2026-08-05
Geometrical origin of the propagating wave
Could a freely traveling wave be an unfolded state of the same underlying wave structure, rather than a separate object introduced alongside a particle? This article considers precisely that geometric possibility. A closed state periodically returns the wave to its center, whereas an open state replaces that return with translational propagation.
Once multilevel idempotent splitting is introduced, phase dynamics, two-sheeted closure, and electric charge must be distinguished. In accordance with the current concept and the article on spin, a closed particle is described by the dynamical operator \(J_\sigma(a,b)\), the closure factor \(Q_\Gamma(a)\), and the independent charge sector \(J_{q,\eta}\). The sign specifying the direction of free propagation will be denoted by \(\varepsilon\). Formal cancellation of the closure factor does not eliminate the charge sector and does not, by itself, physically convert a particle into a photon.
\[\tag{1} \boxed{\begin{aligned} \text{closed particle}&:\quad J_\sigma(a,b)Q_\Gamma(a)J_{q,\eta},\\ \text{free wave}&:\quad \text{transport phase of a neutral outgoing state}. \end{aligned}} \] This article does not describe an ordinary atomic transition in which the electron remains in the atom and the photon carries the energy difference \(E_i-E_f\). Instead, it considers a limiting geometric transition of the entire localized structure. Its physical realization requires another participant or an external system that ensures conservation of charge, momentum, angular momentum, and the other conserved quantities.
The idea of an electron as a closed electromagnetic wave state is also investigated in other approaches. Reference [1] proposes a model of a rotating electromagnetic wave in a toroidal geometry, to which its authors relate the electron’s charge, spin, and magnetic moment. This is a conceptual parallel, rather than independent confirmation of the mechanism considered here. Establishing a correspondence between the toroidal model and the splitting operators requires a separate mathematical construction.
1. The Full Multilevel Operator of a Closed Particle
The initial two-phase operator has the form
\[\tag{2} J(a,b) = \j^a(-\j)^b = \ep e^{i\pi b} + \em e^{i\pi a}. \] The first-level idempotents satisfy the relations
\[\tag{3} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \qquad \ep+\em=1, \qquad \j=\ep-\em. \] An independent pair of deeper-level idempotents is introduced to describe the additional two-sheeted cycle:
\[\tag{4} \pmp^2=\pmp, \qquad \pme^2=\pme, \qquad \pmp\pme=0, \qquad \pmp+\pme=1, \qquad \jp=\pmp-\pme. \] The mathematics of this extension is developed in detail in the following articles:
- “Multilevel Idempotent Splitting of Phase Planes”;
- “The Two-Period Special Case of Multilevel Idempotent Splitting.”
Define the two-sheeted closure factor independently of the charge sector:
\[\tag{5} Q_\Gamma(a)=\jp^{a/2}=\pmp+\pme e^{i\pi a/2},\qquad Q_\Gamma\overline{Q_\Gamma}=1. \] In the reduced description considered here, the full operator of a closed particle is written as
\[\tag{6} \boxed{\mathcal J_{\mathrm{cl}}^{(\sigma,\eta)}(a,b) =J_\sigma(a,b)Q_\Gamma(a)J_{q,\eta},\qquad \sigma=1,2,\quad\eta=\pm1.} \] The index \(\sigma=1,2\) denotes two realizations of the primary channels: \(J_1(a,b)=J(a,b)\) and \(J_2(a,b)=J(b,a)\). When the channels are exchanged, \(a\) remains the internal phase parameter, while \(b\) remains the external motion parameter. The factor \(Q_\Gamma\) specifies the two-sheeted extension; \(J_{q,\eta}\) describes a separate charge sector with \(\eta=\pm1\). The spin interpretation of the realizations is adopted from the corresponding article in the model and is not proved again here.
In (6), the independent factors act on different spaces; their product is understood after extending each factor by identity operators on the remaining factors. This differs from the tensor product of states of different particles. The charge sector is assumed to remain unchanged throughout the internal cycle. For each realization \(\sigma\), the observed velocity of the center must be extracted from the channel containing the parameter \(b\); when \(b=0\), the center is at rest in both cases.
2. Independent Attributes of the Full State
Closure, charge, the realization of the spin channels, and the direction of free propagation have different roles:
\[\tag{7} \boxed{\begin{aligned} Q_\Gamma(a)&\longrightarrow\text{two-sheeted closure},\\ J_{q,\eta}&\longrightarrow\text{charge sector},\\ b&\longrightarrow\text{external motion of the center},\\ J_1(a,b)\longleftrightarrow J_2(a,b)&\longrightarrow\text{two realizations of the spin channels},\\ \varepsilon=\pm1&\longrightarrow\text{two outgoing-wave directions}. \end{aligned}} \] For charge, adopt the convention
\[\tag{8} q_{\eta}=-e\eta. \] The electron corresponds to \(\eta=+1\), and the positron to \(\eta=-1\), with \(e>0\). Each particle allows both realizations \(\sigma=1,2\). The exchange \(J_1\leftrightarrow J_2\) does not change the charge; a transition between charge sectors need not change the closure factor. The specific structure of \(J_{q,\eta}\) and its possible connection to the orientation of the internal contour remain separate physical questions. Below, the index \(\varepsilon=\pm1\) denotes only the direction of the outgoing wave and is identified with neither \(\sigma\) nor \(\eta\).
3. The Closed State and Its Two Periodicities
The primary phase is determined by the internal parameter \(a(t)\). Here, \(t\) is time in the rest frame of the center of the initial particle, and \(\omega_e\) is the angular frequency with respect to that time:
\[\tag{9} a=\varpi t, \qquad \varphi=\pi a, \qquad \omega_e=\pi\dot a. \] For fixed \(b\) and realization \(\sigma\), the finite dynamical operator returns to its initial value when \(a\) increases by two:
\[\tag{10} J_\sigma(a+2,b)=J_\sigma(a,b),\qquad\sigma=1,2. \] The factor \(Q_\Gamma\) has period four in the parameter \(a\). If the charge sector remains unchanged during the internal cycle, the full operator expression has the following periodicity:
\[\tag{11} \boxed{\mathcal J_{\mathrm{cl}}^{(\sigma,\eta)}(a+2,b)\ne\mathcal J_{\mathrm{cl}}^{(\sigma,\eta)}(a,b),\qquad \mathcal J_{\mathrm{cl}}^{(\sigma,\eta)}(a+4,b)=\mathcal J_{\mathrm{cl}}^{(\sigma,\eta)}(a,b).} \] In a physical mapping, the deeper-level idempotents may be associated with two nearby branches \(C_+\) and \(C_-\):
\[\tag{12} C_+ \xrightarrow{\;2\pi\;} C_- \xrightarrow{\;2\pi\;} C_+. \] Sequential traversal of the two sheets belongs to the internal extension of a single state. Periodicity (11) alone does not prove that the overall sign of a spinor changes under a physical rotation through \(2\pi\): a separate mapping into the space of spin states is required. Nor is the internal traversal a translational motion of the center.
4. Cyclic and Unwrapped Interpretations of the Phase
The full closed state retains its position within the two-sheeted cycle, that is, a phase label modulo \(4\pi\). The primary factor alone already repeats after \(2\pi\). For free propagation, introduce an unwrapped transport phase:
\[\tag{13} \boxed{\begin{aligned} \text{full closed cycle:}&\quad\Theta_{\mathrm{cl}}=\pi a\bmod4\pi,\\ \text{unwrapped transport phase:}&\quad\Theta_{\mathrm w}^{(\varepsilon)}=\varepsilon\pi(a_{\mathrm w}-a_{\mathrm w0})\in\mathbb R. \end{aligned}} \] The unwrapped phase can retain the number of completed periods. Converting it into a spatial distance requires an additional mapping, specified below. Neither the cyclic value \(e^{i\pi a}\) nor any algebraic cancellation of a factor determines, by itself, the direction or length of physical transport.
5. Internal and Energy Frequencies
For the electron’s internal geometric scale, adopt the invariant
\[\tag{14} \boxed{ r_e\omega_e=c. } \] The total rest energy is related to the Compton frequency, rather than directly to \(\hbar\omega_e\):
\[\tag{15} \boxed{ E_e^{(0)} = m_ec^2 = \hbar\omega_C = \alpha_{\mathrm{fs}}\hbar\omega_e. } \] Hence,
\[\tag{16} \omega_C = \alpha_{\mathrm{fs}}\omega_e. \] The two outgoing directions are assigned the same positive energy. This condition concerns a symmetric outgoing state and does not follow solely from reversal of the internal phase. The sign \(\varepsilon\) changes the direction of momentum:
\[\tag{17} E_{\mathrm w}^{(+)} = E_{\mathrm w}^{(-)} >0, \qquad \mathbf p_{\mathrm w}^{(-)} = -\mathbf p_{\mathrm w}^{(+)}. \] 6. Removing the Deeper-Level Closure Factor
To analyze the algebraic expression, introduce the inverse closure factor:
\[\tag{18} Q_\Gamma^{-1}(a)=\jp^{-a/2},\qquad Q_\Gamma(a)Q_\Gamma^{-1}(a)=1. \] Multiplying the full particle operator by this factor gives the exact identity:
\[\tag{19} \boxed{\mathcal J_{\mathrm{cl}}^{(\sigma,\eta)}(a,0)Q_\Gamma^{-1}(a) =J_\sigma(a,0)J_{q,\eta}.} \] Only the factor \(Q_\Gamma\) disappears from the expression. The charge sector \(J_{q,\eta}\) remains. The remainder in (19) should therefore not be denoted as a free-photon operator. Schematically, the algebraic operation is:
\[\tag{20} \boxed{\underbrace{J_\sigma(a,0)Q_\Gamma(a)J_{q,\eta}}_{\text{full initial operator}} \xrightarrow{\;\times Q_\Gamma^{-1}(a)\;} \underbrace{J_\sigma(a,0)J_{q,\eta}}_{\text{charge sector preserved}}.} \] 7. Charge Balance of the Full Process
The equality \(Q_\Gamma Q_\Gamma^{-1}=1\) is an identity involving invertible factors. It does not specify a physical evolution operator, prove that a spatial contour opens, or allow an isolated electron to lose its charge. Removing physical closure requires the dynamics of the full interacting state.
Neutral radiation can arise only while the total charge of the entire system is conserved:
\[\tag{21} \boxed{ \sum q_{\mathrm{initial}} = \sum q_{\mathrm{final}}. } \] Equation (19) describes algebraic cancellation, rather than a complete physical transition. For annihilation, for example, one must consider the joint state \(\mathcal S_{\rm in}=\mathcal J_{\rm cl}^{(\sigma_1,+1)}\otimes\mathcal J_{\rm cl}^{(\sigma_2,-1)}\) and the transformation \(\mathcal S_{\rm out}=\mathcal U_{\rm int}\mathcal S_{\rm in}\). In the two-photon channel, the total charge is zero before and after the process. The form of \(\mathcal U_{\rm int}\) and the transformations of closure and polarization remain to be derived. Zero total charge does not mean simply multiplying or removing the individual charge sectors.
8. Frequency and Scale Transformations
Choose a specific case consistent with energy conservation: an electron–positron pair is at rest in the center-of-mass frame, and the outgoing channel contains two photons of equal energy. In the idealized limit of a free pair with negligible kinetic energy and no binding energy, the energy of each photon is
\[\tag{22} E_{\mathrm w} = E_e^{(0)} = m_ec^2. \] Using the Planck relation
\[\tag{23} E_{\mathrm w} = \hbar\omega_{\mathrm w} \] the frequency of the unfolded state is
\[\tag{24} \boxed{ \omega_{\mathrm w} = \omega_C = \alpha_{\mathrm{fs}}\omega_e. } \] For the free branch, define the phase scale \(r_{\mathrm w}=c/\omega_{\mathrm w}\). Together with the assumed internal relation (14), this gives
\[\tag{25} r_e\omega_e = r_{\mathrm w}\omega_{\mathrm w} = c. \] Then
\[\tag{26} \boxed{ r_{\mathrm w} = \frac{r_e}{\alpha_{\mathrm{fs}}} = \frac{c}{\omega_C} = \frac{\hbar}{m_ec} = \overline{\lambda}_C. } \] The full spatial period of the unfolded wave equals the Compton wavelength:
\[\tag{27} \boxed{ \lambda_{\mathrm w} = 2\pi r_{\mathrm w} = \frac{h}{m_ec} = \lambda_C. } \] The phase scale, frequency, and energy are the same for \(\varepsilon=+1\) and \(\varepsilon=-1\). Only the orientation of unfolding changes.
9. Scaling the Two-Branch Geometry
Let the characteristic separation of the internal branches in the closed state be \(d_e\). If the entire transition geometry scales by a single factor, then
\[\tag{28} d_{\mathrm w} = d_e\frac{\omega_e}{\omega_{\mathrm w}} = \frac{d_e}{\alpha_{\mathrm{fs}}}. \] For the previously adopted condition \(d_e=r_e\):
\[\tag{29} \boxed{ d_{\mathrm w}=r_{\mathrm w}, \qquad 2\pi d_{\mathrm w}=\lambda_{\mathrm w}. } \] Equations (28)–(29) express a conditional hypothesis of self-similar scaling of the transition geometry. Here, \(d_{\mathrm w}\) denotes an auxiliary scale of that configuration, rather than an established distance between two material branches of a photon. Such scaling and free propagation do not follow from cancellation of \(Q_\Gamma\).
10. The Topological Transition: Return Becomes Translation
In the closed regime, one revolution of the basic spatial projection returns it to the center. With a two-sheeted factor present, this is not yet a return of the full internal state:
\[\tag{30} (\varphi,z) \longrightarrow (\varphi+2\pi,z). \] After opening, the same phase is reproduced at a neighboring point along the chosen one-dimensional channel. For the two orientations:
\[\tag{31} \boxed{ (\varphi,z) \longrightarrow (\varphi+2\pi,z+\varepsilon\lambda_{\mathrm w}), \qquad \varepsilon=\pm1. } \] One full phase cycle becomes one directed spatial step:
\[\tag{32} \boxed{ \Delta\varphi=2\pi \quad\Longleftrightarrow\quad \Delta z=\varepsilon\lambda_{\mathrm w}. } \] In the proposed spatial mapping, opening means replacing local return with directed translation. The correspondence between internal traversal and the chosen transport direction is a mapping hypothesis, rather than a consequence of the invertibility of \(Q_\Gamma\).
11. Two Transport Coordinates
Let the transition begin at time \(t_0\), with the unwrapped parameter evolving as
\[\tag{33} a_{\mathrm w}(t) = a_{\mathrm w0} + \frac{\omega_{\mathrm w}}{\pi}(t-t_0). \] The two orientations of the primary phase correspond to two transport coordinates:
\[\tag{34} \boxed{ s_{\mathrm w}^{(\varepsilon)}(t) = r_{\mathrm w}\Theta_{\mathrm w}^{(\varepsilon)}(t) = \varepsilon\pi r_{\mathrm w} \left[ a_{\mathrm w}(t)-a_{\mathrm w0} \right]. } \] Using \(r_{\mathrm w}\omega_{\mathrm w}=c\), we obtain
\[\tag{35} \boxed{ s_{\mathrm w}^{(\varepsilon)}(t) = \varepsilon c(t-t_0), \qquad \frac{ds_{\mathrm w}^{(\varepsilon)}}{dt} = \varepsilon c. } \] Once this mapping is adopted, two transport orientations can be assigned to the conjugate powers of the primary operator:
\[\tag{36} \boxed{ \j^{+a_{\mathrm w}} \xrightarrow{\;\text{opening}\;} +c, \qquad \j^{-a_{\mathrm w}} \xrightarrow{\;\text{opening}\;} -c. } \] The signs \(+\) and \(-\) here refer to two orientations along an already chosen one-dimensional channel. Assigning a particular orientation to the positive direction of the coordinate axis remains a convention.
For the outgoing phase, use the notation \(J_{\mathrm w}^{(\varepsilon)}(a_{\mathrm w})=\j^{\varepsilon a_{\mathrm w}}\). The parameter \(a_{\mathrm w}\) belongs to the outgoing branch and is not identified with the internal parameter \(a\) of the initial electron. The operator serves as a phase label in the chosen channel; its relation to the coordinate-space amplitude is established by an additional mapping.
12. Forward and Backward Traveling Waves
The following discussion considers only the longitudinal scalar phase of the free branch. A complete photon state must additionally include transverse polarization. The relation between the initial joint spin state and the polarizations is not determined by the sign \(\varepsilon\) alone.
After choosing a spatial coordinate, the two wave branches can be written in a common form:
\[\tag{37} \boxed{ \Psi_{\varepsilon}(z,t) = A e^{i(\varepsilon k_{\mathrm w}z-\omega_{\mathrm w}t)}, \qquad \varepsilon=\pm1. } \] For \(\varepsilon=+1\) and \(\varepsilon=-1\):
\[\tag{38} \Psi_+(z,t) = A e^{i(k_{\mathrm w}z-\omega_{\mathrm w}t)}, \qquad \Psi_-(z,t) = A e^{i(-k_{\mathrm w}z-\omega_{\mathrm w}t)}. \] The constant-phase condition gives
\[\tag{39} \varepsilon k_{\mathrm w}dz - \omega_{\mathrm w}dt =0, \qquad \boxed{ \frac{dz}{dt} = \varepsilon\frac{\omega_{\mathrm w}}{k_{\mathrm w}} = \varepsilon c. } \] This correspondence is specified for the chosen axis and constitutes a physical mapping rule. The operators \(\j^{\pm a_{\mathrm w}}\) should not be literally identified with the full spacetime amplitude \(\Psi_\pm\). In equation (38), the sign of the spatial wavenumber changes, while the temporal factor \(e^{-i\omega_{\mathrm w}t}\) is the same for both branches. Consequently, the two positive-energy waves are not, in general, complex-conjugate amplitudes.
\[\tag{40} \boxed{ J_{\mathrm w}^{(+)}=\j^{a_{\mathrm w}} \longleftrightarrow \Psi_+, \qquad J_{\mathrm w}^{(-)}=\j^{-a_{\mathrm w}} \longleftrightarrow \Psi_-. } \] This correspondence fixes the chosen orientation of the axis. Reversing the coordinate axis exchanges the labels “forward” and “backward,” while the existence of the two conjugate directions remains invariant.
13. Energy and Momentum of the Two Directions
The phase of each branch is
\[\tag{41} \Phi_{\varepsilon}(z,t) = \varepsilon k_{\mathrm w}z - \omega_{\mathrm w}t. \] Its time derivative gives the same positive energy:
\[\tag{42} \boxed{ E_{\mathrm w}^{(\varepsilon)} = -\hbar\frac{\partial\Phi_{\varepsilon}}{\partial t} = \hbar\omega_{\mathrm w}. } \] The spatial gradient changes sign with \(\varepsilon\):
\[\tag{43} \boxed{ \mathbf p_{\mathrm w}^{(\varepsilon)} = \hbar\nabla\Phi_{\varepsilon} = \varepsilon\hbar k_{\mathrm w}\widehat{\mathbf n}. } \] Upon adopting the massless propagation law \(E=|p|c\):
\[\tag{44} \boxed{ \left|\mathbf p_{\mathrm w}^{(\varepsilon)}\right| = \frac{E_{\mathrm w}}{c} = m_ec, \qquad \mathbf p_{\mathrm w}^{(-)} = -\mathbf p_{\mathrm w}^{(+)}. } \] Thus, reversing the direction changes the momentum, but not the energy.
14. Counterpropagating Waves and Momentum Conservation
A closed system at rest has zero total momentum. A single directed wave with nonzero momentum therefore cannot arise without recoil being transferred to another participant or to the surrounding system.
A natural configuration with balanced momentum contains two counterpropagating waves:
\[\tag{45} \boxed{ \mathbf p_{\mathrm w}^{(+)} + \mathbf p_{\mathrm w}^{(-)} =0. } \] If their energies are equal, then
\[\tag{46} E_+=E_-, \qquad \left|\mathbf p_+\right| = \left|\mathbf p_-\right|, \qquad \mathbf p_+=-\mathbf p_-. \] The pair of primary operators \(\j^{a_{\mathrm w}}\) and \(\j^{-a_{\mathrm w}}\) is used here to label the two transport directions. The number of outgoing waves, their energies, and their polarizations are determined by the full interaction. A two-photon channel cannot be obtained merely from the existence of two conjugate powers: the initial joint spin state and the conservation laws must be taken into account.
15. Compton and de Broglie Scales
After the internal contour opens, its radius does not disappear from the description. It becomes a scale determining the spatial period of the free wave. The expressions for the radius and momentum magnitude imply
\[\tag{47} r_{\mathrm w} = \frac{\hbar}{m_ec} = \frac{\hbar}{\left|\mathbf p_{\mathrm w}\right|}. \] The quantity \(r_{\mathrm w}\) coincides with the electron’s reduced Compton wavelength. It is also the reciprocal of the wavenumber magnitude:
Here, \(r_{\mathrm w}\) is not the radius of the free wave’s trajectory in ordinary space. After the closed geometry opens, it serves as a residual phase scale: \(k_{\mathrm w}r_{\mathrm w}=1\). A phase interval of \(2\pi\) therefore corresponds to a distance \(2\pi r_{\mathrm w}\).
The full spatial period is obtained from the length of the former closed contour. Hence,
\[\tag{48} \boxed{ 2\pi r_{\mathrm w} = \frac{h}{\left|\mathbf p_{\mathrm w}\right|} = \lambda_{\mathrm{dB},\mathrm w} = \lambda_C. } \] In the chosen energy limit, the Compton scale of the initial electron coincides with the spatial phase scale of the outgoing branch. This is a correspondence between scales, rather than proof that a material circle literally turns into a straight wave. The quantity \(r_{\mathrm w}\) is determined by the frequency of the outgoing branch; the spatial period is \(2\pi r_{\mathrm w}\).
The equality \(\lambda_{\mathrm{dB},\mathrm w}=\lambda_C\) applies specifically to the selected free branch, for which the conditions \(E_{\mathrm w}=m_ec^2\) and \(\left|\mathbf p_{\mathrm w}\right|=m_ec\) and the propagation speed \(c\) have been adopted. It should not automatically be extended to the de Broglie wavelength of a massive electron moving at an arbitrary velocity: in general, its momentum is different, and \(\lambda_{\mathrm{dB}}=h/|\mathbf p|\) need not equal the Compton wavelength.
The two branches \(\j^{a_{\mathrm w}}\) and \(\j^{-a_{\mathrm w}}\) have the same energy, momentum magnitude, frequency, and wavelength. The sign in the operator exponent changes the direction in which the spatial phase increases, rather than the positive quantity \(\lambda_{\mathrm{dB},\mathrm w}\). The direction of propagation should therefore be specified by the wavevector:
\[\tag{49} \boxed{ \mathbf k_{\varepsilon} = \varepsilon k_{\mathrm w}\widehat{\mathbf n}, \qquad \varepsilon=\pm1. } \] With the chosen axis orientation and phase-sign convention, the branch \(\j^{a_{\mathrm w}}\) corresponds to \(\varepsilon=+1\), and the branch \(\j^{-a_{\mathrm w}}\) to \(\varepsilon=-1\). Both branches have the same positive wavenumber magnitude \(k_{\mathrm w}=1/r_{\mathrm w}\), but oppositely directed wavevectors:
The signs of the primary operators \(\j^{a_{\mathrm w}}\) and \(\j^{-a_{\mathrm w}}\) label the chosen forward and backward directions in the adopted mapping. They do not denote the sign of electric charge. Charge belongs to the separate sector \(J_{q,\eta}\), while the factor \(Q_\Gamma\) describes two-sheeted closure.
This distinction is particularly important for the open state: the positive wavelength determines the spatial period, while the sign \(\varepsilon\) retains information about the direction in which the corresponding branch propagates.
16. A Finite Wave Packet
A harmonic function extends infinitely through space and describes only the carrier. If the transition lasts a finite time \(\Delta t\), the unfolded state must be represented by a packet:
\[\tag{50} \boxed{ \Psi_{\varepsilon}(z,t) = A\!\left(t-\varepsilon\frac{z}{c}\right) e^{i(\varepsilon k_{\mathrm w}z-\omega_{\mathrm w}t)}. } \] Its longitudinal extent is estimated as
\[\tag{51} L_{\mathrm{packet}} \sim c\Delta t. \] Three scales must be distinguished:
\[\tag{52} \boxed{ r_{\mathrm w} = \frac{c}{\omega_C}, \qquad \lambda_{\mathrm w} = 2\pi r_{\mathrm w}, \qquad L_{\mathrm{packet}} \sim c\Delta t. } \] A finite transition duration leads to a finite spectral width:
\[\tag{53} \Delta\omega\,\Delta t \gtrsim 1. \] 17. A One-Dimensional Channel and Its Orientation in Space
The index \(\varepsilon\) selects one of two orientations along an already specified one-dimensional channel. It does not determine how that channel is oriented in three-dimensional space. The full wavevector is
\[\tag{54} \boxed{ \mathbf k_{\mathrm w}^{(\varepsilon)} = \varepsilon k_{\mathrm w}\widehat{\mathbf n}, \qquad \mathbf p_{\mathrm w}^{(\varepsilon)} = \hbar\mathbf k_{\mathrm w}^{(\varepsilon)}. } \] Here, \(\widehat{\mathbf n}\) specifies the orientation of the channel itself, while \(\varepsilon\) selects one of the two directions along it. The choice of \(\widehat{\mathbf n}\) must be determined by the geometry of the full interaction, an external field, boundary conditions, or another physical law.
18. The Sequence of the Full Geometric Transition
The main relations must be separated into an algebraic operation on the initial state and a physical description of the selected outgoing branch:
\[\tag{55} \boxed{\begin{aligned} \mathcal J_{\mathrm{cl}}^{(\sigma,\eta)}(a,0)Q_\Gamma^{-1}(a)&=J_\sigma(a,0)J_{q,\eta},\\ \mathcal S_{\mathrm{out}}&=\mathcal U_{\mathrm{int}}\mathcal S_{\mathrm{in}},\qquad Q_{\mathrm{tot,in}}=Q_{\mathrm{tot,out}}=0,\\ J_{\mathrm w}^{(\varepsilon)}&:=\j^{\varepsilon a_{\mathrm w}}\quad\text{(outgoing-phase label)},\\ E_{\mathrm w}&=m_ec^2=\hbar\omega_C\quad\text{(chosen energy limit)},\\ r_{\mathrm w}&=r_e/\alpha_{\mathrm{fs}}=\overline\lambda_C,\\ s_{\mathrm w}^{(\varepsilon)}&=\varepsilon c(t-t_0),\\ \mathbf p_{\mathrm w}^{(\varepsilon)}&=\varepsilon(E_{\mathrm w}/c)\widehat{\mathbf n}. \end{aligned}} \] The first line of (55) is an algebraic identity that preserves the charge sector. The remaining lines concern the neutral outgoing branch after a separately specified full interaction. They use the chosen energy limit and the transport mapping. Conversion into a photon, a change of frequency, and spatial translation do not follow from the first line.
19. What Follows from the Equations and What Remains Hypothetical
The adopted algebra implies: the form of \(J_\sigma\); the invertibility of \(Q_\Gamma\); identity (19); the algebraic unitarity of the phase factors; and periodicity (11) when the charge sector is unchanged. The algebraic equality \(J\overline J=1\) must be distinguished from a unit Hilbert-space norm of a state vector and from conservation of physical energy.
Additional physical conditions yield: the transport coordinates \(s_{\mathrm w}^{(\varepsilon)}\) and velocities \(\pm c\); energy and momentum (42)–(44); and the Compton spatial period in the chosen limit. These results require the transport mapping, the phase–energy correspondence, and the massless propagation law, rather than idempotent algebra alone.
The following remain physical hypotheses: the mechanism of spatial closure and opening; the full interaction operator; the connection between internal factors and observed charge and spin; the geometry of the transition region; the correspondence between conjugate powers and propagation directions; and the transformation of the joint spin state into photon polarizations. The number of outgoing waves and the energy distribution cannot be assigned independently of the initial state and the conservation laws.
The algebra shows which conjugate states and balances are possible. Physics must separately establish the processes in which these possibilities are realized and their consistency with all conservation laws.
Conclusion
A closed particle and a free wave are considered different geometric regimes of a common phase structure. However, the full particle operator contains an independent charge sector. Algebraic cancellation of the closure factor leaves this sector intact:
\[\tag{56} \boxed{J_\sigma(a,0)Q_\Gamma(a)J_{q,\eta} \xrightarrow{\;\times Q_\Gamma^{-1}(a)\;} J_\sigma(a,0)J_{q,\eta}.} \] For a free branch produced in a full interaction, a transport interpretation can be assigned to the conjugate powers of the primary operator:
\[\tag{57} \boxed{ \begin{aligned} \j^{+a_{\mathrm w}} &\longrightarrow \text{one traversal and propagation direction},\\ \j^{-a_{\mathrm w}} &\longrightarrow \text{opposite direction}. \end{aligned} } \] Once the spatial mapping is adopted, the sign \(\varepsilon\) specifies one of two transport coordinates. This is a separate rule relating the phase operator to the direction of propagation:
\[\tag{58} \boxed{ s_{\mathrm w}^{(\pm)}(t) = \pm c(t-t_0). } \] Closure, charge, and the spin realization remain distinct attributes of the initial particle:
\[\tag{59} \boxed{\begin{aligned} Q_\Gamma(a)&\longrightarrow\text{two-sheeted closure},\\ J_{q,\eta}&\longrightarrow\text{charge sign }q_\eta=-e\eta,\\ \sigma=1,2&\longrightarrow\text{realization of the spin channels},\\ \varepsilon=\pm1&\longrightarrow\text{direction of free propagation}. \end{aligned}} \] The proposed notation separates internal closure, the charge sector, the realization of the spin channels, and the direction of free propagation. Formal cancellation of \(Q_\Gamma\) preserves the charge sector. The physical transition to neutral radiation concerns the full interacting state and requires conservation of charge, energy, momentum, and angular momentum, together with a separate law governing the transformation of polarizations.
Why the Local Transfer Law Leads to the Wave Equation
Consider a norm-preserving phase mode of the state:
\[\tag{A1} J(z,t) = J_0(z)e^{i\psi(z,t)}, \qquad J\overline J =1. \] Introduce two local phase quantities with dimensions of inverse length. In this appendix, the symbols \(q\) and \(p\) do not denote electric charge and mechanical momentum:
\[\tag{A2} q = \frac{1}{c}\frac{\partial\psi}{\partial t}, \qquad p = \frac{\partial\psi}{\partial z}. \] Equality of mixed partial derivatives and an additional symmetric local transfer law give the system
\[\tag{A3} \frac{\partial p}{\partial t} = c\frac{\partial q}{\partial z}, \qquad \frac{\partial q}{\partial t} = c\frac{\partial p}{\partial z}. \] It decomposes into two characteristic components:
\[\tag{A4} u_+=p-q, \qquad u_-=p+q, \] \[\tag{A5} \boxed{ \frac{\partial u_+}{\partial t} = -c\frac{\partial u_+}{\partial z}, \qquad \frac{\partial u_-}{\partial t} = c\frac{\partial u_-}{\partial z}. } \] Substitution of definitions (A2) gives the wave equation:
\[\tag{A6} \boxed{ \frac{\partial^2\psi}{\partial t^2} - c^2\frac{\partial^2\psi}{\partial z^2} =0. } \] The two characteristic branches propagate with opposite velocities. In the adopted transport mapping, they can be associated with labels provided by the primary operators:
\[\tag{A7} \boxed{ u_+ \longleftrightarrow J_{\mathrm w}^{(+)}=\j^{a_{\mathrm w}}, \qquad u_- \longleftrightarrow J_{\mathrm w}^{(-)}=\j^{-a_{\mathrm w}}. } \] System (A3) gives the characteristic velocities \(\pm c\) after the additional transfer law is adopted. Correspondence (A7) labels these directions using phase operators; it is not an identity between a characteristic amplitude and an algebra element. The wave equation for a scalar phase does not, by itself, specify transverse electromagnetic fields, polarization, or quantization of the photon state.
Materials used
- C. A. M. dos Santos, M. J. J. Fleury, An electromagnetic model of the electron, arXiv:2510.22384, 2025.