2026-07-30
Attraction and repulsion of charges and currents as two projections of operator correlation
Why do like electric charges repel each other, opposite charges attract each other, and the opposite rule applies for parallel currents: like currents attract each other, while opposite currents repel each other? To derive these signs from geometry, it's not enough to compare the positions or phases of two particles. It's necessary to isolate the directional components of their states and construct a correlation that is preserved when the directions coincide and changes sign when one of them reverses.
In a previous version of this paper, the charge sign was related to the reversal of the fundamental internal phase of the operator \(J(a,b)=\j^a(-\j)^b\). After introducing multilevel idempotent splitting, this assumption can be refined. The fundamental factor \(\j^a\) remains the same for the electron and positron, and the charge opposition is transferred to an independent two-sheet factor \(\jp^{\pm a/2}\). As a result, the electric and current correlations have different but consistent sources within the full particle operator.
\[\tag{1} \boxed{ \begin{aligned} \text{charge sign} &\longleftarrow \text{deep two-sheet bypass orientation},\\ \text{current direction} &\longleftarrow \text{charge sign}\times\text{transfer direction}. \end{aligned} } \] 1. Why Correlation, Not Difference, Is Needed
Let \(\xi_1\) and \(\xi_2\) denote two directional components. Their difference is unsuitable for determining the sign of the interaction: when the states are completely identical, it is zero, although the interaction of like charges does not disappear.
\[\tag{2} \xi_1=\xi_2 \quad\Longrightarrow\quad \xi_1-\xi_2=0. \] The product behaves differently. When the directions are the same, it is positive, but when one direction is reversed, it changes sign:
\[\tag{3} \boxed{ \begin{aligned} \xi_1\xi_2>0 &\quad\Longrightarrow\quad \text{same orientations},\\ \xi_1\xi_2<0 &\quad\Longrightarrow\quad \text{opposite orientations}. \end{aligned} } \] Therefore, we use not the distance between operator states, but the product of their directional components. For stationary charges, the orientations of the deep bypass are compared, and for currents, the directions of external motion are also compared.
2. Mathematical Basis of the Second Level
The mathematical construction of the second level is presented in detail in this paper. Here, we use only the results necessary for the physical derivation.
The second independent pair of idempotents satisfies the conditions
\[\tag{4} \pmp^2=\pmp, \qquad \pme^2=\pme, \qquad \pmp\pme=0, \qquad \pmp+\pme=1. \] Their difference forms the second hyperbolic unit:
\[\tag{5} \boxed{ \jp=\pmp-\pme, \qquad \jp^2=1. } \] Its continuous power has the form
\[\tag{6} \jp^y = \pmp+\pme e^{i\pi y}. \] For the two-periodic special case, the exponent \(y=\eta a/2\) is chosen, where \(\eta=\pm1\) specifies the orientation of deep phase accumulation:
\[\tag{7} \boxed{ Q_{\eta}(a) = \jp^{\eta a/2} = \pmp+\pme e^{i\eta\pi a/2}, \qquad \eta=\pm1. } \] The two orientations are mutually conjugate and inverse:
\[\tag{8} Q_{-\eta}(a) = \overline{Q_{\eta}(a)} = Q_{\eta}^{-1}(a), \qquad Q_{\eta}(a)Q_{-\eta}(a)=1. \] Both have unit norm. Therefore, reversing the deep orientation does not change the total norm and does not imply negative energy:
\[\tag{9} Q_{\eta}(a)\overline{Q_{\eta}(a)}=1. \] 3. Geometric mapping of two deep channels
Second-level algebra creates two independent child channels. In the physical model, they are associated with two close branches of a single closed circuit:
\[\tag{10} \pmp\longmapsto C_+, \qquad \pme\longmapsto C_-. \] In radial representation, these branches can be written as
\[\tag{11} r_+=r_0+\frac{\Delta r}{2}, \qquad r_-=r_0-\frac{\Delta r}{2}, \qquad \Delta r=r_+-r_-. \] These are not two separate waves or two charge states. One wave sequentially passes through both branches and the transitions connecting them. After the first spatial revolution, the fundamental phase has already repeated, but the deep leaf has changed; the full configuration is restored after the second revolution.
The sign \(\eta\) determines not the initial branch, which can be changed by shifting the phase zero, but the orientation of the entire deep cycle. If
\[\tag{12} \chi_{\eta}(a) = \eta\frac{\pi a}{2}, \] then the invariant orientation feature is
\[\tag{13} \boxed{ \eta = \operatorname{sgn} \left( \frac{d\chi_{\eta}}{dt} \right). } \] 4. The Complete Electron and Positron Operator
We represent the complete multilevel particle operator as a product of three independent factors:
\[\tag{14} \boxed{ J_{\eta}(a,b) = \j^a(-\j)^b\jp^{\eta a/2}. } \] The factor \(\j^a\) describes the general internal wave dynamics, \((-\j)^b\) describes the external motion of the particle's center, and \(\jp^{\eta a/2}\) describes the orientation of the additional two-sheet closure.
We choose the following correspondence between an electron and a positron:
\[\tag{15} \boxed{ \begin{aligned} J_{e^-}(a,b) &= \j^a(-\j)^b\jp^{a/2},\\ J_{e^+}(a,b) &= \j^a(-\j)^b\jp^{-a/2}. \end{aligned} } \] The usual sign of electric charge is associated with orientation by convention.
\[\tag{16} \boxed{ q_{\eta}=-e\eta. } \] The orientation \(\eta\) should not be identified with the observed spin-up and spin-down states. Charge and spin are independent properties. Both spin states must exist for both the electron and the positron; they are determined by the distribution of the total internal state between the \(J(a,0)\) and \(J(0,a)\) modes, while \(\eta\) distinguishes charge-conjugate second-level round-trips.
5. Charge Correlation of Deep Operators
Consider the deep operators of two particles:
\[\tag{17} Q_k = \pmp+\pme e^{i\chi_k}, \qquad \chi_k = \eta_k\frac{\pi a_k}{2}. \] The product with conjugation contains the deep phase difference:
\[\tag{18} Q_1\overline{Q_2} = \pmp+\pme e^{i(\chi_1-\chi_2)}. \] The direct product contains their sum:
\[\tag{19} Q_1Q_2 = \pmp+\pme e^{i(\chi_1+\chi_2)}. \] Using the identity
\[\tag{20} \cos(x-y)-\cos(x+y) = 2\sin x\sin y. \] The difference between the real parts of the direct and conjugate products removes the position components and leaves the product of the directed deep components:
\[\tag{21} \boxed{ \begin{aligned} \mathcal C^{(q)}_{12} &= \frac12\operatorname{Re} \left( Q_1\overline{Q_2}-Q_1Q_2 \right)\\ &= \pme\sin\chi_1\sin\chi_2. \end{aligned} } \] Thus, charge correlation is not introduced by sign selection. It is distinguished by the same combination of direct and conjugate products that distinguishes the product of two sine components in ordinary trigonometry.
6. The Sign of Interaction of Fixed Charges
For two consistent fundamental cycles, we assume \(a_1=a_2=a\). Then
\[\tag{22} \mathcal C^{(q)}_{12} = \pme\eta_1\eta_2 \sin^2\left(\frac{\pi a}{2}\right). \] The normalized average over a complete two-turn cycle is equal to
\[\tag{23} \boxed{ \sigma_{q,12} = 2\left\langle \sin\chi_1\sin\chi_2 \right\rangle = \eta_1\eta_2. } \] For the same deep bypass orientations, the correlation is positive, and for opposite ones, it is negative:
\[\tag{24} \boxed{ \begin{aligned} \eta_1=\eta_2 &\quad\Longrightarrow\quad \sigma_{q,12}=+1 \quad\Longrightarrow\quad \text{repulsion},\\ \eta_1=-\eta_2 &\quad\Longrightarrow\quad \sigma_{q,12}=-1 \quad\Longrightarrow\quad \text{attraction}. \end{aligned} } \] Since \(q_k=-e\eta_k\), the product of the charges has the same relative sign:
\[\tag{25} q_1q_2 = e^2\eta_1\eta_2, \qquad \boxed{ \operatorname{sgn}(q_1q_2) = \eta_1\eta_2. } \] Thus, the geometry determines the relative interaction rule. The choice of which of the two orientations is called the electron with charge \(-e\) and which the positron with charge \(+e\) remains a convention and does not affect the observed attraction or repulsion.
7. Attaching the Correlation Sign to the Geometrical Origin of Electric Force
The spatial dependence of potential energy and force has already been obtained in this article. There, the difference in projections of two close internal branches leads to the potential kernel \(1/\ell\) in the far field, and its external gradient to the law \(1/\ell^2\). This conclusion is not repeated here: the deep correlation sign \(\sigma_{q,12}=\eta_1\eta_2\) is attached to the already found positive modulus.
\[\tag{26} U_0(\ell) \approx E_*\frac{\Delta r}{\ell}, \qquad F_0(\ell) = -\frac{dU_0}{d\ell} = E_*\frac{\Delta r}{\ell^2}. \] The total potential energy of the pair and the corresponding radial force are given by
\[\tag{27} \boxed{ U_{12}(\ell) \approx \eta_1\eta_2 E_*\frac{\Delta r}{\ell}, } \] \[\tag{28} \boxed{ F_{E,\ell} = -\frac{dU_{12}}{d\ell} \approx \eta_1\eta_2 E_*\frac{\Delta r}{\ell^2}. } \] The direction of increasing distance \(\ell\) is chosen to be positive. Therefore \(F_{E,\ell}>0\) means repulsion, and \(F_{E,\ell}<0\) means attraction.ie.
For elementary particles, the previously obtained relations are used.
\[\tag{29} E_*=m_ec^2, \qquad \Delta r_{\mathrm{eff}}=r_e, \qquad m_ec^2r_e = \alpha_{\mathrm{fs}}\hbar c = \frac{e^2}{4\pi\varepsilon_0}. \] As a result, the electric force is written as
\[\tag{30} \boxed{ F_{E,\ell} = \eta_1\eta_2 \frac{\alpha_{\mathrm{fs}}\hbar c}{\ell^2} = \eta_1\eta_2 \frac{e^2}{4\pi\varepsilon_0\ell^2} = \frac{q_1q_2}{4\pi\varepsilon_0\ell^2}. } \] Thus, the geometry of the two branches determines the magnitude and spatial dependence of the force, and the correlation of the two-sheeted bypasses determines its sign.
8. Moving charge and current direction
The external motion of a particle is determined by the parameter \(b_k\):
\[\tag{31} b_k = \frac{\arcsin\beta_k}{\pi}, \qquad \beta_k = \frac{v_k}{c}, \qquad \sin(\pi b_k)=\beta_k. \] The direction of the current is determined not by velocity alone, but by charge transfer. For an elementary particle
\[\tag{32} q_kv_k = -e\eta_kc\beta_k. \] Let's define the dimensionless orientation of the conventional current:
\[\tag{33} \boxed{ \iota_k = \frac{q_kv_k}{ec} = -\eta_k\beta_k. } \] For an electron, \(\eta=+1\), so the conventional current is directed opposite to the electron's velocity. For a positron, \(\eta=-1\), and the direction of the conventional current coincides with its direction of motion. Therefore, the product \(\beta_1\beta_2\) is insufficient if the interacting flows can contain carriers of different signs.
9. Carrier flow and macroscopic current
Let \(n_k\) be the linear number density of identical carriers. Then the signed linear charge density is
\[\tag{34} \Lambda_k = n_kq_k = -en_k\eta_k. \] The current they create is
\[\tag{35} \boxed{ I_k = \Lambda_kv_k = -en_kc\eta_k\beta_k. } \] If a current is created by several types of carriers, their contributions add up:
\[\tag{36} \boxed{ I = \sum_s n_sq_sv_s. } \] In an electrically neutral conductor, the static contributions of positive and negative charges cancel out on average. However, moving carriers create a non-zero directional current, so the magnetic interaction between the conductors is preserved.
10. Current Correlation and Compliance with Ampere's Law
For two flows of elementary carriers of the same magnitude, the product of the currents is
\[\tag{37} I_1I_2 = e^2n_1n_2c^2 \eta_1\eta_2\beta_1\beta_2. \] Therefore, directional current correlation combines two independent features: charge orientation and the direction of external motion.
\[\tag{38} \boxed{ \sigma_{I,12} \propto \eta_1\eta_2\beta_1\beta_2 = \sigma_{q,12}\sigma_{v,12}. } \] For two parallel conductors, the known Ampere force per unit length is written as
\[\tag{39} \boxed{ \frac{F_{A,\ell}}{L} = -\frac{\mu_0I_1I_2}{2\pi\ell}. } \] The minus sign before the product of the currents means that the magnetic interaction rule is opposite to the electrostatic one:
\[\tag{40} \boxed{ \begin{aligned} \eta_1\eta_2\beta_1\beta_2>0 &\quad\Longrightarrow\quad I_1I_2>0 \quad\Longrightarrow\quad \text{attraction},\\ \eta_1\eta_2\beta_1\beta_2<0 &\quad\Longrightarrow\quad I_1I_2<0 \quad\Longrightarrow\quad \text{repulsion}. \end{aligned} } \] For example, an electron and a positron moving in the same spatial direction have \(\beta_1\beta_2>0\), but \(\eta_1\eta_2=-1\). Their conventional currents are opposite. If the electron and positron move in opposite directions, then their conventional currents are codirectional.
11. Two Correlations of a Complete Multilevel State
The electric and magnetic interactions should no longer be represented as two components of only the first level \(J(a,b)\). The charge correlation is at a deep level, and the direction of transport is in the outer phase of the first level:
\[\tag{41} \boxed{ \begin{aligned} \jp^{\eta a/2} &\longrightarrow \text{charge sign},\\ (-\j)^b &\longrightarrow \text{direction of movement},\\ \jp^{\eta a/2}(-\j)^b &\longrightarrow \text{current direction}. \end{aligned} } \] The general rule of thumb is
\[\tag{42} \boxed{ \begin{aligned} \text{stationary charges:} &\qquad \operatorname{sgn}F_{E,\ell} = \operatorname{sgn}(\eta_1\eta_2),\\ \text{parallel currents:} &\qquad \operatorname{sgn}F_{A,\ell} = -\operatorname{sgn} \left( \eta_1\eta_2\beta_1\beta_2 \right). \end{aligned} } \] In the first case, the orientations of the two-sheet charge cycles are compared. In the second, the directions of transport of these cycles in external space are compared.
12. What follows from mathematics and what remains a physical mapping
The following follow from second-level mathematics: the existence of a pair of independent idempotents \(\pmp,\pme\); two conjugate powers \(\jp^{\pm a/2}\); their unit norm and mutual invertibility; deep phase \(\pm\pi a/2\); the difference in state after the first rotation and its restoration after the second; isolation of the product \(\sin\chi_1\sin\chi_2\) by a combination of the direct and conjugate products.
Within the adopted physical mapping, the following are obtained: a second-level representation by two close branches; the relationship of the orientation \(\eta\) with the sign of the charge; relative rule \(\sigma_{q,12}=\eta_1\eta_2\); combining charge orientation with external motion into current correlation \(\eta_1\eta_2\beta_1\beta_2\).
Additional physical conditions remain: the specific shape of the two spatial branches; the magnitude of their effective separation \(\Delta r_{\mathrm{eff}}=r_e\); associating the chosen orientation with the electron, and the conjugate with the positron; application of operator correlation to the sign of the potential energy; full geometric derivation of the magnetic spatial core for an arbitrary current configuration.
The dependence of the electric force \(1/\ell^2\) is not borrowed here from Coulomb's law and is not re-derived. It was obtained from the gradient of the difference of geometric projections in a separate paper. This paper adds to this modulus a sign arising from the correlation of deep bypasses.
Conclusion
Multilevel idempotent splitting allows one to separate the fundamental internal dynamics of a particle from its charge orientation. The electron and positron share a common factor \(\j^a\), but differ in the conjugate directions of the deep two-sheet cycle \(\jp^{\pm a/2}\).
The direct and conjugate products of deep operators isolate the product of the directed components. After averaging over a full cycle, this yields the correlation.
\[\tag{43} \boxed{ \sigma_{q,12} = \eta_1\eta_2. } \] This value is added to the previously derived geometric modulus of the electric force:
\[\tag{44} \boxed{ F_{E,\ell} = \eta_1\eta_2 \frac{\alpha_{\mathrm{fs}}\hbar c}{\ell^2} = \frac{q_1q_2}{4\pi\varepsilon_0\ell^2}. } \] For a moving charge, the direction of external motion is added to the deep cycle orientation. Therefore, the current correlation contains the product \(\eta_1\eta_2\beta_1\beta_2\), and the force between parallel currents has the opposite sign rule:
\[\tag{45} \boxed{ \frac{F_{A,\ell}}{L} = -\frac{\mu_0I_1I_2}{2\pi\ell}. } \] Thus, a stationary charge is characterized by the orientation of a two-sheeted circuit, and the current is characterized by the transfer of this orientation in external space. The attraction and repulsion of charges and currents become two coordinated manifestations of a multilevel operator correlation.