2026-07-22
Geometric representation of particle interactions
Part 3. Sector restructuring of operator states
In part one, the particle was represented as a multioperator state, and the interaction as a transformation of the complete composite system. In part two, it was shown that the direct and cross assemblies of a single set of commutating split branches preserve the sum and product within the selected operator sector.
Now it is necessary to determine the physical meaning of this transformation. The particle is no longer viewed as a connection between one universal branch of the first plane and one universal branch of the second. Dynamics, internal closure, charge, orbital state, and other features belong to different levels. Therefore, the interaction must change connections only in explicitly specified active sectors, preserving the other levels.
A particle is not a pair of universal branches, but a stable, coordinated assembly of operator sectors. The interaction changes the membership maps of branches \(\kappa_r\) in the active sectors, after which the new assembly must pass all algebraic, physical, topological, and geometric conditions.
\[\tag{1} \boxed{ \mathcal S_{\mathrm{out}} = \left[ \prod_{r\in\mathcal R_{\mathrm{act}}} \mathcal U_r(\kappa_r) \right] \mathcal S_{\mathrm{in}}. } \] 1. Modern Definition of a Particle
The complete operator state of a particle is written as
\[\tag{2} \boxed{ J_P =J_{\mathrm{dyn}}^{(\sigma)}(a,b) Q_\Gamma J_qJ_{\mathrm{orb}}J_fJ_cJ_g\cdots, \qquad \sigma=\pm. } \] The operator \(J_{\mathrm{dyn}}^{(\sigma)}\) contains the internal periodicity, external motion, and orientation of the state. The factor \(Q_\Gamma\) describes the internal closure; \(J_q\) is the charge structure; \(J_{\mathrm{orb}}\) is the orbital sector of the bound state. Other independent features can be added by their own normalized operators.
Therefore, the previous notation
\[\tag{3} J\equiv(A,B) \] can describe one selected split sector, but not the entire particle. A more complete definition is
\[\tag{4} \boxed{ \text{particle} =\text{stable consistent assembly of operator sectors}. } \] 2. Spin Orientation within a Dynamic Operator
In the current construction, a separate factor \(J_s\) for the two spin orientations is not required. They are contained in the choice of dynamic operator:
\[\tag{5} \begin{aligned} J_{\mathrm{dyn}}^{(+)}(a,b) &=\ep e^{i\pi b}+\em e^{i\pi a},\\ J_{\mathrm{dyn}}^{(-)}(a,b) &=\ep e^{i\pi a}+\em e^{i\pi b}. \end{aligned} \] The parameter \(a\) in both lines remains internal, while \(b\) remains external. The external motion projector must yield a single velocity regardless of orientation:
\[\tag{6} \mathcal P_{\mathrm{ext}} [J_{\mathrm{dyn}}^{(+)}] =\mathcal P_{\mathrm{ext}} [J_{\mathrm{dyn}}^{(-)}] =\Sin(\pi b)=\beta. \] A new independent spin operator can be introduced in the future only if additional structure is discovered that is not contained in \(J_{\mathrm{dyn}}^{(\sigma)}\) and the deep two-sheeted cycle.
3. Independent Sector Splits
Each independent operator level \(r\) can be assigned its own pair of complementary projectors:
\[\tag{7} (p_r^+)^2=p_r^+, \qquad (p_r^-)^2=p_r^-, \qquad p_r^+p_r^-=0, \qquad p_r^++p_r^-=1. \] The sector operator has the form
\[\tag{8} J_r =p_r^+A_r+p_r^-B_r. \] We denote the set of levels as
\[\tag{9} r\in\mathcal R =\{ \mathrm{dyn},\Gamma,q, \mathrm{orb},f,c,g,\ldots \}. \] The projectors \(p_r^+,p_r^-\) of different levels are independent. They cannot be automatically identified with the basic pair \(\ep,\em\) of the finite dynamic operator. The principle of splitting the whole into orthogonal components remains common.
4. Structural Signature of a Particle
The physical type of a particle is determined by a complete set of operator and metric features. Let us denote this collection by signature
\[\tag{10} \Lambda(J_P) =\left( \sigma,E_0,Q,N_\Gamma, \mathrm{orb},f,c,g,\ldots \right). \] For the electronic class schematically:
\[\tag{11} \Lambda(J_e) =\left( \sigma,m_ec^2,-e, N_\Gamma=\pm1,\ldots \right). \] For the free photon regime:
\[\tag{12} \Lambda(J_\gamma) =\left( \lambda,E_\gamma,0, N_\Gamma=0,\beta=1,\ldots \right). \] Formulas (11) and (12) are not an exhaustive table of quantum numbers. They show the main thing: one crossed pair of comThe component \((A_1,B_2)\) does not yet define an electron, photon, or other particle.
5. The State of Two Particles
The joint state of two objects is written as a tensor product:
\[\tag{13} \boxed{ \mathcal S_{12} =J_{P_1}\otimes J_{P_2}. } \] The sign \(\otimes\) preserves the belonging of the operator factors to each particle and allows for correlations between them. It differs from the usual product of sector operators within a single state \(J_P\).
Arithmetic sum
\[\tag{14} J_{P_1}+J_{P_2} \] can be used as an additive invariant of a compatible sector, but does not replace the full state \(\mathcal S_{12}\).
6. Branches of a single selected sector
Let two particles in sector \(r\) have operators
\[\tag{15} \begin{aligned} J_{r,1}&=p_r^+A_{r,1}+p_r^-B_{r,1},\\ J_{r,2}&=p_r^+A_{r,2}+p_r^-B_{r,2}. \end{aligned} \] Direct assembly connects components with the same indices:
\[\tag{16} (A_{r,1},B_{r,1}), \qquad (A_{r,2},B_{r,2}). \] The cross-assembly has the form
\[\tag{17} (A_{r,1},B_{r,2}), \qquad (A_{r,2},B_{r,1}). \] These four branches belong only to the \(r\) sector. The charge, closing, and other operators of the two particles do not have to rearrange themselves along with them.
7. Branch Membership Map
The way the components are connected is conveniently specified by the permutation \(\kappa_r\):
\[\tag{18} \kappa_r: \{1,2\}\longrightarrow\{1,2\}. \] We write the general output operator of the selected sector as
\[\tag{19} \boxed{ J_{r,k}^{\mathrm{out}} =p_r^+A_{r,k} +p_r^-B_{r,\kappa_r(k)}. } \] For direct assembly
\[\tag{20} \kappa_r^{\parallel} =\operatorname{id}, \qquad \kappa_r^{\parallel}(1)=1, \qquad \kappa_r^{\parallel}(2)=2. \] For cross-linking
\[\tag{21} \kappa_r^{\times}=(12), \qquad \kappa_r^{\times}(1)=2, \qquad \kappa_r^{\times}(2)=1. \] Thus, the geometry of the connections is described not only by the branch values, but also by a separate map of their membership in whole states.
8. Active and spectator sectors
During interaction, not all levels of the particle must change. Let's divide many sectors:
\[\tag{22} \mathcal R =\mathcal R_{\mathrm{act}} \cup\mathcal R_{\mathrm{sp}}, \qquad \mathcal R_{\mathrm{act}} \cap\mathcal R_{\mathrm{sp}} =\varnothing. \] For the active sector, the connection map changes:
\[\tag{23} \kappa_r^{\mathrm{out}} \ne\kappa_r^{\mathrm{in}}, \qquad r\in\mathcal R_{\mathrm{act}}. \] The spectator sector retains its structure:
\[\tag{24} J_{s,k}^{\mathrm{out}} =J_{s,k}^{\mathrm{in}}, \qquad \kappa_s^{\mathrm{out}} =\kappa_s^{\mathrm{in}}, \qquad s\in\mathcal R_{\mathrm{sp}}. \] For example, when an electron is scattered, the dynamic parameters \(a,b\) may change, while its charge operator and internal closure index are preserved.
9. Sector Swap Operator
Let \(\Pi_r^{\mathrm{swap}}\) swap branches \(B_{r,1}\) and \(B_{r,2}\), and \(I_{\ne r}\) act identically on the remaining levels. Then the cross-swap operator has the form
\[\tag{25} \mathcal U_{\times}^{(r)} =I_{\ne r}\otimes \Pi_r^{\mathrm{swap}}. \] Its action transforms the direct assembly into a cross assembly:
\[\tag{26} \mathcal S_{\times}^{(r)} =\mathcal U_{\times}^{(r)} \mathcal S_{\parallel}^{(r)}. \] A pure permutation is an involution:
\[\tag{27} \left( \mathcal U_{\times}^{(r)} \right)^2=I. \] Therefore, a second permutation restores the original connection. This is algebraic reversibility, not a statement about the equality of probabilities of the forward and reverse physical processes.
10. The Complete Interaction Operator
If the interaction affects several independent sectors, the complete restructuring operator can be represented as a composition:
\[\tag{28} \boxed{ \mathcal U_{\mathrm{int}} =\prod_{r\in\mathcal R_{\mathrm{act}}} \mathcal U_r(\kappa_r). } \] Then
\[\tag{29} \mathcal S_{\mathrm{out}} =\mathcal U_{\mathrm{int}} \mathcal S_{\mathrm{in}}. \] If the sector operators do not commute, the order of the factors in formula (28) becomes part of the definition of the interaction. For commuting independent levels, the order does not affect the result.
11. When a permutation is only an exchange of particles
If the same permutation simultaneously transfers the full set of sector factors, then the twoThe particles only swap places:
\[\tag{30} \kappa_r=\kappa \quad\text{for all }r \quad\Longrightarrow\quad J_{P_1}\otimes J_{P_2} \longrightarrow J_{P_2}\otimes J_{P_1}. \] Such an exchange does not create a new type of particle. A true internal rearrangement occurs if different sectors change their membership differently:
\[\tag{31} \kappa_r\ne\kappa_s \qquad \text{for some }r\ne s. \] Therefore, it is the relative permutation of operator levels that is physically meaningful, not a simple exchange of labels of two whole objects.
12. Sector Double Balance
For each active commutative sector, invariants are defined
\[\tag{32} \Sigma_r =J_{r,1}+J_{r,2}, \qquad \Pi_r =J_{r,1}J_{r,2}. \] For a pure rearrangement of one set of branches
\[\tag{33} \boxed{ \Sigma_r^{\mathrm{in}} =\Sigma_r^{\mathrm{out}}, \qquad \Pi_r^{\mathrm{in}} =\Pi_r^{\mathrm{out}}. } \] A detailed proof of the equality of the sum and product is given in Part Two. Here, double balance is used as a check for sector rearrangement.
13. Pure Permutation and Dynamic Permutation
Two regimes must be distinguished. In a pure permutation, the branch values do not change:
\[\tag{34} A_{r,k}'=A_{r,k}, \qquad B_{r,k}'=B_{r,k}. \] Only the membership map \(\kappa_r\) changes. In this case, the double balance follows directly from preserving the set of branches.
In a dynamic permutation, the parameters themselves also change:
\[\tag{35} A_{r,k}\longrightarrow A_{r,k}', \qquad B_{r,k}\longrightarrow B_{r,k}'. \] Then we cannot claim that the original set of branches is literally preserved. Certain sector invariants must be preserved:
\[\tag{36} \Sigma_r(A,B) =\Sigma_r(A',B'), \qquad \Pi_r(A,B) =\Pi_r(A',B'). \] Formula (36) is a more general interaction hypothesis. Unlike pure permutation, it does not follow solely from the presence of four invariant components.
14. Stability of the Output Assembly
An algebraically admissible compound does not necessarily exist as a stable particle. Let's introduce the stable class projector:
\[\tag{37} \mathcal P_{\mathrm{stable}}[J_P'] =J_P'. \] For a localized wave, geometric closure must also hold. For
\[\tag{38} V(t)=cJ(t) \] the path functional is
\[\tag{39} \mathcal C(T) =\int_0^T V(t)\,dt. \] For a stable particle
\[\tag{40} \boxed{ \mathcal C(T_P)=0. } \] A freely propagating state does not return the path to the starting point:
\[\tag{41} \mathcal C(T)\ne0. \] Therefore, the output assembly must either satisfy the stable closure condition or belong to a certain free wave class.
15. Topological closure sector
The closure factor \(Q_\Gamma\) is characterized by the relative deep phase index:
\[\tag{42} N_\Gamma[Q_\Gamma] =\frac1{2\pi i} \oint_\Gamma q_{\mathrm{rel}}^{-1} \,dq_{\mathrm{rel}}. \] For the continuous normalized evolution of an isolated state, the index is preserved. In a complete interaction, local closures can be redistributed, but
\[\tag{43} \boxed{ \sum_{\mathrm{in}}N_{\Gamma,k} =\sum_{\mathrm{out}}N_{\Gamma,l}. } \] If \(Q_\Gamma\) is a spectator sector, the index of each particle is stored separately. If this sector is active, closures can appear or disappear only in a compensated manner. This principle is discussed in more detail in the article "Multilevel Splitting of an Electron".
16. Full admissibility condition
For the output assembly, we define four sequential filters:
\[\tag{44} \begin{aligned} \mathcal C_{\mathrm{alg}} &=\text{sector double balance},\\ \mathcal C_{\mathrm{phys}} &=\text{physical projection balance},\\ \mathcal C_\Gamma &=\text{topological balance},\\ \mathcal C_{\mathrm{stable}} &=\text{output class stability}. \end{aligned} \] Physically allowed states form an intersection
\[\tag{45} \boxed{ \mathcal C_{\mathrm{allowed}} =\mathcal C_{\mathrm{alg}} \cap\mathcal C_{\mathrm{phys}} \cap\mathcal C_\Gamma \cap\mathcal C_{\mathrm{stable}}. } \] Thus, the crossing permutation is only a candidate. The new operator must preserve the necessary invariants and correspond to a stableparticle, a bound state, or a free wave.
17. When a new type of particle appears
After the restructuring, the signature of the output state is calculated:
\[\tag{46} \Lambda(J_P') =\left( \sigma',E_0',Q',N_\Gamma', \mathrm{orb}',f',c',g',\ldots \right). \] A new physical type arises only if this signature corresponds to a stable class:
\[\tag{47} \boxed{ \Lambda(J_P') =\Lambda_{\mathrm{known}}, \qquad \mathcal P_{\mathrm{stable}}[J_P']=J_P'. } \] If the required eigenvalues, closure, or metric map are missing, the algebraic assembly remains a formal state and should not be called a new particle.
18. Electron-positron annihilation
The two-photon channel is written as a transformation of the full tensor state:
\[\tag{48} \boxed{ J_{e^-}\otimes J_{e^+} \overset{\mathcal U_{\mathrm{ann}}}{\longrightarrow} J_{\gamma_1}\otimes J_{\gamma_2}. } \] Defining photons only by crossed dynamical branches
\[\tag{49} \ep A_-+\em B_+, \qquad \ep A_++\em B_- \] is not enough. In annihilation, at least the dynamic, closing, and charge levels are active, and the spin information must be coordinatedly transferred to the radiation polarization state.
The complete transformation is verified by the conditions
\[\tag{50} \begin{aligned} Q_{e^-}+Q_{e^+}&=0,\\ N_{\Gamma,e^-}+N_{\Gamma,e^+}&=0,\\ E_{e^-}+E_{e^+}&=E_{\gamma_1}+E_{\gamma_2},\\ \mathbf p_{e^-}+\mathbf p_{e^+} &=\mathbf p_{\gamma_1}+\mathbf p_{\gamma_2}. \end{aligned} \] The photon class is determined by the complete output signature: zero charge, no electron closure, velocity \(c\), zero rest mass, and allowed polarization.
19. Inverse Pair Production
The inverse process is as follows
\[\tag{51} J_{\gamma_1}\otimes J_{\gamma_2} \overset{\mathcal U_{\mathrm{pair}}}{\longrightarrow} J_{e^-}\otimes J_{e^+}. \] Here, several sectors are simultaneously reorganized. Two free states form two compensated closures:
\[\tag{52} N_\Gamma: 0\longrightarrow(+1)+(-1)=0, \] and the charge sector creates opposite eigenvalues:
\[\tag{53} Q: 0\longrightarrow(-e)+(+e)=0. \] Therefore, pair production is not a repetition of a single permutation ((12)). It is a coordinated rearrangement of the dynamic, closure, charge, and other active levels while maintaining a full set of invariants.
20. Scattering without changing the particle type
For Compton scattering
\[\tag{54} J_\gamma\otimes J_e \longrightarrow J_\gamma'\otimes J_e' \] The output signatures remain photon and electron. For the electron, the charge and trailing sectors are spectator:
\[\tag{55} J_{q,e}'=J_{q,e}, \qquad N_{\Gamma,e}'=N_{\Gamma,e}. \] The active dynamic sector modifies the internal and external parameters:
\[\tag{56} (a_\gamma,b_\gamma; a_e,b_e) \longrightarrow (a_\gamma',b_\gamma'; a_e',b_e'). \] Therefore, scattering is a dynamic rearrangement that preserves particle classes. It does not require a transfer of the charge or topological sector from the electron to the photon.
21. Transformation of an electron pair into a muon pair
For the reaction
\[\tag{57} e^-+e^+ \longrightarrow \mu^-+\mu^+ \] it is necessary to change the internal energy and classification levels, since
\[\tag{58} E_{0e}=m_ec^2, \qquad E_{0\mu}=m_\mu c^2, \qquad E_{0e}\ne E_{0\mu}. \] A simple cross-permutation of the branches of the fundamental dynamical operator cannot explain the change in mass. The active set must include frequency and family sectors, and the output operators must have a muon signature:
\[\tag{59} \Lambda(J_{\mu^-}), \quad \Lambda(J_{\mu^+}). \] The specific mechanism for such a transition remains a matter for further determination of the family operators and the internal energy scale.
22. Bound State
Positronium is not a new pair of free branches, but a common bound state of an electron and a positron:
\[\tag{60} \boxed{ J_{\mathrm{Ps}} =\mathcal P_{\mathrm{bound}} \left( J_{e^-}\otimes J_{e^+} \right). } \] The \(\mathcal P_{\mathrm{bound}}\) projector identifies the consistent orbital structure, relative phase, and shared spin configuration. During binding, the orbital and dynamical sectors are active, while the charges of the electron and positron are conserved within the composite state.
РThe additive formation of positronium has the form
\[\tag{61} J_{e^-}\otimes J_{e^+} \longrightarrow J_{\mathrm{Ps}}\otimes J_\gamma. \] A free wave or surrounding system carries away the difference between the energy and momentum projections. Parapositronium and orthopositronium differ in their shared spin sector, not just in the direct or cross map of a single set of branches.
23. Permutations for \(n\) Particles
For \(n\) objects, the membership map belongs to the permutation group:
\[\tag{62} \kappa_r\in S_n. \] The output sector operators can be written as
\[\tag{63} J_{r,k}^{\mathrm{out}} =p_r^+A_{r,k} +p_r^-B_{r,\kappa_r(k)}, \qquad k=1,2,\ldots,n. \] For two particles, the group \(S_2\) contains the identity and cross permutations. For \(n>2\), the number of algebraic connections increases, but only assemblies that pass all sector and metric checks remain physically valid.
If the branches also change dynamically, the permutation must be supplemented with a sector transformation:
\[\tag{64} (A_{r,k},B_{r,k}) \overset{\mathcal T_r}{\longrightarrow} (A_{r,k}',B_{r,k}') \overset{\kappa_r}{\longrightarrow} J_{r,k}^{\mathrm{out}}. \] 24. Connection to the Probabilistic Model
The third part defines geometrically and physically permissible permutations:
\[\tag{65} \{\mathcal U_r(\kappa_r)\} \longrightarrow \mathcal C_{\mathrm{allowed}}. \] But the number of possible permutations does not determine the probability of their occurrence. The channel \(c\) requires an amplitude
\[\tag{66} \mathcal A_c =\left\langle \mathcal S_c^{\mathrm{out}} \middle| \mathcal U_{\mathrm{int}} \middle| \mathcal S^{\mathrm{in}} \right\rangle, \] and the probability after normalization has the form
\[\tag{67} P_c =\frac{|\mathcal A_c|^2} {\displaystyle\sum_d|\mathcal A_d|^2}. \] The origin of amplitudes, phase matching, and interference of alternatives are discussed in Part Four. Here, formulas (66) and (67) only mark the boundary of the geometric description.
25. What follows from the construction, and what remains a hypothesis?
The following follow from the idempotent algebra: sector decomposition into orthogonal branches; the possibility of direct and cross connection; reversibility of pure permutation; double balance of one set of commutating branches.
The following follow from the adopted multi-operator architecture: separation of the dynamic, closure, charge, and other levels; allocation of active and spectator sectors; independent \(\kappa_r\) maps; the need to verify the full signature of the output state.
The physical hypotheses are: description of the interaction as a sector rearrangement; a specific set of active levels for each reaction; correspondence of the output assembly to a specific particle; the existence of a stable state projector \(\mathcal P_{\mathrm{stable}}\).
A separate dynamic inference is required: the specific form of \(\mathcal U_{\mathrm{int}}\); the change in branches at \(\mathcal T_r\); energy thresholds; Amplitudes, probabilities, scattering cross sections, and decay times.
Conclusions
A particle in the modern model is determined not by a pair of universal branches of the first two idempotent planes, but by a stable, coordinated assembly of independent operator sectors:
\[\tag{68} \boxed{ J_P =J_{\mathrm{dyn}}^{(\sigma)}(a,b) Q_\Gamma J_qJ_{\mathrm{orb}}J_fJ_cJ_g\cdots. } \] Interaction changes membership maps only in active sectors, preserving spectator levels:
\[\tag{69} \boxed{ \{\kappa_r^{\mathrm{in}}\} \overset{\mathcal U_{\mathrm{int}}}{\longrightarrow} \{\kappa_r^{\mathrm{out}}\}, \qquad r\in\mathcal R_{\mathrm{act}}. } \] An identical permutation of all sectors means only an exchange of whole particles. A new operator configuration arises from the relative rearrangement of different levels. But it becomes a physical state only after satisfying algebraic, physical, topological, and geometric conditions:
\[\tag{70} \boxed{ \mathcal I_{\mathrm{in}} =\mathcal I_{\mathrm{out}}, \qquad \mathcal P_{\mathrm{stable}} [\mathcal S_{\mathrm{out}}] =\mathcal S_{\mathrm{out}}. } \] Thus, the third part answers not the question of why the sum and product are conserved this result was already obtained in the second part but the question of how the active levels of the complete state are reorganized and under what conditions the new assembly becomes a stable particle, a bound state, or a free wave.