2026-07-22
Geometric representation of particle interactions
Part 1. Operator states and transformation invariants
In physics, interaction is usually written as the transition of one set of particles to another. This notation shows the initial and final objects, but does not reveal how their internal states form a single transformation. In the Wave Electricity model, a particle is considered a structured operator state, and interaction is considered a rearrangement of the complete set of its orthogonal components.
The modern operator \(J(a,b)\) differs from the previously used form \(e^{ia}\j^b\). Now, the parameter \(a\) always describes the internal state, and the parameter \(b\) describes the external motion. Charge, internal closure, orbital, and other properties are defined by independent operator levels. Therefore, interaction cannot be reduced to an arbitrary permutation of two phases or to the simple addition of several operators.
The main goal of the first part is to construct a unified language of interactions. The complete composite state is transformed as a whole, and the energy, momentum, charge, angular momentum, and closure index act as different projections of the preserved operator content.
\[\tag{1} \boxed{ \mathcal S_{\mathrm{in}} \overset{\mathcal U_{\mathrm{int}}}{\longrightarrow} \mathcal S_{\mathrm{out}}, \qquad \mathcal I[\mathcal S_{\mathrm{in}}] =\mathcal I[\mathcal S_{\mathrm{out}}]. } \] 1. Four-dimensional i-basis
The model is based on two mutually complementary idempotents. Together with the imaginary unit, they form a four-dimensional basis. \[\tag{2} \left\{ \ep,\;i\ep,\;\em,\;i\em \right\} = (\ep,i\ep)\oplus(\em,i\em). \]
Idempotents satisfy the relations. \[\tag{3} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \qquad \ep+\em=1. \]
The condition \(\ep\em=0\) means that two algebraic channels are orthogonal. However, it does not in itself designate one plane as internal and the other as external. The physical meaning is determined by which parameter and which projector are associated with the corresponding component.
The hyperbolic unit is defined by the difference of idempotents:
\[\tag{4} \j=\ep-\em, \qquad \j^2=1. \] 2. The Modern Finite Operator
The state of internal dynamics and external motion is described by a two-parameter operator
\[\tag{5} \boxed{ J(a,b)=\j^a(-\j)^b =\ep e^{i\pi b}+\em e^{i\pi a}. } \] The parameters have a fixed physical meaning:
\[\tag{6} a=\varpi t, \qquad \pi\varpi=\omega_{\mathrm{int}}, \qquad b=\frac{\arcsin\beta}{\pi}, \qquad \beta=\frac vc. \] Therefore, \(a\) organizes the internal periodicity, while \(b\) defines the magnitude of the external projection of the motion. In the general case, \(a=a(t)\), \(b=b(t)\), and \(v=v(t)\).
The external motion projector extracts the physical velocity from the operator:
\[\tag{7} \mathcal P_{\mathrm{ext}}[J(a,b)] =\Sin(\pi b)=\beta, \qquad \mathbf v=c\beta\widehat{\boldsymbol\tau}. \] The final operator is normalized:
\[\tag{8} J(a,b)\overline{J(a,b)}=1. \] The equality of the norm to unity means the preservation of the overall phase state when it is redistributed between orthogonal components. It does not assert that physical energy is numerically equal to unity. The relationship of the norm to the energy scale requires a separate physical representation.
3. Two orientations of the dynamic state
Permuting the phase planes creates two alternative orientations of the complete dynamic state:
\[\tag{9} \boxed{ \begin{aligned} J_{\mathrm{dyn}}^{(+)}(a,b) &=\ep e^{i\pi b}+\em e^{i\pi a},\\ J_{\mathrm{dyn}}^{(-)}(a,b) &=\ep e^{i\pi a}+\em e^{i\pi b}. \end{aligned} } \] In the second line, the placement of the parameters across the idempotent planes is changed, but their physical labels are not rearranged. The parameter \(a\) remains internal, and \(b\) is external. Therefore, both orientations yield the same velocity of the center:
\[\tag{10} \mathcal P_{\mathrm{ext}} [J_{\mathrm{dyn}}^{(+)}] =\mathcal P_{\mathrm{ext}} [J_{\mathrm{dyn}}^{(-)}] =\beta. \] The two orientations can be associated with spin states relative to the chosen axis, but do not determine the sign of the electric charge. The charge belongs to an independent operator level.
4. Multilevel Particle Operator
Different physical properties of a particle should not be placed in a single parameter. We represent the overall architecture of the state as a product of independent compatible operators:
\[\tag{11} \boxed{ J_P =J_{\mathrm{dyn}}^{(\sigma)}(a,b) Q_\Gamma J_qJ_{\mathrm{orb}}J_fJ_cJ_g\cdots, \qquad \sigma=\pm. } \] Here \(J_{\mathrm{dyn}}^{(\sigma)}\) contains the internal dynamics, external motion, and orientationstate; \(Q_\Gamma\) describes the internal closure; \(J_q\) — the charge structure; \(J_{\mathrm{orb}}\) — the orbital generators of the bound state. The factors \(J_f,J_c,J_g\) can be used for other independent features after defining their own algebra.
The usual multiplication in formula (11) means the combined action of operators within a single extended algebra. Adding a new independent feature should not change the meaning of the already introduced parameters \(a\) and \(b\).
For a missing feature, the identity operator is used:
\[\tag{12} J_{r,0}=1, \qquad J_PJ_{r,0}=J_P. \] 5. Deep State of a Moving Particle
The final operator does not contain the metric depth of the external motion. For a massive particle, it corresponds to a recursive state of orthogonal channels.
\[\tag{13} \boldsymbol\Gamma_\beta =\sum_{n=0}^{\infty} \beta^n\boldsymbol\xi_n, \qquad \boldsymbol\xi_n\boldsymbol\cdot\boldsymbol\xi_m =\delta_{nm}. \] Its norm is
\[\tag{14} \boxed{ \|\boldsymbol\Gamma_\beta\| =\frac1{\sqrt{1-\beta^2}} =\gamma(\beta). } \] After physical mapping, the energy and momentum of the massive state take the form
\[\tag{15} E=\gamma E_0, \qquad \mathbf p =\frac{\gamma E_0}{c} \boldsymbol\beta, \qquad E^2-p^2c^2=E_0^2. \] Thus, when interacting, it is necessary to distinguish between the norm of the finite phase operator and the metric norm of the deep state:
\[\tag{16} J\overline J=1, \qquad \|\boldsymbol\Gamma_\beta\|=\gamma. \] A detailed derivation of the chain of deep projections and the state \(\boldsymbol\Gamma_\beta\) is given in the article "Unified Concept of Wave Electricity". Only the metric projections necessary for interactions are used here.
6. Composite state of several particles
The arithmetic sum of operators is convenient for checking additive balance, but by itself does not describe the complete joint state. For a system of \(n\) particles, we introduce the tensor product:
\[\tag{17} \boxed{ \mathcal S =J_{P_1}\otimes J_{P_2} \otimes\cdots\otimes J_{P_n}. } \] The sign \(\otimes\) preserves the belonging of each component to its own object and allows for correlations between them. It differs from the usual product of operator factors of one particle in formula (11).
The initial and final states of the reaction are of the form
\[\tag{18} \mathcal S_{\mathrm{in}} =\bigotimes_{k=1}^{n}J_k, \qquad \mathcal S_{\mathrm{out}} =\bigotimes_{l=1}^{m}J_l'. \] 7. Interaction Operator
We represent interaction as a restructuring operator for the complete composite state:
\[\tag{19} \boxed{ \mathcal S_{\mathrm{out}} =\mathcal U_{\mathrm{int}} \mathcal S_{\mathrm{in}}. } \] The operator \(\mathcal U_{\mathrm{int}}\) can change the way branches are connected, the internal phases, the velocities, and the composition of the output objects. However, it is not an arbitrary permutation: an admissible transformation must preserve the full set of system invariants.
If the interaction is closed and reversible at the full-state level, a natural condition is the preservation of its norm:
\[\tag{20} \mathcal U_{\mathrm{int}} \overline{\mathcal U}_{\mathrm{int}}=1. \] Normalization alone is not sufficient for choosing a physical response. Additionally, the values of independent operator and metric projectors must be preserved.
8. Rearrangement of Orthogonal Components
The final dynamic operator of each particle can be represented as
\[\tag{21} J_k=\ep A_k+\em B_k, \qquad A_k=e^{i\pi b_k}, \qquad B_k=e^{i\pi a_k}. \] For two original objects, the direct connection of the components is
\[\tag{22} \left\{ \ep A_1+\em B_1, \quad \ep A_2+\em B_2 \right\} \] and the cross rearrangement is
\[\tag{23} \left\{ \ep A_1+\em B_2, \quad \ep A_2+\em B_1 \right\} . \] Orthogonality (\ep\em=0) allows the channels to be separated algebraically. However, not every permutation produces valid particles. After regrouping, it is necessary to check which parameters retained an internal or external physical label, which charge and closing factors were transferred to the new objects, and whether the corresponding metric regime exists.
Regrouping is a geometric possibility, not an automatic physical process. The admissible channel is determined simultaneously by the component algebra, deep indices, and physical projections.
9. A Complete Set of Invariants
Let's denoteres \(\mathcal I[\mathcal S]\) complete set of characteristics of the composite state:
\[\tag{24} \mathcal I[\mathcal S] =\left( \mathcal N, E, \mathbf P, Q, \mathbf M, N_\Gamma, \ldots \right). \] Here \(\mathcal N\) denotes the operator normalization, \(E\) is the total energy, \(\mathbf P\) is the momentum, \(Q\) is the charge, \(\mathbf M\) is the total angular momentum, and \(N_\Gamma\) is the internal closure index.
A physically admissible interaction must satisfy the condition
\[\tag{25} \boxed{ \mathcal I[\mathcal S_{\mathrm{in}}] =\mathcal I[\mathcal S_{\mathrm{out}}]. } \] This equality does not mean that energy, charge, and momentum are a single operator. It states that they all belong to a single complete state and are extracted by their own mappings:
\[\tag{26} \mathcal P_r[\mathcal B_{\mathrm{in}}] =\mathcal P_r[\mathcal B_{\mathrm{out}}], \qquad r=E,p,q,M,\Gamma,\ldots \] Here \(\mathcal B\) denotes the complete operator content of the system, and \(\mathcal P_r\) is the projector or physical mapping of a particular observable.
10. Energy and Momentum as Metric Projections
For a set of massive states, the energy and momentum are constructed from the internal scales \(E_{0k}\) and norms \(\boldsymbol\Gamma_{\beta_k}\):
\[\tag{27} E_{\mathrm{total}} =\sum_k \|\boldsymbol\Gamma_{\beta_k}\|E_{0k}, \qquad \mathbf P_{\mathrm{total}} =\sum_k \frac{\|\boldsymbol\Gamma_{\beta_k}\|E_{0k}}{c} \boldsymbol\beta_k. \] For a closed interacting system, the conservation of these projections is assumed:
\[\tag{28} \sum_{\mathrm{in}}E_k =\sum_{\mathrm{out}}E_l, \qquad \sum_{\mathrm{in}}\mathbf p_k =\sum_{\mathrm{out}}\mathbf p_l. \] These equalities are not derived from \(J\overline J=1\). The energy mapping requires the intrinsic scale \(E_0=\hbar\omega\), the deep-state metric, and the time homogeneity of the complete system. The operator norm preserves the phase structure, and the energy projector translates it into physical balance.
11. Charge and Independent Internal Properties
Electric charge is determined by its own operator and the projector of the observable:
\[\tag{29} \widehat QJ_{q,Q}=QJ_{q,Q}. \] For complete interaction, the charge balance is as follows
\[\tag{30} \sum_{\mathrm{in}}Q_k =\sum_{\mathrm{out}}Q_l. \] Independent projectors of aroma, color, and other internal properties can be introduced using the same principle. The numerical value of an observable quantity is an eigenvalue of the corresponding operator, not an amplitude factor that violates the normalization of the state.
Therefore, the opposite sign of an idempotent component cannot automatically be declared to be the sign of the electric charge. The particle and antiparticle are distinguished by the charge operator, and their spin orientations and directions of internal closure are specified separately.
12. Internal Closure Index
A closed particle and a free wave are distinguished by a topological index. For the relative deep phase \(q_{\mathrm{rel}}\), it is defined as
\[\tag{31} N_\Gamma[Q_\Gamma] =\frac1{2\pi i} \oint_\Gamma q_{\mathrm{rel}}^{-1} \,dq_{\mathrm{rel}}. \] For continuous normalized evolution of an isolated state
\[\tag{32} N_\Gamma=\operatorname{const}. \] In a complete interaction, the local indices of individual objects can be redistributed, but their total value must be preserved:
\[\tag{33} \boxed{ \sum_{\mathrm{in}}N_{\Gamma,k} =\sum_{\mathrm{out}}N_{\Gamma,l}. } \] This condition prohibits the spontaneous opening of a single electron wave and the spontaneous formation of a single closed particle from a free regime. Closures can appear or disappear only in a compensated complete process. A detailed derivation is given in the article "Multilevel Splitting of the Electron".
13. Preliminary Double Balance Rule
For the interaction of two input and two output operators, it is proposed to check two algebraic invariants:
\[\tag{34} \boxed{ J_1+J_2=J_3+J_4, \qquad J_1J_2=J_3J_4. } \] The first equality preserves the additive distribution of the components, the second, their joint multiplicative relationship. For commuting operators, the possible output states are the roots of a quadratic equation. \[\tag{35} X^2-(J_1+J_2)X+J_1J_2=0. \]
The double balance rule does not replace the tensor composite state and does not prove the admissibility of any reaction. It is an additional algebraicA filter for compatible operator channels. Its derivation and application are discussed in detail in part two.
14. Particle and Antiparticle
The states of a particle and antiparticle should be distinguished by several independent features:
\[\tag{36} \begin{aligned} J_P &=J_{\mathrm{dyn}}^{(\sigma)} Q_{\Gamma,P}J_{q,+}J_{r,1}\cdots,\\ J_{\bar P} &=J_{\mathrm{dyn}}^{(\bar\sigma)} Q_{\Gamma,\bar P}J_{q,-}J_{r,1}\cdots. \end{aligned} \] The operators \(J_{q,+}\) and \(J_{q,-}\) define opposite charge states. The parameters \(\sigma\) and \(\bar\sigma\) denote spin orientations, and \(Q_{\Gamma,P}\) and \(Q_{\Gamma,\bar P}\) denote the internal closure structures. These properties may be related in a particular particle, but they are not the same property.
Therefore, the previous forms
\[\tag{37} J_+=\ep A+\em B, \qquad J_-=\ep A-\em B \] can only be used as two algebraic orientations of the component. They are insufficient to fully define the particle and antiparticle.
15. Electron-Positron Annihilation
The initial state of the electron-positron pair is the tensor product:
\[\tag{38} \mathcal S_{\mathrm{in}} =J_{e^-}\otimes J_{e^+}. \] In the two-photon channel, the final state is
\[\tag{39} \mathcal S_{\mathrm{out}} =J_{\gamma_1}\otimes J_{\gamma_2}. \] The full transformation is written as
\[\tag{40} \boxed{ J_{e^-}\otimes J_{e^+} \overset{\mathcal U_{\mathrm{ann}}}{\longrightarrow} J_{\gamma_1}\otimes J_{\gamma_2}. } \] In a compensated channel, independent conditions are satisfied simultaneously.
\[\tag{41} Q_{e^-}+Q_{e^+}=0, \qquad N_{\Gamma,e^-}+N_{\Gamma,e^+}=0. \] The first equality applies to the electric charge, the second to the internal short circuit. They cannot be identified. During interaction, the opposite closures cancel out, and the coupled wave components transition to free propagating states.
In the center-of-mass frame of a pair at rest, the total external momentum is zero:
\[\tag{42} \mathbf p_{\gamma_1} +\mathbf p_{\gamma_2}=0, \qquad \mathbf p_{\gamma_1} =-\mathbf p_{\gamma_2}. \] A single free photon with non-zero energy cannot have zero momentum. Therefore, the simplest solution is formed by a matched pair of oppositely directed photons. This does not exclude other channels, such as the three-photon decay of orthopositronium, for which compensation is achieved by three external projections.
The previously used transformation \((iE+P,iE-P)\to(E+iP,E-iP)\) can be viewed as a visual analogy for component rotation, but not as a fundamental operator law. Energy and momentum must be extracted by the corresponding physical projections, rather than identified directly with the axes of the i-basis.
16. Inverse transformation and pair production
The inverse transformation of two free states into an electron-positron pair is written as
\[\tag{43} J_{\gamma_1}\otimes J_{\gamma_2} \overset{\mathcal U_{\mathrm{pair}}}{\longrightarrow} J_{e^-}\otimes J_{e^+}. \] Free states do not contain an internal electron closure. In the complete process, two opposite closures arise simultaneously:
\[\tag{44} 0\longrightarrow(+1)+(-1)=0. \] Opposite charge states arise simultaneously:
\[\tag{45} 0\longrightarrow(-e)+(+e)=0. \] Therefore, the internal closure and charge do not arise separately and do not disturb the overall balance. For the process to occur, suitable energy and momentum projections of the complete initial state must also exist.
17. Positronium as a Bound State
An electron and positron may not immediately transition to free emission, but form a bound state. This cannot be determined by a simple arithmetic sum of operators. Let's introduce the binding projector:
\[\tag{46} \boxed{ J_{\mathrm{Ps}} =\mathcal P_{\mathrm{bound}} \left( J_{e^-}\otimes J_{e^+} \right). } \] The projector \(\mathcal P_{\mathrm{bound}}\) identifies a consistent state of a pair with a common orbital structure and a specific relative phase. Parapositronium and orthopositronium correspond to different shared spin sectors, and not simply to a zero or nonzero arithmetic sum of internal operators.
The decay of a bound state must preserve the full set of its projectors. Therefore, the difference between two- and three-photon channels is associated with different joint orientations of the original pair and different distribution methods.of the total angular momentum between the output states.
18. Particle Scattering
In elastic or inelastic scattering, the particle types can be conserved, while the parameters of their dynamic states change:
\[\tag{47} J_1(a_1,b_1)\otimes J_2(a_2,b_2) \longrightarrow J_1(a_1',b_1')\otimes J_2(a_2',b_2'). \] A change in \(b_k\) corresponds to a change in the external velocity and momentum. A change in \(a_k\) relates to the internal phase, frequency, or excitation. This corrects the previous interpretation, in which the overall phase \(a\) was associated with external motion, and \(b\) with internal rearrangement.
If the particle types do not change, their classification operators are preserved:
\[\tag{48} J_{q,k}'=J_{q,k}, \qquad J_{f,k}'=J_{f,k}, \qquad J_{c,k}'=J_{c,k}, \quad\ldots \] At the same time, the total metric projections must be preserved:
\[\tag{49} E_1+E_2=E_1'+E_2', \qquad \mathbf p_1+\mathbf p_2 =\mathbf p_1'+\mathbf p_2'. \] 19. Absorption and Emission of a Free Wave
The absorption of a photon by a particle is a transformation of the composite input state:
\[\tag{50} J_P\otimes J_\gamma \overset{\mathcal U_{\mathrm{abs}}}{\longrightarrow} J_P'. \] Emission corresponds to the inverse type of rearrangement:
\[\tag{51} J_P \overset{\mathcal U_{\mathrm{em}}}{\longrightarrow} J_P'\otimes J_\gamma. \] Equality of the total photon operator of the difference of particle states
\[\tag{52} J_\gamma=J_P-J_P' \] cannot be used in general. It does not take into account the difference between tensor and additive levels, nor the possible rearrangement of deep and classification operators. The difference is permissible after applying a specific linear projection:
\[\tag{53} E_\gamma=E_P-E_P', \qquad \mathbf p_\gamma =\mathbf p_P-\mathbf p_P', \qquad Q_\gamma=Q_P-Q_P'=0. \] Thus, a photon is determined by its own free wave state, and not just by the numerical difference between the two states of the particle.
20. Free Photon and the Massive Chain Limit
For the Photon Regime
\[\tag{54} \beta_\gamma=1, \qquad b_\gamma=\frac12, \qquad E_\gamma=c|\mathbf p_\gamma|. \] But the photon cannot be obtained by substituting \(\beta=1\) into a massive deep state, since
\[\tag{55} \|\boldsymbol\Gamma_\beta\| =\frac1{\sqrt{1-\beta^2}} \longrightarrow\infty \qquad(\beta\to1). \] The divergence marks the boundary of the massive metric sector. The photon is a separate free regime with zero electron closure index:
\[\tag{56} N_\Gamma[J_\gamma]=0. \] 21. What is defined and what remains unknown
The following follow directly from finite algebra: expansion over two idempotent planes; orthogonality of components; the modern form \(J(a,b)\); additive law of exponents; unit operator norm; the possibility of direct and cross channel rearrangement.
The adopted architecture of the model is: composite particle operator; separation of dynamic, closure, charge, and other levels; tensor description of a multiparticle system; Metric state \(\boldsymbol\Gamma_\beta\); closure index; double balance rule.
A separate dynamic model requires: a specific form of \(\mathcal U_{\mathrm{int}}\); reaction time; channel amplitudes and probabilities; relationship to measured scattering cross sections and decay widths; The mechanism for selecting a single registered result.
The second part examines the double balance rule, the third part examines the geometry of branch reorganization, and the fourth part examines the probabilities and dynamics of possible channels. These papers should also use the modern meaning of the parameters \(a\) and \(b\).
Conclusions
A modern description of interactions begins with the operator \(J(a,b)=\j^a(-\j)^b\), in which \(a\) specifies the internal state, and \(b\) specifies the external motion. Other properties are not hidden in these two parameters, but are added by independent operator levels.
The state of a single particle is a product of compatible operators, while the state of several particles forms a tensor product:
\[\tag{57} \boxed{ J_P =J_{\mathrm{dyn}}^{(\sigma)}Q_\Gamma J_qJ_{\mathrm{orb}}\cdots, \qquad \mathcal S=\bigotimes_kJ_{P_k}. } \] The interaction isis a restructuring of the complete state by the operator \(\mathcal U_{\mathrm{int}}\). The admissible channel preserves the entire set of independent invariants:
\[\tag{58} \boxed{ \mathcal S_{\mathrm{out}} =\mathcal U_{\mathrm{int}} \mathcal S_{\mathrm{in}}, \qquad \mathcal I[\mathcal S_{\mathrm{out}}] =\mathcal I[\mathcal S_{\mathrm{in}}]. } \] Energy, momentum, charge, angular momentum, and closure index are not a single number. Each of them is obtained by its own projection of the complete operator content. This approach connects known physical balances with the internal geometry of the model and simultaneously reveals which conditions still require independent dynamic inference.
The sum-product conservation rule remains an important algebraic filter:
\[\tag{59} \boxed{ J_1+J_2=J_3+J_4, \qquad J_1J_2=J_3J_4. } \] However, it is now applied within a more complete architecture: with fixed meanings for the parameters \(a\) and \(b\), independent physical feature operators, deep metric states, and topological closure balance.