Previously, a new idempotent basis was proposed that allows one to separate the translational and internal motions of a particle. Based on this, a motion vector was constructed that simultaneously describes the particle's spatial transport and internal periodic rotation. This approach allows one to consider the electron's internal structure as a natural consequence of the geometry of the new basis, without introducing additional spatial dimensions or special transformations.

In this paper, this model is further developed. We consider a hypothesis relating the natural frequency of internal motion to the electron's rest energy, fine structure constant, classical radius, and magnetic moment, which were previously considered in this paper. The proposed model's possible connection with Dirac's zitterbewegung effect is also discussed, and it is shown that the internal charge motion can serve as a unified geometric mechanism that combines several fundamental characteristics of the electron.

The geometric development of the model is based on a new idempotent basis. \[ \tag{1} \left\{ \ep,\;i\ep,\;\em,\;i\em \right\}, \] where \[ \ep^2=\ep,\qquad \em^2=\em,\qquad \ep\em=0. \]

The motion vector is written as the sum of the translational and rotational components. This notation allows us to separate the motion of the particle's center from its internal periodic motion, and then relate the internal frequency to the rest energy, the fine structure constant, and the hypothetical Lorentz factor of the internal dynamics.

Consider the dimensionless vector \[ \tag{2} J(t)=\ep+\em e^{-i\omega t}. \] The first component is directed along the fixed idempotent direction \(\ep\), while the second rotates in the complex plane \[ \left\{\em,\;i\em\right\}. \]

Expanding the complex exponential gives \[ \tag{3} J(t) = \ep + \em\cos\omega t - i\em\sin\omega t. \] Therefore, the end of the second component moves along the unit circle in the plane \((\em,i\em)\), while the first component remains constant.

Multiplying the vector by the speed of light yields the velocity vector \[ \tag{4} V(t) = cJ(t) = c\ep + c\em e^{-i\omega t}. \] In this notation, the component \(c\ep\) describes the translational direction, and the component \[ c\em e^{-i\omega t} \] denotes the internal circular motion in the geometry of the new basis.

Integrating expression (4) over time, we obtain the position vector \[ \tag{5} R(t) = ct\ep + \frac{ic}{\omega}\em e^{-i\omega t} + R_0. \] The sign and initial phase of the rotational component can be changed by choosing the initial instant of time. The radius of the geometric circle is \[ \tag{6} r^{(J)} = \frac{c}{\omega}. \]

Since \[ \tag{7} \omega=2\pi\nu, \] the radius can be represented as \[ \tag{8} r^{(J)} = \frac{c}{2\pi\nu}. \] Thus, the intrinsic frequency directly determines the spatial scale of the periodic motion of the vector \(J(t)\).

We assume that the electron can be represented as an electromagnetic oscillatory system with effective parameters \(L_e\) and \(C_e\). Then its natural frequency is equal to \[ \tag{9} \nu_e = \frac{1}{2\pi\sqrt{L_eC_e}}. \] In the model under consideration, the value is adopted \[ \tag{10} \nu_e \approx 1.69\cdot10^{22}\;\text{Hz}. \]

This frequency corresponds to the angular frequency \[ \tag{11} \omega_e = 2\pi\nu_e \approx 1.06\cdot10^{23}\;\text{s}^{-1}. \] The geometric radius of the rotational component is \[ \tag{12} r_e^{(J)} = \frac{c}{2\pi\nu_e} \approx 2.82\cdot10^{-15}\;\text{m}. \]

The obtained value is close to the classical radius of the electron \[ \tag{13} r_e = \frac{1}{4\pi\varepsilon_0} \frac{e^2}{m_ec^2} \approx 2.818\cdot10^{-15}\;\text{m}, \] where \(e=|q_e|\) is the absolute value of the electron's electric charge.

Further, we will proceed from the hypothesis that the geometric radius \[ r_e^{(J)} = \frac{c}{2\pi\nu_e} \] corresponds to the characteristic electromagnetic scale of the internal motion of the charge and can be identified with the classical radius of the electron: \[ \tag{14} r_e^{(J)} \approx r_e. \] Then the natural frequency \[ \nu_e = \frac{1}{2\pi\sqrt{L_eC_e}} \] acquires a direct geometric meaning.

Let's introduce a hypothetical relationship between the rest energy of an electron and its internal frequency: \[ \tag{15} m_ec^2 = h\nu_e\alpha_{\mathrm{fs}}, \] where \(\alpha_{\mathrm{fs}}\) is the fine structure constant: \[ \tag{16} \alpha_{\mathrm{fs}} \approx \frac{1}{137.036}. \]

From expression (15) it follows \[ \tag{17} \nu_e = \frac{m_ec^2} {h\alpha_{\mathrm{fs}}}. \] Numerically, this gives \[ \tag{18} \nu_e \approx 1.695\cdot10^{22}\;\text{Hz}, \] which corresponds to the accepted value of the natural frequency.

For comparison, the Compton frequency of an electron is \[ \tag{19} \nu_C = \frac{m_ec^2}{h} \approx 1.236\cdot10^{20}\;\text{Hz}. \] Therefore, \[ \tag{20} \nu_e = \frac{\nu_C} {\alpha_{\mathrm{fs}}}. \] That is, the proposed intrinsic frequency is approximately \(137\) times higher than the Compton frequency.

Let's make an additional assumption: \[ \tag{21} \alpha_{\mathrm{fs}} = \frac{1}{\gamma_{\mathrm{int}}}, \] where \(\gamma_{\mathrm{int}}\) is the hypothetical Lorentz factor of the electron's internal motion.

Then \[ \tag{22} \gamma_{\mathrm{int}} = \frac{1}{\alpha_{\mathrm{fs}}} \approx 137.036. \] According to the usual definition of the Lorentz factor, \[ \tag{23} \gamma_{\mathrm{int}} = \frac{1} {\sqrt{1-\beta_{\mathrm{int}}^2}}, \qquad \beta_{\mathrm{int}} = \frac{v_{\mathrm{int}}}{c}. \]

Substituting expression (21), we obtain \[ \tag{24} \alpha_{\mathrm{fs}} = \sqrt{1-\beta_{\mathrm{int}}^2}. \] Hence \[ \tag{25} \beta_{\mathrm{int}} = \sqrt{1-\alpha_{\mathrm{fs}}^2}. \]

Numerically \[ \tag{26} \beta_{\mathrm{int}} \approx 0.99997337, \] therefore \[ \tag{27} v_{\mathrm{int}} \approx 0.99997337c. \]

Thus, within the framework of this hypothesis, the fine structure constant characterizes the small difference between the velocity of internal motion and the speed of light. In this case, the quantity \(\alpha_{\mathrm{fs}}\) receives a geometric interpretation as the inverse internal Lorentz factor.

The geometric radius \(r_e^{(J)}\) is determined by the velocity \(c\), embedded in the vector \(V(t)=cJ(t)\). However, the physical radius of the expected charge motion must be related to the found velocity \(v_{\mathrm{int}}\): \[ \tag{28} r_{\mathrm{int}} = \frac{v_{\mathrm{int}}}{\omega_e} = \frac{v_{\mathrm{int}}} {2\pi\nu_e}. \]

Using Expressions \[ \nu_e = \frac{m_ec^2} {h\alpha_{\mathrm{fs}}} \] And \[ v_{\mathrm{int}} = c\sqrt{1-\alpha_{\mathrm{fs}}^2}, \] we get \[ \tag{29} r_{\mathrm{int}} = \alpha_{\mathrm{fs}} \sqrt{1-\alpha_{\mathrm{fs}}^2} \frac{\hbar}{m_ec}. \]

The classical radius of an electron can be represented as \[ \tag{30} r_e = \alpha_{\mathrm{fs}} \frac{\hbar}{m_ec}. \] Therefore, \[ \tag{31} r_{\mathrm{int}} = r_e \sqrt{1-\alpha_{\mathrm{fs}}^2}. \]

Since \[ \sqrt{1-\alpha_{\mathrm{fs}}^2} \approx 0.99997337, \] we get \[ \tag{32} r_{\mathrm{int}} \approx 0.99997337r_e. \]

Thus, one must distinguish between the geometric radius of the rotational component \[ r_e^{(J)} = \frac{c}{\omega_e} \] and the physical radius of the supposed charge motion \[ r_{\mathrm{int}} = \frac{v_{\mathrm{int}}}{\omega_e}. \] They differ only by the factor \[ \sqrt{1-\alpha_{\mathrm{fs}}^2}, \] which is extremely close to unity.

Substituting expression (21) into formula (15) yields \[ \tag{33} m_ec^2 = \frac{h\nu_e} {\gamma_{\mathrm{int}}}. \] Hence \[ \tag{34} h\nu_e = \gamma_{\mathrm{int}}m_ec^2. \]

The right-hand side of formula (34) has the form of the total relativistic energy. Therefore, the quantity \[ h\nu_e \] can be interpreted as the total energy of the internal periodic process, and the rest energy \[ m_ec^2 \] as its part determined by the factor \[ \frac{1}{\gamma_{\mathrm{int}}} = \alpha_{\mathrm{fs}}. \]

We obtain a system of relations \[ \tag{35} h\nu_e = \gamma_{\mathrm{int}}m_ec^2, \qquad m_ec^2 = \alpha_{\mathrm{fs}}h\nu_e, \qquad \gamma_{\mathrm{int}} = \frac{1}{\alpha_{\mathrm{fs}}}. \]

These equalities allow for a geometric interpretation. If we introduce the angle \(\vartheta\) for which \[ \tag{36} \sin\vartheta = \beta_{\mathrm{int}}, \] then \[ \tag{37} \cos\vartheta = \sqrt{1-\beta_{\mathrm{int}}^2} = \frac{1}{\gamma_{\mathrm{int}}} = \alpha_{\mathrm{fs}}. \]

Let \[ \tag{38} W_e = \gamma_{\mathrm{int}}m_ec^2 = h\nu_e \] denote the total energy of the internal periodic motion. Then the rest energy of the electron is determined by the projection of this energy: \[ \tag{39} m_ec^2 = \frac{W_e} {\gamma_{\mathrm{int}}} = W_e\cos\vartheta = \alpha_{\mathrm{fs}}W_e. \] Thus, the fine structure constant is given byThe geometric interpretation is as the projection coefficient of the total internal energy onto the direction corresponding to the rest energy.

In relativistic quantum mechanics, the zitterbewegung ("trembling motion") effect is well known, arising in the analysis of the Dirac equation. Despite the difference in mathematical formalism, the proposed model exhibits several fundamental similarities.

First, in both cases, a rapid internal periodic motion is considered, accompanying the electron and not directly related to its translational motion in space.

Second, the characteristic velocity of this internal motion turns out to be extremely close to the speed of light. In the proposed model, it is determined by the internal Lorentz factor and is \[ v_{\mathrm{int}} \approx 0.99997337c. \] Therefore, the internal motion naturally acquires a relativistic character.

Third, both models allow for a separation of the electron's external motion and its internal dynamics, which may be related to the origin of the particle's intrinsic magnetic moment and other internal properties.

The main difference lies in the origin of this motion. In Dirac's theory, it arises as a consequence of the structure of the relativistic wave equation, whereas in the present work, the internal rotation follows directly from the geometry of the vector \[ J(t) = \ep+\em e^{-i\omega t}, \] constructed in an idempotent basis.

It can be assumed that the rotational component \[ \em e^{-i\omega t} \] describes the motion of the center of charge, while the translational component \(\ep\) is associated with the motion of the center of energy or inertia. A similar separation of the center of charge and center of mass is also considered in some interpretations of the zitterbewegung models, but in this paper it is introduced as an independent hypothesis.

Thus, the proposed model is not opposed to Dirac's theory, but represents an attempt to provide a geometric interpretation of the internal motion of the electron based on a new basis.

If the rotational component of the vector \[ J(t) = \ep+\em e^{-i\omega_e t} \] describes the periodic motion of an electric charge, then such motion is equivalent to a closed electric current and should create the electron's own magnetic field. Therefore, the second field, discussed in the previous work, can be directly related to the particle's magnetic moment.

For a charge \(e\) rotating in a circular motion with angular frequency \(\omega_e\) along a trajectory of radius \(r\), the classical expression for the magnetic moment is \[ \tag{40} \mu_{\mathrm{circ}} = \frac{e\omega_e r^2}{2}. \]

Within the framework of this model, it is additionally assumed that the observed fundamental magnetic moment of the electron [2] is related to the magnetic moment of the internal circular current via the internal Lorentz factor: \[ \tag{41} \mu_e^{(0)} = \gamma_{\mathrm{int}} \mu_{\mathrm{circ}}. \] Since \[ \gamma_{\mathrm{int}} = \frac{1}{\alpha_{\mathrm{fs}}}, \] we obtain \[ \tag{42} \mu_e^{(0)} = \frac{e\omega_e r^2} {2\alpha_{\mathrm{fs}}}. \]

Expression (42) is a separate postulate of the proposed model. It does not follow solely from the classical formula for circular current and requires further physical justification.

To obtain the fundamental scale of the magnetic moment, we use the geometric radius of the rotational component \[ r=r_e^{(J)}, \] for which, according to expression (6), \[ \tag{43} \omega_e r_e^{(J)} = c. \] Then \[ \tag{44} \mu_e^{(0)} = \frac{ec r_e^{(J)}} {2\alpha_{\mathrm{fs}}}. \]

Under the accepted hypothesis, \[ r_e^{(J)} \approx r_e, \] and the classical radius of the electron is related to the reduced Compton length by the expression \[ \tag{45} r_e = \alpha_{\mathrm{fs}} \frac{\hbar}{m_ec}. \] Substituting it into expression (44), we obtain \[ \tag{46} \mu_e^{(0)} = \frac{ec} {2\alpha_{\mathrm{fs}}} \left( \alpha_{\mathrm{fs}} \frac{\hbar}{m_ec} \right) = \frac{e\hbar}{2m_e}. \]

The right-hand side of expression (46) coincides with the Bohr magneton: \[ \tag{47} \mu_B = \frac{e\hbar}{2m_e}. \] Therefore, within the framework of the proposed model, the internal circular motion of the charge reproduces the fundamental, Dirac scale of the electron magnetic moment: \[ \tag{48} \mu_e^{(0)} = \frac{e\omega_e\left(r_e^{(J)}\right)^2} {2\alpha_{\mathrm{fs}}} = \frac{ec r_e^{(J)}} {2\alpha_{\mathrm{fs}}} \approx \mu_B. \]

The experimental magnetic moment of the electron contains a small anomalous correction, which is not included in the present modelis considered. Therefore, expression (48) should be understood as obtaining the fundamental magnetic scale corresponding to the Bohr magneton.

The fine structure constant serves a dual function in the model. On the one hand, it determines the ratio of the rest energy to the total energy of internal motion, and on the other, it relates the geometric radius of the electron to the fundamental scale of its magnetic moment. In this case, the factor \(\alpha_{\mathrm{fs}}\), contained in the radius \(r_e\), cancels out with the factor \(1/\alpha_{\mathrm{fs}}\), arising from the internal Lorentz factor.

In this interpretation, the electron's second field is the magnetic field created by the charge's motion in the plane \[ \left( \em,i\em \right). \] The \(\ep\) component describes the translational motion of the electron's center, while the rotational component \(\em e^{-i\omega_e t}\) determines its internal electromagnetic dynamics and the fundamental magnetic moment.

The proposed model shows that the vector \[ J(t) = \ep+\em e^{-i\omega t} \] naturally separates the electron's motion into two components. The \(\ep\) component describes the translational motion of the particle's center, while the rotational component in the \((\em,i\em)\) plane defines the internal periodic motion, characterized by a natural frequency, geometric radius, and a velocity close to the speed of light.

Under the proposed hypothesis, the total energy of internal motion is determined by the expression \[ W_e = h\nu_e = \gamma_{\mathrm{int}}m_ec^2, \] and the rest energy of the electron is its geometric projection: \[ m_ec^2 = \frac{W_e} {\gamma_{\mathrm{int}}} = \alpha_{\mathrm{fs}}W_e. \] The fine structure constant is interpreted as the reciprocal of the internal Lorentz factor.

The article distinguishes between the geometric radius of the rotational component \(r_e^{(J)}=c/\omega_e\) and the physical radius of the assumed charge motion \(r_{\mathrm{int}}=v_{\mathrm{int}}/\omega_e\). Their ratio is determined by the factor \(\sqrt{1-\alpha_{\mathrm{fs}}^2}\), which is extremely close to unity. Therefore, both scales practically coincide with the classical electron radius, but have different meanings within the model.

If the rotational component corresponds to the internal motion of the electric charge, the resulting circular current creates its own magnetic field. Under the additional assumption of a relativistic enhancement of the magnetic moment by the internal Lorentz factor, the model reproduces the fundamental scale of the electron's magnetic moment, equal to the Bohr magneton.

Thus, the new idempotent basis allows us to relate the electron's internal motion, its natural frequency, total internal energy, rest energy, fine structure constant, classical radius, and magnetic moment within a single geometric framework. The proposed relationships are hypothetical in nature, but they form a consistent system that can serve as the basis for further development of a geometric model of the electron's internal structure.