2026-07-30
Attraction and repulsion of charges and currents as two projections of operator correlation
Electric and Magnetic Interactions as Two Projections of Operator Correlation
Why do like charges repel each other, opposite charges attract each other, and why is the opposite rule true for parallel currents: like currents attract each other, while opposite currents repel each other? In this article, we won't start with the ready-made laws of Coulomb and Ampere. First, we'll try to find a quantity within the particle operator that can distinguish between like and opposite directions of motion.
The main question here is very simple: what exactly needs to be compared between two particles to determine the sign of their interaction? The difference in state is not suitable for this. If the states of two like charges are completely identical, their difference is zero, but the electrical interaction does not disappear. Therefore, we need a quantity that remains non-zero and positive when the directions coincide, but changes sign when one of the directions reverses.
This quantity is the product of the directed components of motion. Below, we will show that it is not artificially introduced into the model, but is precisely extracted from two natural products of operators: \(J_1\overline J_2\) and \(J_1J_2\).
The distance between interacting objects is denoted throughout by \(\ell\), not by \(r\). This is done to avoid confusion with the radius of a particle, internal motion, or orbit.
1. What parts of motion does the operator contain?
The state of each particle is determined by the operator.
\[ \tag{1} J_k = \j^{a_k}(-\j)^{b_k} = \ep e^{i\pi b_k} + \em e^{i\pi a_k}. \] The parameter \(a_k\) describes the internal state of the particle, and the parameter \(b_k\) describes its external motion. For the external parameter, we take
\[ \tag{2} b_k = \frac{\arcsin\beta_k}{\pi}, \qquad \beta_k = \frac{v_k}{c}. \] Therefore, the sine of the external phase is directly equal to the directional velocity in units of \(c\):
\[ \tag{3} \sin(\pi b_k) = \beta_k. \] The imaginary part of the operator contains the sine components of both phases:
\[ \tag{4} \operatorname{Im}J_k = \ep\beta_k + \em\sin(\pi a_k). \] It is these components that change sign when the direction of motion is reversed. The external component \(\beta_k\) changes sign when \(v_k\) is replaced by \(-v_k\). The internal component \(\sin(\pi a_k)\) changes sign when the internal orientation reverses. Therefore, to determine the relative directionality of two states, it is natural to compare their imaginary parts.
2. Why we need a product, not a difference
Let's denote the directional internal components of two particles by \(\eta_1\) and \(\eta_2\). If the directions coincide, their difference vanishes:
\[ \tag{5} \eta_1=\eta_2 \quad\Longrightarrow\quad \eta_1-\eta_2=0. \] But zero difference does not mean no interaction. Two identical charges should repel each other. Therefore, the difference only shows the degree of difference between the states, but cannot serve as a measure of the sign of the interaction.
The product behaves differently:
\[ \tag{6} \begin{aligned} \eta_1\eta_2\gt0 &\quad\text{for the same directions}, \\ \eta_1\eta_2\lt0 &\quad\text{for opposite directions}. \end{aligned} \] When the states are completely identical, the product does not vanish. When one of the directions reverses, it changes sign. This is precisely the property required of a sign characteristic of pairwise interaction.
Therefore, we seek not the distance between two operator states, but the correlation of their directional components.
3. Why this particular correlation was chosen
Let's first consider a common trigonometric identity:
\[ \tag{7} \cos(x-y)-\cos(x+y) = 2\sin x\sin y. \] Its meaning is especially important here. The cosine of the phase difference contains the sum of two parts: the product of the cosines and the product of the sines. The cosine of the sum contains the same products, but the sine part appears with the opposite sign. Therefore, subtraction cancels the cosine components and leaves only the product of the sines, that is, the product of the directed motions.
In operator form, the phase difference is found in the product with conjugation:
\[ \tag{8} J_1\overline J_2 = \ep e^{i\pi(b_1-b_2)} + \em e^{i\pi(a_1-a_2)}. \] The sum of the phases is found in the direct product:
\[ \tag{9} J_1J_2 = \ep e^{i\pi(b_1+b_2)} + \em e^{i\pi(a_1+a_2)}. \] Subtract the real part of the second product from the real part of the first. By identity (7), the cosine components cancel out, leaving only the products of the sines. This is how correlation arises.
\[ \tag{10} \boxed{ \begin{aligned} \mathcal C_{12} &= \frac12\operatorname{Re} \left( J_1\overline J_2-J_1J_2 \right) \\ &= \operatorname{Im}J_1 \operatorname{Im}J_2. \end{aligned} } \] Thus, this correlation is not specifically tailored to Coulomb's and Ampère's laws. It is the simplest combination of two natural products of operators, removing the position components and leaving the product of the directional components of motion.
In other words, \(J_1\overline J_2\) reports the phase difference, \(J_1J_2\) reports their sum, and the difference between these two quantities highlights how the directions of motion relate to each other.
4. Automatic Separation of Two Interactions
We substitute the imaginary parts of the operators from formula (4) into the correlation. Because the idempotents are orthogonal (ep=0), mixed products vanish:
\[ tag{11} \mathcal C_{12} = ep\beta_1\beta_2 + \em \sin(pi a_1) \sin(pi a_2). \] This formula itself splits into two independent parts:
\[ \tag{12} \begin{aligned} \mathcal C_{12}^{\mathrm{ext}} &= \ep\beta_1\beta_2, \\ \mathcal C_{12}^{\mathrm{int}} &= \em \sin(\pi a_1) \sin(\pi a_2). \end{aligned} \] The external projection compares the directions of translational motion. The internal projection compares the directions of internal motion. They do not mix and therefore can account for two different physical manifestations of the same operator law.
5. Static Charges and Correspondence to Coulomb's Law
For stationary particles, external velocities are zero. Then only internal correlation remains. The model adopts the physical hypothesis: two stable opposite orientations of internal motion correspond to two signs of electric charge.
For periodic motion, it is convenient to use normalized correlation
\[ \tag{13} \sigma_{12} = 2 \left\langle \sin(\pi a_1) \sin(\pi a_2) \right\rangle. \] If the internal motions are coherent and have the same direction, then \(\sigma_{12}=+1\). This corresponds to identical charges. If one internal motion is reversed relative to the other, then \(\sigma_{12}=-1\), which corresponds to opposite charges.
The spatial kernel is not yet derived from the operator, but is taken from the well-known Coulomb law. Assuming that a positive radial force increases the distance \(\ell\), we obtain
\[ \tag{14} F_{E,\ell} = \frac{e^2\sigma_{12}} {4\pi\varepsilon_0\ell^2}. \] Here the rule is read directly from the formula sign:
\[ \tag{15} \begin{aligned} \sigma_{12}\gt0 &\quad\Longrightarrow\quad F_{E,\ell}\gt0 \quad\Longrightarrow\quad \text{repulsion}, \\ \sigma_{12}\lt0 &\quad\Longrightarrow\quad F_{E,\ell}\lt0 \quad\Longrightarrow\quad \text{attraction}. \end{aligned} \] Thus, the internal correlation gives the required sign of \(q_1q_2\). But the factors \(1/(4\pi\varepsilon_0)\) and \(1/\ell^2\) have not yet been derived from the operator here. They demonstrate the correspondence of the found sign to the already known Coulomb law.
6. Moving charges, current, and correspondence to Ampere's law
The external correlation projection contains the product of dimensionless velocities \(\beta_1\beta_2\). For two conductors, the current in each is determined not only by the carrier velocity but also by their linear charge density:
\[ \tag{16} I_k = \Lambda_kv_k = \Lambda_kc\beta_k. \] Therefore
\[ \tag{17} I_1I_2 = \Lambda_1\Lambda_2c^2 \beta_1\beta_2. \] It is the product of the currents, and not just the product of the carrier velocities, that determines the physical direction of interaction between conductors. This is especially important for negative carriers, for which the direction of the conventional current is opposite to the direction of particle motion.
For two parallel conductors, Ampere's well-known law is written in terms of the distance \(\ell\) as follows:
\[ \tag{18} \frac{F_{A,\ell}}{L} = -\frac{\mu_0I_1I_2} {2\pi\ell}. \] The minus sign before the product of the currents means that the Ampère sign rule is opposite to the Coulomb sign rule:
\[ \tag{19} \begin{aligned} I_1I_2\gt0 &\quad\Longrightarrow\quad F_{A,\ell}\lt0 \quad\Longrightarrow\quad \text{attraction}, \\ I_1I_2\lt0 &\quad\Longrightarrow\quad F_{A,\ell}\gt0 \quad\Longrightarrow\quad \text{repulsion}. \end{aligned} \] Consequently, the external correlation yields precisely the signed variable that, after transitioning to currents, enters into Ampère's law. As in the Coulomb case, the spatial factor \(1/(2\pi\ell)\) is still taken from well-known electrodynamics.
7. The Main Rule of Thumb
Now the result can be formulated without repetition inof this conclusion. First, the correlation of the directional components is calculated. Then, its physical projection is selected.
\[ \tag{20} \boxed{ \begin{aligned} \text{static charges:} &\quad \operatorname{sign}F_{E,\ell} = \operatorname{sign}\sigma_{12}, \\ \text{parallel currents:} &\quad \operatorname{sign}F_{A,\ell} = -\operatorname{sign}(I_1I_2). \end{aligned} } \] In all cases, \(F_\ell\gt0\) means an increase in the distance \(\ell\), i.e., repulsion. The value \(F_\ell\lt0\) signifies a decrease in distance, i.e., attraction.
For static charges, the same direction of internal motion results in positive correlation and repulsion; the opposite direction results in negative correlation and attraction. For currents, co-directionality results in a positive product \(I_1I_2\), but due to the minus sign in Ampere's law, the force is directed toward decreasing \(\ell\), i.e., the conductors are attracted.
8. What has been proven and what is accepted as a hypothesis
It has been precisely proven mathematically: the combination of the direct and conjugate products of operators yields the product of their imaginary parts. Thanks to the idempotent basis, this correlation is automatically separated into external and internal projections.
The physical hypothesis of the model is: the correspondence of two stable orientations of internal motion to two signs of electric charge. It is this hypothesis that allows us to link the internal correlation with the sign of the product \(q_1q_2\).
The verification correspondence is: the correct appearance of the product \(\beta_1\beta_2\), which after taking into account the carrier densities turns into \(I_1I_2\), as well as the correct rules of attraction and repulsion when substituting correlations into the well-known Coulomb and Ampere laws.
The following has not yet been deduced: the origin of the spatial dependences \(1/\ell^2\) and \(1/\ell\), the coefficients \(1/(4\pi\varepsilon_0)\) and \(\mu_0/(2\pi)\), as well as the absolute value of the elementary charge. To do this, it is necessary to construct the field generated by the integral of the operator \(J\).
Conclusion
The choice of correlation is determined not by the desire to obtain Coulomb's and Ampère's laws in advance, but by the internal logic of the operator. To determine the sign of the interaction, it is necessary to compare the directions of motion. The difference in states is unsuitable for this purpose, since it disappears when they coincide. The product of the directional components preserves its magnitude when they coincide and changes sign when one of the directions is reversed.
The combination \(J_1\overline J_2-J_1J_2\) is a natural way to extract this product: the first term contains the phase differences, the second contains the phase sums, and subtracting them removes the cosine components and leaves the products of the sine components.
\[ \tag{21} \boxed{ \mathcal C_{12} = \ep\beta_1\beta_2 + \em \sin(\pi a_1) \sin(\pi a_2) } \] As a result, the electric and magnetic interactions share a common correlation source, but remain two distinct physical projections. The internal projection is associated with the sign of the charge, and the external projection is associated with the direction of the current. A complete derivation of Coulomb's and Ampère's laws will require the next step: constructing a spatial field from the integral continuation of the operator.
The mathematical basis of the operator is discussed in the article "Euler's Formula in Split Geometry." The relationship between internal frequency and mass is discussed in the work "Mass as a Geometric Projection", and other applications of the model are collected in the section "Wave Electricity".