2026-07-22
Geometric representation of particle interactions
Part 2. Geometric rule of particle interaction
In the first part, interaction was represented as a transformation of a complete multioperator state. A single particle is described by the product of dynamical, closure, charge, and other operators, while the state of several particles forms a tensor product.
In this part, we investigate the double balance rule. Its mathematical basis is the possibility of connecting the same set of idempotent branches in two different ways. Such assemblies do indeed have the same sum and product. However, this does not necessarily mean that any physical reaction must be a rearrangement of the preserved branches.
It is necessary to distinguish between an exact algebraic result and a physical hypothesis. Conservation of the sum and product strictly follows from the rearrangement of a single set of commuting branches. The applicability of this rearrangement to particle interactions requires a separate physical justification.
\[\tag{1} \boxed{ \text{preservation of a set of branches} \quad\Longrightarrow\quad \begin{cases} J_1+J_2=J_3+J_4,\\ J_1J_2=J_3J_4, \end{cases} } \] 1. Two Levels of Interaction Description
The complete state of a particle has a multioperator structure:
\[\tag{2} J_P =J_{\mathrm{dyn}}^{(\sigma)}(a,b) Q_\Gamma J_qJ_{\mathrm{orb}}J_fJ_cJ_g\cdots. \] For two particles, the composite state is written as
\[\tag{3} \mathcal S_{12} =J_{P_1}\otimes J_{P_2}. \] The double balance rule should not be automatically applied to the entire tensor product. First, it is necessary to select an active commutative sector \(r\) within which the two operators have a split decomposition.
\[\tag{4} J_{r,k} =\ep A_{r,k}+\em B_{r,k}, \qquad k=1,2. \] The index \(r\) can denote a dynamical, charge, orbital, closure, or other independent level. Double balance is a property of the branches of the selected sector, and not an arithmetic operation on the entire many-particle system.
2. Componentwise action of functions
Let
\[\tag{5} J=\ep A+\em B, \] where \(A\) and \(B\) belong to two complex idempotent planes. We use the properties
\[\tag{6} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \qquad \ep+\em=1. \] For any integer \(n\geqslant1\), the mixed terms vanish, therefore
\[\tag{7} (\ep A+\em B)^n =\ep A^n+\em B^n. \] If the function is defined by a convergent power series
\[\tag{8} f(z)=\sum_{n=0}^{\infty}c_nz^n, \] then it acts separately in each idempotent plane:
\[\tag{9} \boxed{ f(\ep A+\em B) =\ep f(A)+\em f(B). } \] Formula (9) is a precise consequence of idempotent algebra. If a function is single-valued, it produces one result in each plane. If a function is multivalued, its branches arise independently in two channels.
3. The Square Root as an Example of Multivaluedness
Let values \(A_0\) and \(B_0\) be chosen that satisfy the conditions
\[\tag{10} A_0^2=A, \qquad B_0^2=B. \] Then
\[\tag{11} (\ep A_0+\em B_0)^2 =\ep A+\em B. \] In each complex plane, the square root has two branches:
\[\tag{12} \sqrt A\in\{A_0,-A_0\}, \qquad \sqrt B\in\{B_0,-B_0\}. \] Since the choices are independent, the complete set of roots contains four split values:
\[\tag{13} \sqrt{\ep A+\em B} \in \left\{ \begin{aligned} &\ep A_0+\em B_0,\quad \ep A_0-\em B_0,\\ &-\ep A_0+\em B_0,\quad -\ep A_0-\em B_0 \end{aligned} \right\}. \] The four-branch property arises as the Cartesian product of two sets:
\[\tag{14} 2\times2=4. \] These four values are branches of a single split expression. By themselves, they do not yet constitute four physical particles or four observable reaction channels.
4. General Four-Branch Construction
Let's denote the two branches of the first component by \(A_1,A_2\), and the second by \(B_1,B_2\). Then, four combinations arise:
\[\tag{15} \begin{aligned} J_{11}&=\ep A_1+\em B_1, &J_{12}&=\ep A_1+\em B_2,\\ J_{21}&=\ep A_2+\em B_1, &J_{22}&=\ep A_2+\em B_2. \end{aligned} \] The complete set of branches is
\[\tag{16} \mathcal R =\{A_1,A_2\}\times\{B_1,B_2\}. \] Each branch of the first plane can be connected to every branch of the second. This combinatorial property applies to a single selected split sector and does not depend on the physical meaning subsequently assigned to it.
5. Two Complementary Pairs
The four states can be assembled into two pairs such that each pair uses all the values \(A_1,A_2,B_1,B_2\) once.
Direct assembly:
\[\tag{17} \mathcal C_{\parallel} =\{J_{11},J_{22}\}. \] Cross assembly:
\[\tag{18} \mathcal C_{\times} =\{J_{12},J_{21}\}. \] In a direct pair, the original correspondence of \(A_1\leftrightarrow B_1\) and \(A_2\leftrightarrow B_2\) is preserved. In a crossed pair, the relationships change:
\[\tag{19} A_1\leftrightarrow B_2, \qquad A_2\leftrightarrow B_1. \] No branches disappear or duplicate. Only the way a single set of components is combined changes.
6. Sum Preservation Theorem
The sum of a direct pair is
\[\tag{20} J_{11}+J_{22} =\ep(A_1+A_2) +\em(B_1+B_2). \] For a crossed pair, we obtain
\[\tag{21} J_{12}+J_{21} =\ep(A_1+A_2) +\em(B_2+B_1). \] Since addition of components is commutative, the sums are the same:
\[\tag{22} \boxed{ J_{11}+J_{22} =J_{12}+J_{21}. } \] Thus, the preservation of the sum is a precise consequence of the fact that both pairs are assembled from the same set of branches.
7. The Product Preservation Theorem
Due to the orthogonality of idempotents, the product of a direct pair is
\[\tag{23} J_{11}J_{22} =\ep A_1A_2 +\em B_1B_2. \] The product of a crossed pair is
\[\tag{24} J_{12}J_{21} =\ep A_1A_2 +\em B_2B_1. \] If the components of the selected sector commute,
\[\tag{25} B_1B_2=B_2B_1, \] then the products of the two assemblies are the same:
\[\tag{26} \boxed{ J_{11}J_{22} =J_{12}J_{21}. } \] The commutativity condition is essential. If the branches are non-commuting matrix operators, the order of the factors must be preserved or a symmetrized product must be introduced. Formula (26) is then inapplicable without additional definition.
8. The Double Balance Theorem
The results of the two previous sections can be combined.
If two pairs of commuting split operators are constructed by different combinations of the same complete set of idempotent branches, then their sums and products are the same.
\[\tag{27} \boxed{ \begin{aligned} J_1+J_2&=J_3+J_4,\\ J_1J_2&=J_3J_4. \end{aligned} } \] In this formulation, double balance is not an arbitrarily chosen algebraic postulate. It is proven for two ways of assembling a single set of branches. But the physical assertion that real interactions preserve precisely this set of branches is an additional hypothesis.
\[\tag{28} \boxed{ \begin{aligned} &\text{rearrangement of preserved branches} \Longrightarrow\text{double balance},\\ &\text{physical interaction} \Longrightarrow\text{rearrangement of branches} \quad\text{— model hypothesis}. \end{aligned} } \] 9. A Quadratic Equation as a Corollary
For the selected sector, we introduce the total sum and product:
\[\tag{29} \Sigma=J_1+J_2, \qquad \Pi=J_1J_2. \] Each operator in a valid pair is a root of the equation
\[\tag{30} \boxed{ X^2-\Sigma X+\Pi=0. } \] If
\[\tag{31} X=\ep X_A+\em X_B, \qquad \Sigma=\ep\Sigma_A+\em\Sigma_B, \qquad \Pi=\ep\Pi_A+\em\Pi_B, \] then the equation splits into two independent complex equations:
\[\tag{32} X_A^2-\Sigma_AX_A+\Pi_A=0, \qquad X_B^2-\Sigma_BX_B+\Pi_B=0. \] The first gives the roots \(A_1,A_2\), the second — \(B_1,B_2\). Their independent combination creates four split roots \(J_{ij}\).
10. Why does a quadratic equation have four split roots?
In a normal number field, a quadratic equation has no more than two roots. Split algebra contains zero divisors:
\[\tag{33} \ep\ne0, \qquad \em\ne0, \qquad \ep\em=0. \] Therefore, the roots of the two idempotent components are chosen independently:
\[\tag{34} \{A_1,A_2\} \times \{B_1,B_2\} =\{J_{11},J_{12},J_{21},J_{22}\}. \] The quadratic equation here does not create a physical interaction. It compactly restores all algebraic assemblies that have the same sector invariants \(\Sigma\) and \(\Pi\).
11. Active sector of a multioperator state
Let the total operator of a particle be represented by the product of sector operators:
\[\tag{35} J_{P_k}=\prod_{r\in\mathcal R}J_{r,k}. \] If the interaction rearranges only the \(r\) sector, it is convenient to isolate it:
\[\tag{36} J_{P_k} =J_{r,k} \prod_{s\ne r}J_{s,k}. \] The remaining levels are specialctatory and are preserved:
\[\tag{37} J_{s,k}'=J_{s,k}, \qquad s\ne r. \] If several sectors are active, each one has its own invariants:
\[\tag{38} \Sigma_r =J_{r,1}+J_{r,2}, \qquad \Pi_r =J_{r,1}J_{r,2}. \] 12. Cross-swap operator
We denote the direct multiparticle state as
\[\tag{39} \mathcal S_{\parallel} =J_{P_1}\otimes J_{P_2}. \] Let \(\Pi_r^{\mathrm{swap}}\) swap branches between two particles only in the active sector \(r\), while \(I_{\ne r}\) preserves the remaining levels. Then
\[\tag{40} \mathcal U_{\times}^{(r)} =I_{\ne r}\otimes\Pi_r^{\mathrm{swap}}, \] \[\tag{41} \mathcal S_{\times} =\mathcal U_{\times}^{(r)} \mathcal S_{\parallel}. \] A double permutation returns the original assembly:
\[\tag{42} \left(\mathcal U_{\times}^{(r)}\right)^2=I. \] This equality expresses the algebraic invertibility of direct and cross joins. It does not imply equality of probabilities for the forward and reverse physical processes.
13. Sector Double Balance
For each active commutative sector, the rule is as follows
\[\tag{43} \boxed{ \Sigma_r^{\mathrm{in}} =\Sigma_r^{\mathrm{out}}, \qquad \Pi_r^{\mathrm{in}} =\Pi_r^{\mathrm{out}}. } \] The complete algebraic set of interaction invariants for two particles can be represented as
\[\tag{44} \mathcal I_{\mathrm{alg}} =\left\{ (\Sigma_r,\Pi_r) \right\}_{r\in\mathcal R_{\mathrm{act}}}. \] Preserving double balance in one sector does not guarantee equality of the sum of the complete particle operators. Products of different levels can be cross-correlated, so the transition from sector equalities to the complete operator requires an explicit assembly rule.
Double balance should be applied to explicitly specified compatible sectors. A notation without a sector index is admissible only when the entire operator under consideration belongs to a single commutative split algebra.
14. Algebraic, Physical, and Topological Admissibility
The algebraic rearrangement is only the first level of selection. We denote:
\[\tag{45} \begin{aligned} \mathcal C_{\mathrm{alg}} &=\text{sector double balance channels},\\ \mathcal C_{\mathrm{phys}} &=\text{physical projector channels},\\ \mathcal C_{\Gamma} &=\text{topological balance channels}. \end{aligned} \] The complete set of admissible transformations is their intersection:
\[\tag{46} \boxed{ \mathcal C_{\mathrm{allowed}} =\mathcal C_{\mathrm{alg}} \cap\mathcal C_{\mathrm{phys}} \cap\mathcal C_{\Gamma}. } \] Therefore, the presence of four split roots does not necessarily imply the existence of four observables. Some algebraic assemblies may not have the required energy scale, charge sector, stable closure, or correct metric mapping.
15. Physical Projectors
Energy, momentum, charge, and angular momentum must be obtained by separate projectors of the full state. For a massive particle, the energy and momentum projections are associated with the deep state \(\boldsymbol\Gamma_\beta\):
\[\tag{47} E_k =\|\boldsymbol\Gamma_{\beta_k}\|E_{0k}, \qquad \mathbf p_k =\frac{\|\boldsymbol\Gamma_{\beta_k}\|E_{0k}}{c} \boldsymbol\beta_k. \] For full interaction, balances are checked
\[\tag{48} \begin{aligned} \sum_{\mathrm{in}}E_k &=\sum_{\mathrm{out}}E_l,\\ \sum_{\mathrm{in}}\mathbf p_k &=\sum_{\mathrm{out}}\mathbf p_l,\\ \sum_{\mathrm{in}}Q_k &=\sum_{\mathrm{out}}Q_l,\\ \sum_{\mathrm{in}}\mathbf M_k &=\sum_{\mathrm{out}}\mathbf M_l. \end{aligned} \] These equalities are not the same condition and do not follow from the norm \(J\overline J=1\) alone. They represent different physical projections of the full operator content.
16. Topological Balance of Closures
For processes in which localized particles transform into free waves or vice versa, the internal closure index must be taken into account:
\[\tag{49} N_\Gamma[Q_\Gamma] =\frac1{2\pi i} \oint_\Gamma q_{\mathrm{rel}}^{-1} \,dq_{\mathrm{rel}}. \] In the complete interaction, the total index is preserved:
\[\tag{50} \boxed{ \sum_{\mathrm{in}}N_{\Gamma,k} =\sum_{\mathrm{out}}N_{\Gamma,l}. } \] This condition prevents a single closed state from spontaneously becoming a free wave. Closures can disappear or arise only in a compensated process. A detailed justification is given in the article "Multilevel Splitting of the Electron".
17. The Modern Meaning of Dynamic Branches
The dynamic sector uses the modern operator
\[\tag{51} J_k(a_k,b_k) =\ep e^{i\pi b_k} +\em e^{i\pi a_k}. \] Therefore, the components are
\[\tag{52} A_k=e^{i\pi b_k}, \qquad B_k=e^{i\pi a_k}. \] Direct and cross assemblies take the form
\[\tag{53} \begin{aligned} J_{11}&=\ep e^{i\pi b_1} +\em e^{i\pi a_1},\\ J_{22}&=\ep e^{i\pi b_2} +\em e^{i\pi a_2},\\ J_{12}&=\ep e^{i\pi b_1} +\em e^{i\pi a_2},\\ J_{21}&=\ep e^{i\pi b_2} +\em e^{i\pi a_1}. \end{aligned} \] During cross-assembly, the parameter \(b_1\) does not become internal, and \(a_2\) does not become external. Physical labels are preserved regardless of their new connection. However, the new operator must pass the external velocity projector, internal frequency, and stability conditions.
18. Electron-Positron Pair Annihilation
The two-photon annihilation channel is written as a transformation of tensor states:
\[\tag{54} J_{e^-}\otimes J_{e^+} \overset{\mathcal U_{\mathrm{ann}}}{\longrightarrow} J_{\gamma_1}\otimes J_{\gamma_2}. \] Sector rearrangement may be part of this transformation, but it does not automatically determine photon states. It is necessary to simultaneously check
\[\tag{55} \begin{aligned} Q_{e^-}+Q_{e^+}&=0,\\ N_{\Gamma,e^-}+N_{\Gamma,e^+}&=0,\\ E_{e^-}+E_{e^+}&=E_{\gamma_1}+E_{\gamma_2},\\ \mathbf p_{e^-}+\mathbf p_{e^+} &=\mathbf p_{\gamma_1}+\mathbf p_{\gamma_2}. \end{aligned} \] The first equality pertains to the charge sector, the second to the closure topology, and the last two to metric projections. Only the intersection of these conditions with the sector double balance creates an admissible channel.
19. Inverse Pair Production
The inverse process has the form
\[\tag{56} J_{\gamma_1}\otimes J_{\gamma_2} \overset{\mathcal U_{\mathrm{pair}}}{\longrightarrow} J_{e^-}\otimes J_{e^+}. \] The algebraic permutation of branches is reversible, but this is not sufficient for physical reversibility. The process must simultaneously generate compensated closures and charge states:
\[\tag{57} N_\Gamma: 0\longrightarrow(+1)+(-1)=0, \qquad Q: 0\longrightarrow(-e)+(+e)=0. \] Furthermore, the initial free states must have sufficient energy and momentum projections. Therefore, algebraic admissibility specifies a possible assembly but does not eliminate the physical reaction threshold.
20. Compton Scattering
Compton scattering is written as
\[\tag{58} J_\gamma\otimes J_e \longrightarrow J_\gamma'\otimes J_e'. \] The particle types before and after the reaction are conserved. The dynamic sector is predominantly active, while the charge and closing operators of the electron remain spectator:
\[\tag{59} J_{q,e}'=J_{q,e}, \qquad N_{\Gamma,e}'=N_{\Gamma,e}. \] The double balance here should be checked for the active dynamic components. The photon and electron should not be declared roots of a single universal quadratic equation along with all their classification operators.
To quantitatively derive the Compton dependence, it is necessary to relate the sector invariants to the frequencies and directions of the metric projections. Conservation of branches alone does not yield an angular dependence of scattering.
21. Transformation of an electron pair into a muon pair
Consider the reaction
\[\tag{60} e^-+e^+ \longrightarrow \mu^-+\mu^+. \] The initial and final pairs have the same total charge, but different internal energy scales:
\[\tag{61} E_{0e}\ne E_{0\mu}. \] Therefore, simply rearranging the dynamical branches is not enough. Additional internal and classification sectors must be reorganized. Equalities
\[\tag{62} J_{e^-}+J_{e^+} =J_{\mu^-}+J_{\mu^+}, \qquad J_{e^-}J_{e^+} =J_{\mu^-}J_{\mu^+} \] can only be considered as a testable hypothesis after explicitly defining all the factors of the complete operators. These equalities do not follow from the four-branch nature of the dynamic sector alone.
22. Bound State
The formation of positronium leads not to two free output objects, but to a single composite system. Therefore, it is described by the binding projector:
\[\tag{63} \boxed{ J_{\mathrm{Ps}} =\mathcal P_{\mathrm{bound}} \left( J_{e^-}\otimes J_{e^+} \right). } \] Double balance can be applied to the internal channels of the pair before binding, but the transition itself additionally requires a common orbital structure, phasematching, choosing a shared spin sector, and removing excess energy and momentum projections.
Parapositronium and orthopositronium cannot be identified only with the direct and crossed assemblies. For such a correspondence, it is necessary to show how both assemblies are projected onto the singlet and triplet states.
23. Generalization to \(n\) States
For two commuting operators, the sum and product are elementary symmetric invariants:
\[\tag{64} \sigma_1=J_1+J_2, \qquad \sigma_2=J_1J_2. \] For \(n\) operators, a complete set arises
\[\tag{65} \begin{aligned} \sigma_1&=\sum_iJ_i,\\ \sigma_2&=\sum_{i<j}J_iJ_j,\\ &\;\vdots\\ \sigma_n&=\prod_iJ_i. \end{aligned} \] The corresponding characteristic equation is of the form
\[\tag{66} X^n-\sigma_1X^{n-1} +\sigma_2X^{n-2} -\cdots+(-1)^n\sigma_n=0. \] Thus, double balance is a special case of the more general preservation of symmetric invariants in the commutative sector. For non-commuting operators, such a generalization requires ordered products and a separate study.
24. Admissibility Does Not Determine Probability
The intersection of algebraic, physical, and topological conditions forms a set of admissible channels, but does not determine the frequency of their realization. Even if direct and cross-assemblies exist, their probabilities need not be equal.
For channels \(r\), the amplitudes \(\mathcal A_r\) must be determined, after which the normalized probabilities can have the form
\[\tag{67} P_r =\frac{|\mathcal A_r|^2} {\displaystyle\sum_s|\mathcal A_s|^2}. \] The origin of the amplitudes, the branch restructuring operator, and the interference of alternatives are discussed in Part Four. In this article, formula (67) is used only to denote the boundary of the algebraic rule.
25. What has been proven and what is additionally accepted
The following immediately follow from idempotent algebra: componentwise action of analytic functions; independent branching of two planes; four split roots for two branches in each plane; equality of the sums of direct and cross pairs; equality of products under commutativity; a quadratic equation with sector operator coefficients.
The algebraic architecture of the model is: separation of active and spectator sectors; cross-transformation operator; application of double balance separately to compatible levels; a set of sector invariants \((\Sigma_r,\Pi_r)\); generalization through symmetric functions.
The physical hypotheses remain: preservation of the full set of branches in real interactions; the possibility of describing the reaction as their rearrangement; The relationship of a specific assembly to specific particles; the applicability of the double balance to all active physical sectors.
The following require separate dynamic derivation: the type of operator \(\mathcal U_{\mathrm{int}}\); the choice of active sectors; reaction thresholds; channel amplitudes and probabilities; scattering cross sections; decay times and the detection mechanism.
Conclusions
The double balance rule has a rigorous algebraic basis. If two output statements are obtained by cross-connecting the same set of commuting idempotent branches as two input statements, their sum and product are preserved:
\[\tag{68} \boxed{ \begin{aligned} J_{11}+J_{22}&=J_{12}+J_{21},\\ J_{11}J_{22}&=J_{12}J_{21}. \end{aligned} } \] In the modern multi-operator model, this result applies not to the unshared full state, but to each explicitly chosen active sector:
\[\tag{69} \boxed{ \mathcal I_{\mathrm{alg}} =\left\{ (\Sigma_r,\Pi_r) \right\}_{r\in\mathcal R_{\mathrm{act}}}. } \] A physically feasible channel must additionally preserve the metric, charge, momentum, and topological projections:
\[\tag{70} \boxed{ \mathcal I_2 =\mathcal I_{\mathrm{alg}} \cup \left\{ E,\mathbf P,Q,\mathbf M,N_\Gamma,\ldots \right\}, \qquad \mathcal I_2^{\mathrm{in}} =\mathcal I_2^{\mathrm{out}}. } \] Therefore, the double balance retains its central place, but its status is being clarified. This is a proven theorem about two ways to assemble a single set of branches and, simultaneously, a hypothesis that real interactions can be represented by such a sector rearrangement. The next task is to construct a specific interaction operator that determines the active sectors and the amplitudes of admissible transitions.