In this article, we consider the second level, since it is for this level that the ratio \(r_e/R\) yields a small correction of the order of \(\alpha_{\mathrm{fs}}\).
4. Radially split inner contour
Let the inner contourThe electron's orbital path has two closely spaced branches with radii \(R_+\) and \(R_-\). They are symmetrically located with respect to the mean radius \(R\), and the distance between them is equal to the intrinsic electromagnetic scale:
\[\tag{10} R_\pm=R\pm\frac{r_e}{2}, \qquad R_+-R_-=r_e. \] If we consider these branches as two independent circular motions and simply average their radii, the linear correction disappears. Indeed, the mean value is again equal to \(R\). Therefore, the mere presence of two radii does not yet create a first-order term in \(r_e/R\).
A new possibility arises if the two branches belong not to two independent waves, but to a single continuous path. The wave passes through the outer branch, crosses to the inner branch, crosses it, and then returns to the original branch. The complete state is closed only after passing both circles and both transitions.
In the simplest geometry, transitions are considered radial. Each of them has a length of \(r_e\). Therefore, the complete path consists of four parts: two circles and two connecting segments.
\[\tag{11} L_\Gamma=2\pi R_++2\pi R_-+2r_e. \] The sum of the radii is \(2R\), so the expression immediately cancels out:
\[\tag{12} \boxed{ L_\Gamma=4\pi R+2r_e }. \] The first term represents two complete circular paths. The second occurs only because of the need to transition between branches twice. It is this small addition that will subsequently create a first-order frequency lag in \(\alpha_{\mathrm{fs}}\).
The radial form of transitions is the simplest approximation. If the transition occurs along a curved trajectory, its length may differ from \(r_e\), and the numerical correction factor will change. Therefore, obtaining \(1/(2\pi)\) is associated with a specific geometric assumption: the total additional length of a complete cycle is equal to \(2r_e\).
5. Effective Radius of a Complete Cycle
A complete two-branch circuit contains two circular revolutions and corresponds to a geometric phase of \(4\pi\). It is convenient to replace it with an equivalent circuit with an effective radius of \(R_{\mathrm{eff}}\). We define this radius not in terms of area, but only in terms of the length of the full path:
\[\tag{13} L_\Gamma=4\pi R_{\mathrm{eff}}. \] Comparing formulas (12) and (13), we obtain
\[\tag{14} \boxed{ R_{\mathrm{eff}}=R+\frac{r_e}{2\pi} }. \] Using relation (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 the third internal orbit of the electron. It is also not introduced as the radius of the area swept by the classical current. It is only the radius of a uniform \(4\pi\)-cycle, which has the same length and the same transit time as the original two-branch path.
This definition is important for the sign of the result. Additional transitions increase the path length, so \(R_{\mathrm{eff}}>R\). With constant velocity, the wave requires more time to fully close, and the corresponding frequency must decrease.
6. Frequency of Two-Branch Closure
To relate spatial scale and frequency, we use the invariant of wave motion. For the initial radius and the full contour, it is written the same way:
\[\tag{16} \Omega_0R=c, \qquad \Omega_\Gamma R_{\mathrm{eff}}=c. \] The notation \(\Omega\) is chosen intentionally. The quantities \(\Omega_0\) and \(\Omega_\Gamma\) are the internal geometric frequencies of the model; At this stage, they are not yet identified with the experimental \(\omega_s\) and \(\omega_c\).
From (16), it follows that an increase in the effective radius decreases the frequency of the full closure:
\[\tag{17} \Omega_\Gamma =\frac{\Omega_0} {1+\alpha_{\mathrm{fs}}/(2\pi)} <\Omega_0. \] Let's clarify the meaning of this frequency. Without transitions, the length of two round trips is \(4\pi R\), and the time it takes to complete them is \(4\pi R/c\). Since the full geometric phase of such a cycle is \(4\pi\), the rate of phase change is \(c/R=\Omega_0\). After adding the transitions, the total time increases to \(4\pi R_{\mathrm{eff}}/c\), and the phase velocity decreases to \(c/R_{\mathrm{eff}}=\Omega_\Gamma\).
Now we find the relative lag of the new frequency. Due to the same invariant, the expression is reduced without an approximate expansion:
\[\tag{18} \frac{\Omega_0-\Omega_\Gamma}{\Omega_\Gamma} =\frac{R_{\mathrm{eff}}-R}{R}. \] The difference in radii is already known from (14). Therefore, we finally get
\[\tag{19} \boxed{ \frac{\Omega_0-\Omega_\Gamma}{\Omega_\Gamma} =\frac{r_e}{2\pi R} =\frac{\alpha_{\mathrm{fs}}}{2\pi} }. \] Formula (19)is the central geometric result of the paper. It is accurate within the accepted conditions: the distance between branches is \(r_e\), the full path contains two transitions of length \(r_e\), and the same frequency and radius invariant holds for the original and effective circuits.
The positive sign is also obtained automatically. The increased path produces a lower frequency, so the difference \(\Omega_0 - \Omega_\Gamma\) is positive. Thus, the geometry provides not only the required modulus but also the direction of the frequency shift.
7. From Internal Frequencies to Experiment
At this stage, a separate physical assumption must be made. 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 \(\omega_c\), in contrast, depend on the field \(B\). Therefore, a direct numerical equality of these pairs is impossible.
Suppose that the magnetic field projects both internal frequencies into the observed dynamics with the same coefficient \(K(B)\):
\[\tag{20} \omega_s=K(B)\Omega_0, \qquad \omega_c=K(B)\Omega_\Gamma. \] The general coefficient can include the magnetic field strength, charge, mass, and geometry of the external motion. For the ratio under consideration, its specific form is not required: the same factor cancels out.
\[\tag{21} \frac{\omega_s-\omega_c}{\omega_c} = \frac{\Omega_0-\Omega_\Gamma}{\Omega_\Gamma}. \] The left-hand side, according to the experimental definition (6), is equal to \(a_e\), and the right-hand side has already been found in (19). Therefore, the frequency projection of the two-branch geometry reproduces the first term of the anomaly:
\[\tag{22} \boxed{ a_e^{(1)}=\frac{\alpha_{\mathrm{fs}}}{2\pi} }. \] The corresponding approximation for the magnetic factor is
\[\tag{23} g^{(1)}=2\left(1+\frac{\alpha_{\mathrm{fs}}}{2\pi}\right). \] The overall coefficient \(K(B)\) has not yet been derived from the state operator. Therefore, formula (20) is a hypothesis of the same magnetic projection, and not an already proven consequence of split geometry. At the same time, the very structure of the experiment makes this hypothesis testable: the magnetic field does indeed cancel out with respect to the cyclotron and anomalous frequencies, since the electron is used as its own magnetometer.
8. Writing the Internal Frequency via the \(J\) Operator
The complete state of a particle in split geometry is given by the operator
\[\tag{24} J(a,b)=\jmath^a(-\jmath)^b =\mathfrak{e}e^{i\pi b} +\bar{\mathfrak{e}}e^{i\pi a}, \qquad |J|=1. \] The parameter \(a\) describes the internal state, and the parameter \(b\) describes the external motion of the particle. To consider its own internal geometry, the electron's center can be assumed to be at rest. Then \(b=0\), and the operator cancels out:
\[\tag{25} J(t)=\jmath^{a(t)}, \qquad a(t)=\frac{\Omega t}{\pi}. \] For the basic contour in (25), \(\Omega=\Omega_0\) is used, and for the complete two-branch closure, \(\Omega=\Omega_\Gamma\). Reducing the frequency does not violate the normalization \(|J|=1\): the rate of passage of the internal phase changes, but not the absolute value of the complete state.
However, the value of \(J(t)\) alone is not sufficient to distinguish between the two geometric branches. After the first rotation, the operator's phase may repeat, although the wave is already on a different branch. Therefore, the complete internal state must additionally contain a discrete branch label \(\eta=+1\) or \(\eta=-1\):
\[\tag{26} \mathcal S(t)=\bigl(J(t),\eta(t)\bigr). \] After one rotation, the branch changes, and after the second, both the phase and the label are restored. In this limited sense, the two-branch circuit has a period of \(4\pi\). Formula (26) is not yet a complete spinor derivation of the electron rotation by \(720^\circ\); It only shows what additional information needs to be added to the operator \(J\) to describe the sequential traversal of two branches.
The next task should be to obtain the label \(\eta\) not as an external notation, but as an eigenstate of the extended operator. At the same time, it is necessary to deduce from \(J(a,b)\) how the external magnetic field creates the overall projection coefficient \(K(B)\).
9. Why the frequency approach is preferable to area calculations
For a classical closed current, the magnetic moment is determined by the product of the current and the oriented area. However, in the two-branch model, an ambiguity immediately arises: radial transitions increase the time of a complete round-trip, but do not themselves add oriented area, since in such a section the radius vector and the path element are parallel.
If we mechanically substitute the increased period intoIf we use the classical formula, preserving only the areas of two circles, the linear correction may have the wrong sign. However, if we declare \(R_{\mathrm{eff}}\) to be the radius of the magnetic area in advance, we will obtain a positive answer, but this will require additional geometric justification.
The frequency path does not encounter this problem. In it, \(R_{\mathrm{eff}}\) is determined solely by the length and duration of the complete circuit. Frequencies are then compared—precisely the quantities whose ratio is used in the experiment. Therefore, the conclusion (19) is independent of the choice of the conventional area of the transition sections.
This does not mean that the question of current distribution and magnetic area becomes unnecessary. It remains important for the complete construction of the magnetic moment. However, for the frequency interpretation of the first term of the anomaly, it can be separated from the geometric result already obtained.
10. Boundaries of the Obtained Result
Within the adopted model, the following statements were consistently obtained:
1. The internal branches are separated by \(r_e\), and the ratio of this scale to the reduced Compton length is \(\alpha_{\mathrm{fs}}\).
2. One wave successively passes through both branches and two radial transitions. Therefore, the total path is \(4\pi R+2r_e\).
3. The equivalent radius of the period is \(R+r_e/(2\pi)\).
4. The invariant \(\Omega R=c\) transforms the additional length into an exact relative frequency lag \(\alpha_{\mathrm{fs}}/(2\pi)\).
5. For identical projections of two internal frequencies by an external magnetic field, this ratio coincides with the first term of the electron anomaly.
However, this article does not yet provide a complete derivation of the magnetic moment. It is necessary to separately derive the interaction of the operator \(J\) with the magnetic field, justify the overall coefficient \(K(B)\), construct the internal label of the branch, and explain the subsequent terms of the expansion in powers of \(\alpha_{\mathrm{fs}}). It also remains to verify how the proposed geometry agrees with all the already constructed properties of the electron: mass, charge, spin, and Coulomb interaction.
Conclusion
The electron's anomalous magnetic moment manifests itself experimentally as a small difference between two frequencies. This allows us to search for its geometric origin not through the conventional rotation of a charged body, but through the difference in the closure times of two internal wave states.
In the radially split model, the full path of one wave includes two circles and two transitions between them. 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 retains the value \(c\), the frequency of the complete two-branch bypass is slightly less than the base frequency.
The relative lag of these frequencies is
\[\tag{27} \boxed{ \frac{\Omega_0-\Omega_\Gamma}{\Omega_\Gamma} =\frac{\alpha_{\mathrm{fs}}}{2\pi} }. \] Thus, the factor \(1/(2\pi)\) arises from converting the total additional length of the two transitions to the effective radius of the complete \(4\pi\)-cycle, and the fine structure constant arises from the ratio of the internal scales \(r_e/R\).
If the external magnetic field projects the base frequency and the full closure frequency equally, formula (27) directly transforms into the first term of the anomalous magnetic moment. The geometric part of this result is already closed. The physical part requires the next step constructing the magnetic projection mechanism directly from the operator \(J(a,b)\).
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).