2026-07-22
Geometric representation of particle interactions
Part 1. States of particles and their transformations in the i-basis
Introduction
In physics, the interaction of particles is usually written as the transition of one set of objects to another. For example, an electron and a positron can annihilate to form two photons, and two photons, if the necessary conditions are met, can participate in the creation of a particle-antiparticle pair.
This notation shows the initial and final particles well, but says almost nothing about the internal geometry of the transformation itself. It remains unclear whether there is a single mathematical object that could describe both the initial and final states.
This article proposes to consider each particle as a geometric state in a four-dimensional u-basis. Then the interaction can be represented as a transformation of several states occurring simultaneously in two connected complex planes.
At this stage we will not introduce probabilities and will not prove the general law of interaction. First, we will show how the states of particles and the simplest reactions can be represented using split geometry.
1. Four-dimensional i-basis
The model is based on two idempotents \(\ep\) and \(\em\). Together with the imaginary unit they form a four-dimensional basis
\[\tag{1} \left\{ \ep,\; i\ep,\; \em,\; i\em \right\}. \] Idempotents satisfy the conditions
\[\tag{2} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \qquad \ep+\em=1. \] This basis can be represented as the sum of two independent complex planes:
\[\tag{3} (\ep,i\ep) \qquad\text{and}\qquad (\em,i\em). \] The main property is that two idempotent parts do not mix when multiplied. The mixed product disappears due to the condition \(\ep\em=0\). Therefore, the state can be divided into two complex components and studied independently.
In the future, the first plane can be conditionally associated with the general, or external, component of the state, and the second - with the internal one. This is not a necessary physical interpretation of algebra, but it turns out to be convenient for describing particles and their transformations.
2. State operator
The state of an individual object will be represented by an operator
\[\tag{4} J(a,b)=e^{ia}\j^b. \] The \(a\) parameter specifies the overall phase of the object, while the \(b\) parameter specifies the relative rotation between two complex planes.
In the i-basis, the operator is expanded as
\[\tag{5} J(a,b) = \ep e^{ia} + \em e^{i(a+\pi b)}. \] Both parts of the state have a common phase \(a\), however, the second plane is additionally rotated by an angle \(\pi b\). Therefore, the parameter \(a\) describes the coordinated rotation of the entire state, and the parameter \(b\) changes the internal relationship between its two parts.
The same construction can be written through split cosine and split sine:
\[\tag{6} J(a,b) = \operatorname{cs}(a,b) + i\operatorname{sn}(a,b). \] where
\[\tag{7} \operatorname{cs}(a,b) = \ep\cos a + \em\cos(a+\pi b). \] \[\tag{8} \operatorname{sn}(a,b) = \ep\sin a + \em\sin(a+\pi b). \] Formula (6) is an extension of Euler's formula. In the ordinary complex plane, the state is described by a single circle. Here two circles exist simultaneously, connected by a common phase and a relative angle.
If \(b=0\), both planes have the same phase, and the operator goes to the usual complex exponential:
\[\tag{9} J(a,0)=e^{ia}. \] If \(a=0\), only internal rotation remains:
\[\tag{10} J(0,b)=\j^b. \] Thus, ordinary complex rotation and internal split rotation are two special cases of one object. Thus, each particle is described not by four independent coordinates, but by two angular parameters a and b, which determine the position of the state in two connected complex planes.
3. What does interaction mean
Consider a process in which two initial states transform into two final states:
\[\tag{11} J_1+J_2 \longrightarrow J_3+J_4. \] At this stage, the arrow only denotes transformation. We are not yet claiming that the sum on the left is necessarily equal to the sum on the right. This condition will be considered in the second part of the work.
Each state contains two idempotent components. Therefore, interaction can be understood as changing the way in which these components are combined in individual objects.
Let the four states have the form
\[\tag{12} J_k=\ep A_k+\em B_k. \] Then the interaction scheme is written as followsway:
\[\tag{13} \left( \ep A_1+\em B_1 \right) + \left( \ep A_2+\em B_2 \right) \longrightarrow \left( \ep A_3+\em B_3 \right) + \left( \ep A_4+\em B_4 \right). \] This entry shows that transformation can affect not only the general phases of objects, but also the way their components are connected.
Before interaction, the component \(A_1\) can be combined with the component \(B_1\), and \(A_2\) - with \(B_2\). After the interaction, another union may arise: \(A_1\) with \(B_2\), and \(A_2\) with \(B_1\).
Consequently, even if the complete set of components is preserved, there may be different ways to form the final objects.
4. Particle and antiparticle
For a geometric model, it is convenient to consider a particle and an antiparticle as two mutually related states. Their components may have the same moduli, but differ in relative orientation, phase sign, or direction of internal rotation.
In the simplest symbolic form, such a pair can be represented as
\[\tag{14} J_{+}=\ep A+\em B, \qquad J_{-}=\ep A-\em B. \] The sign in front of the second component here should not be automatically identified with electric charge. It only shows the opposite orientation of one of the parts of the state.
The physical definition of charge must be introduced as a separate condition. Nevertheless, formula (14) shows that a particle and an antiparticle can be represented as different configurations of the same geometric structure.
We can also consider a more general version, in which the opposite is specified not by a sign, but by a phase shift:
\[\tag{15} J_{+}=e^{ia}\j^b, \qquad J_{-}=e^{ia}\j^{-b}. \] In this case, the two states have one common phase, but opposite internal phases.
5. Annihilation of an electron and a positron
Consider the reaction
\[\tag{16} e^-+e^+ \longrightarrow \gamma_1+\gamma_2. \] In the usual physical interpretation, the electron and positron disappear, and their energy and momentum are transferred to two photons.
In the proposed geometric model, we are not talking about the disappearance of a mathematical state, but about transforming the way of combining its components.
Let us denote the states of an electron, a positron and two photons as
\[\tag{17} J_{e^-}, \qquad J_{e^+}, \qquad J_{\gamma_1}, \qquad J_{\gamma_2}. \] Then the geometric scheme of the reaction takes the form
\[\tag{18} J_{e^-}+J_{e^+} \longrightarrow J_{\gamma_1}+J_{\gamma_2}. \] Before interaction, the components of the two complex planes are combined in the electron and positron states. After interaction, the same geometric components should form two photonic states.
You can use the transformation as a visual diagram
\[\tag{19} \left( iE+P,\; iE-P \right) \longrightarrow \left( E+iP,\; E-iP \right). \] On the left side, the quantity \(E\) is located primarily along the imaginary direction, and the quantity \(P\) is located primarily along the real one. On the right side, their geometric roles change.
Such a transformation can be interpreted as a rotation of the state structure, in which a bound particle-antiparticle pair transforms into a pair of propagating states.
Formula (19) is a geometric scheme. It does not replace the standard laws of physics. To describe a real reaction, the laws of conservation of energy, momentum, electric charge and angular momentum must additionally be satisfied.
6. Why two photons appear
One of the important questions is why the annihilation of a resting electron and positron produces at least two photons.
From the point of view of standard physics, one photon cannot simultaneously retain the total energy and zero total momentum of the original system. Therefore, two photons appear, the impulses of which are directed in opposite directions.
In a geometric model, this means that the final state must also be closed by a pair of mutually complementary objects.
Conditionally, such a pair can be represented as
\[\tag{20} J_{\gamma_1}=E+iP, \qquad J_{\gamma_2}=E-iP. \] The spatial components of the two states have opposite signs and are mutually compensated, while their energy parts add up.
Therefore, two photons can be considered not as two completely independent results, but as one matched pair of output states.
7. Reverse transformation: birth of a couple
The reverse process can be written as
\[\tag{21} \gamma_1+\gamma_2 \longrightarrow e^-+e^+. \] In gein metric form this corresponds to an inverse rearrangement of the components:
\[\tag{22} \left( E+iP,\; E-iP \right) \longrightarrow \left( iE+P,\; iE-P \right). \] Two propagating states form a bound particle-antiparticle pair. The geometric structure does not disappear, but moves from one method of organization to another.
Consequently, annihilation and the birth of a pair can be considered as mutually inverse transformations of the same system of components.
At the same time, formula (22) does not mean that any two photons necessarily create an electron and a positron. For the birth of a pair, energy and kinematic conditions must be met.
In real processes, an additional object may also be required, for example a core, which receives part of the impulse and ensures the fulfillment of conservation laws.
8. Positronium formation
An electron and a positron do not necessarily annihilate immediately. They can form a bound state - positronium.
Geometrically, this means that the two initial states are closed into a single consistent system:
\[\tag{23} J_{e^-}+J_{e^+} \longrightarrow J_{\mathrm{Ps}}. \] Here \(J_{\mathrm{Ps}}\) should be understood not as the usual arithmetic sum of two numbers, but as a composite state in which the phases and internal components of the electron and positron are consistent with each other.
Different forms of positronium can be tentatively associated with different relative orientations of the initial states.
In one case, internal directions can cancel each other out:
\[\tag{24} J_{e^-}^{(\mathrm{int})}+J_{e^+}^{(\mathrm{int})}=0. \] Alternately, they can form a consistent non-zero internal orientation:
\[\tag{25} J_{e^-}^{(\mathrm{int})} + J_{e^+}^{(\mathrm{int})} \ne 0. \] Such a difference can be further compared to parapositronium and orthopositronium. However, a strict connection with quantum spin will require a separate analysis.
At this stage, the only important thing is that the same pair of initial objects can form different connected states depending on the relative orientation of the components.
9. Particle Scattering
The proposed approach is not limited to the processes of creation and destruction of particles. It can be applied to interactions in which particle types are preserved but their states are changed.
For example, the scattering of two particles can be represented as
\[\tag{26} J_1(a_1,b_1) + J_2(a_2,b_2) \longrightarrow J_1(a_3,b_3) + J_2(a_4,b_4). \] Before and after interaction, the same objects are present, but the parameters of their states change.
A change in the overall phase \(a\) can be associated with external motion or direction of propagation. A change in the relative phase \(b\) may reflect a restructuring of the internal state.
In this representation, scattering is not a collision of geometrically unchanged points, but a transformation of the phase structures of two objects.
10. Photon absorption
The absorption of a photon by a particle can be written as
\[\tag{27} J_{\mathrm{particle}} + J_\gamma \longrightarrow J_{\mathrm{particle}}^{\prime}. \] In the usual interpretation, a photon transfers energy and momentum to a particle. In the geometric model, its state is included in the new state of the particle.
In this case, both the general phase of the particle and the relative phase between the two u-basis planes can change.
Schematically you can write
\[\tag{28} e^{ia_1}\j^{b_1} + e^{ia_2}\j^{b_2} \longrightarrow e^{ia_3}\j^{b_3}. \] Formula (28) is not the usual rule for adding exponentials. It shows the transition of a composite system to a new bound state.
11. Photon emission
The inverse transformation corresponds to the emission of a photon:
\[\tag{29} J_{\mathrm{particle}} \longrightarrow J_{\mathrm{particle}}^{\prime}+J_{\gamma}. \] In this case, part of the structure of the initial state is separated into an independent propagating state.
The difference between the initial and final states of the particle determines the state of the emitted photon:
\[\tag{30} J_{\gamma} = J_{\mathrm{particle}} - J_{\mathrm{particle}}^{\prime}. \] Here the equal sign should be understood as a geometric condition for the balance of states. Its strict meaning will be discussed in the second part of the article.
12. Single transformations
Some physical processes outwardly look like the transformation of one particle into another. However, almost always, along with a new particle, additional volumes appearcts or fields that ensure the fulfillment of conservation laws.
The general scheme of such a process can be written as
\[\tag{31} J_A \longrightarrow J_B+J_C+\ldots \] In a geometric model, this means that one initial state breaks down into several mutually consistent components.
The reverse process looks like
\[\tag{32} J_B+J_C+\ldots \longrightarrow J_A. \] Thus, the separation and union of states can be considered as two directions of one geometric transformation.
13. General interaction scheme
Summarizing the examples considered, the interaction of several objects can be represented as a transformation of a set of states:
\[\tag{33} J_1+J_2+\ldots+J_n \longrightarrow J_1^{\prime} + J_2^{\prime} + \ldots + J_m^{\prime}. \] This entry does not yet determine which reactions are allowed. It creates a unified language in which each particle is described by an object \(J\), and interaction means the transformation and rearrangement of such objects.
A physically permissible reaction must additionally satisfy the laws of conservation of energy, momentum, charge, angular momentum and other quantum numbers.
Consequently, the geometric model does not cancel the known laws. It adds to them a description of the internal structure of the state.
14. What does split view do?
The main advantage of the i-basis is the division of an object into two complex components.
The usual complex exponential describes rotation in one plane:
\[\tag{34} e^{ia} = \cos a+i\sin a. \] The split operator contains two planes:
\[\tag{35} e^{ia}\j^b = \ep e^{ia} + \em e^{i(a+\pi b)}. \] Therefore, one part of the state can describe the overall rotation, and the other - the internal phase cycle.
Interaction then represents not only a change in external motion, but also a restructuring of the mutual orientation of the two parts of the state.
It is especially important that due to the condition \(\ep\em=0\) the two components can be redistributed independently. In the future, it is this property that will allow us to obtain several acceptable options for the final state.
15. Geometric rearrangement
Consider two source objects:
\[\tag{36} J_1=\ep A_1+\em B_1, \qquad J_2=\ep A_2+\em B_2. \] The first obvious option for the final pair preserves the original combination of components:
\[\tag{37} \left\{ \ep A_1+\em B_1, \; \ep A_2+\em B_2 \right\}. \] However, cross rearrangement is also possible:
\[\tag{38} \left\{ \ep A_1+\em B_2, \; \ep A_2+\em B_1 \right\}. \] In the second case, the complete set of components remains the same, but they form different objects.
It is precisely this rearrangement that can be tentatively considered as a geometric mechanism for transforming some particles into others without changing the full set of components of the system.
For now we are not saying that both configurations are necessarily valid. This requires an additional condition, which must take into account not only the sum, but also the mutual relationship of the states.
16. Limits of the geometric model
The given transformations should be considered as geometric schemes. They show a possible way to represent interactions, but do not yet answer several fundamental questions.
Firstly, a single mathematical condition has not yet been defined that allows one to distinguish a valid transformation from an invalid one.
Secondly, it is not shown which properties of the initial pair must be preserved in the final pair.
Thirdly, it has not yet been explained why different final results can arise for the same initial states.
Finally, at this stage nothing is said about the probability of individual outcomes.
To answer the first and second questions, the next part will introduce the double balance rule. It will require preserving two characteristics of a pair of states: their sum and product.
\[\tag{39} J_1+J_2=J_3+J_4. \] \[\tag{40} J_1J_2=J_3J_4. \] It will be shown that these two conditions lead to a quadratic equation whose roots are the possible final states of the interaction.
After this, the examples discussed in this article can be studied not only as visual geometric diagrams, but also as consequences of the general rule.
Conclusions
The article proposes to represent the state of a particle by the operator \(J=e^{ia}\j^b\), acting in a four-dimensional i-basis.
This operator combines two complex planes and allows you to simultaneously take into account the overall phase of the object and the relative phase between the two parts of its state.
The interaction of particles is considered as a transformation and rearrangement of geometric states. Using examples of the annihilation of an electron and a positron, the birth of a pair, the formation of positronium, scattering, absorption and emission of a photon, it is shown that different physical processes can be represented in a single language.
Within the framework of this interpretation, a particle, an antiparticle and a wave state are not necessarily fundamentally different mathematical objects. They can correspond to different ways of combining and orienting components of the same i-basis.
However, a geometric reaction scheme alone is not enough. You need a rule that defines valid end states.
In the second part, the rule of double balance of the sum and product of states will be introduced. It will be shown that it is these two invariants that make it possible to reconstruct the possible final states of the interaction and lead to a quadratic transformation equation.