2026-08-14
Transition of electron and positron spin into photon polarization
During the annihilation of an electron and positron, the initial localized state of the pair is transformed into the photon state, with the total angular momentum of the system being conserved. Below, we consider the geometric scheme of the transition from a two-plane spin basis, defined by the states \(J(a,0)\) and \(J(0,a)\), to a correlated polarization structure of two photons.
Introduction
In the first part of the work, a geometric interpretation of electron spin was proposed. Since the electron and positron have the same spin of 1/2, in this paper we assume that the two-plane spin structure is also carried over to the positron. The possible geometric difference between the particle and antiparticle is considered separately from the spin basis. Its basis is two internal states.
\[\tag{1} J_1(a)=J(a,0), \qquad J_2(a)=J(0,a), \] which correspond to two mutually perpendicular planes of internal motion:
\[\tag{2} J(a,0)\longleftrightarrow\Pi_1, \qquad J(0,a)\longleftrightarrow\Pi_2, \qquad \Pi_1\perp\Pi_2. \] These states are not two spatially separated orbits and do not differ in opposite directions of rotation. They form two components of a single internal geometry. The transition between them results in the internal state not returning to its original state after a rotation of \(2\pi\), and the full cycle is completed after a rotation of \(4\pi\).
The observed spin-up and spin-down states do not arise as direct designations of the \(\Pi_1\) and \(\Pi_2\) planes, but as two projections of the internal two-component state relative to the chosen direction in physical three-dimensional space.
In the singlet annihilation channel under consideration, the electron and positron transform into two photons. A photon does not have spin-up and spin-down states in the same sense as a massive particle. Its intrinsic angular momentum manifests itself through two helicities corresponding to right- and left-handed circular polarizations.
The main goal of this work is to establish how the joint spin state of an electron and positron, built on the internal states \(J(a,0)\) and \(J(0,a)\), is transformed into a joint polarization state of two photons.
\[\tag{3} \boxed{ |\Psi_s\rangle_{e^-e^+} \;\longrightarrow\; |\Psi_{\mathrm{pol}}\rangle_{\gamma_1\gamma_2} } \] To construct the transition, it is necessary to separate the spin structure from the charge, form a joint state of the electron and positron, determine the admissible helicities of the two photons, and verify the conservation of energy, momentum, and total angular momentum.
The proposed scheme does not replace the quantum electrodynamic description of annihilation. It defines a geometric framework in which the spin of a massive particle and the polarization of a photon are considered different observable manifestations of a two-component internal state.
1. Two internal states of a particle
We define the internal state of one particle as a two-component object
\[\tag{4} |\psi\rangle = c_1|J(a,0)\rangle +c_2|J(0,a)\rangle, \qquad |c_1|^2+|c_2|^2=1. \] The basis states correspond to two mutually perpendicular internal planes. The coefficients \(c_1\), \(c_2\) and their relative phase determine the specific orientation of the observed state after mapping the internal geometry into physical space.
The sequence of the complete internal cycle is as follows
\[\tag{5} J(a,0) \xrightarrow{\;2\pi\;} J(0,a) \xrightarrow{\;2\pi\;} J(a,0). \] Therefore, the geometric state has spinor periodicity:
\[\tag{6} 2\pi:\quad|\psi\rangle\longrightarrow-|\psi\rangle, \qquad 4\pi:\quad|\psi\rangle\longrightarrow|\psi\rangle. \] 2. From Intrinsic Geometry to Observed Spin
The states \(J(a,0)\) and \(J(0,a)\) form an internal basis, but should not be directly referred to as spin-up and spin-down states. We write the internal state as a spinor. \[\tag{7} |\psi\rangle = \begin{pmatrix} c_\_2 \end{pmatrix}. \]
It is associated with a unit spatial vector. \[\tag{8} \mathbf n = \langle\psi|\boldsymbol{\sigma}|\psi\rangle, \qquad \boldsymbol{\sigma}=(\sigma_x,\sigma_y,\sigma_z). \]
The spin operator is defined by the expression
\[\tag{9} \widehat{\mathbf S} = \frac{\hbar}{2}\boldsymbol{\sigma}, \] For an arbitrarily chosen unit axis \(\mathbf u\), the spin projection operator and its eigenvaluesThe values are of the form
\[\tag{10} \widehat S_{\mathbf u} = \frac{\hbar}{2}\,\boldsymbol{\sigma}\cdot\mathbf u, \qquad S_{\mathbf u}=\pm\frac{\hbar}{2}. \] If \(\mathbf u=\mathbf n\), then the state \(|\psi\rangle\) is an eigenstate with projection \(+\hbar/2\). The minus sign corresponds to an orthogonal spinor.
Thus,
\[\tag{11} \boxed{ \left\{J(a,0),J(0,a)\right\} \longrightarrow |\psi\rangle \longrightarrow \mathbf n, \qquad \operatorname{spec}(\widehat S_{\mathbf u}) = \left\{-\frac{\hbar}{2},+\frac{\hbar}{2}\right\} }. \] Mutually perpendicular planes are the geometric basis of the internal state, while spin up and spin down are observable projections of this state relative to the chosen axis.
3. Electron and Positron
The electron and positron have the same spin \(1/2\). Therefore, their internal spin spaces have the same structure:
\[\tag{12} \mathcal H_s^{(e^-)} = \mathcal H_s^{(e^+)} = \operatorname{span} \left\{|J(a,0)\rangle,|J(0,a)\rangle\right\}. \] The difference between an electron and a positron cannot be attributed to the opposite orientation of their spin planes. The sign of the charge is determined by individual states.
\[\tag{13} J_{q,-}, \qquad J_{q,+}. \] The complete states can be schematically represented as products of independent components:
\[\tag{14} J_{e^-} = J_{\mathrm{dyn}}^{(e)}J_{q,-}J_s^{(e^-)}, \qquad J_{e^+} = J_{\mathrm{dyn}}^{(e)}J_{q,+}J_s^{(e^+)}. \] This separation does not allow us to transfer the charge opposition to the relative positions of the internal planes.
In this paper, the identical spin basis notation for the electron and positron is an accepted continuation of the model. The geometric mechanism of charge conjugation has not yet been established. In particular, the correspondence \(e^+: J(-a,0),J(0,-a)\) is not postulated here, since it has not yet been shown that the substitution \(a\to-a\) changes the electric charge itself, and not just the direction of internal motion or phase.
4. Joint spin state of the pair
The pair state belongs to the tensor product of two spin spaces:
\[\tag{15} |\Psi_s\rangle_{e^-e^+} \in \mathcal H_s^{(e^-)}\otimes\mathcal H_s^{(e^+)}. \] Next, we consider the initial \(S\)-wave singlet configuration with \(S=0\), corresponding to the ground state of parapositronium. For this particular state, the two-photon channel is the leading one:
\[\tag{16} |S=0,m=0\rangle = \frac{1}{\sqrt2} \left( |\uparrow\downarrow\rangle -|\downarrow\uparrow\rangle \right). \] This is a single coherent superposition, not a statistical mixture of two variants. The total spin of the state is zero:
\[\tag{17} \left( \widehat{\mathbf S}_{e^-} +\widehat{\mathbf S}_{e^+} \right) |S=0,m=0\rangle =0. \] Geometrically, two two-plane structures enter into a common state without a preferred resulting direction of the total spin.
5. Photon Polarization Space
A free photon has two physical transverse polarizations. In a circular basis, they correspond to states
\[\tag{18} |+\rangle_\gamma, \qquad |-\rangle_\gamma, \] with helicities \(\lambda=+1\) and \(\lambda=-1\). The projection of the photon's intrinsic angular momentum onto the direction of propagation is equal to
\[\tag{19} \mathbf S_\gamma\cdot\widehat{\mathbf k} = \lambda\hbar. \] The polarization space is of the form
\[\tag{20} \mathcal H_{\mathrm{pol}}^{(\gamma)} = \operatorname{span}\left\{|+\rangle_\gamma,|-\rangle_\gamma\right\}. \] Linear polarization is a superposition of two helicity states:
\[\tag{21} |\mathrm{lin}\rangle = \frac{1}{\sqrt2} \left( |+\rangle+e^{i\varphi}|-\rangle \right). \] 6. Correspondence of Two-Component Spaces
The spin space of a massive particle and the physical polarization space of a photon are both two-dimensional:
\[\tag{22} \dim\mathcal H_s^{(e)}=2, \qquad \dim\mathcal H_{\mathrm{pol}}^{(\gamma)}=2. \] The equality of dimensions allows for a structural comparison.
\[\tag{23} \dim\mathcal H_s^{(e)} = \dim\mathcal H_{\mathrm{pol}}^{(\gamma)} =2. \] However, the equality of dimensions does not imply a physical mapping from the state of one electron to the state of one photon: the annihilation transition acts on the shared space of the pair.
However, individual basis states cannot be identified without an additional derivation:
\[\tag{24} J(a,0)\not\equiv|+\rangle_\gamma, \qquad J(0,a)\not\equiv|-\rangle_\gamma. \] The states \(J(a,0)\), \(J(0,a)\) are the internal geometric basis, and the circular polarizations are the physical basis of the propagating photon. The desired operator must map the joint spin space of the pair to the joint polarization space of the photons and take into account the directions of their emission.
7. Three Independent Annihilation Balances
The electron–positron transition to photons should be divided into charge, energy-momentum, and spin-polarization balances.
\[\tag{25} Q_{e^-}+Q_{e^+}=-e+e=0. \] \[\tag{26} P_{e^-}^{\mu}+P_{e^+}^{\mu} = K_{\gamma_1}^{\mu}+K_{\gamma_2}^{\mu}. \] \[\tag{27} |\Psi_s\rangle_{e^-e^+} \longrightarrow |\Psi_{\mathrm{pol}}\rangle_{\gamma_1\gamma_2}. \] The complete diagram looks like this
\[\tag{28} \boxed{ \begin{aligned} J_{q,-}\otimes J_{q,+}&\longrightarrow J_{q,0},\\ P_{e^-}^{\mu}+P_{e^+}^{\mu} &\longrightarrow K_{\gamma_1}^{\mu}+K_{\gamma_2}^{\mu},\\ |\Psi_s\rangle_{e^-e^+} &\longrightarrow|\Psi_{\mathrm{pol}}\rangle_{\gamma_1\gamma_2}. \end{aligned} } \] Here \(J_{q,0}\) denotes the neutral final charge state. This notation expresses the law \(-e+e=0\), but does not introduce the previously undefined algebraic equality \(J_{q,-}J_{q,+}=1\). Charge compensation does not imply mutual annihilation or rotation of the planes \(J(a,0)\) and \(J(0,a)\).
8. Energy and Momentum of Two Photons
In the center-of-mass system of a slow pair
\[\tag{29} \mathbf p_{e^-}+\mathbf p_{e^+}=0. \] A single photon with nonzero energy cannot conserve zero momentum, since \(E_\gamma=c|\mathbf p_\gamma|\). For two photons, the condition is satisfied when
\[\tag{30} \mathbf p_{\gamma_1}=-\mathbf p_{\gamma_2}. \] If the kinetic energy of the particles can be neglected,
\[\tag{31} E_{\mathrm{in}}=2m_ec^2, \qquad E_{\gamma_1}=E_{\gamma_2}=m_ec^2=511\, \text{keV}. \] Therefore,
\[\tag{32} \boxed{ \omega_\gamma = \frac{m_ec^2}{\hbar} = \omega_C }, \qquad \lambda_\gamma=\frac{h}{m_ec}. \] The energy Compton frequency \(\omega_C\) should not be automatically identified with the frequency of the complete geometric cycle between \(J(a,0)\) and \(J(0,a)\), unless such an equality has been separately derived.
9. Transformation of a Singlet State
The singlet state must transform into a two-photon state with zero total angular momentum:
\[\tag{33} |S=0,m=0\rangle_{e^-e^+} \longrightarrow |\Psi_0\rangle_{\gamma_1\gamma_2}. \] Let the photons propagate in opposite directions:
\[\tag{34} \widehat{\mathbf k}_2=-\widehat{\mathbf k}_1. \] The projection of the total angular momentum onto the expansion axis is zero if the helicities defined relative to the proper directions of photon motion are the same:
\[\tag{35} \lambda_1=\lambda_2=\lambda, \] \[\tag{36} \mathbf S_{\gamma_1}+\mathbf S_{\gamma_2} = \lambda\hbar \left( \widehat{\mathbf k}_1+\widehat{\mathbf k}_2 \right) =0. \] For the complete state with \(J=0\), taking into account the rotational symmetry and discrete symmetries of the singlet channel, two components remain:
\[\tag{37} |\Psi_0\rangle_{\gamma_1\gamma_2} = A|+,+\rangle+B|-,-\rangle. \] In the absence of a preferred orientation, the amplitude moduli are:
\[\tag{38} |A|=|B|=\frac{1}{\sqrt2}. \] Then
\[\tag{39} |\Psi_0\rangle_{\gamma_1\gamma_2} = \frac{1}{\sqrt2} \left( |+,+\rangle +e^{i\delta}|-,-\rangle \right). \] The relative phase \(\delta\) is not determined by the angular momentum conservation law alone. In the standard quantum electrodynamic description, it is fixed by symmetries and the annihilation amplitude; its specific sign also depends on the phase agreements for the transverse bases of two oppositely directed photons. The geometric model has yet to reproduce this result with its own transition operator.
10. Geometrical meaning of the transition
In the internal model of the electron, the states \(J(a,0)\) and \(J(0,a)\) define two mutually perpendicular planes. Their superposition forms the observed spin direction after mapping into three-dimensional space. For a photon, the field is transverse to the direction of propagation, and the two independent components can be represented by two linear or two circular polarizations.
This allows us to propose a geometric principle
\[\tag{40} \boxed{ \text{two-plane internal state of a fermion} \longrightarrow \text{two-component transverse state of a photon} }. \] In a localized state, the two-component nature manifests itself through the spinor periodicity \(4\pi\) and projections \(\pm\hbar/2\). In a propagating photonic state, it manifests itself through two polarizations and helicities \(\pm\hbar\).
However, it cannot be claimed that one electron plane literally transforms into the first photon, and the other into the second. The joint state of the entire pair is transformed:
\[\tag{41} \boxed{ \left( \mathcal H_s^{(e^-)} \otimes \mathcal H_s^{(e^+)} \right)_{S=0} \longrightarrow \left( \mathcal H_{\mathrm{pol}}^{(\gamma_1)} \otimes \mathcal H_{\mathrm{pol}}^{(\gamma_2)} \right)_{J=0} }. \] 11. Polarization Correlation
State (39) is not represented by a product of independent single-photon states:
\[\tag{42} |\Psi_0\rangle_{\gamma_1\gamma_2} \ne |\gamma_1\rangle\otimes|\gamma_2\rangle. \] In the geometric interpretation, this relationship arises because both photons are formed not from two independently defined spins, but from a single shared singlet state of the electron-positron pair.
The structural correspondence can be written as
\[\tag{43} \boxed{ \frac{|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle}{\sqrt2} \longrightarrow \frac{|+,+\rangle+e^{i\delta}|-,-\rangle}{\sqrt2} }. \] Formula (43) does not directly transform each arrow into a specific helicity. Calculating measurement probabilities requires a transition operator between the input and output baselines.
12. Linear Polarization under Test
The helicity states in formula (39) describe the same physical transverse polarization that can be represented in a linear basis. To convert to analyzer readings, it is necessary to determine the relative phase \(\delta\) and match the transverse axes of the two local baselines. In one standard basis matching, the quantum electrodynamic result for a singlet two-photon channel is written as
\[\tag{44} \boxed{ |\Psi_{\mathrm{lin}}\rangle = \frac{1}{\sqrt2} \left( |H\rangle_1|V\rangle_2 - |V\rangle_1|H\rangle_2 \right) }, \] where \(H\) and \(V\) denote two mutually perpendicular linear polarizations. With a different matching of the axes or phases of the basis vectors, the sign between the components may change; the physical probabilities under a matched transformation of the analyzers remain the same. Formula (44) does not indicate the presence of pre-selected polarizations for individual photons, but rather the correlation between the results of their combined measurement.
Let the axes of two linear analyzers form angles \(\alpha\) and \(\beta\) with a common reference axis. The transmission state of one analyzer is as follows
\[\tag{45} |\alpha\rangle = |H\rangle\cos\alpha + |V\rangle\sin\alpha, \qquad |\beta\rangle = |H\rangle\cos\beta + |V\rangle\sin\beta. \] The amplitude of the joint passage of two analyzers is
\[\tag{46} \langle\alpha|_1 \langle\beta|_2 \Psi_{\mathrm{lin}}\rangle = \frac{1}{\sqrt2} \sin(\beta-\alpha). \] Therefore, the probability of joint passage is
\[\tag{47} \boxed{ P_{++}(\alpha,\beta) = \frac12 \sin^2(\alpha-\beta) }. \] For parallel analyzers, we get
\[\tag{48} P_{++}(\alpha,\alpha)=0, \] and for mutually perpendicular ones —
\[\tag{49} P_{++} \left( \alpha, \alpha+\frac{\pi}{2} \right) = \frac12. \] The factor \(1/2\) in formula (49) is related to the probability of passing the first analyzer. If the first photon is already detected behind the analyzer, the conditional probability of the second photon passing through a perpendicularly mounted analyzer is equal to unity:
\[\tag{50} P \left( 2:\alpha+\frac{\pi}{2} \,\middle|\, 1:\alpha \right) =1. \] Thus, formula (47) defines an idealized testable dependence for linear projection analyzers. It is still the target result that the geometric model must derive, rather than an already obtained independent prediction of the model. Annihilation photons have an energy of \(51\ext{keV}\), so conventional optical polarizing filters are not applicable to them. Their polarization correlation is studied using the joint azimuthal dependence of two Compton scatterings. The Compton setup is not an ideal transmission filter, so a directly measured distribution requires convolution of the state with a polarization-dependent scattering cross section.
Formulas (44)–(50) are a testable result that the geometric model must reproduce. However, the two-plane structure itself \(J(a,0)\), \(J(0,a)\) does not yet automatically derive the relative phase. To complete the modelIt is necessary to obtain state (44) by applying the annihilation transition operator.
13. Transition Operator
We introduce the annihilation transformation operator \(\widehat{\mathcal A}\):
\[\tag{51} \widehat{\mathcal A}: \mathcal H_s^{(e^-)}\otimes\mathcal H_s^{(e^+)} \longrightarrow \mathcal H_{\mathrm{pol}}^{(\gamma_1)} \otimes \mathcal H_{\mathrm{pol}}^{(\gamma_2)}. \] For a singlet state,
\[\tag{52} \widehat{\mathcal A}|S=0,m=0\rangle = |\Psi_0\rangle_{\gamma_1\gamma_2}. \] The transition operator must match the eigenvalues of the conserved total quantities in the input and output spaces:
\[\tag{53} \widehat Q_{\mathrm{out}}\widehat{\mathcal A} = \widehat{\mathcal A}\widehat Q_{\mathrm{in}}, \qquad \widehat P^{\mu}_{\mathrm{out}}\widehat{\mathcal A} = \widehat{\mathcal A}\widehat P^{\mu}_{\mathrm{in}}, \qquad \widehat{\mathbf J}_{\mathrm{out}}\widehat{\mathcal A} = \widehat{\mathcal A}\widehat{\mathbf J}_{\mathrm{in}}. \] Furthermore, it must take into account the discrete symmetries of the initial state and determine the relative phase \(\delta\). Constructing the explicit form of \(\widehat{\mathcal A}\) is the next mathematical step of the model.
14. Two-photon and three-photon channels
The above scheme pertains to the singlet configuration:
\[\tag{54} (e^-e^+)_{S=0}\longrightarrow2\gamma. \] For the triplet configuration with \(S=1\), the three-photon channel is the main one:
\[\tag{55} (e^-e^+)_{S=1}\longrightarrow3\gamma. \] The difference in channels is determined not by the arrangement of the planes \(J(a,0)\), \(J(0,a)\), but by the symmetry of the shared state of the pair. For three photons, polarization, energy, momentum, and angular momentum are distributed between the three states, so the two-photon mapping (41) is not directly applicable to this channel.
15. The Inverse Process
The inverse transition has the form
\[\tag{56} \gamma_1+\gamma_2\longrightarrow e^-+e^+. \] In operator form, it means the transformation of a joint polarization state into a joint spin state of two particles with opposite charges:
\[\tag{57} |\Psi_{\mathrm{pol}}\rangle_{\gamma_1\gamma_2} \longrightarrow |\Psi_s\rangle_{e^-e^+}, \qquad J_{q,0}\longrightarrow J_{q,-}\otimes J_{q,+}. \] For pair production, the invariant energy of the system must satisfy the condition
\[\tag{58} s=(K_1+K_2)^2\ge4m_e^2c^2, \qquad E_{\mathrm{cm}}\ge2m_ec^2. \] For two photons with energies \(E_1\), \(E_2\) and an angle \(\theta\) between the propagation directions, this is equivalent to the condition
\[ 2E_1E_2(1-\cos\theta) \ge 4m_e^2c^4. \] A single photon in free space cannot produce a pair with simultaneous conservation of energy and momentum. A second photon or an external body that receives part of the momentum is required.
16. Verification of Conservation Laws
| Magnitude | Before Annihilation | After Annihilation |
|---|---|---|
| Energy | \(2m_ec^2\) | \(2\hbar\omega_C\) |
| Charge | \(-e+e=0\) | \(0+0=0\) |
| Momentum in the Center of Mass System | \(0\) | \(\mathbf p-\mathbf p=0\) |
| Total singlet momentum | \(0\) | \(0\) |
| Internal basis | \(J(a,0),J(0,a)\) | two polarizations |
| Joint state | spin correlation | polarization correlation |
The summary diagram is as follows
\[\tag{59} \boxed{ \begin{aligned} \text{opposite charges} &\longrightarrow\text{neutral state},\\ \text{rest energy of the pair} &\longrightarrow\text{energy of two photons},\\ \text{singlet spin correlation} &\longrightarrow\text{polarization correlation}. \end{aligned} } \] 17. Results and Limits of the Model
The constructed scheme leads to the following results.
1. The states \(J(a,0)\) and \(J(0,a)\) correspond to mutually perpendicular planes and form the geometric basis for the spin of each particle.
2. In the present extension of the model, the same two-plane spin basis is adopted for the electron and positron. Their difference is represented by separate charge states; The geometric mechanism of charge conjugation still needs to be deduced.
3. During annihilation, the joint spin state of the pair is transformed, rather than the spin of each particle being transformed independently.
4. The singlet state transforms into a correlated two-photon state with identical helicities relative to p.opposite directions of propagation.
5. A hypothesis is proposed according to which the two-component nature of a localized fermion state is geometrically related to the two-component nature of the photon polarization space.
Further derivation requires: an explicit operator \(\widehat{\mathcal A}\); the relationship of the planes \(J(a,0)\), \(J(0,a)\) with the transverse basis of the photon for an arbitrary radiation direction; the relative phase of the polarization components; derivation of the law of polarization correlations from the geometric operator; annihilation dynamics and geometry of the three-photon channel.
It is necessary to separately construct the charge conjugation operator \(\mathcal C\) and establish its effect on geometric states. At least two schemes are possible:
\[ \mathcal C: \begin{cases} J_{q,-}\,J(a,0)\longrightarrow J_{q,+}\,J(a,0),\\ J_{q,-}\,J(0,a)\longrightarrow J_{q,+}\,J(0,a), \end{cases} \] or, if a connection between the sign of the parameter \(a\) and the sign of the charge can be proven,
\[ \mathcal C: \begin{cases} J(a,0)\longrightarrow J(-a,0),\\ J(0,a)\longrightarrow J(0,-a). \end{cases} \] In this paper, the first scheme with a separate charge state is used. The choice between these options should follow from the geometric derivation of charge and magnetic moment, and not be introduced solely by the requirement that the electron and positron be opposite.
Conclusion
This paper proposes a geometric scheme for the transition from the spin state of an electron-positron pair to the polarizations of two photons. The spin state is based on two mutually perpendicular internal planes, represented by the states \(J(a,0)\) and \(J(0,a)\). They form a two-component basis; the corresponding spin projection operator has eigenvalues \(\pm\hbar/2\).
In this extension of the model, the same spin geometric basis is adopted for the electron and positron. The particle opposites are represented by individual charge states and are not identified with the arrangement of internal planes. Therefore, during annihilation, charge compensation and spin state transformation are described separately. The geometric origin of charge conjugation remains an open question.
\[\tag{60} \boxed{ \begin{aligned} J_{q,-}\otimes J_{q,+}&\longrightarrow J_{q,0},\\ 2m_ec^2&\longrightarrow \hbar\omega_{\gamma_1}+\hbar\omega_{\gamma_2},\\ |S=0\rangle_{e^-e^+}&\longrightarrow |\Psi_0\rangle_{\gamma_1\gamma_2}. \end{aligned} } \] The central correspondence of the model takes the form
\[\tag{61} \boxed{ \left\{J(a,0),J(0,a)\right\} \longrightarrow \text{spin space} \longrightarrow \text{joint state of the pair} \longrightarrow \text{photon polarization} }. \] Thus, in the proposed hypothesis, the spin two-component nature of a massive particle and the polarization two-component nature of a photon are considered as structurally related forms of geometry. In a localized state, this structure is expressed through spinor periodicity and spin projections, and in a propagating state, through two helicities and a correlation of the polarizations of two photons. The explicit geometric operator proving this correspondence remains the subject of further derivation.