2026-08-14
Electron spin as a consequence of the two-sheet cycle of a closed wave
Electron spin is not the mechanical rotation of a charged sphere. In the Wave Electricity model, the electron is treated as a localized wave mode with an internal two-sheeted extension. This geometry motivates a half-angle internal phase, but the transition to physical spin rotations requires a separately defined mapping.
Internal sheets, realizations of the dynamical channels, and spin space must be distinguished. The two sheets belong to a single internal cycle and are specified by \(Q_\Gamma\). The operators \(J_1(a,b)=J(a,b)\) and \(J_2(a,b)=J(b,a)\) denote two realizations of the principal channels. Their connection to orthogonal spin states is a physical mapping hypothesis. Electric charge is specified by a separate sector \(J_{q,\eta}\) and is derived neither from the choice of sheet nor from the interchange of channels.
When the two circular branches are traversed sequentially and the lengths of the connecting transitions are neglected, the total path is \(L_J=4\pi R\). For the fundamental mode, which accumulates a phase of \(2\pi\) along this path, a half-angle phase arises relative to the internal geometric angle \(\theta_{\mathrm{int}}=s/R\):
\[\tag{1} \boxed{L_J\simeq4\pi R\quad\Longrightarrow\quad k_J\simeq\frac1{2R}\quad\Longrightarrow\quad\chi\simeq\frac{\theta_{\mathrm{int}}}{2}.} \] 1. What needs to be explained
For a particle with spin \(1/2\), measuring its projection onto an arbitrarily chosen axis \(\mathbf n\) yields two outcomes:
\[\tag{2} S_{\mathbf n}=+\frac{\hbar}{2}, \qquad S_{\mathbf n}=-\frac{\hbar}{2}. \] If the state is prepared with a positive projection onto the original axis, the outcome probabilities for an analyzer axis at an angle \(\theta\) have a half-angle dependence. This is the angle between physical axes, rather than the angle of internal traversal. Separately, a spinor representative changes its overall sign under a physical rotation through \(2\pi\) and returns to its original value after \(4\pi\):
\[\tag{3} P_+(\theta)=\cos^2\frac{\theta}{2}, \qquad P_-(\theta)=\sin^2\frac{\theta}{2}, \] \[\tag{4} \Psi(2\pi)=-\Psi(0), \qquad \Psi(4\pi)=\Psi(0). \] The model must establish a connection between the internal geometry and these observable properties. The two-sheeted path yields an internal phase coefficient of \(1/2\). However, it does not by itself imply the group of spin rotations in three dimensions, the orthogonality of the two spin states, the scale \(\hbar\), or the probability rule. Below, the standard spinor representation is introduced explicitly, and its correspondence to the internal channels is identified as a separate hypothesis.
The logical sequence distinguishes two constructions: two-sheeted geometry \(\longrightarrow\) half-angle internal phase; spin mapping \(\longrightarrow\) the \(SU(2)\) representation of rotations \(\longrightarrow\) measurable projections and, once the Born rule is adopted, probabilities.
2. The finite-level state operator
At the finite operator level, two complementary idempotents are used:
\[\tag{5} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \qquad \ep+\em=1. \] The state operator takes the form
\[\tag{6} \boxed{ J(a,b)=\j^a(-\j)^b =\ep e^{i\pi b}+\em e^{i\pi a}. } \] Here, the parameters have different physical roles:
\[\tag{7} a=\varpi t, \qquad \pi\varpi=\omega_{\mathrm{int}}, \qquad b=\frac{\arcsin\beta}{\pi}, \qquad \beta=\frac{v}{c}. \] The parameter \(a\) describes the internal phase, while \(b\) describes the external motion of the center. In the example of a uniform internal cycle, \(t\) is the time in the rest frame of the center, and the frequency is defined with respect to that time. The equality \(J\overline J=1\) expresses algebraic unitarity; it must be distinguished from the Hilbert-space norm of the state, energy, and probability.
3. The two-sheeted extension of the internal contour
The finite-level operator \(J(a,b)\) specifies the state at the level of the two principal idempotent channels. Describing two-sheeted closure requires a deeper splitting of the internal channel. Let us introduce an additional hyperbolic unit and its idempotents:
\[\tag{8} \j_{\!p}=\pmp-\pme, \qquad \pmp^2=\pmp, \qquad \pme^2=\pme, \qquad \pmp\pme=0, \qquad \pmp+\pme=1. \] Let \(Q_\Gamma\) denote the minimal internal factor that distinguishes the two successive sheets:
\[\tag{9} Q_\Gamma(a)=\left(\pmp-\pme\right)^{a/2} =\pmp+\pme e^{i\pi a/2}. \] The subscript \(\Gamma\) denotes the internal two-sheeted extension. Relating this phase factor to physical spatial closure requires a geometric mapping. It neither selects a spin realization nor determines charge. For one realization in the rest frame, the extended operator describing dynamics and closure takes the following form; the charge sector is omitted for now and will be added in Section 11:
\[\tag{10} \mathcal J_{\Gamma}(a)=J(a,0)Q_\Gamma(a). \] The shift \(a\to a+2\) returns the finite-level part \(J(a,0)\) to the same value, but takes the deeper component to a different representative. Only the shift \(a\to a+4\) restores the entire extended operator:
\[\tag{11} J(a+2,0)=J(a,0), \qquad \mathcal J_{\Gamma}(a+2)\ne\mathcal J_{\Gamma}(a), \qquad \mathcal J_{\Gamma}(a+4)=\mathcal J_{\Gamma}(a). \] The two sheets belong to a deeper internal extension of a single wave. They are not two electrons, two simultaneously existing orbits, or the states "spin up" and "spin down".
4. Geometric mapping of the two sheets
In a physical mapping, the two deeper channels can be represented by two nearby branches of a single closed contour:
\[\tag{12} \pmp\longleftrightarrow C_+, \qquad \pme\longleftrightarrow C_-, \qquad R_\pm=R\pm\frac{\Delta R}{2}, \qquad \Delta R\ll R. \] The branches can lie in the same physical plane and differ by a small radial displacement. They need not be placed in mutually perpendicular spatial or phase planes. The same wave traverses them sequentially:
\[\tag{13} C_+\xrightarrow{\;2\pi\;}C_- \xrightarrow{\;2\pi\;}C_+. \] To leading order, the branch lengths are
\[\tag{14} L_+=2\pi R_+, \qquad L_-=2\pi R_-, \qquad L_++L_-=4\pi R. \] In the approximation of two circular traversals, the coefficient \(1/2\) arises from the total path length and the choice of the fundamental mode. The radial separation \(\Delta R\) and the connecting transitions introduce separate corrections. The resulting half-angle is an internal geometric angle; its connection to physical rotations of the particle still needs to be established.
5. The full cycle and the internal half-angle
The fundamental mode on the full contour must accumulate a phase of \(2\pi\) over the length \(L_J=4\pi R\). Its internal wave number is therefore
\[\tag{15} k_J=\frac{2\pi}{L_J}=\frac{1}{2R}. \] If \(s\) is the distance traveled and \(\theta_{\mathrm{int}}=s/R\) is the internal geometric angle, then
\[\tag{16} \chi(s)=k_Js=\frac{s}{2R}=\frac{\theta_{\mathrm{int}}}{2}. \] After the internal parameter increases by two, the finite-level operator returns to its original value, but only the second component of \(Q_\Gamma\) changes sign. This is not a change in the overall sign of the entire operator:
\[\tag{17} \boxed{\begin{aligned} Q_\Gamma(a+2)&=\pmp-\pme e^{i\pi a/2}\ne-Q_\Gamma(a),\\ Q_\Gamma(a+4)&=Q_\Gamma(a). \end{aligned}} \] The scalar phase factor \(e^{i\chi}\) does change sign when \(\theta_{\mathrm{int}}\) increases by \(2\pi\). However, this factor is not automatically the complete electron state. The overall spinor sign in (4) requires a separate law of physical rotation; it cannot be derived solely from the periodicity in (11) or (17). This distinction is consistent with the article on deep splitting.
6. Channel interchange and the spin-mapping hypothesis
Interchanging the principal idempotent channels defines two realizations of the dynamical operator. Let us denote this operation by \(\mathsf S_\Pi\):
\[\tag{18} \mathsf S_\Pi:\quad \ep\longleftrightarrow\em, \qquad \mathsf S_\Pi^2=I. \] The two possible realizations of the finite-level operator then take the form
\[\tag{19} \boxed{ \begin{aligned} J_1(a,b)&=J(a,b) =\ep e^{i\pi b}+\em e^{i\pi a},\\ J_2(a,b)&=\mathsf S_\Pi J_1(a,b) =J(b,a) =\ep e^{i\pi a}+\em e^{i\pi b}. \end{aligned} } \] In the second expression, the channels are interchanged, while the physical roles of the parameters are preserved: \(a\) remains internal and \(b\) external. The interchange itself does not establish the orthogonality of spin states. For example, the two operators coincide when \(a=b\). Their correspondence to two distinguishable spin states requires additional information about the complete state and an explicit mapping.
In the rest frame, \(b=0\), the realizations are written as
\[\tag{20} J_1(a,0)=J(a,0), \qquad J_2(a,0)=J(0,a). \] The notation \(J(a,0\mid0,a)\) denotes a choice between two realizations, rather than their arithmetic sum or the existence of two independent electron waves. Each realization includes the entire two-sheeted contour \(C_+\cup C_-\) and the same factor \(Q_\Gamma(a)\):
\[\tag{21} \mathcal J_{e,1}(a,b) =J_1(a,b)Q_\Gamma(a), \qquad \mathcal J_{e,2}(a,b) =J_2(a,b)Q_\Gamma(a). \] Sheets and spin states serve different functions. \(Q_\Gamma\) specifies the internal two-sheeted periodicity, while \(J_1,J_2\) label the dynamical realizations. For a spin description, normalized orthogonal states \(|+\rangle,|-\rangle\) with \(\langle+|-\rangle=0\) are introduced separately. They are not identified with the coefficient vectors of \(J_1,J_2\) without a separate mapping.
Let us adopt a spin mapping that connects the complete internal realizations to the states \(|+\rangle,|-\rangle\). A rotation about a chosen physical axis is described by the angle \(\Theta\), distinct from the internal angle \(\theta_{\mathrm{int}}\). In this representation:
\[\tag{22} \Psi_1(\Theta)=e^{-i\Theta/2}|+\rangle,\qquad\Psi_2(\Theta)=e^{i\Theta/2}|-\rangle, \] For the rotation generator
\[\tag{23} \widehat S_{\mathbf n}=i\hbar\frac{\partial}{\partial\Theta}, \] the eigenvalues of the spin projection are
\[\tag{24} \widehat S_{\mathbf n}\Psi_1 =+\frac{\hbar}{2}\Psi_1, \qquad \widehat S_{\mathbf n}\Psi_2 =-\frac{\hbar}{2}\Psi_2. \] Equation (24) follows from the adopted rotation law (22) and the definition of the generator (23). It does not derive this law from \(Q_\Gamma\). The scale \(\hbar\) and the physical correspondence of the generator to measurable angular momentum are specified separately. The two-sheeted geometry remains a possible internal basis whose connection to the full spinor representation requires proof.
7. The spinor representation
We now introduce the standard spin space \(\mathbb C^2\) and the action of physical rotations. This provides a consistent description of observed spin, but does not prove that it follows uniquely from the commutative algebra of \(J\) and \(Q_\Gamma\).
Let us choose the standard two-dimensional complex space of spin states with an orthonormal basis:
\[\tag{25} |+\rangle= \begin{pmatrix}1\\ 0\end{pmatrix}, \qquad |-\rangle= \begin{pmatrix}0\\ 1\end{pmatrix}. \] In the adopted representation of rotations, the state with a positive projection onto the axis \(\mathbf n\), with polar angle \(\theta\) and azimuth \(\varphi\), is written as
\[\tag{26} |+\mathbf n\rangle = \begin{pmatrix} \cos(\theta/2)\\ e^{i\varphi}\sin(\theta/2) \end{pmatrix}. \] The orthogonal state takes the form
\[\tag{27} |-\mathbf n\rangle = \begin{pmatrix} -e^{-i\varphi}\sin(\theta/2)\\ \cos(\theta/2) \end{pmatrix}. \] For a physical rotation through an angle \(\Theta\) about the axis \(\mathbf n\) and for a spin-projection measurement, let us adopt the standard operators:
\[\tag{28} U_{\mathbf n}(\Theta)=\exp\left(-\frac{i\Theta}{2}\boldsymbol\sigma\cdot\mathbf n\right),\qquad \widehat S_{\mathbf n}=\frac{\hbar}{2}\boldsymbol\sigma\cdot\mathbf n. \] Here, \(\boldsymbol\sigma\) denotes the Pauli matrices; these must be distinguished from the realization index \(\sigma=1,2\). The algebra of the adopted representation implies \([\widehat S_i,\widehat S_j]=i\hbar\epsilon_{ijk}\widehat S_k\) and \(\widehat{\mathbf S}^{\,2}=3\hbar^2I/4\). Thus, \(s=1/2\), the projections are \(\pm\hbar/2\), and the magnitude of the quantum angular momentum is \(\sqrt{s(s+1)}\hbar=\sqrt3\hbar/2\). The law (28) ensures spinor periodicity:
\[\tag{29} U_{\mathbf n}(2\pi)=-I, \qquad U_{\mathbf n}(4\pi)=I. \] 8. Changing the measurement axis
Suppose the state is prepared with a positive projection onto the original axis, and the new analyzer axis makes an angle \(\theta\) with it. The inner products of the spinors (26)–(27) yield amplitude magnitudes of \(\cos(\theta/2)\) and \(\sin(\theta/2)\). Once the Born rule is adopted:
\[\tag{30} \boxed{ P_{+\to+}(\theta)=\cos^2\frac{\theta}{2}, \qquad P_{+\to-}(\theta)=\sin^2\frac{\theta}{2}. } \] The probabilities sum to unity, and the mean projection onto the new axis is
\[\tag{31} P_{+\to+}+P_{+\to-}=1, \qquad \langle S_{\mathbf n}\rangle =\frac{\hbar}{2}\cos\theta. \] The half-angle in these probabilities follows from the chosen spinor representation and the Born rule. Its correspondence to the internal half-angle in (16) is not proved merely by the matching coefficient \(1/2\). The contour geometry and measurement statistics must be connected by an explicit physical mapping.
9. External motion preserves the meaning of the parameters
For a moving electron, the parameter \(b\) retains its meaning as external motion in both orientations. Using the definitions of Section 6, this requirement can be written compactly as
\[\tag{32} J_1(a,b)=J(a,b), \qquad J_2(a,b)=J(b,a). \] In the second expression, the placement of the parameters in the channels changes, but their physical labels are preserved. The external functional must read the channel containing \(b\) according to the realization. Both realizations then specify the same velocity:
\[\tag{33} \mathcal P_{\mathrm{ext}}^{(1)}[J_1]=\mathcal P_{\mathrm{ext}}^{(2)}[J_2]=\sin(\pi b)=\beta,\qquad\mathbf v=c\beta\widehat{\boldsymbol\tau}. \] The spin axis and the tangent \(\widehat{\boldsymbol\tau}\) to the motion of the center are independent geometric directions. Their alignment is possible only in a particular regime and must not be built into the definition of spin.
10. What the Stern–Gerlach experiment measures
An idealized spin analyzer in an inhomogeneous magnetic field connects magnetic-moment projections to spatial channels. Below, the magnetic interaction energy and the corresponding gradient force are given; the approximation assumes a distinguished \(z\) axis:
\[\tag{34} U=-\boldsymbol{\mu}\cdot\mathbf B, \qquad F_z=-\frac{\partial U}{\partial z} \simeq\mu_z\frac{\partial B_z}{\partial z}. \] For the electron, the relation between magnetic moment and spin is written as
\[\tag{35} \boldsymbol{\mu}_e =-g\frac{e}{2m_e}\mathbf S. \] If the state contains two amplitudes before entering the analyzer, interaction with the field correlates them with two diverging wave packets:
\[\tag{36} \left(A_+|+\mathbf n\rangle+A_-|-\mathbf n\rangle\right)\Phi_0 \longrightarrow A_+|+\mathbf n\rangle\Phi_+ +A_-|-\mathbf n\rangle\Phi_-. \] Over repeated trials, the ratio of channel intensities is determined by the squared magnitudes of the amplitudes:
\[\tag{37} I_+:I_-=|A_+|^2:|A_-|^2. \] The spatial channels of the analyzer are not images of the internal sheets. Relation (37) follows from the adopted rule for detecting amplitudes. For a free charged electron, a complete analysis must also account for the electric and magnetic Lorentz forces and the apparatus geometry; (34) does not give the total force in every regime. The dynamics of an individual detection and the origin of the \(g\)-factor remain separate problems.
11. Spin and charge belong to different levels
The direction of internal traversal cannot simultaneously define the sign of spin and electric charge. In the adopted architecture, \(J_1,J_2\) denote the dynamical realizations, the common factor \(Q_\Gamma\) denotes the two-sheeted extension, and \(J_{q,\eta}\) denotes the charge sector. A spin state is obtained only after an additional mapping of the complete state:
\[\tag{38} \boxed{\mathcal J_{\mathrm{particle}}^{(\sigma,\eta)}=J_\sigma(a,b)Q_\Gamma(a)J_{q,\eta}\cdots,\qquad\sigma=1,2,\quad\eta=\pm1.} \] The index \(\sigma\) selects the placement of the internal state in the principal channels, while \(\eta\) selects the charge sector. Thus, the interchange \(J_1\leftrightarrow J_2\) does not turn an electron into a positron, and charge conjugation \(J_{q,-}\leftrightarrow J_{q,+}\) need not change the spin orientation or the two-sheeted cycle.
12. Connection to the spin-to-polarization transition
For the selected singlet two-photon annihilation channel of an electron–positron pair, the joint spin information of the localized states can be mapped onto the joint polarization state of two propagating photons:
\[\tag{39} |S=0\rangle_{e^-e^+} \longrightarrow |\Psi_{\mathrm{pol}}\rangle_{\gamma_1\gamma_2}. \] One internal sheet does not become one photon while the other sheet becomes the second. The complete joint state of the pair undergoes the transformation. This issue is discussed separately in the article "Transition of Electron and Positron Spin into Photon Polarization".
13. Scope of the result
The adopted geometry and the choice of the fundamental mode imply \(L_J\simeq4\pi R\), \(k_J\simeq1/(2R)\), and the internal phase \(\chi\simeq\theta_{\mathrm{int}}/2\) in the approximation that neglects transition lengths. The algebra of \(Q_\Gamma\) implies the periodicity in (11) and (17). The overall spinor sign, noncommuting rotations about different axes, the projections \(\pm\hbar/2\), and the angular probabilities follow from the additionally adopted spin representation and detection rule, rather than directly from the contour length.
In the article on internal motion and magnetic moment, the principal scale \(\mu_B\) is reproduced after adopting the charge contour, effective current, and magnetic mapping. Comparing it with the projection \(\hbar/2\) gives \(g_0=2\). The additional length \(2\Delta r\), with \(\Delta r=r_e\), has the relative scale \(\alpha_{\mathrm{fs}}/(2\pi)\). However, the transition from this additional length to a positive magnetic-moment anomaly requires a separate frequency relation to the external field, as stated in the article on the anomaly. Matching the scale of the correction does not constitute a complete dynamical derivation of \(a_e\).
A mapping of complete internal states onto orthogonal spinors must still be constructed, and the action of physical rotations, coupling to a magnetic field, and detection rule must be justified. The dynamics of a single outcome, a consistent derivation of the principal and anomalous magnetic moments, and higher-order corrections remain separate tasks.
The two-sheeted geometry provides an internal phase scale and periodicity. The standard spin space, rotations, probabilities, and magnetic mapping are introduced separately. A complete derivation of spin from the internal geometry must connect these levels without inserting the required rotation law as an already proven result.
A related conceptual idea is explored by C. A. M. dos Santos and M. J. J. Fleury in "An electromagnetic model of the electron" [1], which considers a toroidal electromagnetic wave state. This is a separate model, rather than independent verification of the two-sheeted mechanism proposed here. The correspondence between the two constructions requires a separate analysis.
Conclusion
In the geometric hypothesis considered here, two successive sheets of a single internal contour can be mapped onto nearby radial branches \(R_+,R_-\). In the approximation that neglects the lengths of the connecting transitions, the fundamental internal mode has a half-angle phase:
\[\tag{40} \boxed{\begin{gathered} C_+\to C_-\to C_+\quad\Longrightarrow\quad L_J\simeq4\pi R\quad\Longrightarrow\quad\chi\simeq\theta_{\mathrm{int}}/2,\\ \text{separate spinor mapping:}\quad U_{\mathbf n}(2\pi)=-I,\qquad U_{\mathbf n}(4\pi)=I. \end{gathered}} \] The two-sheeted internal cycle and the spinor law of physical rotations have different definitions. The two dynamical realizations can label spin states only after the corresponding mapping has been justified. The chosen standard spinor representation yields the projections \(\pm\hbar/2\), the \(4\pi\) periodicity, and, once the Born rule is adopted, the angular probabilities. External motion and electric charge retain independent roles. Connecting this entire structure to the internal geometry remains the central task of a further derivation.
Materials used
- C. A. M. dos Santos, M. J. J. Fleury, An electromagnetic model of the electron, arXiv:2510.22384, 2025.