2026-07-16
A Unified Geometric Model of Wave and Particle Motion
Particle as a localized rotational state of a wave
Waves and particles are usually described as different physical entities: a wave is distributed in space, while a particle is localized. However, quantum phenomena show that the same state can exhibit both properties. In Wave Electricity, this contradiction is resolved by the initial assumption: the normalized wave state is considered primary, and its stable closed mode is called a particle.
To prevent this idea from confusing internal dynamics with the observed motion of a body, it is necessary to distinguish three operations. The operator \(J(a,b)\) defines the phase state, the external projection of this state determines the velocity of the center, and the velocity of the center constructs a trajectory in physical space:
\[\tag{1} \boxed{ J(a,b) \;\longrightarrow\; \mathcal P_{\mathrm{ext}}[J] \;\longrightarrow\; \mathbf v \;\longrightarrow\; \mathbf r(s) }. \] A different, independent logic is at work at the same time: the closure applies not to the path of the particle's center, but to its internal wave. Therefore, a freely flying electron can move along an open trajectory and still remain a localized particle.
The main thesis of the article: the observed motion of the center and the internal wave geometry are two different representations of a single state. They cannot be identified with each other or with the coordinates of idempotent planes.
1. A Single State for Wave and Particle
The proposed model does not assume that a separate wave resides within an existing material particle. On the contrary, the stable localized wave itself is observed as a particle. If the internal cycle retains its shape, frequency, and closure, the state has a finite localization region. If cyclic return is replaced by spatial continuation, a free propagation regime arises.
\[\tag{2} \boxed{ \begin{aligned} \text{internal closure}&\;\longleftrightarrow\;\text{particle},\\ \text{spatial opening}&\;\longleftrightarrow\;\text{free wave}. \end{aligned}} \] This distinction is a distinction between regimes of the same wave nature, not a contrast between two primary entities. Moreover, the presence of an external field does not necessarily mean the particle is open: the field can be a continuation of its internal gradient, while the internal contour remains closed.
2. Four Model Spaces
For a clear physical interpretation, it is useful to distinguish four levels. The first is the ordinary three-dimensional space in which the coordinates of devices and bodies are measured. The second is the one-dimensional coordinate \(s\), measured along a specific trajectory. Even if this trajectory is curved, the position on it is specified by a single number.
The third level is formed by two orthogonal complex planes of the finite operator \(J\). They describe the components of the phase state, not the two physical directions of particle motion. The fourth level is a deeply split space with a sequence of mutually orthogonal channels. It is this metric length that will be related to the Lorentz factor.
The two planes of the operator \(J\) are not two planes of the laboratory. They belong to the state space. The transition to physical \(3D\) space is accomplished by a separate mapping and requires specifying the direction of the trajectory.
Therefore, the postulate of one-dimensional motion should be understood not as a prohibition of curvilinear trajectories, but as a statement about local kinematics: at each moment, the observed motion occurs along a single tangent \(\widehat{\boldsymbol\tau}(s)\).
More details: The geometric meaning of one-dimensional motion and its mapping into physical space are discussed in the current concept of the model.
3. Finite State Operator
The algebraic foundation is built on two mutually complementary idempotents:
\[\tag{3} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \qquad \ep+\em=1. \] The hyperbolic unit is defined as \(\j=\ep-\em\), which yields \(\j^2=1\). After complex expansion, the product of two fractional powers takes the divided form:
\[\tag{4} \boxed{ J(a,b)=\j^a(-\j)^b =\ep e^{i\pi b}+\em e^{i\pi a} }. \] Parameters \(a\) and \(b\) play strictly different physical roles. The internal parameter \(a\) specifies the phase of the particle's state, and the external parameter \(b\) encodes its instantaneous motion relative to the chosen reference frame:
\[\tag{5} a=\varpi t, \qquad omega_{\mathrm{int}}=\pi\varpi, \qquad b=\frac{\arcsin\beta}{\pi}, \qquad \beta=\frac{v_s}{c}. \] After substitution, we obtain two orthogonal phase components of the same state: the external \(\ep e^{i\arcsin\beta}\) and the internal \(\em e^{i\omega_{\mathrm{int}}t}\). Their orthogonality follows from the equality \(\ep\em=0\), but the physical interpretation of these components is introduced additionally by the model.
More details: The derivation of the powers of the hyperbolic unit is given in the article "From Euler's Formula to Split Geometry".
4. Phase Dynamics of Constant Norm
The state operator can be multiplied by the limiting velocity \(c\):
\[\tag{6} \boxed{V_J(t)=cJ(t).} \] The quantity \(V_J\) describes the complete phase-geometric dynamics. It is convenient as a generator of intrinsic parametric geometry, but it is not the laboratory velocity of the particle's center, the four-velocity, or the sum of two ordinary velocities.
For the conjugate state,
\[\tag{7} J\overline J=1, \qquad V_J\overline{V_J}=c^2. \] This means that the final phase norm is constant. It does not mean that the observed energy, momentum, or velocity of the center are constant. During acceleration, the parameter \(b(t)\), velocity \(\beta(t)\), factor \(\gamma(t)\), and energy of the particle can change, although the normalization of the final operator is preserved.
The formula \(V_J=cJ\) belongs to the phase space of the operator. The observed velocity arises only after choosing an external projection and a direction in physical space.
For more information: The normalization of the rotational state and its relationship to orthogonality are discussed in the article "Rotation as a Consequence of Norm Conservation".
5. External Projection and Observed Velocity
The external component of the finite operator contains two circular projections. One is equal to \(\beta\), the other to \(1/\gamma\). For observed spatial motion, the first is used:
\[\tag{8} \mathcal P_{\mathrm{ext}}[J] =\sin(\pi b) =\beta, \qquad \boxed{ \mathbf v =c\beta\widehat{\boldsymbol\tau}(s) }. \] The parameter \(b\) specifies the magnitude and sign of the velocity along the selected coordinate \(s\), but does not itself contain a three-dimensional direction. The direction is specified by the unit tangent vector \(\widehat{\boldsymbol\tau}(s)\). Therefore, the same scalar quantity \(b\) can describe motion along a straight line, a circle, or an arbitrary smooth curve.
This separates the phase geometry from the kinematics of the measured center. The coordinates of the \(\ep\) and \(\em\) planes do not replace the physical axes \(x,y,z\), and the external parameter does not select a direction in the laboratory without additional representation.
More details: The definition of the external parameter and the limitations of its physical interpretation are given in the concept of Wave Electricity.
6. Physical Trajectory of the Center
After specifying the tangential direction, the observed motion is constructed using ordinary kinematic integration. First, the coordinate traveled along the curve is determined, then it is mapped into three-dimensional space:
\[\tag{9} \frac{ds}{dt}=c\beta(t), \qquad \mathbf r(t) =\mathbf r(0) +\int_0^t c\beta(\tau) \widehat{\boldsymbol\tau}\bigl(s(\tau)\bigr)\,d\tau. \] This is the laboratory trajectory of the localization center. It can be open, curved, and non-periodic. Its shape is determined not only by the particle's state, but also by external fields, initial conditions, and the interaction geometry.
The phase quantity \(V_J=cJ\) can be integrated separately:
\[\tag{10} \mathbf R_J(t) =\mathbf R_J(0) +\int_0^t V_J(\tau)\,d\tau. \] The result \(\mathbf R_J\) should be called the internal phase-geometric curve. It helps to see the periodic and open components of the state, but does not provide the coordinates of the center in the laboratory. This preserves the useful idea of \(cJ\) integration, but eliminates its previous, overly literal spatial interpretation.
More details: The transition from phase dynamics to observable motion is systematized in the reference article on the model postulates.
7. Deep Splitting and the Lorentz Factor
The finite external phase yields circular projections \(\beta\) and \(\sqrt{1-\beta^2}\), but by itself does not explain why the total relativistic magnitude grows like \(\gamma\). For this, the model uses aA deeper level of splitting, containing a sequence of orthogonal channels:
\[\tag{11} \boldsymbol\Gamma_\beta =\boldsymbol\xi_0 +\beta\boldsymbol\xi_1 +\beta^2\boldsymbol\xi_2 +\cdots. \] Each subsequent channel repeats the structure of the previous one with a coefficient of \(\beta\). Due to orthogonality, the lengths add up quadratically. The self-similarity of the entire chain leads to the relation \(\|\boldsymbol\Gamma_\beta\|^2=1+\beta^2\|\boldsymbol\Gamma_\beta\|^2\), whence
\[\tag{12} \boxed{ \|\boldsymbol\Gamma_\beta\| =\frac{1}{\sqrt{1-\beta^2}} =\gamma =\frac{1}{\cos(\pi b)} }. \] Thus, the deep space creates a full metric length \(\gamma\), and the finite operator compactly encodes the inverse projection \(1/\gamma\) and the velocity \(\beta\). These are two consistent levels of description, not two independent derivations of the same norm.
More details: The full derivation is given in the article "The Origin of the Lorentz Factor from Infinite Idempotent Splitting".
8. Two Norms and the Energy Map
At this stage, it is especially important not to confuse the two different norms. The equality \(J\overline J=1\) refers to the finite phase operator. The equality \(\|\boldsymbol\Gamma_\beta\|=\gamma\) refers to the state length in a deeply split metric space.
The connection with physical energy and momentum arises after a separate mapping:
\[\tag{13} \frac{E_0}{E}=\cos(\pi b)=\frac1\gamma, \qquad \frac{pc}{E}=\sin(\pi b)=\beta. \] From this follow the familiar relations \(E=\gamma E_0\), \(pc=\gamma\beta E_0\), and the external relativistic invariant:
\[\tag{14} \boxed{E^2-p^2c^2=E_0^2.} \] The unit norm \(J\) therefore does not conflict with the increase in the energy of a moving particle. The first characterizes the shape of the normalized phase state, while the second physical quantity takes into account the deep length \(\gamma\) and the work done on the particle.
More details: The early geometric notation of the energy invariant is discussed in the article "Origin of the Lorentz Factor and the Energy Invariant from the Hyperbolic Unit", and the deep mechanism of \(\gamma\) is discussed in a subsequent paper on splitting.
9. Internal Closure and Particle Formation
Consider an internal phase with a constant frequency \(\omega_{\mathrm{int}}\). Integrating the corresponding component \(cJ\) creates a bounded periodic curve. Its characteristic radius is determined by the simple relation:
\[\tag{15} \boxed{ r_{\mathrm{int}} =\frac{c}{\omega_{\mathrm{int}}}, \qquad r_{\mathrm{int}}\omega_{\mathrm{int}}=c }. \] Over a full internal period, the wave returns to its original configuration. This closure allows energy to repeatedly circulate within a finite region without necessarily being transferred outward. In the working physical interpretation, it is the stability of this periodic return that creates a localized object.
\[\tag{16} \boxed{ \text{particle} =\text{stable closed mode of a normalized wave} }. \] The formula is a definition of the mode within the model, not a consequence of idempotent algebra alone. The algebra admits periodic internal geometry; The assertion that such a geometry physically manifests as a particle is the next level of hypothesis.
More details: The internal rotation and structure of the electron are developed in the articles "Geometric Model of Electron Structure" and "Capacitance and Inductance of the Electron".
10. A Moving Particle Maintains Internal Closure
If \(|\beta|<1\), the localization center can move relative to the observer without breaking the internal loop. The external parameter \(b(t)\) describes the motion of the center, and the parameter \(a(t)\) continues to measure the internal phase. Their distinct roles do not disappear even when both are included in a single operator.
Therefore, an open trajectory of the center is not a sign of a free wave. An electron flying through the laboratory in a straight line still remains a particle if its internal wave structure is closed and stable.
The closure of the internal wave is not the same as the closure of the trajectory of the center. The particle's center can move arbitrarily far, while the internal state continues to periodically return to itself.
External acceleration changes \(b(t)\), \(\beta(t)\), \(\gamma(t)\), energy, and momentum. But the very existence of a particle requires the maintenance of consistency in its internal regime, not the immobility of its center.
More details: The separation of the internal state and external motion is enshrined in the model concept.
11. Internal Periodicity and Mass
Internal periodicity corresponds to the energy \(E_{\mathrm{int}}=\hbar\omega_{\mathrm{int}}\). However, in the electron model, the observed rest energy is not identified with the total internal energy. It is considered as its geometric projection with the coefficient \(\alpha_{\mathrm{fs}}\):
\[\tag{17} \boxed{ m_ec^2 =\alpha_{\mathrm{fs}}\hbar\omega_{\mathrm{int}} }. \] This internal mass projection should be strictly distinguished from the external Lorentz factor. The constant \(\alpha_{\mathrm{fs}}\) relates the electron's intrinsic frequency to its rest energy, while \(\gamma(\beta)\) describes the state of a particle already in motion:
\[\tag{18} \boxed{ \alpha_{\mathrm{fs}} \;:\;\text{internal mass projection}, \qquad \gamma(\beta) \;:\;\text{external length of the state of motion} }. \] The Lorentz factor does not create mass and does not explain the rest energy. It acts on an already formed state: \(E=\gamma m_ec^2\). Thus, the origin of mass and the relativistic change in energy are separated into different geometric levels.
More details: The internal projection hypothesis is detailed in the article "Mass as a Geometric Projection of a Closed Wave".
12. Opening and Free Propagation
A free wave does not arise because the center of a localized state begins to move. It requires a qualitative restructuring of the internal topology: instead of returning to the starting point after a complete phase cycle, a continuation in space arises.
\[\tag{19} (\varphi,z) \;\longrightarrow\; (\varphi+2\pi,z+\lambda). \] This notation means that after a phase change of \(2\pi\), the state does not return to its previous spatial position, but is transferred to a wavelength of \(\lambda\). The closed circle transforms into an open helical or equivalent propagating structure.
The field and radiation cannot be considered the same regime. As long as the electron is conserved, its internal wave remains closed, even if there is an electric field around it or a photon is emitted from the system. Complete opening of the electron state is possible only in a process where the localized configuration itself disappears, for example, during the annihilation of an electron and positron.
For more information: The open-branch mechanism is discussed in the article "The Geometric Origin of the Propagating Wave".
13. The Light Limit Does Not Automatically Transform into a Photon
For the light state, the kinematic conditions \(|\beta|=1\), \(E_0=0\), and \(E=|p|c\) are satisfied. However, a single limiting value of the external parameter is not sufficient to completely define the photon.
A photon must simultaneously possess zero rest energy, an open propagation structure, energy and momentum transport, and transverse polarization. Therefore, a massive particle does not transform into a photon simply by increasing \(b\). While maintaining a nonzero mass, achieving \(|\beta|=1\) would require infinite energy.
The light limit is a kinematic condition, while the open limit is a topological condition. In the photon model, both conditions must be satisfied in concert, but they are not identical.
The \(b\) parameter is also not an accumulated spatial phase. It encodes the instantaneous velocity. The spatial phase of the propagating state is introduced through the action:
\[\tag{20} d\mathcal S=p_s\,ds-E\,dt. \] It is the accumulation of action along various admissible extensions that allows us to describe interference and diffraction, while \(b(t)\) is responsible only for local kinematics.
More details: The spatial phase is clearly used in the papers "Particle Diffraction by a Slit" and "Interference during Passage through Two Slits".
14. Energy balance includes the source.
A constant phase norm does not mean that an external force only redistributes the existing energy within the particle. If the source accelerates the particle, it performs a workFrom here, and the laboratory energy of the particle increases. The conservation law should be applied to the complete interacting system:
\[\tag{21} \boxed{ E_{\mathrm{int}} +E_{\mathrm{motion}} +E_{\mathrm{source}} =\mathrm{const} }. \] For positive source work, \(\Delta E_{\mathrm{particle}}=-\Delta E_{\mathrm{source}}\). Within the operator description, we can talk about a change in phase projections, but the physical energy increase cannot be obtained from the normalization (J\overline J=1\) alone.
This refinement preserves the geometric structure of the model and simultaneously aligns it with the standard energy accounting. When accelerated, the particle's energy becomes \(E=\gamma E_0\), and the required difference comes from an external source.
More details: The relativistic energy-momentum mapping is compared with the finite operator in the article on the energy invariant.
15. What follows from mathematics, and what remains a hypothesis
The algebraic and physical parts of the model must be explicitly separated. The following directly follow from the chosen idempotent basis and the definition of the operator:
- decomposition of unity into two orthogonal idempotents;
- two independent first-level complex phase planes;
- the form and composition rules of the operator \(J(a,b)\);
- the unit norm of the final phase state;
- the possibility of sequential splitting into orthogonal channels;
- the metric sum of an orthogonal self-similar chain, equal to \(\gamma\), if its normalization and similarity coefficient \(\beta\) are accepted.
Additional physical hypotheses of the model are:
- the primacy of the normalized wave;
- the mapping of the parameter \(b\) onto the observable velocity;
- interpretation of the integral of \(cJ\) as internal phase geometry;
- particle as a stable closed regime;
- opening as a free wave mechanism;
- internal energy projection as the origin of mass;
- a concrete physical embodiment of deep spallation channels.
This distinction does not weaken the model. On the contrary, it shows which results are rigorous consequences of the adopted algebra, and which require further physical justification, experimental verification, or more complete dynamics.
Conclusions
The final operator \(J(a,b)\) combines the internal phase and external state of motion, but is not a direct coordinate of the particle. Its multiplication by \(c\) defines the normalized phase-geometric dynamics. The observed velocity is obtained only after the external projection, and the physical trajectory is obtained after choosing the tangent direction and integrating this velocity.
The deeply split state complements the finite operator: its metric length is \(\gamma\), while the finite circular projection encodes \(1/\gamma\) and \(\beta\). These norms refer to different spaces and therefore do not contradict each other.
Finally, the distinction between a particle and a free wave is determined not by the shape of the path of the observed center, but by the organization of the internal state. A closed stable cycle manifests itself as a particle; Spatial continuation, instead of cyclic return, creates a freely propagating wave.
\[\tag{22} \boxed{ \begin{gathered} J(a,b) \longrightarrow \mathcal P_{\mathrm{ext}}[J] \longrightarrow \mathbf v \longrightarrow \mathbf r(s),\\ \text{closure} \longrightarrow \text{particle}, \qquad \text{opening} \longrightarrow \text{free wave}. \end{gathered}} \] A particle and a free wave are considered as two modes of a single normalized wave state. Phase dynamics describes their general mathematical framework, external projection describes the observed motion, and closure or opening describes the physical organization of the state.