2026-07-26
The Birth of Matter from Split Operators
Part 1. Construction of Known Particles from Normalized Operators
Throughout the history of science, mathematics has served as a language for describing physical phenomena. However, there is a deeper perspective: mathematical structures can not simply describe nature, but define its fundamental properties. In this case, the laws of physics are a consequence of the intrinsic geometry of the state space, and observable quantities are manifestations of mathematical regularities.
In previous works in this series, split geometry was developed, based on the operator \(J(t)=\jmath^a(-\jmath)^b\). It enabled geometric interpretations of motion, energy, mass, wave processes, and a number of well-known physical relations. This work takes this idea a step further: if each independent physical property corresponds to its own normalized split operator, then matter itself can be represented as a combination of these properties. In other words, elementary particles are viewed not as primary objects, but as the result of combining fundamental mathematical operators. 
In earlier works, the state of a point or particle was described by an operator containing two independent factors: its internal state and its external motion. However, known particles differ not only in mass and velocity. They also possess charge, flavor, color, and other quantum characteristics. Therefore, a natural question arises: is it possible to represent each such property by a separate operator and then assemble the complete state of the particle from these operators?
Within the proposed model, the answer is that each independent characteristic is defined by its own normalized operator \(J\), and the total state of the particle is obtained by their product. However, the fundamental invariant is preserved: the absolute value of the total state remains equal to unity.
1. Basic Operators of Internal State and Motion
The initial state of the particle is defined by two operators
\[ \tag{1} J_a(t)=\jmath^{a(t)}, \qquad J_b(t)=(-\jmath)^{b(t)}. \] Here, parameter \(a\) describes the internal state of the particle, and parameter \(b\) describes its external motion:
\[ \tag{2} a=\varpi t, \qquad \varpi\pi=\omega, \] \[ \tag{3} b=\frac{\arcsin\beta}{\pi}, \qquad \beta=\frac{v}{c}. \] Operators \(J_a\) and \(J_b\) are the fundamental building blocks of the model. The first describes the internal evolution of the particle's state, the second its motion relative to external space. This pair of operators alone allows us to separate internal and external processes without violating the general normalization condition. In the future, new split operators responsible for the remaining physical properties will be successively added to them.
The basic state operator has the form
\[ \tag{4} J_0(t)=J_a(t)J_b(t) =\jmath^{a(t)}(-\jmath)^{b(t)}. \] It satisfies the normalization condition
\[ \tag{5} \left|J_0(t)\right|=1. \] This condition means that a change in the internal state or external motion does not alter the overall energy scale of the state, but only redistributes it between independent components.
2. Adding New Operator Classes
These two operators are already sufficient to describe motion, but they still do not allow us to distinguish between an electron, a quark, or a photon. To do this, additional independent properties must be taken into account. In the proposed model, each such property is described by its own split operator, and the full state of the particle is constructed as their product. By normalizing each factor, the absolute value of the full state remains unchanged regardless of the number of operators used.
\[ \tag{6} J_P(t) = J_a(t) J_b(t) J_{q,Q} J_{f,F} J_{c,C}. \] The first index indicates the class of the property, and the second indicates its specific value. For example, \(J_{q,-1}\) denotes the charge state of the electron, \(J_{f,u}\) denotes the flavor of the \(u\) quark, and \(J_{c,r}\) denotes the red color state.
Each partial operator must have unit modulus:
\[ \tag{7} |J_a|=|J_b|=|J_{q,Q}|=|J_{f,F}|=|J_{c,C}|=1. \] Then the modulus of the full state is automatically preserved:
\[ \tag{8} |J_P| = |J_a| |J_b| |J_{q,Q}| |J_{f,F}| |J_{c,C}| =1. \] Therefore, the numerical value of the charge cannot be directly substituted into the state as an amplitude factor. For example, a factor of \(2/3\) would decrease the absolute value of the state. The charge value must be the label of the normalized charge operator or an eigenvalue of the corresponding operator of the observable.
\[ \tag{9} \widehat Q J_{q,Q}=QJ_{q,Q},\qquad |J_{q,Q}|=1. \] 3. Quark Color as a Three-Phase System
Of all the additional properties, the most convenient place to start is with the color state of quarks [1]. It consists of three discrete states and has a beautiful geometric interpretation: each color can be assigned a phase on the unit circle. Thanks to this, the properties of color operators are derived directly from geometry.
The most illustrative example of a separate class is provided by the three color states of quarks. Let us associate them with three equally spaced phases of the unit circle:
\[ \tag{10} J_{c,r}=1, \qquad J_{c,g}=e^{i2\pi/3}, \qquad J_{c,b}=e^{i4\pi/3}. \] Each color phase has a unit modulus:
\[ \tag{11} |J_{c,r}|=|J_{c,g}|=|J_{c,b}|=1. \] Color operators can also be represented as powers of a single normalized operator:
\[ \tag{12} J_{c,r}=\jmath^{0}, \qquad J_{c,g}=\jmath^{2/3}, \qquad J_{c,b}=\jmath^{4/3}, \] if the chosen power representation \(\jmath\) reproduces the corresponding phase rotations. In this case, it is not the names of the colors that are physically important, but their relative positions: the three states are separated by the same angle \(2\pi/3\).
4. Sum of Color Phases
The first fundamental property of the three color operators is related to their sum:
\[ \tag{13} J_{c,r}+J_{c,g}+J_{c,b} = 1+e^{i2\pi/3}+e^{i4\pi/3}. \] Let's expand the exponents using Euler's formula:
\[ \tag{14} e^{i2\pi/3} = \cos\frac{2\pi}{3} +i\sin\frac{2\pi}{3} = -\frac12+i\frac{\sqrt3}{2}, \] \[ \tag{15} e^{i4\pi/3} = \cos\frac{4\pi}{3} +i\sin\frac{4\pi}{3} = -\frac12-i\frac{\sqrt3}{2}. \] Substituting these expressions into the sum, we obtain
\[ \tag{16} \begin{aligned} 1+e^{i2\pi/3}+e^{i4\pi/3} &= 1+ \left(-\frac12+i\frac{\sqrt3}{2}\right) + \left(-\frac12-i\frac{\sqrt3}{2}\right) \\[1mm] &= 1-\frac12-\frac12 +i\frac{\sqrt3}{2} -i\frac{\sqrt3}{2} \\[1mm] &=0. \end{aligned} \] Therefore, the three color phases form a closed system:
\[ \tag{17} \boxed{ J_{c,r}+J_{c,g}+J_{c,b}=0. } \] Geometrically, this means that the three unit vectors, directed at angles \(0\), \(2\pi/3\), and \(4\pi/3\), completely cancel each other out. Their vector sum returns the system to the center. Within the model, this can be viewed as an additive condition of color neutrality.
Note that the resulting equality is independent of the specific choice of the notations red, green, and blue. Only the existence of three uniformly spaced phases is essential. Therefore, additive closure is a consequence of the geometry of the operators, not the color naming convention.
5. Product of Color Phases
The second fundamental property arises when multiplying the same three operators:
\[ \tag{18} J_{c,r}J_{c,g}J_{c,b} = 1\cdot e^{i2\pi/3}\cdot e^{i4\pi/3}. \] When multiplying the exponentials, their phases are added together:
\[ \tag{19} \begin{aligned} J_{c,r}J_{c,g}J_{c,b} &= e^{i\left(0+2\pi/3+4\pi/3\right)} \\[1mm] &= e^{i2\pi} \\[1mm] &=1. \end{aligned} \] Therefore
\[ \tag{20} \boxed{ J_{c,r}J_{c,g}J_{c,b}=1. } \] The product of the three color phases makes a full rotation and returns to the unit state. If the sum expresses the mutual compensation of color directions, then the product expresses their multiplicative closure.
Unlike the sum, which characterizes the mutual compensation of color states, the product describes their joint evolution. Returning to unity means that the full cycle of three phases is closed without changing the normalization. Thus, the additive and multiplicative properties are independent characteristics of the same color structure.
Thus, for a complete set of colors, two independent conditions are simultaneously satisfied:
\[ \tag{21} \boxed{ J_{c,r}+J_{c,g}+J_{c,b}=0, \qquad J_{c,r}J_{c,g}J_{c,b}=1. } \] The first condition is additive, the second is multiplicative. Together, they show that the three color states are not arbitrary labels: they form a strictly closed phase configuration.
6. Anticolors
The anticolor is naturally defined by the inverse phase:
\[ \tag{22} J_{c,\bar C}=J_{c,C}^{-1}. \] Since the modulus of the color operator is unity, the inverse operator coincides with the complex conjugate:
\[ \tag{23} J_{c,C}^{-1}=J_{c,C}^{*}. \] Therefore, the color and correspondingThe existing anticolors form a neutral pair:
\[ \tag{24} J_{c,C}J_{c,\bar C}=1. \] This allows us to describe two types of color closures in the same way: three different colors in the baryon system and a color with an anticolor in the meson system.
7. Unclosed Trajectories of Color Operators
Unlike operators describing observable free particles, color operators have an important geometric feature. With continuous changes in internal parameters, the corresponding trajectory does not form a closed loop.
This means that an individual color operator cannot exist as an independent stable state. An unclosed trajectory indicates the need to combine several color operators into a single system, the resulting trajectory of which becomes closed.
This is why quarks are observed only in hadrons. For baryons, closure is achieved by combining three different color operators, and for mesons, by a pair of color-anticolor operators. In both cases, the resulting geometric trajectory becomes closed, corresponding to a stable observable state.
8. Charge and Flavor Operators
Charge and flavor states are introduced using the same principle. We denote the charge operator as \(J_{q,Q}\), where \(Q\) is the observed charge value:
\[ \tag{25} J_{q,0}, \qquad J_{q,-1}, \qquad J_{q,+2/3}, \qquad J_{q,-1/3}. \] All of them are normalized:
\[ \tag{26} |J_{q,Q}|=1. \] We denote the flavor operator as \(J_{f,F}\):
\[ \tag{27} J_{f,e}, \quad J_{f,\nu_e}, \quad J_{f,u}, \quad J_{f,d}, \quad J_{f,s}, \quad J_{f,c}, \quad J_{f,b}, \quad J_{f,t}. \] In this article, these operators are used as normalized classification states. Their internal algebra must be defined separately. It is only important that their addition does not violate the general invariant:
\[ \tag{28} |J_{f,F}|=1. \] 9. Examples of Elementary Particles
After defining the basic split operators, we can construct the first states of known particles. The examples below do not claim to be a complete description of all quantum numbers. Their purpose is to demonstrate the principle of operator assembly, in which each new physical characteristic corresponds to the addition of another normalized operator.
Now we can write several simple states. For the missing property, the identity operator is used:
\[ \tag{29} J_{x,0}=1. \] The electron is colorless. Its state can be represented as
\[ \tag{30} J_{e^-}(t) = J_a^{(e)}(t) J_b^{(e)}(t) J_{q,-1} J_{f,e}. \] For an electron neutrino, the charge and color operators are unity:
\[ \tag{31} J_{\nu_e}(t) = J_a^{(\nu_e)}(t) J_b^{(\nu_e)}(t) J_{q,0} J_{f,\nu_e}. \] A photon in the minimal scheme has no electric charge, flavor, or color:
\[ \tag{32} J_{\gamma}(t) = J_a^{(\gamma)}(t) J_b^{(\gamma)}(t), \qquad J_{q,0}=J_{f,0}=J_{c,0}=1. \] For the red u-quark, we get
\[ \tag{33} J_{u_r}(t) = J_a^{(u)}(t) J_b^{(u)}(t) J_{q,+2/3} J_{f,u} J_{c,r}. \] For the green d-quark:
\[ \tag{34} J_{d_g}(t) = J_a^{(d)}(t) J_b^{(d)}(t) J_{q,-1/3} J_{f,d} J_{c,g}. \] In all cases, the same condition is satisfied:
\[ \tag{35} |J_{e^-}| =|J_{\nu_e}| =|J_{\gamma}| =|J_{u_r}| =|J_{d_g}| =1. \] 10. The Simplest Composite States
Up to this point, we have considered the states of individual elementary particles. However, the same operator approach naturally extends to composite systems. Their state is constructed from the operators of the constituent particles, with additional closure conditions arising automatically due to the properties of the color phases.
A composite particle is constructed from the states of its components. For the proton, the quark composition \(uud\) is used, and for the neutron, \(udd\). At the level of full multicomponent notation, it is convenient to use the tensor product:
\[ \tag{36} J_p = J_{u_r}\otimes J_{u_g}\otimes J_{d_b}, \] \[ \tag{37} J_n = J_{u_r}\otimes J_{d_g}\otimes J_{d_b}. \] In both cases, the color part contains the full set of phases \(r\), \(g\), \(b\). Therefore, it simultaneously satisfies the conditions
\[ \tag{38} J_{c,r}+J_{c,g}+J_{c,b}=0, \] \[ \tag{39} J_{c,r}J_{c,g}J_{c,b}=1. \] The proton charge is determined by the sum of the eigenvalues of the charge operators:
\[ \tag{40} \frac23+\frac23-\frac13=1. \] The neutron charge is
\[ \tag{41} \frac23-\frac13-\frac13=0. \] For the meson dResidual color-anticolor pairs. For example, for the state \(u\bar d\):
\[ \tag{42} J_{c,C}J_{c,\bar C}=1. \] Thus, a composite particle is obtained not by a simple mixture of characteristics, but by a coordinated combination of normalized operators, fulfilling the corresponding closure rules.
11. The First Table of Operator Composition
The examples discussed above show that each particle can be represented as a combination of several independent split operators. Each operator is responsible for only one physical property—internal state, motion, charge, aroma, or color. The complete state of the particle is determined by the combination of these operators, and not by any single characteristic.
For clarity, we summarize the obtained results in a single table. It shows the minimal operator composition of several known particles and demonstrates the general principle of construction: the difference between particles is determined not by changes in the basic structure, but by the choice of specific split operators for each independent property.
| Particle | Internal State | Motion | Charge | Aroma | Color |
| \(\gamma\) | \(J_a^{(\gamma)}\) | \(J_b^{(\gamma)}\) | \(J_{q,0}\) | \(J_{f,0}\) | \(J_{c,0}\) |
| \( e^-\) | \(J_a^{(e)}\) | \(J_b^{(e)}\) | \(J_{q,-1}\) | \(J_{f,e}\) | \(J_{c,0}\) |
| \(\nu_e\) | \(J_a^{(\nu_e)}\) | \(J_b^{( \nu_e)}\) | \(J_{q,0}\) | \(J_{f,\nu_e}\) | \(J_{c,0}\) |
| \(u_r\) | \(J_a^{(u)}\) | \(J_b^{(u)}\) | \(J_{q,+2/3}\) | \(J_{f ,u}\) | \(J_{c,r}\) |
| \(d_g\) | \(J_a^{(d)}\) | \(J_b^{(d)}\) | \(J_{q,-1/3}\) | \(J_{f,d}\) | \(J_{c,g}\) |
This table contains only a few examples so far. Its purpose is to demonstrate the principle: a particle is defined not by a single label, but by an ordered set of normalized operators. An expanded table of known particles will be constructed separately after the charge, flavor, and color operators are finally determined.
12. Summary of Part I
A unified method for constructing particle states has been introduced: each independent property is represented by its own operator \(J\), and the complete state is obtained by multiplying these operators.
\[ \tag{43} \boxed{ J_P(t) = J_a(t) J_b(t) J_{q,Q} J_{f,F} J_{c,C}, \qquad |J_P(t)|=1. } \] The color class was examined in greatest detail. Three color phases form a closed configuration for which the additive and multiplicative conditions are simultaneously satisfied:
\[ \tag{44} \boxed{ J_{c,r}+J_{c,g}+J_{c,b}=0, \qquad J_{c,r}J_{c,g}J_{c,b}=1. } \] These equalities show that color neutrality can be described not only verbally, but also as an exact phase closure. In the next section, this construction will be generalized: any new class of physical states can be introduced as a new normalized operator without changing the existing structure.
The main result of the first section is the transition from the description of individual physical quantities to the description of independent state split operators. In this picture, the particle is no longer considered an indivisible object. It becomes a combination of normalized operators, each responsible for only one physical property. Thanks to this, expanding the model does not require changing the existing structure: discovering a new quantum number simply means adding a new split operator. In the second part, this principle will be applied to construct a more complete operator table of known elementary particles.
Materials used
- Wikipedia. Quark.