2026-07-22
Geometric representation of particle interactions
Part 2. Geometric rule of particle interaction
In the first part, the idea of double balance was proposed for the interaction of two states. However, the balance rule itself becomes much clearer if you start not with the quadratic equation, but with the more general property of split numbers.
The main feature of split analysis is that the function of the split number is split into two independent parts. Therefore, the multivalued function arises separately in each idempotent plane. If a function has two branches in each plane, then the complete split expression already has four branches.
It is this four-branch structure that leads to four possible states. These four states are then naturally combined into two pairs. It turns out that both pairs preserve the same sum and the same product. This is where the double balance rule comes from.
1. Split number and component-by-component action of functions
Any state in the basis under consideration can be written as
\[ \tag{1} J=\ep A+\em B. \] Here \(A\) refers to the first complex plane, and \(B\) to the second. Idempotents satisfy the conditions
\[ \tag{2} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \qquad \ep+\em=1. \] The condition \(\ep\em=0\) means that the two parts of the split number do not mix when multiplied. Therefore, any degrees of a state are calculated independently in two idempotent planes.
For any integer \(n\geqslant1\) is satisfied
\[ \tag{3} (\ep A+\em B)^n = \ep A^n+\em B^n. \] When the degree is expanded, all mixed terms contain the product \(\ep\em\) and become zero. In this case, the proper degrees of idempotents do not change: \(\ep^n=\ep\) and \(\em^n=\em\).
Now consider a function that is given by a convergent power series
\[ \tag{4} f(x) = \sum_{n=0}^{\infty} c_nx^n. \] Substitute the split number \(J=\ep A+\em B\) into this series. Thanks to property (3), each member of the series splits into two independent components. As a result we get
\[ \tag{5} f(\ep A+\em B) = \ep f(A)+\em f(B). \] Thus, the same function acts separately in the first and second complex planes. This is not an additional assumption, but a direct consequence of the idempotent relations.
It is property (5) that is the starting point for further reasoning. If the function is unambiguous, the result simply breaks down into two components. If the function is multivalued, then the branches arise separately in each plane and then are connected in all allowable combinations.
2. Square root as the first multivalued example
A particularly important example is the square root. Unlike the usual analytical function, the square root is multi-valued, therefore formula (5) must be understood not as a choice of one main branch, but as a rule for constructing a complete set of roots.
Choose one root value in each complex plane and denote them by \(A_0\) and \(B_0\):
\[ \tag{6} A_0^2=A, \qquad B_0^2=B. \] Then the number \(\ep A_0+\em B_0\) is indeed the square root of the split number. This can be verified by direct squaring:
\[ \tag{7} (\ep A_0+\em B_0)^2 = \ep A_0^2+\em B_0^2 = \ep A+\em B. \] Consequently, the square root is calculated independently in two idempotent planes:
\[ \tag{8} \sqrt{\ep A+\em B} = \ep\sqrt{A} + \em\sqrt{B}. \] Formula (8) is a short representation of the set of roots. The symbols \(\sqrt{A}\) and \(\sqrt{B}\) here denote all valid complex values, not just pre-selected main branches.
There are two values for the first component:
\[ \tag{9} \sqrt{A}=\pm A_0, \qquad A_0^2=A. \] For the second component there are also two values:
\[ \tag{10} \sqrt{B}=\pm B_0, \qquad B_0^2=B. \] Since the two idempotent planes are independent, the sign in the first component does not have to coincide with the sign in the second. Therefore, two choices in the first plane and two choices in the second form four combinations.
3. Four branches as a property of split analysis
The complete set of square roots of a split number is
\[ \tag{11} \sqrt{\ep A+\em B} \in \left\{ \ep A_0+\em B_0, \; \ep A_0-\em B_0, \; -\ep A_0+\em B_0, \; -\ep A_0-\em B_0 \right\}. \] Thus, the ordinary complex square root has two branches, and the square root of a split number has four.
The reason is simple: a split number contains two independent complex parts. Each part gives its own choice of sign. Therefore, the total number of branches is equal to the product of the number of branches in each plane.
In this case you getm:
\[ \tag{12} 2\times2=4. \] Consequently, four-branchness is not a feature of one quadratic equation. This is a more general property of split analysis. The quadratic equation only manifests it in a specific form.
4. Transition from four branches to four states
To describe the interaction, it is more convenient to designate two possible values of the first component as \(A_1\) and \(A_2\), and two possible values of the second component as \(B_1\) and \(B_2\).
Then the four branches take the form
\[ \tag{13} J_{11}=\ep A_1+\em B_1, \qquad J_{12}=\ep A_1+\em B_2, \] \[ \tag{14} J_{21}=\ep A_2+\em B_1, \qquad J_{22}=\ep A_2+\em B_2. \] These states are not introduced artificially. They appear automatically because each branch of the first plane can be combined with each branch of the second.
Here the fundamental feature of split geometry manifests itself: one polysemantic expression generates not two, but four compound branches.
5. Why four states form two pairs
The interaction involves two objects, so the final result must consist of two states. The four branches must be divided into pairs so that each pair uses both branches of the first plane and both branches of the second plane.
This can be done in two natural ways.
The first method preserves the original correspondence of the components:
\[ \tag{15} \mathcal C_{\parallel} = \left\{ J_{11}, J_{22} \right\}. \] The second method changes the correspondence between two planes:
\[ \tag{16} \mathcal C_{\times} = \left\{ J_{12}, J_{21} \right\}. \] In the first case, the component \(A_1\) corresponds to \(B_1\), and the component \(A_2\) corresponds to \(B_2\). In the second case, the connections intersect: \(A_1\) connects to \(B_2\), and \(A_2\) connects to \(B_1\).
At the same time, the four components themselves do not change. Only the way they are combined into two whole states changes.
6. Saving amount
Consider the sum of states in the first pair:
\[ \tag{17} J_{11}+J_{22} = \ep(A_1+A_2) + \em(B_1+B_2). \] For the cross pair we get exactly the same amount:
\[ \tag{18} J_{12}+J_{21} = \ep(A_1+A_2) + \em(B_1+B_2). \] Hence, both pairs save the same total split vector.
This is the first balance condition:
\[ \tag{19} J_1+J_2=J_3+J_4. \] But the amount alone is not enough. Different pairs can have the same amount, so it does not completely determine the internal structure of the pair.
7. Saving a work
Now consider the product of states. Thanks to the condition \(\ep\em=0\) it is calculated componentwise.
For the first pair:
\[ \tag{20} J_{11}J_{22} = \ep A_1A_2 + \em B_1B_2. \] For cross pair:
\[ \tag{21} J_{12}J_{21} = \ep A_1A_2 + \em B_2B_1. \] Since ordinary complex components commute, \(B_2B_1=B_1B_2\). Therefore, the products of both pairs are the same.
We get the second balance condition:
\[ \tag{22} J_1J_2=J_3J_4. \] Thus, saving the work is not introduced arbitrarily. It follows directly from the fact that the two pairs are constructed from the same branches of the split function.
8. Double balance rule
Both valid pairs preserve both the sum and the product. Therefore, the natural rule for transforming two states is
\[ \tag{23} J_1+J_2=J_3+J_4, \] \[ \tag{24} J_1J_2=J_3J_4. \] We will call this system the double balance rule.
The first equality preserves the common split vector of the pair. The second preserves its internal algebraic structure. Together they make it possible to distinguish a valid rearrangement of components from an arbitrary pair with the same sum.
The main thing here is the order of reasoning. Double balance is not invented in advance. First, four branches emerge from the polysemy of the split function. These branches then form two pairs. And after this it is discovered that both pairs retain the same sum and the same product.
9. Quadratic equation as a consequence
If we denote the total sum of a pair by \(S\), and the total product by \(P\), then any state of the pair satisfies the equation
\[ \tag{25} J^2-SJ+P=0. \] This equation is convenient for calculations, but it is not the primary source of the four states.
After expansion into idempotents, it splits into two ordinary complex quadratic equations. Each of them gives two branches. Independent connectionThe connection of these branches again creates four states.
Therefore, the quadratic equation only repeats the already found structure:
\[ \tag{26} 2\, \text{branches of the first plane} \times 2\, \text{branches of the second plane} = 4\, \text{split states}. \] It is in this sense that four-branchness is a fundamental property of split analysis, and the quadratic equation is one of its manifestations.
10. General meaning of the result
In ordinary complex analysis, the square root creates two branches. In split analysis, each of the two independent complex planes creates its own branches. Therefore, the number of general solutions is multiplied.
This property is not limited to the square root. If a function has several branches in the first plane and several branches in the second, then the complete split expression contains all possible combinations of these branches.
For the interaction of two objects, the simplest case is especially important: two branches in each plane. It creates four states and two natural ways to assemble them into a final pair.
The first pair preserves the original component connection. The second corresponds to their cross rearrangement. Both pairs have the same sum and product, which results in a double balance.
11. Physical interpretation
At the geometric level, interaction can be understood as a change in the way the components of two independent planes are connected.
The original components do not disappear and do not appear again. They just move from one assembly to another. Therefore, the first pair and the cross pair describe two different ways of forming two entire states from the same set of internal components.
In this case, double balance is only a geometric condition of admissibility. For a real physical process, the laws of conservation of energy, momentum, charge, angular momentum and other quantities must additionally be satisfied.
The general condition can be written as the intersection of geometric and physical constraints:
\[ \tag{27} \mathcal C_{\mathrm{total}} = \mathcal B_2 \cap \mathcal C_{\mathrm{phys}}. \] Here \(\mathcal B_2\) denotes the set of states that satisfy the double balance, and \(\mathcal C_{\mathrm{phys}}\) denotes the set of physically allowed states.
12. Main conclusion
The main result can be formulated as follows.
The split-number function acts independently in two idempotent planes. Therefore, the polysemy of the function also arises separately in each plane.
For a square root, each plane has two branches. Their independent connection creates four split states.
These four states form two natural pairs: direct and cross. Both pairs save the same sum and the same product.
This directly follows the double balance rule:
\[ \tag{28} J_1+J_2=J_3+J_4, \qquad J_1J_2=J_3J_4. \] Thus, double balance is not an external postulate, but a consequence of the four-branch structure of split functions.
The quadratic equation \(J^2-SJ+P=0\) gives the same four solutions, but acts as an algebraic form and a test of a more general property of split analysis.
13. Particle interaction rule
The result obtained allows us to formulate a general rule for the interaction of two particles within the framework of split geometry.
Each particle is represented by a split state
\[ \tag{29} J_k=e^{ia_k}\j^{b_k} = \ep A_k+\em B_k. \] When two particles interact, two branches arise in the first idempotent plane and two branches in the second. These branches are not required to maintain the original connection. They can be reassembled into two final states.
Therefore, we will understand the interaction of two particles as an admissible rearrangement of the four branches of the two initial split states.
The general diagram looks like this
\[ \tag{30} J_1+J_2 \longrightarrow J_3+J_4. \] Here, the final states must be assembled from the same complete set of idempotent branches that were present before the interaction.
If the initial states are written as
\[ \tag{31} J_1=\ep A_1+\em B_1, \qquad J_2=\ep A_2+\em B_2, \] then geometry allows two ways to assemble a finite pair.
The first method preserves the original component connection:
\[ \tag{32} \left\{ J_3,J_4 \right\} = \left\{ \ep A_1+\em B_1, \; \ep A_2+\em B_2 \right\}. \] This is a direct channel. At the geometric level, it means that the belonging of components to individual states has not changeds.
The second method changes the component connection:
\[ \tag{33} \left\{ J_3,J_4 \right\} = \left\{ \ep A_1+\em B_2, \; \ep A_2+\em B_1 \right\}. \] This is a cross channel. In it, each final particle receives one component from one initial state and a second component from another.
Thus, the rule of interaction can be formulated in the words:
the interaction of two particles is a rearrangement of the four branches of the split function with the formation of two new composite states.
Such a rearrangement automatically preserves the sum and product of states. Therefore, for any admissible two-particle channel, a double balance is performed:
\[ \tag{34} J_1+J_2=J_3+J_4, \qquad J_1J_2=J_3J_4. \] In this formulation, double balance is no longer the original postulate. It acts as an algebraic consequence of preserving the complete set of branches and their admissible rearrangement.
The result obtained allows us to interpret the interaction of particles in a new way. In the proposed model, it is not the particles themselves that are preserved, but the complete set of their internal branches. Particles represent different ways of combining these branches into entire states. Therefore, the interaction of two particles can be considered as a restructuring of the structure of compounds with a constant set of internal components. It is this restructuring that leads to the emergence of new final states.
14. What geometry determines and what physics determines
Split geometry determines only the possible structure of the final pair. It shows which ways of connecting branches are algebraically permissible.
However, from geometry alone it is impossible to determine whether a specific reaction will occur, at what energy it will begin, and with what probability a particular channel will be realized.
For a real process, the usual physical conservation laws must simultaneously be satisfied:
\[ \tag{35} E_1+E_2=E_3+E_4, \qquad \mathbf p_1+\mathbf p_2 = \mathbf p_3+\mathbf p_4. \] In addition to energy and momentum, electric charge, angular momentum and other quantum numbers related to the reaction in question must be conserved.
Therefore, the complete rule can be represented as a sequence of two checks.
The geometry first defines the valid branch assemblies. Then the physical conditions select from them the actually possible processes.
\[ \tag{36} \text{four branches} \longrightarrow \text{two geometric pairs} \longrightarrow \text{physically allowed channel}. \] Consequently, the direct and cross channels should not be identified in advance with specific particles. Such a correspondence must be obtained from an additional physical model relating the parameters \(A,B\) or \(a,b\) with mass, charge, momentum, spin and other characteristics of particles.
15. Electron-positron annihilation
Consider the well-known two-particle process
\[ \tag{37} e^-+e^+ \longrightarrow \gamma_1+\gamma_2. \] In the split representation, states are associated with the original particles
\[ \tag{38} J_{e^-}=\ep A_-+\em B_-, \qquad J_{e^+}=\ep A_++\em B_+. \] Cross rearrangement creates two new states:
\[ \tag{39} J_{\gamma_1} = \ep A_-+\em B_+, \qquad J_{\gamma_2} = \ep A_++\em B_-. \] Within the framework of the proposed interpretation, photon states are considered as a possible new assembly of electron and positron components.
This entry is not yet a derivation of photon properties from split geometry. For such a conclusion it is necessary to additionally show that the new states have zero mass, zero charge, corresponding spin and propagate at the speed of light.
Nevertheless, the geometric diagram correctly conveys the general nature of the process: the two initial particles disappear as separate assemblies, and the same complete set of components forms two states of a different type.
16. Birth of an electron-positron pair
The reverse process can be written as
\[ \tag{40} \gamma_1+\gamma_2 \longrightarrow e^-+e^+. \] In this case, two photonic assemblies are rearranged into electronic and positron states.
Geometric invertibility follows from the fact that a cross permutation performed twice returns the original connection of the components.
\[ \tag{41} (A_1,B_1)+(A_2,B_2) \longleftrightarrow (A_1,B_2)+(A_2,B_1). \] Physical reversibility does not mean the same probability of processes. For the birth of a pair, it is necessary that the collision energy be sufficient to form the masses of the electron and positron, and all kinematic conditions must be met.
It is important to distinguish between two-photon pair production and the transformation of one photon into a pair. One free photon in empty space cannot simultaneously conserve energy and momentum when producing \(e^-e^+\). A single-photon process requires an additional participant, such as an atomic nucleus.
17. Compton scattering
Compton scattering has the form
\[ \tag{42} \gamma+e^- \longrightarrow \gamma'+e'^-. \] Before and after interaction, the types of particles are preserved: in the initial and final states there is one photon and one electron. However, their energy, momentum and phase change.
In split geometry, such a process can be interpreted as a rearrangement of components, after which new assemblies still belong to the electronic and photonic classes.
This is an important example of the fact that a cross channel does not necessarily mean the transformation of particles into particles of a different kind. It can accommodate the exchange of internal components while preserving the end object classes.
For a quantitative description of the Compton effect, it is necessary to relate the parameters of the split state to energy and momentum and obtain from the double balance the known kinematic relationship between frequency and scattering angle. For now, such communication remains a separate task.
18. Transformation of an electron pair into a muon pair
A particularly clear example of the transformation of one pair into another is the reaction
\[ \tag{43} e^-+e^+ \longrightarrow \mu^-+\mu^+. \] Here the initial and final pairs have the same total electric charge, but consist of particles of different types and masses.
In the proposed geometric picture, the electron and positron define four initial branches. After interaction, these branches form two new assemblies, which must satisfy the conditions of a muon and an antimuon.
Double balance specifies the algebraic continuity of the transformation:
\[ \tag{44} J_{e^-}+J_{e^+} = J_{\mu^-}+J_{\mu^+}, \] \[ \tag{45} J_{e^-}J_{e^+} = J_{\mu^-}J_{\mu^+}. \] However, the reaction becomes physically possible only when the collision energy exceeds the threshold for the formation of a muon pair.
This example shows the potentially wide application of the rule: it can describe not only the permutation of states of the same type, but also the transition from one pair of particles to another pair while maintaining the overall split structure.
19. Positronium formation
An electron and a positron can form a bound state - positronium. However, this process requires separate consideration, since the final result contains one bound system, and not two free particles.
Radiative formation of positronium can be symbolically written as
\[ \tag{46} e^-+e^+ \longrightarrow Ps+\gamma. \] An additional photon carries away excess energy and momentum. In a substance, this role can also be played by the environment.
Positronium exists in two main spin states: singlet parapositronium and triplet orthopositronium. These states differ in the mutual orientation of the electron and positron spins and have different main annihilation channels.
Within split geometry, it is natural to investigate whether two geometric assemblies can be associated with two ways of matching internal phases or spins. But such a correspondence cannot be considered already proven.
To substantiate it, it is necessary to include the spin structure in the \(J\) state and show that one assembly leads to a singlet state, and the other to a triplet state.
Therefore, positronium is not a ready-made confirmation of the model, but an important example for its further verification.
20. Classes of two-particle interactions
The considered examples allow us to identify several general types of processes.
The first type is scattering. Particles retain their type, but change energy, momentum or phase. An example is Compton scattering.
The second type is annihilation and reverse birth of a pair. Two initial assemblies are replaced by two assemblies of a different class. An example is the processes \(e^-+e^+\leftrightarrow\gamma+\gamma\).
The third type is the transformation of one material pair into another. An example is the reaction \(e^-+e^+\to\mu^-+\mu^+\).
The fourth type is the formation of a bound state. Here part of the energy must be removed by an additional object or environment. An example is the formation of positronium.
In all cases, split geometry offers a single first step: decompose the initial states into independent branches and test possible ways to reassemble them.
21. General rule
Based on the structure considered, we can propose the following general scheme of two-particle interaction:
\[ \tag{47} \left( \ep A_1+\em B_1 \right) + \left( \ep A_2+\em B_2 \right) \longrightarrow \left( \ep A_1+\em B_2 \right) + \left( \ep A_2+\em B_1 \right). \] This expression represents a pure geometric rearrangement. It shows the transition from direct to cross-assembly of components.
The corresponding double balance rule is
\[ \tag{48} J_1+J_2=J_3+J_4, \qquad J_1J_2=J_3J_4. \] To apply to specific particles, you need to add a display to it
\[ \tag{49} J_k \longrightarrow \left( E_k, \mathbf p_k, q_k, s_k, m_k, \ldots \right), \] which connects the parameters of the split state with the physical characteristics of the particle.
Only after constructing such a mapping can one strictly determine which of the four branches corresponds to an electron, photon, muon or other object.
Thus, the regrouping rule specifies the general geometric framework of the interaction, and the physical model must fill this framework with measurable quantities.
22. Probability of channel implementation
The presence of two geometrically feasible pairs does not mean that both of them are realized with the same probability.
Double balance defines a set of possible assemblies, but does not specify the amplitude of the transition between them.
For a probabilistic description, it is necessary to introduce an additional quantity depending on phases, energy, spin and other interaction parameters.
The general structure can be represented as
\[ \tag{50} P_{\parallel}+P_{\times}=1, \] where \(P_{\parallel}\) is the probability of a direct channel, and \(P_{\times}\) is the probability of a cross channel.
The task of the next stage is to obtain these probabilities not by external assignment, but from the geometry of states, for example, through phase matching or the scalar product of the initial and final branches.
Conclusions
The split-number function acts independently in two idempotent planes. Therefore, its polysemy also arises separately in each plane.
The square root has two branches in the first plane and two branches in the second. Their independent connection creates four split states.
The four states form two complementary pairs: direct and cross. Both pairs are built from one complete set of branches.
Therefore, both pairs have the same sum and the same product. This implies the double balance rule
\[ \tag{51} J_1+J_2=J_3+J_4, \qquad J_1J_2=J_3J_4. \] Double balance is not an external postulate, but an algebraic consequence of the four-branch nature of split functions and the permissible rearrangement of their branches.
On this basis, the interaction of two particles is proposed to be understood as the reassembly of four branches of two initial split states into two final states.
This approach allows us to consider scattering, annihilation, pair creation, and the transformation of one pair of particles into another in a unified way.
Examples of electron-positron annihilation, Compton scattering and the transformation of an electron pair into a muon pair show possible areas of application of the general rule:
\[ \tag{52} e^-+e^+ \leftrightarrow \gamma+\gamma, \qquad \gamma+e^- \to \gamma'+e'^-, \qquad e^-+e^+ \to \mu^-+\mu^+. \] The formation of positronium demonstrates a more complex case, in which it is necessary to take into account the bound state, spin, and the removal of energy and momentum.
At the same time, geometric feasibility is not equal to physical feasibility. A specific reaction must additionally satisfy the laws of conservation of energy, momentum, charge, angular momentum and other quantum numbers.
The next task is to construct an explicit correspondence between the parameters of the split state and the observed characteristics of the particles, as well as to derive the probabilities of the direct and cross channels.