2026-07-22
Geometric representation of particle interactions
Part 4. Probabilities and dynamics of branch restructuring
In the third part, the interaction of particles was interpreted as a rearrangement of internal branches. The two original particles contain four branches, which can form a direct or crossed pair of compound states.
Such geometry defines the set of valid finite assemblies, but does not answer the question of which of them is implemented in a separate event. To move from the geometry of possible interactions to probabilistic dynamics, it is necessary to represent each assembly as a state and determine the amplitudes of transitions between such states.
The main hypothesis of this part is that the probability of interaction is associated with the transition between different ways of connecting internal branches. To describe such transitions, it is convenient to use the braquet formalism of quantum mechanics, filling it with specific geometric content.
1. State as a connection of branches
Let an individual particle be defined by a split state
\[ \tag{1} J = \ep A + \em B. \] In the third part, this state was presented as an ordered connection of two internal branches:
\[ \tag{2} J \equiv (A,B). \] Now let’s compare the ket state to this assembly
\[ \tag{3} |J\rangle = |A:B\rangle. \] The colon emphasizes that we are talking not just about a pair of numbers, but about the connection between the branch \(A\) of the first idempotent plane and the branch \(B\) of the second plane.
We denote the dual state as
\[ \tag{4} \langle J| = \langle A:B|. \] The normalized state must satisfy the condition
\[ \tag{5} \langle J|J\rangle = 1. \] In this interpretation, the bra-quet symbolism does not describe the abstract vector itself, but the geometric assembly of internal branches.
2. State of two particles
Let the two initial particles have the form
\[ \tag{6} J_1 = \ep A_1+\em B_1, \qquad J_2 = \ep A_2+\em B_2. \] Write the joint initial state as
\[ \tag{7} |J_{\mathrm{in}}\rangle = |A_1:B_1,\;A_2:B_2\rangle. \] This state contains the full set of four branches
\[ \tag{8} A_1, \qquad A_2, \qquad B_1, \qquad B_2, \] but at the same time fixes the way they are combined into two initial particles.
The direct output state has the form
\[ \tag{9} |J_{\parallel}\rangle = |A_1:B_1,\;A_2:B_2\rangle, \] a cross state
\[ \tag{10} |J_{\times}\rangle = |A_1:B_2,\;A_2:B_1\rangle. \] Thus, geometrically feasible channels become possible final ket states.
3. Branch Rearrangement Operator
We introduce the interaction operator \(\hat V\), which changes the way the branches are connected.
For direct channel
\[ \tag{11} \hat V_{\parallel} |A_1:B_1,\;A_2:B_2\rangle = |A_1:B_1,\;A_2:B_2\rangle. \] For cross channel
\[ \tag{12} \hat V_{\times} |A_1:B_1,\;A_2:B_2\rangle = |A_1:B_2,\;A_2:B_1\rangle. \] The \(\hat V_{\times}\) operator does not create or destroy branches. It only changes the correspondence between the components of two idempotent planes.
Therefore, the interaction can be represented as the action of the rebuild operator on the source assembly:
\[ \tag{13} |J_k\rangle = \hat V_k|J_{\mathrm{in}}\rangle. \] 4. Transition amplitude
The amplitude of the transition from the initial state \(|J_{\mathrm{in}}\rangle\) to the output state \(|J_k\rangle\) is determined by the matrix element of the interaction operator:
\[ \tag{14} \mathcal A_k = \langle J_k| \hat V |J_{\mathrm{in}}\rangle. \] For direct and cross channels
\[ \tag{15} \mathcal A_{\parallel} = \langle J_{\parallel}| \hat V |J_{\mathrm{in}}\rangle, \qquad \mathcal A_{\times} = \langle J_{\times}| \hat V |J_{\mathrm{in}}\rangle. \] If the interaction operator is not explicitly highlighted, the amplitude can be written as the overlap of the source and destination assemblies:
\[ \tag{16} \mathcal A_{J_{\mathrm{in}}\to J_{\mathrm{out}}} = \langle J_{\mathrm{out}}|J_{\mathrm{in}}\rangle. \] However, formula (16) is an abbreviated form. In general, the physical dynamics must be contained in the operator \(\hat V\).
5. Geometric meaning of the scalar product
In ordinary quantum mechanics, the quantity \(\langle J_{\mathrm{out}}|J_{\mathrm{in}}\rangle\) is the inner product of states. In split geometry it can be given an additional meaning: it measures the degree of compatibility of two ways of connecting branches.
If the final assembly is completely identical to the original one, theno
\[ \tag{17} \langle J_{\mathrm{in}}|J_{\mathrm{in}}\rangle = 1. \] If two assemblies are incompatible in the selected state space, then
\[ \tag{18} \langle J_m|J_n\rangle = 0, \qquad m\ne n. \] In general
\[ \tag{19} 0 \leq \left| \langle J_{\mathrm{out}}|J_{\mathrm{in}}\rangle \right| \leq 1. \] The inner product thereby becomes a measure of the geometric coincidence or permissibility of a transition between two assemblies.
6. Channel probability
The probability of the final state being realized is determined by the square of the amplitude modulus:
\[ \tag{20} P_{i\to f_k} = \left| \langle J_k| \hat V |J_{\mathrm{in}}\rangle \right|^2. \] For two main channels
\[ \tag{21} P_{\parallel} = \left| \langle J_{\parallel}| \hat V |J_{\mathrm{in}}\rangle \right|^2, \] \[ \tag{22} P_{\times} = \left| \langle J_{\times}| \hat V |J_{\mathrm{in}}\rangle \right|^2. \] If the direct and cross channels form a complete set of possible outcomes, then
\[ \tag{23} P_{\parallel} + P_{\times} = 1. \] Thus, probability does not refer to the particle as an indivisible object, but to the transition between two ways of connecting the same set of internal branches.
7. Superposition of possible assemblies
Before registering the final result, the system can be represented as a superposition of geometrically feasible assemblies:
\[ \tag{24} |\Psi_{\mathrm{out}}\rangle = \mathcal A_{\parallel} |J_{\parallel}\rangle + \mathcal A_{\times} |J_{\times}\rangle. \] State normalization requires
\[ \tag{25} \langle \Psi_{\mathrm{out}} | \Psi_{\mathrm{out}} \rangle = 1. \] If the direct and cross assemblies are orthogonal, it follows from formula (25)
\[ \tag{26} \left| \mathcal A_{\parallel} \right|^2 + \left| \mathcal A_{\times} \right|^2 = 1. \] Superposition in this model means the presence of several possible geometries for connecting branches before fixing a specific final state.
8. Full set of channels
If there are more than two valid leaf assemblies, the full output state is
\[ \tag{27} |\Psi_{\mathrm{out}}\rangle = \sum_k \mathcal A_k |J_k\rangle. \] For an orthonormal set of final states
\[ \tag{28} \langle J_m|J_n\rangle = \delta_{mn}. \] Then the normalization condition takes the form
\[ \tag{29} \sum_k \left| \mathcal A_k \right|^2 = 1. \] The probability of an individual channel is
\[ \tag{30} P_k = \left| \mathcal A_k \right|^2. \] 9. Geometric feasibility
Not every formally written assembly must belong to the space of admissible states. Let us introduce the projector \(\hat P_{\mathrm{split}}\), which identifies geometrically permissible rearrangements.
\[ \tag{31} \hat P_{\mathrm{split}} |J_k\rangle = \begin{cases} |J_k\rangle, & |J_k\rangle\in\mathcal H_{\mathrm{split}}, \\ 0, & |J_k\rangle\notin\mathcal H_{\mathrm{split}}. \end{cases} \] Here \(\mathcal H_{\mathrm{split}}\) denotes the space of admissible split assemblies.
The geometrically forbidden channel has zero amplitude:
\[ \tag{32} \hat P_{\mathrm{split}}|J_k\rangle = 0 \quad \Longrightarrow \quad \mathcal A_k = 0. \] 10. Physical admissibility
Even a geometrically permissible state may not satisfy the physical laws of conservation. Therefore, we introduce the projector of physical constraints \(\hat P_{\mathrm{phys}}\).
\[ \tag{33} \hat P_{\mathrm{phys}} |J_k\rangle = \begin{cases} |J_k\rangle, & \text{if the physical conservation laws are satisfied}, \\ 0, & \text{if at least one mandatory condition is violated}. \end{cases} \] For non-zero amplitude, at least the following must be fulfilled
\[ \tag{34} E_{\mathrm{in}} = E_{\mathrm{out}}, \qquad \mathbf p_{\mathrm{in}} = \mathbf p_{\mathrm{out}}, \] \[ \tag{35} q_{\mathrm{in}} = q_{\mathrm{out}}, \qquad \mathbf J_{\mathrm{in}} = \mathbf J_{\mathrm{out}}. \] The complete selection operator can be represented as
\[ \tag{36} \hat P_{\mathrm{allowed}} = \hat P_{\mathrm{phys}} \hat P_{\mathrm{split}}. \] Then the amplitude of the allowed transition takes the form
\[ \tag{37} \mathcal A_k = \langle J_k| \hat P_{\mathrm{allowed}} \hat V |J_{\mathrm{in}}\rangle. \] 11. What does the interaction operator depend on
The operator \(\hat V\) cannot be arbitrary. It should depend on the internal parameters of the branches and the physical conditions of the process.
In generalThis form can be written
\[ \tag{38} \hat V = \hat V \left( A_1, A_2, B_1, B_2 \right). \] If branches are characterized by phases, frequencies, energies and spins, then
\[ \tag{39} \hat V = \hat V \left( \phi_1, \phi_2, \omega_1, \omega_2, E, \mathbf p, s, \ldots \right). \] The most natural factors are phase matching, frequency resonance, relative orientation, spin compatibility and conservation laws.
So far, an explicit form of the operator \(\hat V\) has not been obtained from split geometry. Therefore, braquet formalism specifies the structure of the model, but does not yet complete it quantitatively.
12. Phase matching of branches
Let the branches have complex phases:
\[ \tag{40} A_k = |A_k|e^{i\alpha_k}, \qquad B_k = |B_k|e^{i\beta_k}. \] For direct assembly, the phase differences are equal
\[ \tag{41} \Delta_{\parallel,1} = \alpha_1-\beta_1, \qquad \Delta_{\parallel,2} = \alpha_2-\beta_2. \] For cross assembly
\[ \tag{42} \Delta_{\times,1} = \alpha_1-\beta_2, \qquad \Delta_{\times,2} = \alpha_2-\beta_1. \] It can be assumed that the matrix elements of the interaction operator depend on the degree of phase matching:
\[ \tag{43} \langle J_{\parallel}| \hat V |J_{\mathrm{in}}\rangle = F_{\parallel} \left( \Delta_{\parallel,1}, \Delta_{\parallel,2} \right), \] \[ \tag{44} \langle J_{\times}| \hat V |J_{\mathrm{in}}\rangle = F_{\times} \left( \Delta_{\times,1}, \Delta_{\times,2} \right). \] The specific form of the functions \(F_{\parallel}\) and \(F_{\times}\) must be derived from the dynamics of the branches.
13. Possible role of split sine and split cosine
Since the internal states of the branches are formed in split geometry, it is natural to look for matrix elements of the interaction operator through split sine and split cosine:
\[ \tag{45} cs(a,b) = \ep\cos a + \em\cos(a+\pi b), \] \[ \tag{46} sn(a,b) = \ep\sin a + \em\sin(a+\pi b). \] The main formula looks like
\[ \tag{47} e^{ia}\j^b = cs(a,b) + i sn(a,b). \] If the parameters \(a\) and \(b\) characterize the relative phase and type of connection of the branches, then the amplitude can be constructed as a matrix element of some split operator:
\[ \tag{48} \mathcal A_k = \langle J_k| \hat V \left( cs, sn \right) |J_{\mathrm{in}}\rangle. \] At the level of a working hypothesis, the operator can be represented as a combination of direct and cross reconstructions:
\[ \tag{49} \hat V = g_{\parallel} \hat V_{\parallel} + g_{\times} \hat V_{\times}, \] where the coefficients \(g_{\parallel}\) and \(g_{\times}\) are functions of split sine, split cosine and internal parameters of the branches.
14. Interference of alternative rearrangements
If the same output state can be obtained by several indistinguishable sequences of rearrangements, the amplitudes of these paths are added.
Let the output state \(|J_{\mathrm{out}}\rangle\) be achieved by two operators \(\hat V_1\) and \(\hat V_2\). Then
\[ \tag{50} \mathcal A_f = \langle J_{\mathrm{out}}| \left( \hat V_1+\hat V_2 \right) |J_{\mathrm{in}}\rangle. \] Or
\[ \tag{51} \mathcal A_f = \mathcal A_f^{(1)} + \mathcal A_f^{(2)}. \] Probability is equal
\[ \tag{52} P_f = \left| \mathcal A_f^{(1)} + \mathcal A_f^{(2)} \right|^2. \] After expansion, the modulus square contains the interference term:
\[ \tag{53} P_f = |\mathcal A_f^{(1)}|^2 + |\mathcal A_f^{(2)}|^2 + 2\operatorname{Re} \left( \mathcal A_f^{(1)} \overline{\mathcal A_f^{(2)}} \right). \] In the geometric interpretation, interference occurs between various indistinguishable ways of restructuring the same internal branches.
15. Transition matrix
Let \(|J_i\rangle\) and \(|\mathcal C_f\rangle\) denote possible assemblies. Then the elements of the transition matrix are defined as
\[ \tag{54} T_{fi} = \langle \mathcal C_f | \hat V | \mathcal C_i \rangle. \] For direct and cross assemblies
\[ \tag{55} T = \begin{pmatrix} \langle J_{\parallel}|\hat V|J_{\parallel}\rangle & \langle J_{\parallel}|\hat V|J_{\times}\rangle \\ \langle J_{\times}|\hat V|J_{\parallel}\rangle & \langle J_{\times}|\hat V|J_{\times}\rangle \end{pmatrix}. \] If the evolution of a closed system preserves the norm, the transition operator must be unitary:
\[ \tag{56} \hat V^{\dagger} \hat V = I. \] This implies conservation of total probability:
\[ \tag{57} \langle \Psi_{\mathrm{out}} | \Psi_{\mathrm{out}} \rangle = \langle \Psi_{\mathrm{in}} | \Psi_{\mathrm{in}} \rangle. \] 16. Sequential interactions
If the system goes through several successive stages, each stage has its own rearrangement operator.
For chain
\[ \tag{58} |J_{\mathrm{in}}\rangle \longrightarrow |J_m\rangle \longrightarrow |J_{\mathrm{out}}\rangle \] the amplitude through the intermediate state is
\[ \tag{59} \mathcal A_{J_{\mathrm{in}}\to J_m\to J_{\mathrm{out}}} = \langle J_{\mathrm{out}}| \hat V_2 |J_m\rangle \langle J_m| \hat V_1 |J_{\mathrm{in}}\rangle. \] If intermediate states form a complete set, then
\[ \tag{60} \mathcal A_{J_{\mathrm{in}}\to J_{\mathrm{out}}} = \sum_m \langle J_{\mathrm{out}}| \hat V_2 |J_m\rangle \langle J_m| \hat V_1 |J_{\mathrm{in}}\rangle. \] Using the completeness condition
\[ \tag{61} \sum_m |J_m\rangle \langle J_m| = I, \] get
\[ \tag{62} \mathcal A_{J_{\mathrm{in}}\to J_{\mathrm{out}}} = \langle J_{\mathrm{out}}| \hat V_2 \hat V_1 |J_{\mathrm{in}}\rangle. \] This is how braquet formalism naturally describes the composition of successive rearrangements of branches.
17. Example: electron and positron annihilation
We denote the initial state of the electron-positron pair as
\[ \tag{63} |J_{\mathrm{in}}\rangle = |e^-,e^+\rangle. \] One of the possible final channels preserves electron and positron assemblies:
\[ \tag{64} |J_1\rangle = |e'^-,e'^+\rangle. \] The other channel corresponds to two photon states:
\[ \tag{65} |J_2\rangle = |\gamma_1,\gamma_2\rangle. \] The corresponding probabilities are equal
\[ \tag{66} P_{e^-e^+} = \left| \langle e'^-,e'^+| \hat V |e^-,e^+\rangle \right|^2, \] \[ \tag{67} P_{\gamma\gamma} = \left| \langle \gamma_1,\gamma_2| \hat V |e^-,e^+\rangle \right|^2. \] Formulas (66)–(67) define the structure of the probabilistic choice, but to calculate the numerical values it is necessary to know the explicit interaction operator and the exact representation of electronic and photonic assemblies through branches.
18. Multiple end states
With sufficient energy, the same initial pair can transform into several final states:
\[ \tag{68} |e^-,e^+\rangle \longrightarrow |\gamma,\gamma\rangle, \] \[ \tag{69} |e^-,e^+\rangle \longrightarrow |\mu^-,\mu^+\rangle, \] \[ \tag{70} |e^-,e^+\rangle \longrightarrow |Ps,\gamma\rangle. \] Channel probabilities are determined by matrix elements
\[ \tag{71} P_{\gamma\gamma} = \left| \langle \gamma,\gamma| \hat V |e^-,e^+\rangle \right|^2, \] \[ \tag{72} P_{\mu\mu} = \left| \langle \mu^-,\mu^+| \hat V |e^-,e^+\rangle \right|^2, \] \[ \tag{73} P_{Ps\gamma} = \left| \langle Ps,\gamma| \hat V |e^-,e^+\rangle \right|^2. \] If all available channels are listed, then
\[ \tag{74} P_{\gamma\gamma} + P_{\mu\mu} + P_{Ps\gamma} + \ldots = 1. \] 19. Relationship to Observable Frequencies
The probability of a particular channel should be manifested in the statistics of a large number of equally prepared interactions.
If \(N\) events are carried out and the output state \(|J_k\rangle\) is registered \(N_k\) times, then for large \(N\)
\[ \tag{75} P_k \approx \frac{N_k}{N}. \] Thus, matrix element
\[ \tag{76} \langle J_k| \hat V |J_{\mathrm{in}}\rangle \] must be associated with the observed frequency of implementation of the corresponding physical process.
20. What is certain and what remains unknown
The braquet representation allows us to formulate a probabilistic model of interactions in a more rigorous form.
First, each stable assembly of internal branches is associated with a ket state.
Secondly, the interaction is described by an operator that changes the way the branches are connected.
Thirdly, the amplitude of the transition is determined by the matrix element of this operator between the source and final assemblies.
Fourth, the probability is equal to the square of the amplitude modulus, and the preservation of the total probability is associated with the unitarity of the transition operator.
Fifth, indistinguishable tuning paths interfere at the amplitude level.
However, three key elements have not yet been obtained: an explicit inner product for split assemblies, an accurate representation of physical particles through states \(|A:B\rangle\) and a specific operator \(\hat V\), calculated from split sine, split cosine, phases, frequencies and other itemsbranch parameters.
Without these elements, the model specifies the mathematical architecture of the probabilistic description, but does not yet allow calculating the probabilities of specific reactions.
21. Basic principle of the probabilistic model
The resulting construction can be formulated as a general principle.
Each particle or composite system is represented by a ket state, which fixes the way the internal branches are connected.
The interaction is described by a rearrangement operator that transforms the initial assembly into a superposition of geometrically and physically feasible final assemblies.
The probability of a particular channel is equal to the square of the modulus of the matrix element of the interaction operator between the initial and final states.
In short form:
\[ \tag{77} |J_{\mathrm{in}}\rangle \xrightarrow{\hat V} |\Psi_{\mathrm{out}}\rangle = \sum_k \langle J_k| \hat V |J_{\mathrm{in}}\rangle |J_k\rangle, \] \[ \tag{78} P_k = \left| \langle J_k| \hat V |J_{\mathrm{in}}\rangle \right|^2. \] Formulas (77)–(78) connect the geometry of branches with the usual probabilistic apparatus of quantum mechanics, but give it a different physical meaning: states are ways of connecting branches, and operators are rules for their rearrangement.
Conclusions
The fourth part supplements the geometric model of braquet interactions with a representation of probabilistic dynamics.
An individual particle is written as a state \(|A:B\rangle\), where the colon symbol denotes a stable connection of branches of two idempotent planes.
The state of two particles contains four branches and simultaneously fixes the way they are combined into two component systems.
The interaction is described by the operator \(\hat V\), which does not create branches out of nothing, but changes the correspondence between them.
Geometrically feasible straight and cross assemblies become possible final ket states.
The transition amplitude is determined by the matrix element \(\langle J_{\mathrm{out}}|\hat V|J_{\mathrm{in}}\rangle\), and the probability is determined by the square of its modulus.
Superposition corresponds to the simultaneous presence of several possible ways of connecting branches before registering the final result.
The unitarity of the interaction operator ensures the preservation of the full probability, and the completeness condition allows us to describe successive transitions through intermediate assemblies.
If the output state is achieved by several indistinguishable rearrangements, their amplitudes add up and create an interference term.
Geometric and physical projectors separate permissible states from forbidden ones, after which the interaction operator determines the relative amplitudes of the allowed channels.
The main unsolved problem remains the derivation of the inner product and operator \(\hat V\) directly from split sine, split cosine and the dynamics of internal branches.
Thus, the braket formalism does not replace split geometry, but becomes its probabilistic language: the ket describes the assembly of branches, the operator describes their rearrangement, the braket describes the verification of the final configuration, and the square of the matrix element describes the probability of the observed interaction.