2026-08-05
Geometrical origin of the propagating wave
What happens to the internal wave state of a particle at the moment of emission? How does a closed geometry bound to a source transform into a free wave with frequency, energy, momentum, and a specific direction of propagation? This paper proposes to trace a possible geometric mechanism for such a transition from the difference in the particle's internal states to the formation of a bounded wave packet.
This paper continues the geometric investigation of the electron's wave state. In previous articles on the geometric origin of the Bohr atom's parameters and geometric origin of the electric force, the electron's internal electromagnetic scale was related to its frequency by a velocity invariant, and the interaction force was interpreted as a manifestation of an energy gradient. Here, these principles are applied to the emission process. The aim of this work is to construct a consistent hypothesis for the transition from a closed internal wave to a free electromagnetic wave packet.
The basic idea is that when an electron transitions between stationary states, a difference wave component with frequency \(\omega_\gamma\) is released. Maintaining the relation \(r\omega=c\) defines its new spatial scale, changing the two-branch geometry disrupts the local closure, and the internal energy gradient receives an external projection and determines the direction of propagation.
\[\tag{1} \boxed{ \text{closed state} \;\longrightarrow\; \omega_\gamma \;\longrightarrow\; r_\gamma=\frac{c}{\omega_\gamma} \;\longrightarrow\; \text{non-closure and gradient} \;\longrightarrow\; \text{free wave} } \] 1. Initial Assumptions of the Model
The geometric model is based on a normalized split operator that combines the internal periodicity of a particle and its external motion:
\[\tag{2} \boxed{ J(a,b)=\j^a(-\j)^b = \ep e^{i\pi b}+\em e^{i\pi a} } \] Here, two mutually complementary idempotents are used: \(\ep\) and \(\em\):
\[\tag{3} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \qquad \ep+\em=1, \qquad \j=\ep-\em. \] Parameter \(a\) describes the internal periodicity, and parameter \(b\) describes the external motion:
\[\tag{4} a=\varpi t, \qquad \pi\varpi=\omega_e, \qquad b=\frac{\arcsin\beta}{\pi}, \qquad \beta=\frac{v}{c}. \] The complete state preserves the unit norm. In the physical interpretation, this corresponds to the conservation of total energy when redistributed between the internal and external components:
\[\tag{5} J\overline J=1, \qquad E_{\mathrm{ext}}+E_{\mathrm{int}}=\mathrm{const}. \] For the internal electromagnetic scale of the electron, a frequency-spatial invariant is assumed.
\[\tag{6} \boxed{r_e\omega_e=c}. \] The quantity \(r_e\) is the radius of the internal wave state, while the length of one complete phase transition is equal to the circumference of the corresponding circle:
\[\tag{7} \boxed{\lambda_e=2\pi r_e}. \] Further, it is necessary to strictly distinguish between the radius of the wave state \(r\) and its full spatial period \(\lambda=2\pi r\). This distinction eliminates the ambiguity when moving from the electron scale to the radiation scale.
2. Internal Wave Closure
A bound state exists if, after traversing a geometric trajectory, the wave phase changes by an integer number of complete periods:
\[\tag{8} \oint_\Gamma k\,ds=2\pi n, \qquad n\in\mathbb Z. \] For one phase period, the local closure condition can be written as
\[\tag{9} (\varphi,z) \;\longrightarrow\; (\varphi+2\pi,z). \] The phase and spatial position simultaneously return to their original values. Therefore, the wave interferes with itself and forms a stable bound state. In this case, the period of the complex phase can be equal to \(2\pi\), while returning to the full two-branch state, taking into account the branch number, may require changing the parameter to \(4\pi\). These two types of periodicity should not be mixed.
Let two branches of the internal geometry have different radii \(R_+\) and \(R_-\). If the branch energy depends on the radius, then an energy difference arises between the branches:
\[\tag{10} \Delta U = U(R_+)-U(R_-) \simeq \frac{dU}{dR}\,\Delta R, \qquad \Delta R=R_+-R_-. \] However, in a stable closed state, internal changes are compensated for over a full cycle:
\[\tag{11} \oint_\Gamma dU=0. \] Therefore, the difference in radii creates an internal energy gradient, butdoes not, by itself, cause radiation. For the wave to emerge, the local closure condition must be violated.
3. The difference component during the transition
Consider the transition of an electron from the stationary state \(i\) to the state \(f\), for which \(E_i>E_f\). The released energy is
\[\tag{12} \boxed{ E_\gamma=E_i-E_f=\hbar\omega_\gamma }. \] Hence, the radiation frequency is determined by the energy difference:
\[\tag{13} \boxed{ \omega_\gamma=\frac{E_i-E_f}{\hbar} }. \] If each level is pre-assigned a frequency according to a single rule \(E=\hbar\omega\), then expression (13) can be written as \(\omega_\gamma=\omega_i-\omega_f\). However, the energy equality (12) remains primary for the transition.
Not the entire electron transitions to the radiative state. The electron is preserved and ends up in the final state \(f\), and only the difference component of its internal wave state with energy \(E_\gamma\) and frequency \(\omega_\gamma\) becomes free.
4. Increasing the geometric scale
Assume that the frequency-spatial invariant \(r\omega=c\) is also preserved for the released wave component. Then the following equalities hold for the initial and radiative states
\[\tag{14} r_e\omega_e=c, \qquad r_\gamma\omega_\gamma=c. \] The second equality immediately implies a new radius:
\[\tag{15} \boxed{ r_\gamma = \frac{c}{\omega_\gamma} = r_e\frac{\omega_e}{\omega_\gamma} }. \] Therefore, the spatial scale changes inversely proportional to the frequency:
\[\tag{16} \boxed{ \frac{r_\gamma}{r_e} = \frac{\omega_e}{\omega_\gamma} }. \] At the same time, the length of the complete phase shift increases in the same proportion:
\[\tag{17} \lambda_\gamma = 2\pi r_\gamma = \frac{2\pi c}{\omega_\gamma}. \] Therefore, the radius of the radiative state is equal to the reduced wavelength:
\[\tag{18} \boxed{ r_\gamma = \frac{\lambda_\gamma}{2\pi} = \overline\lambda_\gamma }. \] The transition between the original and radiative scales is symmetrical:
\[\tag{19} \boxed{ \frac{r_\gamma}{r_e} = \frac{\lambda_\gamma}{\lambda_e} = \frac{\omega_e}{\omega_\gamma} }. \] The increase in radius \(r_e\to r_\gamma\) is not a separate, arbitrary postulate. It follows from the conservation of the invariant \(r\omega=c\). The new assumption remains the applicability of this invariant to the released difference component during the transition.
5. Scaling of Two-Center Geometry
Let the distance between the centers of the two wave branches in the internal state be equal to the radius of the zeroth orbit:
\[\tag{20} d_e=r_e. \] If the entire two-center structure is scaled by the same factor during the transition, then
\[\tag{21} d_\gamma = d_e\frac{\omega_e}{\omega_\gamma} = r_\gamma. \] Thus, the new distance between the centers is not equal to the full wavelength, but to its reduced scale:
\[\tag{22} \boxed{ d_\gamma=r_\gamma, \qquad 2\pi d_\gamma=\lambda_\gamma }. \] In one complete phase revolution, the local scale \(d_\gamma\) forms a complete spatial period \(\lambda_\gamma\). Equality (22) preserves the original ratio \(d_e=r_e\) and does not require an additional geometric factor.
6. Loss of Local Closure
After the transition, the electron's final state no longer contains the previous trajectory on which the \(\omega_\gamma\) component could close again. Therefore, phase reversal is no longer accompanied by a return to the same spatial center.
The decoupling hypothesis can be expressed by the translational phase condition
\[\tag{23} \boxed{ (\varphi,z) \;\longrightarrow\; (\varphi+2\pi,z+\lambda_\gamma) }. \] In other words, one phase revolution turns into one translational spatial step:
\[\tag{24} \boxed{ \Delta\varphi=2\pi \quad\Longleftrightarrow\quad \Delta z=\lambda_\gamma }. \] This condition corresponds to a traveling harmonic wave.
\[\tag{25} \Psi(z,t) = A e^{i(k_\gamma z-\omega_\gamma t)}, \qquad k_\gamma = \frac{2\pi}{\lambda_\gamma} = \frac{\omega_\gamma}{c}. \] When shifted by one wavelength, the phase repeats:
\[\tag{26} \Psi(z+\lambda_\gamma,t)=\Psi(z,t). \] However, this is no longer a local closure. The same phase is reproduced at the next point in space. Therefore, the expression "the wave does not have time to close" should be understood not as an indication of the insufficient duration of the transition, but as the disappearance of the wave itself.geometrical possibility of returning to the previous trajectory.
The bound wave is closed both in phase and in position. The free wave retains phase periodicity but replaces spatial closure with translation to \(\lambda_\gamma\).
7. Energy Gradient and Direction of Propagation
The difference in the radii of the two branches creates an energy difference according to expression (10). As long as the geometry is closed, the corresponding changes cancel each other out. When opened, the internal asymmetry can be projected onto the external direction \(z\).
If the energy along the unfolded trajectory depends on the phase, then its effective spatial derivative has the form
\[\tag{27} \frac{\partial U}{\partial z} = \frac{\partial U}{\partial\varphi} \frac{\partial\varphi}{\partial z}, \qquad \frac{\partial\varphi}{\partial z}=k_\gamma. \] The corresponding longitudinal force is directed toward decreasing potential energy:
\[\tag{28} \boxed{ F_z=-\frac{\partial U}{\partial z} }. \] In this interpretation, the gradient serves two functions. It transforms the internal difference between the branches into a directional change in energy and determines the orientation of the wave's translational continuation. However, the difference in radii alone is not sufficient to select an axis in three-dimensional space: the axis must arise from a specific asymmetry in the transition, the relative orientation of the two branches, or external conditions.
The wave moves toward \(-\nabla U\) if \(U\) is understood as potential energy. If the gradient of the released energy is considered, then the direction of propagation coincides with the growth of this released quantity. The choice of definition must remain constant throughout the calculation.
8. Gradient Work and Photon Momentum
Let the work of the longitudinal force on the wave formation region be equal to the released energy:
\[\tag{29} \int F_z\,dz=E_\gamma. \] If the forming disturbance continues the wave state with the limiting velocity \(c\), then \(dz=c\,dt\). The momentum imparted to the wave by the gradient is equal to
\[\tag{30} p_\gamma = \int F_z\,dt = \frac{1}{c}\int F_z\,dz. \] Taking into account expressions (12) and (29), we obtain
\[\tag{31} \boxed{ p_\gamma = \frac{E_\gamma}{c} = \frac{\hbar\omega_\gamma}{c} = \hbar k_\gamma }. \] For a formation region one spatial period long, the average force can be written as
\[\tag{32} \overline F_z = \frac{E_\gamma}{\lambda_\gamma}, \qquad T_\gamma = \frac{\lambda_\gamma}{c} = \frac{2\pi}{\omega_\gamma}. \] Then the momentum over one period is
\[\tag{33} \overline F_zT_\gamma = \frac{E_\gamma}{\lambda_\gamma} \frac{\lambda_\gamma}{c} = \frac{E_\gamma}{c}. \] The equality \(p_\gamma=E_\gamma/c\) demonstrates the consistency of the gradient hypothesis with the standard photon momentum. However, this calculation uses the relation \(dz=c\,dt\), so it is not an independent derivation of the speed of light.
9. Relationship with the de Broglie Wavelength
From expressions (15) and (31) it follows
\[\tag{34} r_\gamma = \frac{c}{\omega_\gamma} = \frac{\hbar}{p_\gamma}. \] Consequently, the geometric radius of the free component coincides with the reduced de Broglie wavelength:
\[\tag{35} \boxed{ r_\gamma = \overline\lambda_{\mathrm{dB},\gamma} = \frac{\hbar}{p_\gamma} }. \] The total phase-transition length is equal to the usual de Broglie wavelength and the wavelength of the electromagnetic wave:
\[\tag{36} \boxed{ 2\pi r_\gamma = \frac{h}{p_\gamma} = \lambda_{\mathrm{dB},\gamma} = \lambda_\gamma }. \] Thus, the radius and wavelength are related by a single geometric principle:
\[\tag{37} \boxed{ r=\frac{\hbar}{p}, \qquad 2\pi r=\frac{h}{p} }. \] In the bound state, the quantity \(2\pi r\) describes a closed phase transition. In the free state, the same phase period manifests itself as a longitudinal spatial step. Therefore, the similarity with the de Broglie wave relates not to the conversion of an electron into a photon, but to the general way in which the momentum is related to the geometric period of the wave state.
10. Formation of a Bounded Wave Packet
The monochromatic function (25) is infinite in space and serves only as a local description of the carrier. The actual radiation occurs in a finite time \(\Delta t\), so it must be represented by a wave packet:
\[\tag{38} \Psi(z,t) = A\!\left(t-\frac{z}{c}\right) e^{i(k_\gamma z-\omega_\gamma t)}. \] Its longitudinal extent is determined by the transition duration:
\[\tag{39} \boxed{ L_{\mathrm{packet}}\sim c\Delta t }. \] Therefore, in the geometric description, it is necessary to distinguish three quantities:
\[\tag{40} \boxed{ r_\gamma=\frac{c}{\omega_\gamma}, \qquad \lambda_\gamma=2\pi r_\gamma, \qquad L_{\mathrm{packet}}\sim c\Delta t }. \] The radius \(r_\gamma\) specifies the reduced geometric scale, \(\lambda_\gamma\) is the carrier wave period, and \(L_{\mathrm{packet}}\) is the total length of the photon packet. In general, a packet can contain many periods, so its length does not necessarily coincide with \(\lambda_\gamma\).
The finite transition time also leads to a finite spectral width. At the estimation level, the relation holds
\[\tag{41} \Delta\omega\,\Delta t\gtrsim 1. \] After the transition is complete, the source gradient disappears, but the already formed packet continues to propagate as a free solution of the wave equation:
\[\tag{42} \frac{\partial^2\Psi}{\partial t^2} - c^2\frac{\partial^2\Psi}{\partial z^2} =0. \] 11. The electron is not the only source of the wave.
The proposed mechanism is not tied exclusively to the electron. A potential source of free wave excitation can be any particle or bound system in which a closed internal wave state, conjugate branches, an energy difference between them, and the possibility of a transition coupling a change in the internal geometry to an external field exist.
The splitting of the internal state itself does not necessarily imply radiation. In a stationary configuration, the gradients created by the branches are compensated for over a complete geometric cycle, and the system maintains its bound state. A free wave arises only when the splitting changes over time, the previous condition of local closure is violated, and the released difference component receives an external continuation channel.
The type of wave that emerges is determined not only by the internal geometry of the particle but also by the field that causes its change. Electromagnetic radiation arises if the transition alters the electric or magnetic moment of the system. Therefore, electromagnetic waves can be formed not only by electrons, but also by other charged particles, as well as neutral composite systems with variable electric or magnetic moments.
For other types of internal splitting, the external channel may correspond to a different field and a different type of propagating disturbance. Thus, two-branch geometry is considered a general mechanism for the generation of an energy difference, while the nature of the observed wave is determined by the mode of coupling of the internal transition with external space. This generalization sets the agenda for further research and does not yet constitute proof of the existence of new interactions.
Therefore, a split internal wave state is a necessary element of the proposed mechanism, but by itself is insufficient for radiation. The formation of a free wave additionally requires a transition between states and coupling of the released component with an external field.
12. Geometric Transition Sequence
The proposed mechanism can be reduced to the following chain:
\[\tag{43} \boxed{ \begin{aligned} E_\gamma&=E_i-E_f=\hbar\omega_\gamma, \\ r_\gamma&=\frac{c}{\omega_\gamma} =r_e\frac{\omega_e}{\omega_\gamma}, \\ \lambda_\gamma&=2\pi r_\gamma, \\ d_\gamma&=r_\gamma, \qquad 2\pi d_\gamma=\lambda_\gamma, \\ (\varphi,z)&\longrightarrow (\varphi+2\pi,z+\lambda_\gamma), \\ F_z&=-\frac{\partial U}{\partial z}, \qquad p_\gamma=\frac{E_\gamma}{c}=\hbar k_\gamma. \end{aligned} } \] This sequence describes three related transforms. The frequency transform isolates the difference component. The scale transform increases its radius inversely proportional to the frequency. The topological transform replaces the closed phase tour with a translational continuation of the wave.
13. What is deduced and what is accepted as a hypothesis?
From the previously adopted invariant \(r\omega=c\), Planck's energy relation, and the definition of the spatial period, the following directly follow:
1. Radius of the radiative component \(r_\gamma=c/\omega_\gamma\).
2. Scale ratio \(r_\gamma/r_e=\omega_e/\omega_\gamma\).
3. Wavelength \(\lambda_\gamma=2\pi r_\gamma=2\pi c/\omega_\gamma\).
4. For proportional scaling of the two-center design, the equality is \(d_\gamma=r_\gamma\).
5. When using \(E_\gamma=\hbar\omega_\gamma\) and velocity \(c\), the standard relationships are \(p_\gamma=E_\gamma/c=\hbar k_\gamma\).
New positionsstatements, which at this stage remain geometric hypotheses:
1. The difference component of the internal state preserves the invariant \(r\omega=c\) during the separation process.
2. The local closure is transformed into the translational phase condition \((\varphi,z)\to(\varphi+2\pi,z+\lambda_\gamma)\).
3. The internal gradient caused by the difference in radii receives an external longitudinal projection.
4. The asymmetry of the two-branch transition selects a specific propagation axis and direction of the pulse.
5. After completion of formation, the wave retains the obtained direction as a free state.
6. The same mechanism applies to other particles and bound systems if their split internal state has a communication channel with the corresponding external field.
This distinction is fundamental: the scale and wavelength formulas follow from the already adopted invariant, while the decoupling mechanism, the choice of spatial direction, and the conversion of the internal gradient into external motion require further geometric derivation.
Conclusion
This paper considers a geometric model of the transition from a closed internal wave state of a particle to a freely propagating wave. The initial event of such a transition is a change in the internal frequency of the particle. The difference in the energies of the initial and final states determines the radiation energy:
\[ E_\gamma=\Delta E=\hbar\omega_\gamma. \] Within the accepted wave invariant, the radiation frequency corresponds to a geometric scale.
\[ r_\gamma=\frac{c}{\omega_\gamma}, \qquad r_\gamma\omega_\gamma=c. \] This scale differs significantly from the radius of the particle's internal state and characterizes not the localized motion of the source, but the forming free wave. Upon transition to a propagating state, the energy acquires an outward directional projection. For a massless state, it follows from the model relations.
\[ p_\gamma=\frac{E_\gamma}{c} =\frac{\hbar\omega_\gamma}{c} =\frac{\hbar}{r_\gamma}. \] The corresponding spatial wavelength is
\[ \lambda_\gamma=2\pi r_\gamma =\frac{h}{p_\gamma}, \] which links the geometric radius of the wave state with the de Broglie relation.
Thus, the frequency, energy, momentum, and spatial scale of the radiation form a single chain of interconnected quantities in the model. The free wave is considered not as a separate object arising outside the source, but as a new geometric state of energy released during the restructuring of the particle's internal motion.
At the same time, the opening of the internal trajectory, the transformation of the internal energy gradient into external motion, the choice of the propagation direction, and the mechanism for spatially limiting the wave packet remain geometric hypotheses. To transform the proposed scheme into a complete dynamic mechanism, these transitions must be derived directly from the state operator and its norm conservation law.
The proposed construction is therefore not a definitive proof of the radiation mechanism, but it does provide a consistent geometric framework in which the transition from the internal state of a particle to a free wave packet can be studied as a continuous transformation of a single wave structure.
Why the local transmission law leads to the wave equation
Consider a norm-preserving phase mode of the complete state. If the unperturbed state satisfies the condition \(J_0\overline J_0=1\), then such a mode can be represented as
\[\tag{A1} J(z,t)=J_0(z)e^{i\psi(z,t)}. \] The factor \(e^{i\psi}\) changes the phase, but not the absolute value of the state:
\[ J\overline J = J_0\overline J_0 e^{i\psi}e^{-i\psi} =1. \] We introduce two local quantities with the dimension of the inverse length:
\[\tag{A2} q=\frac{1}{c}\frac{\partial\psi}{\partial t}, \qquad p=\frac{\partial\psi}{\partial z}. \] Since \(p\) and \(q\) are derivatives of the same phase function, the coincidence of the mixed derivatives automatically yields
\[ \frac{\partial p}{\partial t} = c\frac{\partial q}{\partial z}. \] To describe the free propagation ofWe introduce an additional local law of the model: the temporal change in \(q\) is determined by the spatial change in \(p\) at the same speed \(c\):
\[\tag{A3} \frac{\partial p}{\partial t} = c\frac{\partial q}{\partial z}, \qquad \frac{\partial q}{\partial t} = c\frac{\partial p}{\partial z}. \] System (A3) has a local conservation law:
\[ \frac{\partial}{\partial t} \left( \frac{p^2+q^2}{2} \right) + \frac{\partial}{\partial z} \left( -cpq \right) =0. \] Therefore, the quantity \((p^2+q^2)/2\) is neither created nor destroyed between adjacent points, and its change within a region is determined by the flow \(-cpq\) through the boundaries.
The system naturally decomposes into two oppositely propagating phase components:
\[ u_+=p+q, \qquad u_-=p-q, \] \[ \frac{\partial u_+}{\partial t} = c\frac{\partial u_+}{\partial z}, \qquad \frac{\partial u_-}{\partial t} = -c\frac{\partial u_-}{\partial z}. \] This shows that system (A3) describes two wave branches propagating in opposite directions with velocity \(c\).
Substituting the definitions of (A2) into the second equation of system (A3) yields
\[\tag{A4} \frac{1}{c} \frac{\partial^2\psi}{\partial t^2} = c\frac{\partial^2\psi}{\partial z^2}, \qquad \boxed{ \frac{\partial^2\psi}{\partial t^2} - c^2\frac{\partial^2\psi}{\partial z^2} =0 }. \] Thus, the wave equation is not introduced separately: it is a consequence of the phase representation and the local transmission system (A3). However, the second equation of this system remains an additional dynamical law. For a complete geometric derivation, it is necessary to show why the two-branch structure \(J\) leads to precisely the symmetric transmission of changes with velocities \(+c\) and \(-c\).