2026-07-22
Geometric representation of particle interactions
Part 3. Branches as the basis for particle interaction
In the previous part it was shown that a multivalued function of a split number creates independent branches in two idempotent planes. In the simplest case, two branches of the first plane and two branches of the second plane form four composite split states.
These four states allow two natural ways of pairing: direct and cross. Both pairs save the same amount and the same product, resulting in the double balance rule.
Now let's look at the physical meaning of this structure. The main hypothesis is that a particle can be understood not as an indivisible algebraic object, but as a certain way of connecting two internal branches.
1. Particle as a connection of two branches
Let the particle state be specified by a split number
\[ \tag{1} J = \ep A + \em B. \] The component \(A\) belongs to the first complex plane, and the component \(B\) belongs to the second. Together they form one whole state \(J\).
In this notation, the particle is determined not only by the values of \(A\) and \(B\), but also by the very fact of their connection into one construction:
\[ \tag{2} J \equiv (A,B). \] Consequently, the same set of components can form different composite states if the correspondence between the branches of two planes is changed.
This allows us to consider the particle as a stable assembly of two internal components:
\[ \tag{3} \text{particle} = \text{branch of the first plane} + \text{branch of the second plane}. \] This entry does not mean the usual addition of two physical objects. It expresses a structural principle: a complete state contains two independent but united idempotent parts.
2. Two particles and four internal branches
Consider two initial states:
\[ \tag{4} J_1 = \ep A_1 + \em B_1, \qquad J_2 = \ep A_2 + \em B_2. \] Together they contain four internal branches:
\[ \tag{5} A_1, \qquad A_2, \qquad B_1, \qquad B_2. \] Before interaction, these branches are connected in a direct way:
\[ \tag{6} (A_1,B_1), \qquad (A_2,B_2). \] The first state combines \(A_1\) with \(B_1\), and the second - \(A_2\) with \(B_2\).
However, the full set of branches allows another connection:
\[ \tag{7} (A_1,B_2), \qquad (A_2,B_1). \] With such an assembly, the first branch of one particle is connected to the second branch of another particle. Two new composite states arise.
3. Interaction as connection changes
Based on this structure, the interaction of two particles can be interpreted as a change in the way the internal branches are connected.
The original pair has the form
\[ \tag{8} \left( \ep A_1+\em B_1 \right) + \left( \ep A_2+\em B_2 \right). \] After cross-rearrangement, a new pair is formed:
\[ \tag{9} \left( \ep A_1+\em B_2 \right) + \left( \ep A_2+\em B_1 \right). \] Therefore, the general geometric transition can be written as
\[ \tag{10} (A_1,B_1) + (A_2,B_2) \longrightarrow (A_1,B_2) + (A_2,B_1). \] In this interpretation, the original particles disappear not because their constituent branches disappear, but because the original way of connecting these branches is destroyed.
New particles appear when the same set of internal components forms other stable assemblies.
4. What is saved during interaction
The main hypothesis of the proposed model is that geometric interaction preserves the full set of internal branches:
\[ \tag{11} \left\{ A_1,A_2,B_1,B_2 \right\}_{\mathrm{in}} = \left\{ A_1,A_2,B_1,B_2 \right\}_{\mathrm{out}}. \] In this case, the assignment of branches to individual particles can change.
Consequently, it is not necessarily the particles themselves that are preserved and not necessarily their original assemblies. The deeper structure is preserved—the complete set of components from which the initial and final states are formed.
This idea can be expressed by a diagram
\[ \tag{12} \text{saving branches} \quad + \quad \text{change connections} \quad = \quad \text{interaction}. \] This approach allows us to describe in a unified way processes in which the original particles retain their type, transform into other particles, or form a connected system.
5. Direct and cross channels
For four branches, there are two main ways to form two states.
Direct channel retains the original component mapping:
\[ \tag{13} \mathcal C_{\parallel} = \left\{ (A_1,B_1), (A_2,B_2) \right\}. \] Cross channel changes this mapping:
\[ \tag{14} \mathcal C_{\times} = \left\{ (A_1,B_2), (A_2,B_1) \right\}. \] A direct channel does not necessarily mean there is no physical interaction. Branch parameters may change in phase, energy or momentum, even if the general connection type is maintained.
Cross channel means a deeper restructuring, in which the belonging of internal components to constituent states changes.
Both channels are geometrically valid. However, the third part only considers the structure of these channels and does not determine which of them is implemented in a separate event.
6. Double balance as a consequence of maintaining branches
Consider the sum of the original pair:
\[ \tag{15} J_1+J_2 = \ep(A_1+A_2) + \em(B_1+B_2). \] For a cross pair we get
\[ \tag{16} J_3+J_4 = \ep(A_1+A_2) + \em(B_2+B_1). \] Since the addition of components is commutative, the sums are the same:
\[ \tag{17} J_1+J_2 = J_3+J_4. \] Similarly, for the product of initial states we have
\[ \tag{18} J_1J_2 = \ep A_1A_2 + \em B_1B_2. \] For cross states
\[ \tag{19} J_3J_4 = \ep A_1A_2 + \em B_2B_1. \] If the internal components are commutative, we get
\[ \tag{20} J_1J_2 = J_3J_4. \] Thus, the double balance rule
\[ \tag{21} J_1+J_2 = J_3+J_4, \qquad J_1J_2 = J_3J_4 \] is an algebraic consequence of preserving the full set of branches and changing only the way they are connected.
7. The meaning of the appearance of new particles
In the proposed model, a new particle does not arise from nothing. It appears as a new stable configuration of already existing internal branches.
If the original particles have a structure
\[ \tag{22} J_1\equiv(A_1,B_1), \qquad J_2\equiv(A_2,B_2), \] states may arise after interaction
\[ \tag{23} J_3\equiv(A_1,B_2), \qquad J_4\equiv(A_2,B_1). \] The physical type of each state must be determined by the properties of the resulting pair of components.
In other words, an electron, photon, muon or other particle must meet certain conditions on \(A\) and \(B\): their phases, frequencies, norms, orientations and other parameters.
Until such a correspondence is constructed, split geometry sets the general mechanism for the appearance of new assemblies, but does not allow one to unambiguously determine their physical type.
8. Annihilation of an electron and a positron
Consider the process
\[ \tag{24} e^-+e^+ \longrightarrow \gamma_1+\gamma_2. \] Let the initial states have the form
\[ \tag{25} J_{e^-} = \ep A_-+\em B_-, \qquad J_{e^+} = \ep A_++\em B_+. \] Cross assembly leads to states
\[ \tag{26} J_{\gamma_1} = \ep A_-+\em B_+, \qquad J_{\gamma_2} = \ep A_++\em B_-. \] In this scheme, the electron and positron disappear as the original means of connecting components. Instead, two new assemblies arise, which, if additional conditions are met, can be interpreted as photonic states.
For such an interpretation, it is necessary to separately show that the new states have zero electric charge, zero rest mass, corresponding spin, and propagate at the speed of light.
Therefore, formula (26) specifies the geometric structure of the transformation, but is not a complete physical conclusion of the annihilation process.
9. Reverse birth of a couple
The reverse process looks like
\[ \tag{27} \gamma_1+\gamma_2 \longrightarrow e^-+e^+. \] Within the split geometry, it corresponds to the reverse rearrangement of the same internal branches:
\[ \tag{28} (A_1,B_2) + (A_2,B_1) \longrightarrow (A_1,B_1) + (A_2,B_2). \] Geometric invertibility follows from the fact that repeated cross-permutation returns the original correspondence of the components.
However, geometric reversibility does not mean the same physical realizability of the direct and inverse processes. For the creation of an electron-positron pair, threshold and kinematic conditions must be met.
10. Scattering without changing particle type
Not every interaction results in the creation of particles of a different type. For example, with Compton scattering
\[ \tag{29} \gamma+e^- \longrightarrow \gamma'+e'^- \] before and after the interaction there is one electron and one photon.
In this case, the restructuring of branches may be accompanied by a change in their parameters, but the final assemblies continue to satisfyb conditions of electronic and photon states.
Therefore, the general transition should be written in a broader form:
\[ \tag{30} (A_1,B_1) + (A_2,B_2) \longrightarrow (A'_1,B'_2) + (A'_2,B'_1). \] The strokes indicate that phases, frequencies, directions of movement, energy and other parameters of the branches can change during interaction.
Consequently, saving branches does not necessarily mean that all their numerical characteristics are unchanged. More precisely, we are talking about preserving the structure and balance connections between components.
11. Converting one pair of particles to another
Consider the reaction
\[ \tag{31} e^-+e^+ \longrightarrow \mu^-+\mu^+. \] In the geometric interpretation, the initial electronic states form four branches, which, after interaction, are assembled into the muon and antimuon states.
For such a process must be executed
\[ \tag{32} J_{e^-}+J_{e^+} = J_{\mu^-}+J_{\mu^+}, \] \[ \tag{33} J_{e^-}J_{e^+} = J_{\mu^-}J_{\mu^+}. \] However, double balance in itself does not explain the difference in the masses of the electron and muon. To do this, it is necessary to relate the internal parameters of the branches with rest energy and other observable characteristics.
Different masses must correspond to different frequencies, phase relationships or modes of internal rotation of the resulting assemblies. The specific mechanism of such correspondence requires a separate study.
12. Linked state
A special case arises when, after interaction, the branches do not form two independent free particles, but remain in one common connected system.
Such a transition can be designated as
\[ \tag{34} J_1+J_2 \longrightarrow J_{\mathrm{bound}}+J_{\mathrm{out}}. \] Here \(J_{\mathrm{bound}}\) represents the bound configuration, and \(J_{\mathrm{out}}\) is the state that carries away excess energy and momentum.
For example, the radiative formation of positronium has the form
\[ \tag{35} e^-+e^+ \longrightarrow Ps+\gamma. \] In this case, the electron and positron components form a common stable configuration, and the photon ensures the fulfillment of the laws of conservation of energy and momentum.
Parapositronium and orthopositronium can be associated with various methods of phase or spin matching of internal branches. However, such a correspondence requires a separate conclusion.
13. Geometric and physical conditions
Regrouping of branches is only a geometric condition for the admissibility of the process. Real interaction must simultaneously satisfy the physical laws of conservation.
For a two-particle process, at least the following conditions are required
\[ \tag{36} E_1+E_2 = E_3+E_4, \] \[ \tag{37} \mathbf p_1+\mathbf p_2 = \mathbf p_3+\mathbf p_4. \] In addition, electric charge, angular momentum and other quantum numbers must be conserved:
\[ \tag{38} q_1+q_2 = q_3+q_4. \] The complete set of allowed channels is the intersection of geometric and physical conditions:
\[ \tag{39} \mathcal C_{\mathrm{allowed}} = \mathcal C_{\mathrm{split}} \cap \mathcal C_{\mathrm{phys}}. \] Split geometry defines possible ways to connect internal components. Physical laws select from them processes that can be realized in nature.
14. What should the further model determine
To apply the four-branch rule to specific particles quantitatively, it is necessary to construct a mapping of the split state to a set of observable physical quantities:
\[ \tag{40} J \longrightarrow \left( E, \mathbf p, m, q, s, \ldots \right). \] Such a mapping should answer the following questions.
What branch parameters determine the particle mass? How is electric charge related to the relative sign or phase of the components? How is spin represented in a split state? What conditions distinguish a photon assembly from an electron or muon assembly?
Only after answering these questions will it be possible to establish which final particles correspond to the direct and cross channels.
Until such a mapping is constructed, the proposed scheme remains a general geometric model of interaction, and not a complete theory of elementary particles.
15. Basic principle
The resulting interpretation can be formulated as a general principle.
In split geometry, a particle is considered as a way of connecting two internal branches belonging to two independent idempotent planes.
The interaction of two particles is destroyedformation of initial compounds and the formation of new compounds from a complete set of four branches.
In short form this principle is written as
\[ \tag{41} (A_1,B_1) + (A_2,B_2) \longrightarrow (A_1,B_2) + (A_2,B_1). \] At the same time double balance
\[ \tag{42} J_1+J_2 = J_3+J_4, \qquad J_1J_2 = J_3J_4 \] acts as an algebraic expression for preserving the complete set of internal branches.
16. Border of the third part
The third part answers the question of what constitutes interaction in split geometry. It specifies the set of geometrically permissible rearrangements of branches and shows what values are preserved during such a rearrangement.
However, the presence of several valid channels does not mean which channel is implemented in a separate event.
In other words, the interaction geometry determines the possible final assemblies, but does not yet determine the rule for choosing between them.
This task relates to the next level of the model - to constructing the probabilities and dynamics of transitions between admissible assemblies.
Conclusions
The four branches arising from the polysemy of the split function receive a natural physical interpretation when considering the interaction of two particles.
Each particle is represented by the connection of one branch of the first idempotent plane and one branch of the second plane.
Two particles contain four internal branches. These branches can form a direct or cross pair of compound states.
Interaction is interpreted as a change in the way branches are connected. The original particles disappear as initial assemblies, and new particles arise as new stable assemblies of the same internal components.
With such a restructuring, the full set of branches is preserved, and therefore the sum and product of states are automatically preserved.
Double balance is an algebraic reflection of this more general principle.
The proposed interpretation allows us to consider scattering, annihilation, pair creation, transformation of one pair of particles into another and the formation of a bound state in a single scheme.
However, geometric feasibility does not guarantee the physical implementation of the process. Additionally, the laws of conservation of energy, momentum, charge, angular momentum and other quantum numbers must be satisfied.
The main unsolved problem remains the construction of an explicit correspondence between the parameters of the split state and the observed characteristics of particles.
Thus, the third part formulates the geometry of interactions: particles are stable connections of internal branches, and interactions are rearrangements of these connections while maintaining a common set of components.
The question of which of the permissible channels is implemented and with what probability should be considered separately in the probabilistic model of interactions.