2026-08-05
Geometrical origin of the propagating wave
Could a free traveling wave be not a separate object pre-added to a particle, but an unfolded state of the same fundamental wave structure? This paper considers precisely this geometric possibility. A closed state periodically returns the wave to its center, while an open state replaces this return with a translational translation.
After introducing multilevel idempotent splitting, a closed particle should be described more precisely. Its state contains the fundamental wave \(\j^{\pm a}\) and an additional two-sheet factor \(\jp^{\pm a/2}\). The first determines the direction of the main phase evolution, and the second determines the charge orientation of the deep closure. Upon opening, the main wave is preserved and transforms into a transport wave, while the deep multiplier can only cancel within a complete interaction that preserves charge and other physical quantities.
\[\tag{1} \boxed{ \begin{aligned} \text{closed particle} &= \text{main wave} \times \text{two-sheet closure},\\ \text{free wave} &= \text{unfolded main phase}. \end{aligned} } \] This work does not describe the usual atomic transition, in which the electron remains in the atom and the photon acquires an energy difference of \(E_i - E_f\). The limiting geometric transition of the entire localized structure is considered. Its physical implementation requires a second participant or an external system that ensures the conservation of charge, momentum, angular momentum, and other conserved characteristics.
The concept of the electron as a confined electromagnetic wave state is also being independently developed within other approaches. In [1], the electron is represented as a rotating electromagnetic wave localized within a toroidal geometry. The authors associate the electron's charge, spin, and magnetic moment with this configuration and characterize it as a "coiled light" mode. Although their model does not address the operator-based mechanism proposed here for lifting the double-sheeted closure and reforming the localized particle, it independently supports the underlying idea that the electron and a free electromagnetic wave may represent distinct topological states of a single wave structure.
1. Complete multilevel operator of a closed particle
The original two-phase operator has the form
\[\tag{2} J(a,b) = \j^a(-\j)^b = \ep e^{i\pi b} + \em e^{i\pi a}. \] First-level idempotents satisfy the relations
\[\tag{3} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \qquad \ep+\em=1, \qquad \j=\ep-\em. \] To describe the additional two-sheet cycle, an independent pair of deep idempotents is introduced:
\[\tag{4} \pmp^2=\pmp, \qquad \pme^2=\pme, \qquad \pmp\pme=0, \qquad \pmp+\pme=1, \qquad \jp=\pmp-\pme. \] The mathematics of this extension is developed in detail in the following works:
- "Multilevel Idempotent Splitting of Phase Planes",
- "Two-periodic special case of multilevel idempotent splitting".
The deep operator is written as
\[\tag{5} Q_{\eta}(a) = \jp^{\eta a/2} = \pmp+\pme e^{i\eta\pi a/2}, \qquad \eta=\pm1. \] The complete operator of a closed particle is defined by the expression
\[\tag{6} \boxed{ J_{\mathrm{cl}}^{(\sigma,\eta)}(a,b) = \j^{\sigma a}(-\j)^b \jp^{\eta a/2}, \qquad \sigma,\eta=\pm1. } \] Here \(\sigma\) specifies the direction of the fundamental internal phase, \(\eta\) specifies the charge orientation of the deep two-sheet bypass, and the parameter \(b\) describes the external motion of the center of the localized particle.
2. Independent Features of the Complete State
The direction of the fundamental wave, the sign of the charge, the motion of the center, and the spin orientation should not be encoded with a single symbol. In the accepted model, they are separated:
\[\tag{7} \boxed{ \begin{aligned} \j^{\pm a} &\longrightarrow \text{two directions of the fundamental wave},\\ \jp^{\pm a/2} &\longrightarrow \text{two charge orientations},\\ (-\j)^{\pm b} &\longrightarrow \text{direction of motion of the center},\\ J(a,0)\longleftrightarrow J(0,a) &\longrightarrow \text{two spin modes}. \end{aligned} } \] The following convention is adopted for charge
\[\tag{8} q_{\eta}=-e\eta. \] An electron and a positron can have the same direction of the fundamental phase \(\sigma\), but opposite values of \(\eta\). Likewise, each of these particles can be in any allowed spin state.
3. Closed State and Two Periodicities
The fundamental phase is determined by the internal parameter \(a(t)\):
\[\tag{9} a=\varpi t, \qquad \varphi=\pi a, \qquad \omega_e=\pi\dot a. \] The main factor returns after one rotation:
\[\tag{10} \j^{\sigma(a+2)} = \j^{\sigma a}. \] The deep phase changes twice as slowly. Therefore, the complete two-sheet state after the first rotation does not yet coincide with the original, and after the second it is restored:
\[\tag{11} \boxed{ J_{\mathrm{cl}}(a+2) \ne J_{\mathrm{cl}}(a), \qquad J_{\mathrm{cl}}(a+4) = J_{\mathrm{cl}}(a). } \] In the physical representation, deep idempotents can be associated with two close branches \(C_+\) and \(C_-\):
\[\tag{12} C_+ \xrightarrow{\;2\pi\;} C_- \xrightarrow{\;2\pi\;} C_+. \] The sign of \(\sigma\) determines the direction of traversal of this closed configuration. As long as the wave returns to its previous center, this phase reversal does not create translational motion.
4. Cyclic and Expanded Phase Reading
The same continuous exponent \(a(t)\) can be read in two ways. In the closed state, only the phase within the current period is stored, while in the expanded state, the total accumulated number of revolutions is stored:
\[\tag{13} \boxed{ \begin{aligned} \text{closed state:} &\qquad \Theta_{\mathrm{cl}} = \pi a\bmod 2\pi,\\ \text{expanded state:} &\qquad \Theta_{\mathrm w}^{(\sigma)} = \sigma\pi(a_{\mathrm w}-a_{\mathrm w0}) \in\mathbb R. \end{aligned} } \] It is the expanded value of the exponent that stores the direction and length The current cyclic value \(e^{i\pi a}\) alone does not allow us to reconstruct the number of revolutions already completed.
5. Internal and Energy Frequencies
The invariant is adopted for the internal geometric scale of the electron
\[\tag{14} \boxed{ r_e\omega_e=c. } \] The total rest energy is not directly related to \(\hbar\omega_e\), but to the Compton frequency:
\[\tag{15} \boxed{ E_e^{(0)} = m_ec^2 = \hbar\omega_C = \alpha_{\mathrm{fs}}\hbar\omega_e. } \] From here
\[\tag{16} \omega_C = \alpha_{\mathrm{fs}}\omega_e. \] The two directions of the fundamental phase have the same positive energy. The sign of \(\sigma\) must appear in the direction of the momentum, not in the sign of the energy:
\[\tag{17} E_{\mathrm w}^{(+)} = E_{\mathrm w}^{(-)} >0, \qquad \mathbf p_{\mathrm w}^{(-)} = -\mathbf p_{\mathrm w}^{(+)}. \] 6. Deep Circuit Removal
A free unwrapped wave must not preserve the additional two-sheet cycle of a localized charged particle. The inverse deep operator is
\[\tag{18} Q_{\eta}^{-1}(a) = \jp^{-\eta a/2}. \] Formal cancellation of the deep factor yields
\[\tag{19} \boxed{ J_{\mathrm{cl}}^{(\sigma,\eta)}(a,0) \jp^{-\eta a/2} = \j^{\sigma a} = J_{\mathrm w}^{(\sigma)}(a). } \] The fundamental wave does not disappear and does not change direction. Only the additional internal short circuit is eliminated:
\[\tag{20} \boxed{ \underbrace{\j^{\sigma a}}_{\text{fundamental wave}} \underbrace{\jp^{\eta a/2}}_{\text{charge short circuit}} \quad \xrightarrow{\;\times\jp^{-\eta a/2}\;} \quad \underbrace{\j^{\sigma a}}_{\text{free wave}}. } \] 7. Charge Balance of the Complete Process
The equality \(Q_{\eta}Q_{\eta}^{-1}=1\) is an exact algebraic identity, but it does not yet allow a single electron to spontaneously lose charge. The inverse operator must appear as part of a complete interaction with another participant or an external system.
If a localized charged state transitions to a neutral free wave, the charge orientation must compensate:
\[\tag{21} \boxed{ \sum q_{\mathrm{initial}} = \sum q_{\mathrm{final}}. } \] Therefore, formula (19) describes the geometric part of the transition, not the entire physical process. A complete system must simultaneously conserve energy, momentum, angular momentum, charge, and other relevant properties. It is most natural to consider such a transition for a charge-balanced interaction, for example, for a pair of conjugate particles, rather than for an isolated electron.
8. Frequency and Scale Transformations
Suppose that the energy of one output wave branch in the limit process under consideration is equal to the rest energy of the original particle:
\[\tag{22} E_{\mathrm w} = E_e^{(0)} = m_ec^2. \] With Planck's relation
\[\tag{23} E_{\mathrm w} = \hbar\omega_{\mathrm w} \] the frequency of the expanded state is
\[\tag{24} \boxed{ \omega_{\mathrm w} = \omega_C = \alpha_{\mathrm{fs}}\omega_e. } \] Let the invariant \(r\omega=c\) be preserved between two geometric modes:
\[\tag{25} r_e\omega_e = r_{\mathrm w}\omega_{\mathrm w} = c. \] 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 unwrapped wave is equal to the Compton length:
\[\tag{27} \boxed{ \lambda_{\mathrm w} = 2\pi r_{\mathrm w} = \frac{h}{m_ec} = \lambda_C. } \] The radius, frequency, and energy are the same for \(\sigma=+1\) and \(\sigma=-1\). Only the unfolding orientation changes.
9. Scaling Two-Branch Geometry
Let the characteristic distance between the internal branches in the closed state be \(d_e\). If the entire transition geometry is scaled by the same 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 \(d_e=r_e\):
\[\tag{29} \boxed{ d_{\mathrm w}=r_{\mathrm w}, \qquad 2\pi d_{\mathrm w}=\lambda_{\mathrm w}. } \] This scaling applies to the entire configuration in the transition region. After the deep operator cancellation, the two internal branches no longer read as a local two-sheet contour, while the main phase continues to unfold into the transport coordinate.
10. Topological Transition: Return Becomes Translation
Before uncoupling, one phase revolution returns the wave to the original center:
\[\tag{30} (\varphi,z) \longrightarrow (\varphi+2\pi,z). \] After uncoupling, the same phase is reproduced at the adjacent point of the selected one-dimensional channel. For two orientations:
\[\tag{31} \boxed{ (\varphi,z) \longrightarrow (\varphi+2\pi,z+\sigma\lambda_{\mathrm w}), \qquad \sigma=\pm1. } \] One complete phase revolution is converted into one directed spatial step:
\[\tag{32} \boxed{ \Delta\varphi=2\pi \quad\Longleftrightarrow\quad \Delta z=\sigma\lambda_{\mathrm w}. } \] The opening should be understood not as a break in the wave, but as a topological replacement of local return by translation. The direction of the previous circuit becomes the direction of translational transfer.
11. Two transport coordinates
Let the transition begin at time \(t_0\), and the expanded exponent changes as
\[\tag{33} a_{\mathrm w}(t) = a_{\mathrm w0} + \frac{\omega_{\mathrm w}}{\pi}(t-t_0). \] Two orientations of the main phase correspond to two transport coordinates:
\[\tag{34} \boxed{ s_{\mathrm w}^{(\sigma)}(t) = r_{\mathrm w}\Theta_{\mathrm w}^{(\sigma)}(t) = \sigma\pi r_{\mathrm w} \left[ a_{\mathrm w}(t)-a_{\mathrm w0} \right]. } \] Using \(r_{\mathrm w}\omega_{\mathrm w}=c\), we get
\[\tag{35} \boxed{ s_{\mathrm w}^{(\sigma)}(t) = \sigma c(t-t_0), \qquad \frac{ds_{\mathrm w}^{(\sigma)}}{dt} = \sigma c. } \] Thus, the conjugate powers of the basic operator acquire a distinct transport meaning:
\[\tag{36} \boxed{ \j^{+a} \xrightarrow{\;\text{opening}\;} +c, \qquad \j^{-a} \xrightarrow{\;\text{opening}\;} -c. } \] The signs \(+\) and \(-\) here refer to two orientations along the already chosen one-dimensional channel. Assigning a specific orientation to the positive direction of the coordinate axis remains a matter of convention.
12. Forward and backward traveling waves
After choosing the spatial coordinate, the two wave branches can be written in a single form:
\[\tag{37} \boxed{ \Psi_{\sigma}(z,t) = A e^{i(\sigma k_{\mathrm w}z-\omega_{\mathrm w}t)}, \qquad \sigma=\pm1. } \] For \(\sigma=+1\) and \(\sigma=-1\):
\[\tag{38} \Psi_+(z,t) = A e^{i(k_{\mathrm w}z-\omega_{\mathrm w}t)}, \qquad \Psi_-(z,t) = A e^{i(-k_{\mathrm w}z-\omega_{\mathrm w}t)}. \] The constant phase condition gives
\[\tag{39} \sigma k_{\mathrm w}dz - \omega_{\mathrm w}dt =0, \qquad \boxed{ \frac{dz}{dt} = \sigma\frac{\omega_{\mathrm w}}{k_{\mathrm w}} = \sigma c. } \] The operator and coordinate descriptions are related by a correspondence.
\[\tag{40} \boxed{ J_{\mathrm w}^{(+)}=\j^a \longleftrightarrow \Psi_+, \qquad J_{\mathrm w}^{(-)}=\j^{-a} \longleftrightarrow \Psi_-. } \] This correspondence fixes the chosen orientation of the axis. When the coordinate axis is simultaneously inverted, the names "direct" and "inverse" swap, but the existence of two conjugate directions remains invariant.
13. Energy and momentum of two directions
The phase of each branch is equal
\[\tag{41} \Phi_{\sigma}(z,t) = \sigma k_{\mathrm w}z - \omega_{\mathrm w}t. \] Its time derivative gives the same positive energy:
\[\tag{42} \boxed{ E_{\mathrm w}^{(\sigma)} = -\hbar\frac{\partial\Phi_{\sigma}}{\partial t} = \hbar\omega_{\mathrm w}. } \] The spatial gradient changes sign along with \(\sigma\):
\[\tag{43} \boxed{ \mathbf p_{\mathrm w}^{(\sigma)} = \hbar\nabla\Phi_{\sigma} = \sigma\hbar k_{\mathrm w}\widehat{\mathbf n}. } \] Assuming the massless propagation law \(E=|p|c\):
\[\tag{44} \boxed{ \left|\mathbf p_{\mathrm w}^{(\sigma)}\right| = \frac{E_{\mathrm w}}{c} = m_ec, \qquad \mathbf p_{\mathrm w}^{(-)} = -\mathbf p_{\mathrm w}^{(+)}. } \] Thus, reversing the direction changes the momentum, but not the energy.
14. Counterpropagating Waves and Conservation of Momentum
A closed system at rest has zero total momentum. Therefore, a single directed wave with nonzero momentum cannot arise without transferring to another participant or the surrounding system.
The natural momentum-balanced version 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 operators \(J(a)\) and \(J(-a)\) provides an algebraic language for two such directions. However, the number of output waves and the energy distribution between them must be determined by the complete interaction and conservation laws, and not just by the existence of two adjoint solutions.
15. Compton and de Broglie Scales
After the inner contour is opened, its radius does not disappear from the description. It becomes a scale that determines the spatial period of the free wave. From the expressions for the radius and the modulus of the momentum, it follows
\[\tag{47} r_{\mathrm w} = \frac{\hbar}{m_ec} = \frac{\hbar}{\left|\mathbf p_{\mathrm w}\right|}. \] The quantity \(r_{\mathrm w}\) coincides with the reduced Compton length of the electron. It is also the inverse modulus of the wave number:
Here \(r_{\mathrm w}\) is not the radius of the free wave trajectory in ordinary space. After the closed geometry opens, it acts as a residual phase scale: \(k_{\mathrm w}r_{\mathrm w}=1\). Therefore, the phase \(2\pi\) corresponds to the distance \(2\pi r_{\mathrm w}\).
The full spatial period is formed from the length of the previous closed contour. Therefore
\[\tag{48} \boxed{ 2\pi r_{\mathrm w} = \frac{h}{\left|\mathbf p_{\mathrm w}\right|} = \lambda_{\mathrm{dB},\mathrm w} = \lambda_C. } \] Thus, the Compton and de Broglie scales in the limiting state under consideration turn out to be two ways of describing the same geometric quantity. While the wave is confined within the particle, \(r_{\mathrm w}\) characterizes the inner radius of its phase motion. After the wave is uncoupled, the length of the previous contour \(2\pi r_{\mathrm w}\) becomes the spatial period of the propagating wave.
The equality \(\lambda_{\mathrm{dB},\mathrm w}=\lambda_C\) applies specifically to the selected free branch, for which the conditions \(E_{\mathrm w}=m_ec^2\), \(\left|\mathbf p_{\mathrm w}\right|=m_ec\) and the propagation velocity \(c\) are adopted. It should not be automatically transferred to the de Broglie length of a massive electron moving with an arbitrary velocity: in general, its momentum is different, and therefore \(\lambda_{\mathrm{dB}}=h/|\mathbf p|\) does not necessarily coincide with the Compton length.
Both branches \(J(a)\) and \(J(-a)\) have the same energy, momentum magnitude, frequency, and wavelength. The sign in the operator exponent changes not the positive value of \(\lambda_{\mathrm{dB},\mathrm w}\), but the direction of increase of the spatial phase. Therefore, the direction of propagation should be written in terms of the wave vector:
\[\tag{49} \boxed{ \mathbf k_{\sigma} = \sigma k_{\mathrm w}\widehat{\mathbf n}, \qquad \sigma=\pm1. } \] For the chosen axis orientation and phase sign convention, the branch \(J(a)\) corresponds to \(\sigma=+1\), and the branch \(J(-a)\) corresponds to \(\sigma=-1\). Both branches have the same positive absolute value of the wavenumber \(k_{\mathrm w}=1/r_{\mathrm w}\), but oppositely directed wave vectors:
Therefore, the signs in \(J(a)\) and \(J(-a)\) are conveniently used to distinguish between the forward and backward waves. They specify the direction of phase and momentum transfer, but do not by themselves indicate the sign of the electric charge. The charge orientation refers to an independent two-sheet factor of the total particle operator.
This distinction is especially important for the open state: the positive wavelength determines the spatial period, while the sign (sigma) preserves information about the direction in which the corresponding branch propagates.
16. Bounded Wave Packet
The harmonic function is infinite in spaceness and describes only the carrier. If the transition lasts a finite time \(\Delta t\), the unwrapped state must be represented by a packet:
\[\tag{50} \boxed{ \Psi_{\sigma}(z,t) = A\!\left(t-\sigma\frac{z}{c}\right) e^{i(\sigma k_{\mathrm w}z-\omega_{\mathrm w}t)}. } \] Its longitudinal extent is estimated as
\[\tag{51} L_{\mathrm{packet}} \sim c\Delta t. \] Three scales must be distinguished:
\[\tag{52} \boxed{ r_{\mathrm w} = \frac{c}{\omega_C}, \qquad \lambda_{\mathrm w} = 2\pi r_{\mathrm w}, \qquad L_{\mathrm{packet}} \sim c\Delta t. } \] The finiteness of the transition time leads to a finite spectral width:
\[\tag{53} \Delta\omega\,\Delta t \gtrsim 1. \] 17. One-dimensional channel and its orientation in space
The index \(\sigma\) selects one of two orientations along a given one-dimensional channel. It does not determine how this channel is located in three-dimensional space. The full wave vector is of the form
\[\tag{54} \boxed{ \mathbf k_{\mathrm w}^{(\sigma)} = \sigma k_{\mathrm w}\widehat{\mathbf n}, \qquad \mathbf p_{\mathrm w}^{(\sigma)} = \hbar\mathbf k_{\mathrm w}^{(\sigma)}. } \] Here \(\widehat{\mathbf n}\) specifies the orientation of the channel itself, and \(\sigma\) is one of two directions along it. The choice of \(\widehat{\mathbf n}\) should be determined by the geometry of the complete interaction, the external field, boundary conditions, or another physical law.
18. Sequence of a complete geometric transition
The main relations of the model are summarized in the following chain:
\[\tag{55} \boxed{ \begin{aligned} J_{\mathrm{cl}}^{(\sigma,\eta)} &= \j^{\sigma a}\jp^{\eta a/2},\\ J_{\mathrm{cl}}^{(\sigma,\eta)} \jp^{-\eta a/2} &= \j^{\sigma a} = J_{\mathrm w}^{(\sigma)},\\ E_{\mathrm w} &= m_ec^2 = \hbar\omega_C,\\ r_{\mathrm w} &= \frac{r_e}{\alpha_{\mathrm{fs}}} = \overline{\lambda}_C,\\ (\varphi,z) &\longrightarrow (\varphi+2\pi,z+\sigma\lambda_{\mathrm w}),\\ s_{\mathrm w}^{(\sigma)} &= \sigma c(t-t_0),\\ \mathbf p_{\mathrm w}^{(\sigma)} &= \sigma\frac{E_{\mathrm w}}{c}\widehat{\mathbf n}. \end{aligned} } \] The sequence includes four different transformations: deep closure removal, frequency rescaling, spatial rescaling, and topological replacement of return with directed translation.
19. What follows from the formulas and what remains a hypothesis
Following from multilevel algebra are: the existence of ground and deep factors; conjugate states of \(\j^{\pm a}\); mutual invertibility of \(\jp^{\eta a/2}\) and \(\jp^{-\eta a/2}\); norm conservation; Two-turn nature of the complete closed state.
The following follow from the accepted geometric mappings: two transport coordinates \(s_{\mathrm w}^{(\sigma)}\); velocities \(\pm c\); equal positive energy in both directions; opposite wave vectors and momenta; the possibility of a momentum-balanced pair of counterpropagating waves.
The following remain physical hypotheses: the opening mechanism itself; the appearance of an inverse charge operator in a specific interaction; the transformation \(\omega_e\to\omega_C\); preservation of the invariant \(r\omega=c\) between the two modes; the condition \(E_{\mathrm w}=m_ec^2\) for one output branch; choice of spatial axis \(\widehat{\mathbf n}\); number of output waves; energy distribution; transformation of spin and other internal characteristics.
Algebra shows which conjugate states and balances are possible. Physics must separately establish in which processes these possibilities are realized and how they are consistent with all conservation laws.
Conclusion
A closed particle and a free wave are considered as two geometric regimes of a single fundamental phase structure. The closed state contains an additional two-sheet factor, and the free state preserves the fundamental phase without local return:
\[\tag{56} \boxed{ \j^{\sigma a}\jp^{\eta a/2} \quad \xrightarrow{\;\text{deep closure lifting}\;} \quad \j^{\sigma a}. } \] Conjugate principal powers acquire a clear geometric meaning:
\[\tag{57} \boxed{ \begin{aligned} \j^{+a} &\longrightarrow \text{one direction of traversal and propagation},\\ \j^{-a} &\longrightarrow \text{opposite direction}. \end{aligned} } \] In a closed state, these directions distinguish the orientations of traversal of a single contour. After opening, the cyclic phase unfolds, and the same difference turns into two transport coordinates:
\[\tag{58} \boxed{ s_{\mathrm w}^{(\pm)}(t) = \pm c(t-t_0). }\] However, the deep sign has a different function:
\[\tag{59} \boxed{ \jp^{\pm a/2} \longrightarrow \text{charge orientation of the two-sheet closure}. } \] Thus, the direction of the fundamental wave, the sign of the charge, the motion of the center, and the spin are no longer mixed. The new notation simultaneously shows how a closed circuit transforms into a forward or backward traveling wave, and why the physical transition requires complete charge and momentum balance.
Why the local transmission law leads to the wave equation
Consider the norm-preserving phase mode of the state:
\[\tag{A1} J(z,t) = J_0(z)e^{i\psi(z,t)}, \qquad J\overline J =1. \] We introduce two local quantities of inverse length dimension:
\[\tag{A2} q = \frac{1}{c}\frac{\partial\psi}{\partial t}, \qquad p = \frac{\partial\psi}{\partial z}. \] The coincidence of the mixed derivatives and the additional symmetric local transmission law yield 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 is decomposed into two characteristic components:
\[\tag{A4} u_+=p-q, \qquad u_-=p+q, \] \[\tag{A5} \boxed{ \frac{\partial u_+}{\partial t} = -c\frac{\partial u_+}{\partial z}, \qquad \frac{\partial u_-}{\partial t} = c\frac{\partial u_-}{\partial z}. } \] Substituting the definitions (A2) leads to 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. Now they obtain an explicit operator correspondence:
\[\tag{A7} \boxed{ u_+ \longleftrightarrow J_{\mathrm w}^{(+)}=\j^a, \qquad u_- \longleftrightarrow J_{\mathrm w}^{(-)}=\j^{-a}. } \] Thus, the two branches of the wave equation and the two conjugate powers of the fundamental operator describe the same structure at different levels: the local transmission system gives the velocities \(\pm c\), and the operator \(\j^{\pm a}\) stores the orientation of the unwrapped phase.
Materials used
- C. A. M. dos Santos, M. J. J. Fleury, An electromagnetic model of the electron, arXiv:2510.22384, 2025.