2026-08-14
Electron spin as a manifestation of a two-plane cycle of a closed wave
Electron spin is usually introduced as an internal quantum property that cannot be reduced to the ordinary rotation of a material particle. However, its three main features two possible projections, a half-angle law, and the restoration of the state only after two full rotations indicate a specific geometric structure.
In the proposed model, the primary object is not a particle within which something rotates, but a single normalized wave. A particle is conventionally defined as a stable, localized mode of this wave. Inside the electron, it forms a closed path, split into two branches of equal radius, located in mutually perpendicular phase planes. The wave traverses these branches sequentially, so one full rotation does not return it to its original internal state.
The main result of the article will be derived directly from the length of this path. Each branch has a length of \(2\pi R\), and the full contour is \(4\pi R\). Therefore, the fundamental wave on it accumulates phase twice as slowly as the usual orbital angle:
\[\tag{1} \boxed{ L_J=4\pi R \quad\Longrightarrow\quad k_J=\frac{1}{2R} \quad\Longrightarrow\quad \chi=\frac{\theta}{2}. } \] It is this geometric relationship that will lead to the phase factor \(e^{\pm i\theta/2}\), the state recovery after \(4\pi\), and two eigenvalues of the angular momentum projection \(\pm\hbar/2\). The standard spinor notation will appear only after this as a compact representation of the already obtained two-plane geometry.
1. Statement of the Problem
For a particle with spin, one half-dimensional projection onto an arbitrarily chosen axis \(\mathbf n\) yields only two results:
\[\tag{2} S_{\mathbf n}=+\frac{\hbar}{2}, \qquad S_{\mathbf n}=-\frac{\hbar}{2}. \] If the measurement axis is rotated relative to the prepared state by an angle of \( heta\), the probabilities of the two outcomes are equal.
\[\tag{3} P_+(\theta)=\cos^2\frac{\theta}{2}, \qquad P_-(\theta)=\sin^2\frac{\theta}{2}. \] Furthermore, the spinor changes sign after a rotation by \(2\pi\) and is fully restored only after a rotation by \(4\pi\):
\[\tag{4} \Psi(2\pi)=-\Psi(0), \qquad \Psi(4\pi)=\Psi(0). \] The task is not to rewrite these properties using ready-made Pauli matrices. We must first construct the inner wave contour, obtain the half-angle from its length, and then show why the generator of two opposing modes has eigenvalues \(\pm\hbar/2\).
2. Primary Wave and State Operator
The model is based on two mutually complementary idempotents:
\[\tag{5} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \qquad \ep+\em=1. \] The complete state is written by the operator
\[\tag{6} \boxed{ J(a,b)=\jmath^a(-\jmath)^b =\ep e^{i\pi b}+\em e^{i\pi a}. } \] Parameter \(a\) describes the internal state, and parameter \(b\) describes the external motion of the center of the localized wave:
\[\tag{7} a=\varpi t, \qquad \pi\varpi=\omega_{\mathrm{int}}, \qquad b=\frac{\arcsin\beta}{\pi}, \qquad \beta=\frac{v}{c}. \] In the rest frame of the center, \(b=0\). But the internal rotation can be represented in each of the two idempotent planes. Let's introduce two basis states:
\[\tag{8} J_1(a)=J(a,0)=\ep+\em e^{i\pi a}, \] \[\tag{9} J_2(a)=\ep e^{i\pi a}+\em \equiv J(0,a). \] In formula (9), the notation \(J(0,a)\) is used as a shorthand notation for the internal phase transferred to the second idempotent plane. It does not mean that the physical external parameter \(b\) has become equal to the internal parameter \(a\). For the complete moving state, the roles of \(a\) and \(b\) remain unchanged.
The states \(J_1\) and \(J_2\) should also not be called spin-up and spin-down a priori. They form an internal geometric basis from which the observed projections arise only after the physical axis of measurement is chosen.
3. Splitting of a Closed Orbit into Two Planes
Let the original closed path \(C\) split into two branches \(C_1\) and \(C_2\). They have the same radius, but belong to two mutually perpendicular phase planes:
\[\tag{10} C\longrightarrow C_1\cup C_2, \qquad C_1\subset\pi_1, \qquad C_2\subset\pi_2, \qquad \pi_1\perp\pi_2. \] \[\tag{11} R_1=R_2=R. \] These are not two spatially separated orbits, nor are they two independente waves. \(C_1\) and \(C_2\) are two consecutive sections of a single internal path. The same wave first passes through branch \(C_1\), then moves to plane \(\pi_2\), passes through branch \(C_2\), and only then returns to the original plane:
\[\tag{12} \boxed{ C_1\xrightarrow{\ 2\pi\ }C_2 \xrightarrow{\ 2\pi\ }C_1. } \] The transition between planes is considered as a change in the internal state sheet at a common point of the contour. It does not form an additional extended section of the path. This condition is essential: the exact half-factor occurs when the total length consists of exactly two circles of the same radius.
Perpendicularity applies primarily to internal phase planes. It does not mean that a particle instantly changes direction in ordinary three-dimensional space. The observed spatial orientation arises after mapping the complete two-component state.
4. A complete two-plane cycle
We denote the initial position by the triple \((\pi_1,0,0)\), where the first element indicates the phase plane, the second is the accumulated orbital angle \(\theta\), and the third is the wave phase \(\chi\). After passing through the first branch, the orbital angle increases by \(2\pi\), but the wave ends up on the second sheet of the internal geometry:
\[\tag{13} (\pi_1,0,0) \xrightarrow{\ 2\pi\ } (\pi_2,2\pi,\pi). \] After the second revolution, both the angular position and the internal plane number are restored:
\[\tag{14} (\pi_2,2\pi,\pi) \xrightarrow{\ 2\pi\ } (\pi_1,4\pi,2\pi). \] Therefore, the angular extent of the full internal path is
\[\tag{15} \boxed{\Theta_J=4\pi.} \] The usual angular coordinate is repeated at \(2\pi\), but the full state also contains the phase plane number. Therefore, each observed angular position corresponds to two internal representatives:
\[\tag{16} (\theta,\pi_1), \qquad (\theta,\pi_2). \] The split contour thus forms a two-sheeted covering of the ordinary angular cycle. One spatial rotation transfers the wave to the second sheet, and two rotations complete the full path.
5. Geometric Origin of the Half Angle
Now the half angle can be obtained directly from the contour length. The length of each branch is
\[\tag{17} L_1=L_2=2\pi R. \] The total length of a single two-plane path is
\[\tag{18} \boxed{ L_J=L_1+L_2=4\pi R. } \] Let \(s\) be the length traveled along this path. The fundamental mode must accumulate phase \(2\pi\) in one complete closed loop. Therefore, its internal wavenumber is
\[\tag{19} \boxed{ k_J=\frac{2\pi}{L_J} =\frac{1}{2R}. } \] The phase of the wave after traveling a length \(s\) is equal to
\[\tag{20} \chi(s)=k_Js=\frac{s}{2R}. \] The accumulated orbital angle is related to the path length by the usual relation
\[\tag{21} \theta=\frac{s}{R}. \] Substituting \(s=R\theta\) into formula (20), we obtain the central result:
\[\tag{22} \boxed{ \chi=\frac{\theta}{2}. } \] Thus, the coefficient \(1/2\) is not introduced as a known property of the spinor. It arises because the phase rotation \(2\pi\) is distributed over a full path of length \(4\pi R\). Splitting the original orbit into two successive branches doubles the closure length and halves the phase accumulation rate with respect to the orbital angle:
\[\tag{23} \boxed{ L_J=4\pi R \quad\Longrightarrow\quad \chi=2\pi\frac{s}{L_J} =\frac{\theta}{2}. } \] For higher harmonics on the full contour, \(k_{J,n}=n/(2R)\) is possible. In this model, the electron spin state is associated with a fundamental odd mode \(n=1\), which changes sign after traveling half the full distance.
6. Two Directions of Spin Propagation and Projections
A single two-plane contour allows two opposite directions of propagation. We denote the corresponding modes by \(|J_+\rangle\) and \(|J_-\rangle\). They apply to the entire contour \(C_1\cup C_2\), not to its individual planes.
Taking into account formula (22), their phase representatives have the form
\[\tag{24} \Psi_+(\theta)=e^{-i\theta/2}|J_+\rangle, \qquad \Psi_-(\theta)=e^{+i\theta/2}|J_-\rangle. \] The generator of the state change with respect to the angle \(\theta\) is defined by the expression
\[\tag{25} \widehat S_{\mathbf n} =i\hbar\frac{\partial}{\partial\theta}. \] Its action on two opposite modes yields
\[\tag{26} \widehat S_{\mathbf n}\Psi_+ =+\frac{\hbar}{2}\Psi_+, \] \[\tag{27} \widehat S_{\mathbf n}\Psi_- =-\frac{\hbar}{2}\Psi_-. \] Therefore, the geometry of the full contour distinguishes two proper projections:
\[\tag{28} \boxed{ S_{\mathbf n}=\pm\frac{\hbar}{2}. } \] The factor \(1/2\) in these values has the same origin as the half-phase: it is the ratio of one phase revolution to two consecutive orbital revolutions.
7. Rotations by \(2\pi\) and \(4\pi\)
After passing the first branch
\[\tag{29} \theta=2\pi, \qquad \chi=\pi. \] Therefore, both phase factors become equal to minus one:
\[\tag{30} e^{\pm i\pi}=-1. \] When projected onto the same physical angular position, the internal representative of the state changes sign:
\[\tag{31} \boxed{ \Psi(2\pi)=-\Psi(0). } \] Geometrically, the wave is on the second branch of \(C_2\), so the complete internal path is not yet closed. After the second rotation
\[\tag{32} \theta=4\pi, \qquad \chi=2\pi, \qquad e^{\pm i2\pi}=1, \] and the state is fully restored:
\[\tag{33} \boxed{ \Psi(4\pi)=\Psi(0). } \] The overall sign does not change the probability of an individual measurement, since the square of the modulus remains the same. However, the relative sign can appear during interference with a state that has not undergone the corresponding rotation.
8. Two-Component State and the Standard Spinor
After the geometric derivation, the internal state can be represented as a superposition of two basal planes:
\[\tag{34} |\psi\rangle =c_1|J_1\rangle+c_2|J_2\rangle, \qquad |c_1|^2+|c_2|^2=1. \] Here \(|J_1\rangle\) and \(|J_2\rangle\) correspond to the planes \(\pi_1\) and \(\pi_2\). They are an internal basis, but not fixed "up" and "down" states. Specific measured projections are defined relative to the chosen direction \(\mathbf n\) in physical space.
A normalized two-component state is associated with a unit spatial vector.
\[\tag{35} \mathbf n =\langle\psi|\boldsymbol{\sigma}|\psi\rangle, \qquad \boldsymbol{\sigma} =(\sigma_x,\sigma_y,\sigma_z). \] If the direction \(\mathbf n\) is specified by the polar angle \(\theta\) and the azimuth \(\varphi\), the normalized state representative has the form
\[\tag{36} \boxed{ |+\mathbf n\rangle_J = \begin{pmatrix} \cos(\theta/2)\\ e^{i\varphi}\sin(\theta/2) \end{pmatrix}. } \] The half-angle in formula (36) is no longer an initial postulate: its phase scale is already obtained from the length of the split contour in formula (22). The two-component notation only maps this result to an arbitrarily oriented physical direction.
An orthogonal state with the opposite projection can be chosen in the form
\[\tag{37} |-\mathbf n\rangle_J = \begin{pmatrix} -e^{-i\varphi}\sin(\theta/2)\\ \cos(\theta/2) \end{pmatrix}. \] The spatial rotation operator of this two-component state is written as
\[\tag{38} \boxed{ U_{\mathbf n}(\theta) =\exp\left( -\frac{i\theta}{2} \boldsymbol{\sigma}\cdot\mathbf n \right). } \] Thus, the standard map \(SU(2)\to SO(3)\) appears here as the matrix form of the already constructed two-sheeted cycle:
\[\tag{39} U_{\mathbf n}(2\pi)=-I, \qquad U_{\mathbf n}(4\pi)=I. \] Comparing formula (38) with the general rotation operator \(U=\exp(-i\theta\widehat S_{\mathbf n}/\hbar)\), we obtain
\[\tag{40} \boxed{ \widehat S_{\mathbf n} =\frac{\hbar}{2} \boldsymbol{\sigma}\cdot\mathbf n. } \] 9. Changing the Measurement Axis
Let the state be prepared with a positive projection relative to the original axis, and the new analyzer axis be rotated by an angle \(\theta\). In the basis of the new analyzer, the state is decomposed into two proper channels:
\[\tag{41} |+\mathbf z\rangle =\cos\frac{\theta}{2}|+\mathbf n\rangle -e^{-i\varphi}\sin\frac{\theta}{2}|-\mathbf n\rangle. \] If the statistical weight of a result is determined by the square of the absolute value of the corresponding amplitude, we obtain
\[\tag{42} \boxed{ P_{+\to+}(\theta) =\cos^2\frac{\theta}{2}, \qquad P_{+\to-}(\theta) =\sin^2\frac{\theta}{2}. } \] The total probability is preserved:
\[\tag{43} P_{+\to+}+P_{+\to-}=1. \] The average spin projection onto the new axis is
\[\tag{44} \langle S_{\mathbf n}\rangle =\frac{\hbar}{2} \left( \cos^2\frac{\theta}{2} -\sin^2\frac{\theta}{2} \right) =\frac{\hbar}{2}\cos\theta. \] Thus, the angular dependence of the probabilities is a continuation of the same geometric coefficient \(1/2\), which initially arose from doubling the length of the internal path.
10. The Stern-Gerlach Experiment
The Stern-Gerlach experiment demonstrates the discreteness of the magnetic moment projection. The historical setup used a beam of neutral silver atoms. A non-uniform magnetic field associates two possible projections of the magnetic moment with opposite spatial deviations.
The potential energy of the magnetic moment in the field and the corresponding force along the z-axis are equal.
\[\tag{45} U=-\boldsymbol{\mu}\cdot\mathbf B, \qquad F_z=-\frac{\partial U}{\partial z} \simeq\mu_z\frac{\partial B_z}{\partial z}. \] For an electron, the magnetic moment is opposite to the spin due to its negative charge:
\[\tag{46} \boldsymbol{\mu}_e =-g\frac{e}{2m_e}\mathbf S. \] Therefore, the two eigenvalues \(S_{mathbf n}=pmhbar/2\) correspond to opposite forces. Instead of a continuous band, two spatially separated channels arise.
Before the analyzer, the state relative to its axis can contain both projections:
\[\tag{47} |\psi\rangle =A_+|+\mathbf n\rangle +A_-|-\mathbf n\rangle. \] An inhomogeneous field connects them with two diverging wave packets:
\[\tag{48} |\psi\rangle\Phi_0 \longrightarrow A_+|+\mathbf n\rangle\Phi_+ +A_-|-\mathbf n\rangle\Phi_-. \] After the accumulation of a large number of events, the intensity ratio of the two spots tends to
\[\tag{49} \boxed{ I_+:I_-=|A_+|^2:|A_-|^2. } \] The magnet does not create two new internal entities and does not directly select one of the phase planes \(\pi_1\), \(\pi_2\). It defines the physical axis relative to which the two-component information of the complete state is mapped into two spatially diverging channels:
\[\tag{50} \boxed{ \text{two-component internal information} \longrightarrow \text{two spatial channels}. } \] If the second analyzer is oriented along the same axis, the channel prepared by the first magnet is retraced with unit probability. If the axis of the second analyzer is perpendicular to the first, formula (42) yields two equally probable results. Thus, the "up" or "down" result refers not to a fixed internal plane, but to the projection of the complete state onto the current analyzer axis.
Two-plane geometry defines two amplitudes and their angular dependence. However, converting the superposition into a single recorded result requires a separate dynamic model of the wave interaction with the magnet and the screen. In this article, the correspondence between the square of the amplitude modulus and the statistical recording frequency is used.
11. What is derived from geometry and what is adopted additionally
It is useful to separate the immediate results of the construction and further physical correspondences.
From the adopted geometry of the split contour, the following follows:
1. two perpendicular phase planes of the internal state;
2. a single sequential path \(C_1 o C_2 o C_1\);
3. the total length of the fundamental contour \(L_J=4\pi R\);
4. the internal wavenumber \(k_J=1/(2R)\);
5. Half-phase \(\chi=\theta/2\);
6. Change of sign of the representative after \(2\pi\) and recovery after \(4\pi\);
7. Two opposite values of the generator \(\pm\hbar/2\).
The physical mappings or additional conditions remain:
1. Identification of these two values with the observed electron spin;
2. Mapping of the internal idempotent basis onto an arbitrary direction in physical three-dimensional space;
3. Quadratic probability rule;
4. Relationship of spin to magnetic moment and the magnitude of the \(g\)-factor;
5. The dynamics of selecting a single result during registration.
This separation does not weaken the model, but rather reveals the precise boundary of the already constructed geometry. It is particularly important that the factor \(1/2\) arises before invoking the standard spin operator from the length of a single two-plane path.
12. The Relationship with the Transition of Spin to Polarization
The two-plane structure does not disappear when a localized wave transitions to a propagating state. The next paper considers the annihilation of an electron-positron pair, in which the joint spin correlation transforms into a polarization correlation of two photons.
\[\tag{51} \boxed{ |S=0\rangle_{e^-e^+} \longrightarrow |\Psi_{\mathrm{pol}}\rangle_{\gamma_1\gamma_2}. } \] In this case, one cannot assume that the \(\pi_1\) plane literally transforms into one photon, and the \(\pi_2\) plane into the second. The joint state of the entire pair is transformed, and the internal two-component information of the localized regime is continued in the two-component transverse state of free emission. This transition is discussed in detail in the article "Transition of Electron and Positron Spin into Photon Polarization".
Conclusion
In the proposed model, spin does not arise from the rotation of the material ball and is not introduced as an independent property of the final particle. Its geometric basis is a closed primary wave, the path of which is split between two mutually perpendicular phase planes.
Each branch has a length of \(2\pi R\), but the complete state returns to the original plane only after passing through both branches. Therefore, the closure length is \(4\pi R\). The fundamental wave on such a contour has a wavenumber of \(1/(2R)\), as a result of which its internal phase is equal to half the accumulated orbital angle:
\[\tag{52} \boxed{ C_1\perp C_2 \quad\Longrightarrow\quad L_J=4\pi R \quad\Longrightarrow\quad \chi=\frac{\theta}{2} \quad\Longrightarrow\quad S_{\mathbf n}=\pm\frac{\hbar}{2}. } \] After one revolution, the wave moves to the second sheet of the internal state, and its phase representative changes sign. After the second revolution, both the phase and the original plane are restored. The standard \(4\pi\) periodicity of the spinor turns out to be a matrix mapping of this complete two-plane cycle.
When changing the measurement axis, the two internal components form two amplitudes with coefficients \(\cos(\theta/2)\) and \(\sin(\theta/2)\). An inhomogeneous magnetic field maps them into two spatial channels, preserving the norm and information of the complete state. Thus, the split orbit links the half-phase, the \(\pm \hbar/2\) projections, the \(4\pi\) periodicity, and the observed two-channel splitting in a single sequence.