2026-07-26
The Birth of Matter from Split Operators
Part 2. Operator Classification of Matter
In Part One, it was shown that internal state, motion, charge, flavor, and color can be represented by separate normalized operators. The complete state of a particle then arises as a product of independent factors, each responsible for only one physical property.
Now we move from individual examples to a more general construction. Our goal is to define a unified system of operators, establish the basic rules for combining them, and show how elementary and composite particles arise as particular realizations of this system.
Terminology Rule. All operators considered below are normalized split operators. Therefore, the word "split" will be omitted where the operator's belonging to the general system is already clear from the context.
1. General System of Operators
Let each independent physical property correspond to its own operator \(J_k\). Then the complete state of the object is determined by the ordered product of all operators necessary to describe it:
\[ \tag{1} J_P = \prod_{k=1}^{N}J_k. \] The index \(k\) numbers independent classes of properties, not particles. In a minimal system, we can distinguish internal state, external motion, charge, flavor, color, generation, and spin:
\[ \tag{2} J_P(t) = J_a(t) J_b(t) J_q J_f J_c J_g J_s. \] Here \(J_a\) and \(J_b\) define the dynamic basis of the state, and the remaining factors perform a classification or orientation function. The specific set of operators depends on the object under consideration. If a property is absent or not considered in a given problem, the corresponding operator is taken equal to one.
\[ \tag{3} J_{k,0}=1. \] Therefore, the general formula does not require that each particle possess all possible properties. It only defines a single set of positions, each of which can contain a nontrivial operator or a neutral unitary state.
2. Dynamic and Classification Operators
All operators can be conveniently divided into two large groups. Dynamic operators describe the evolution of a state over time and its external motion. In the model under consideration, they are given by the expressions
\[ \tag{4} J_a(t)=\jmath^{a(t)}, \qquad J_b(t)=(-\jmath)^{b(t)}. \] The parameters of these operators have different physical meanings:
\[ \tag{5} a=\varpi t, \qquad \varpi\pi=\omega, \] \[ \tag{6} b=\frac{\arcsin\beta}{\pi}, \qquad \beta=\frac{v}{c}. \] The operator \(J_a\) describes the internal state, and \(J_b\) describes the external motion. Their product forms the basic dynamical operator of the particle:
\[ \tag{7} J_0(t) = J_a(t)J_b(t) = \jmath^{a(t)}(-\jmath)^{b(t)}. \] Classification operators do not have to directly specify motion. They determine which charge, flavor, color, or other state the object belongs to:
\[ \tag{8} J_{q,Q}, \qquad J_{f,F}, \qquad J_{c,C}, \qquad J_{g,G}, \qquad J_{s,S}. \] The second index denotes a specific property value. For example, \(Q\) indicates the observable charge, \(F\) the flavor, \(C\) the color, \(G\) the generation, and \(S\) the spin state. The observable numbers themselves should not replace normalized operators with amplitude coefficients.
3. The Normalization Principle
The main requirement for each operator is a unit modulus:
\[ \tag{9} |J_k|=1. \] Then the product of any finite number of independent operators also remains normalized:
\[ \tag{10} |J_P| = \left|\prod_{k=1}^{N}J_k\right| = \prod_{k=1}^{N}|J_k| =1. \] This property is the main mechanism for model extensibility. Adding a new state class does not change the overall scale of the object if the new operator is also normalized. It adds new structure but does not create additional amplitude.
The observed value of a physical quantity is determined not by the modulus of an operator, but by the eigenvalue of the corresponding operator of the observed quantity:
\[ \tag{11} \widehat A J_{A,\alpha} = \alpha J_{A,\alpha}, \qquad |J_{A,\alpha}|=1. \] Thanks to this, a charge of \(-1\), \(+2/3\), or \(-1/3\) can be associated with a normalized state without decreasing or increasing its modulus. A similar approach can be applied to other discrete characteristics.
4. Independence of Operators
If two operators describe different, independent properties, changing one should not automatically change the other. In the simplest version of the model, this is expressed by their separate parameterization:
\[ \tag{12} J_k=J_k(\xi_k), \qquad J_m=J_m(\xi_m), \qquad k\ne m. \] For independent classes, we can adopt a commutative composition:
\[ \tag{13} J_kJ_m=J_mJ_k. \] Then, the order of the factors does not affect the overall state, and the product serves as the union of independent properties. However, this rule should not be considered mandatory for any future version of the algebra. If the interaction of two properties turns out to be directed or the order of operations becomes physically significant, the corresponding operators may form a non-commutative subsystem.
Therefore, the general construction admits two levels: a commutative union of independent characteristics and a special algebra for those classes whose interaction depends on the order.
5. Inverse and Conjugate Operators
For any normalized operator, there is an inverse state:
\[ \tag{14} J_k^{-1}J_k=1. \] If the operator is represented by a unit phase, then the inverse state coincides with the complex conjugate:
\[ \tag{15} J_k^{-1}=J_k^{*}. \] This state is naturally interpreted as the opposite orientation in the corresponding operator class. This is precisely how anticolors were previously introduced:
\[ \tag{16} J_{c,\bar C}=J_{c,C}^{-1}. \] A similar operation can be applied to other classes if the physical meaning of the anti-state is related to the inversion of the corresponding phase. However, charge, flavor, and spatial inversions do not necessarily represent the same transformation. For each class, it is necessary to separately determine which operator corresponds to the physical antiparticle.
6. Composition of States
A simple combination of independent properties of a single particle is performed by the usual product of operators:
\[ \tag{17} J_P=J_1J_2\cdots J_N. \] If several independent particles are combined, information about each component must be stored. For this purpose, it is convenient to use the tensor product:
\[ \tag{18} J_{AB}=J_A\otimes J_B. \] The ordinary product answers the question of what independent properties a single state consists of. The tensor product answers a different question: what independent states a multicomponent system consists of. Mixing these two operations would lead to a loss of distinction between a particle property and an individual particle.
For a system of \(M\) components, we obtain
\[ \tag{19} J_{\mathrm{sys}} = \bigotimes_{n=1}^{M}J_{P_n}. \] 7. Additive and Multiplicative Conditions
Not all physical constraints are expressed by the same type of operation. In a system of operators, two different classes of conditions naturally arise: additive and multiplicative.
The additive condition describes the compensation of directions or eigenvalues:
\[ \tag{20} \sum_{n=1}^{M}J_n=0. \] The multiplicative condition describes the closure of a complete cycle or the mutual cancellation of inverse phases:
\[ \tag{21} \prod_{n=1}^{M}J_n=1. \] These conditions are not interchangeable. The sum indicates that the resulting direction is absent, and the product indicates that the combined phase returns to unity. The most illustrative example is provided by the full set of color operators:
\[ \tag{22} J_{c,r}+J_{c,g}+J_{c,b}=0, \qquad J_{c,r}J_{c,g}J_{c,b}=1. \] The color-anticolor pair has a multiplicative closure:
\[ \tag{23} J_{c,C}J_{c,\bar C}=1. \] Thus, operator neutrality can take different forms. In each physical class, it is necessary to establish which specific condition expresses the absence of the observed resulting property.
8. Interaction as a Transformation of Operator Composition
Particle interaction can be viewed as a transition from one set of operators to another. In this case, the initial and final states do not necessarily contain the same number of particles or the same set of factors. It is important that certain invariants of the system are preserved.
\[ \tag{24} J_{\mathrm{in}} \longrightarrow J_{\mathrm{out}}. \] For a multicomponent process, this notation takes the form
\[ \tag{25} J_1\otimes J_2\otimes\cdots \longrightarrow J_1'\otimes J_2'\otimes\cdots. \] The most general requirement is to preserve the full normalized scale:
\[ \tag{26} |J_{\mathrm{in}}| = |J_{\mathrm{out}}| =1. \] However, normalization alone is not sufficient to define an admissible process. Additionally, the eigenvalues of those observables for which the corresponding conservation law applies must be preserved. In operator notation, this can be expressed as
\[ \tag{27} \sum_{n\in\mathrm{in}}\alpha_n = \sum_{m\in\mathrm{out}}\alpha_m. \] Here \(\alpha\) can denote charge, lepton or baryon number, or another additive characteristic if included in the model. Therefore, the operator transformation is not an arbitrary substitution of factors: it is constrained by the simultaneous normalization, phase closure, and balance of observed values.
9. Double Balance Rule
The previously introduced double balance rule can be used for the interaction of two input and two output states. It requires the simultaneous conservation of the sum and product of states:
\[ \tag{28} J_1+J_2=J_3+J_4, \] \[ \tag{29} J_1J_2=J_3J_4. \] The first equality fixes additive balance, the second, multiplicative closure. Together, they constrain the possible final states more strongly than either condition alone.
If the sum \(S\) and product \(P\) are known, the possible states are the roots of the operator quadratic equation
\[ \tag{30} X^2-SX+P=0. \] From this, we formally obtain two branches:
\[ \tag{31} X_{\pm} = \frac{S\pm\sqrt{S^2-4P}}{2}. \] This construction shows that the same initial balance can admit several transformation channels. However, the physical probability of each channel requires a separate law and is not determined solely by the algebraic possibility of the root's existence.
10. The Birth and Disappearance of Pairs
If two states are mutually inverse, their product is equal to one:
\[ \tag{32} J_PJ_{\bar P}=1. \] This allows us to interpret the particle-antiparticle pair as a closed multiplicative configuration. During annihilation, the operator composition of the pair can transition to another set of states while maintaining full balance:
\[ \tag{33} J_P\otimes J_{\bar P} \longrightarrow J_{\gamma_1}\otimes J_{\gamma_2}. \] The inverse process describes the creation of a pair:
\[ \tag{34} J_{\gamma_1}\otimes J_{\gamma_2} \longrightarrow J_P\otimes J_{\bar P}. \] These entries only define the structure of the transformation for now. For a complete physical description, it is necessary to further define the energy, momentum, polarization, and admissibility conditions of the process. However, the operator approach shows how different types of particles can be viewed as different decompositions of a single, general, normalized state.
11. General Formula of an Elementary Particle
Taking into account the introduced classes, the state of an elementary particle can be represented in an expanded form:
\[ \tag{35} J_P(t) = J_a^{(P)}(t) J_b^{(P)}(t) J_{q,Q_P} J_{f,F_P} J_{c,C_P} J_{g,G_P} J_{s,S_P}. \] The superscript \((P)\) in dynamical operators emphasizes that the internal frequencies and kinematic parameters of different particles may differ. Subscripts in classification operators indicate specific eigenstates.
If a class is missing, 1 is used. For example, for the colorless state:
\[ \tag{36} J_{c,0}=1. \] Thanks to this, the same formula is applicable to leptons, quarks, photons, and other objects. The difference between them is determined not by a change in the overall structure, but by the choice of specific operators and the rules for their compatibility.
12. Examples of Lepton States
An electron in an extended system can be represented as a product of dynamic operators, charge state, flavor, generation, and spin:
\[ \tag{37} J_{e^-}(t) = J_a^{(e)}(t) J_b^{(e)}(t) J_{q,-1} J_{f,e} J_{g,1} J_{s,S_e}. \] The color operator for an electron is unity. For an electron neutrino, the charge is also neutral:
\[ \tag{38} J_{\nu_e}(t) = J_a^{(\nu_e)}(t) J_b^{(\nu_e)}(t) J_{q,0} J_{f,\nu_e} J_{g,1} J_{s,S_{\nu}}. \] The muon and tau lepton retain the general structure of the electron, but differ in flavor and generation:
\[ \tag{39} J_{\mu^-} = J_a^{(\mu)}J_b^{(\mu)}J_{q,-1}J_{f,\mu}J_{g,2}J_{s,S_{\mu}}, \] \[ \tag{40} J_{\tau^-} = J_a^{(\tau)}J_b^{(\tau)}J_{q,-1}J_{f,\tau}J_{g,3}J_{s,S_{\tau}}. \] These examples demonstrate an important principle: particles of different generations can have the same charge and spin classes, but differ in their flavor, generation, and internal dynamics operators.
13. Examples of Quark States
For a quark, a nontrivial color operator is added to the general construction. For example, the red \(u\)-quark of the first generation has the form
\[ \tag{41} J_{u_r}(t) = J_a^{(u)}(t) J_b^{(u)}(t) J_{q,+2/3} J_{f,u} J_{c,r} J_{g,1} J_{s,S_u}. \] The green \(d\)-quark is written analogouslyexactly:
\[ \tag{42} J_{d_g}(t) = J_a^{(d)}(t) J_b^{(d)}(t) J_{q,-1/3} J_{f,d} J_{c,g} J_{g,1} J_{s,S_d}. \] Changing color does not require changing charge or aroma. Therefore, three color variants of a single quark are formed by replacing just one factor:
\[ \tag{43} J_{u_C} = J_u^{(0)}J_{c,C}, \qquad C\in\{r,g,b\}, \] where \(J_u^{(0)}\) contains all the operators of the \(u\)-quark except color. This factorization makes the independence of the color class explicit.
14. Photon and Neutral Carriers
In the minimal scheme, the photon carries no electric charge, flavor, or color. Its state is determined by dynamical operators and an additional polarization or spin operator:
\[ \tag{44} J_{\gamma}(t) = J_a^{(\gamma)}(t) J_b^{(\gamma)}(t) J_{s,S_{\gamma}}. \] Neutrality does not mean the absence of state, but rather the unity of the corresponding classification factors:
\[ \tag{45} J_{q,0}=J_{f,0}=J_{c,0}=1. \] Other interaction carriers can be constructed using the same principle. However, for them, it is necessary to separately define the internal algebra of spin, polarization, and charges of the corresponding fields. The general formula specifies the position of the operator in the system but does not replace its physical derivation.
15. Composite Particles
The state of a composite particle preserves the individuality of its components and is therefore constructed by a tensor product. For a proton and neutron in quark notation, we obtain
\[ \tag{46} J_p = J_{u_r}\otimes J_{u_g}\otimes J_{d_b}, \] \[ \tag{47} J_n = J_{u_r}\otimes J_{d_g}\otimes J_{d_b}. \] The color part of both systems contains three uniformly spaced phases and satisfies double closure:
\[ \tag{48} J_{c,r}+J_{c,g}+J_{c,b}=0, \qquad J_{c,r}J_{c,g}J_{c,b}=1. \] The charge eigenvalues of the components are summed. For the proton and neutron, respectively:
\[ \tag{49} \frac23+\frac23-\frac13=1, \] \[ \tag{50} \frac23-\frac13-\frac13=0. \] The meson contains a quark and an antiquark. Its color part is closed by a pair of mutually inverse operators:
\[ \tag{51} J_M = J_{q_C}\otimes J_{\bar q_{\bar C}}, \qquad J_{c,C}J_{c,\bar C}=1. \] Thus, a composite particle is defined not only by the set of components but also by the conditions under which their operators are compatible. It is the closure rules that distinguish a stable neutral configuration from an arbitrary set of factors.
16. First Operator Classification of Matter
The resulting system allows one to consider a particle as a row of operator composition. A specific operator or unit is specified for each position. In schematic form, such a line can be written as
\[ \tag{52} \mathcal{C}(P) = \bigl( J_a, J_b, J_q, J_f, J_c, J_g, J_s \bigr)_P. \] Two particles coincide in this classification only if all operators included in their lines coincide:
\[ \tag{53} \mathcal{C}(P_1)=\mathcal{C}(P_2) \quad\Longleftrightarrow\quad J_k^{(P_1)}=J_k^{(P_2)} \\text{for all }k. \] If only some of the operators match, the particles belong to the same class based on the corresponding properties. For example, the electron, muon, and tau lepton have the same charge class but differ in their flavor, generational, and dynamical operators.
After introducing a common system of operators, individual particles can be arranged in a single table. This table is not a list of names, but rather the first operator classification of matter: each row specifies a specific combination of independent states, and each column corresponds to a separate physical property.
| Particle | Internal State | Motion | Charge | Aroma | Color | Generation | Spin | Class |
|---|---|---|---|---|---|---|---|---|
| Electron \(e^-\) | \(J_a^{(e)}\) | \(J_b^{(e)}\) | \(J_{q,-1}\) | \(J_{f,e}\) | \(1\) | \(J_{g,1}\) | \(J_{s,1/2}\) | lepton |
| Electron neutrino \(\nu_e\) | \(J_a^{(\nu_e)}\) | \(J_b^{(\nu_e)}\) | \(J_{q,0}\) | \(J_{f,\nu_e}\) | \(1\) | \(J_{g,1}\) | \(J_{s,1/2}\) | lepton |
| Muon \(\mu^-\) | \(J_a^{(\mu)}\) | \(J_b^{(\mu)}\) | \(J_{q,-1}\) | \(J_{f,\mu}\) | \(1\) | \(J_{g,2}\) | \(J_{s,1/2}\) | lepton |
| Tau lepton \(\tau^-\) | \(J_a^{(\tau)}\) | \(J_b^{(\tau)}\) | \(J_{q,-1}\) | \(J_{f,\tau}\) | \(1\) | \(J_{g,3}\) | \(J_{s,1/2}\) | lepton |
| Quark \(u_C\) | \(J_a^{(u)}\) | \(J_b^{(u)}\) | \(J_{q,+2/3}\) | \(J_{f,u}\) | \(J_{c,C}\) | \(J_{g,1}\) | \(J_{s,1/2}\) | quark |
| Quark \(d_C\) | \(J_a^{(d)}\) | \(J_b^{(d)}\) | \(J_{q,-1/3}\) | \(J_{f,d}\) | \(J_{c,C}\) | \(J_{g,1}\) | \(J_{s,1/2}\) | quark |
| Photon \(\gamma\) | \(J_a^{(\gamma)}\) | \(J_b^{(\gamma)}\) | \(J_{q,0}\) | \(1\) | \(1\) | \(1\) | \(J_{s,1}\) | photon |
| Proton \(p\) | \(J_{u_r}\otimes J_{u_g}\otimes J_{d_b}\) | \(+1\) | \(uud\) | closed | \(J_{g,1}\) | \(J_{s,1/2}\) | baryon | |
| Neutron \(n\) | \(J_{u_r}\otimes J_{d_g}\otimes J_{d_b}\) | \(0\) | \(udd\) | closed | \(J_{g,1}\) | \(J_{s,1/2}\) | baryon | |
| Meson \(q_C\bar q_{\bar C}\) | \(J_{q_C}\otimes J_{\bar q_{\bar C}}\) | depends on composition | \(q\bar q\) | \(J_{c,C}J_{c,\bar C}=1\) | depends on composition | composite | meson | |
In the table, the unit \(1\) denotes the neutral state of the corresponding class, not the absence of a mathematical object. The symbol \(C\) denotes one of three colors, and for physically neutral composite states, the color operators must satisfy closure conditions.
This classification is still a preliminary scheme, not a complete table of all particles. Its fundamental advantage is its extensibility: new particles, additional quantum properties, and refined operators can be added as new rows and columns, preserving the existing structure.
Thus, the table becomes an intermediate link between the general operator algebra and concrete particle physics. It shows which properties coincide, which differ, and which closure conditions are necessary for the existence of composite states.
17. Boundaries of the constructed scheme
It is important to separate the general architecture of the model from the physics already derived. Normalization and composition rules determine the form of the operator system, but by themselves do not specify the numerical masses of particles, decay probabilities, interaction constants, or field dynamics.
To complete the model, it is necessary to separately construct the internal algebra of charge, flavor, generation, and spin operators, determine their relationship to observable quantities, and derive admissible transformations. Therefore, the given examples should be viewed as an operator framework, not as a definitive replacement for the existing theory of elementary particles.
The strength of this construction lies elsewhere: it offers a unified way to organize various physical properties and allows us to formulate interactions as transformations of normalized operator compositions.
18. Summary of Part Two
In Part Two, a general system of normalized operators was constructed. Each independent property is represented by its own multiplier, and the complete state of an elementary particle has the form
\[ \tag{54} \boxed{ J_P(t) = \prod_{k=1}^{N}J_k, \qquad |J_k|=1, \qquad |J_P|=1. } \] It was shown that the ordinary product combines the independent properties of a single particle, while the tensor product preserves the independence of the components of a composite system:
\[ \tag{55} \boxed{ J_P=\prod_kJ_k, \qquad J_{\mathrm{sys}}=\bigotimes_nJ_{P_n}. } \] The interaction was represented as a transformation of the operator composition, constrained by normalization, eigenvalue balance, and phase closure conditions. For processes with two input and two output states, the double balance rule was introduced:
\[ \tag{56} \boxed{ J_1+J_2=J_3+J_4, \qquad J_1J_2=J_3J_4. } \] In this picture, known particles are not the original indivisible notations, but specific combinations of operators of internal state, motion, charge, flavor, color, generation, and spin. Matter arises as a consistent composition of normalized states, and interaction is an admissible transformation of one operator structure into another.
The presented operator model has a natural extension potential. Each fundamental property of a particle is described by its own independent operator, and the complete state is their product.
Therefore, the discovery of any new fundamentalThe introduction of a new property, a new class of particles, or a new law related to the internal structure of matter, mathematically simply means introducing an additional operator and incorporating it into the general product. In this case, the previously constructed system does not require revision, but rather is naturally expanded.
Thus, the operator notation is not a fixed model of known particles, but an open mathematical construct that allows for further development as new experimental data emerges.