2026-07-30
Attraction and repulsion of charges and currents as two projections of operator correlation
Why do like electric charges repel and opposite charges attract, while parallel currents obey the opposite rule: currents flowing in the same direction attract, and those flowing in opposite directions repel? To obtain these signs from geometry, comparing only the positions or phases of two particles is insufficient. One must identify signed components of their states and construct a correlation that is preserved for matching orientations and changes sign when one orientation is reversed.
In the current formulation of the model, internal phase dynamics, two-sheeted closure, and charge must be specified separately. The dynamical operator \(J_\sigma(a,b)\) describes the internal state and external motion in one of two channel realizations; \(Q_\Gamma(a)\) specifies the two-sheeted extension, and \(J_{q,\eta}\) represents an independent charge sector. Here, \(\sigma=1,2\) labels the realization of the spin channels, while \(\eta=\pm1\) is the charge label. This notation is consistent with the concept, the article on spin, and the revised description of the propagating wave.
\[\tag{1} \boxed{\begin{aligned} \text{charge sign}&\longleftarrow\text{charge sector }J_{q,\eta},\\ \text{two-sheeted closure}&\longleftarrow Q_\Gamma(a),\\ \text{current direction}&\longleftarrow\text{charge sign}\times\text{transport direction}. \end{aligned}} \] 1. Why a Correlation Is Needed Rather Than a Difference
Let \(\xi_1\) and \(\xi_2\) denote nonzero real signed components. Their simple difference is unsuitable as the sole criterion for the sign of an interaction: it vanishes when the components coincide, although like charges can interact. This observation does not exclude other functions of the distance between states; rather, it motivates the choice of a bilinear criterion:
\[\tag{2} \xi_1=\xi_2 \quad\Longrightarrow\quad \xi_1-\xi_2=0. \] The product behaves differently. It is positive for matching orientations and changes sign when one orientation is reversed:
\[\tag{3} \boxed{ \begin{aligned} \xi_1\xi_2>0 &\quad\Longrightarrow\quad \text{matching orientations},\\ \xi_1\xi_2<0 &\quad\Longrightarrow\quad \text{opposite orientations}. \end{aligned} } \] The phase correlation of the internal branches must henceforth be distinguished from the charge correlation. Charge labels are used for stationary charges; transport velocities are additionally used for currents. The algebraic product of signs does not yet establish which sign of the potential energy corresponds to attraction or repulsion.
2. Mathematical Foundation of the Second Level
The mathematical construction of the second level is given in this article. Here, it is used to describe closure and phase correlations, without identifying that level with electric charge.
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 defines a 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-period special case, the exponent \(y=a/2\) is chosen. The closure factor has the form
\[\tag{7} \boxed{Q_\Gamma(a)=\jp^{a/2}=\pmp+\pme e^{i\pi a/2}.} \] The conjugate factor is the inverse:
\[\tag{8} \overline{Q_\Gamma(a)}=Q_\Gamma^{-1}(a)=\jp^{-a/2},\qquad Q_\Gamma(a)Q_\Gamma^{-1}(a)=1. \] Algebraic unitarity holds. It must be distinguished from the scalar norm of a state vector and from conservation of physical energy. Passing to the inverse factor is not, by itself, charge conjugation:
\[\tag{9} Q_\Gamma(a)\overline{Q_\Gamma(a)}=1. \] 3. Geometric Mapping of the Two Deeper-Level Channels
The second-level algebra creates two independent child channels. In the physical model, they are mapped to two nearby branches of a single closed contour:
\[\tag{10} \pmp\longmapsto C_+, \qquad \pme\longmapsto C_-. \] In a 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 neither two separate waves nor two charge states. A single wave sequentially traverses both branches and the transitions connecting them. After the first spatial revolution, the primary phase has already repeated, but the deeper-level sheet has changed; the full configuration is restored after the second revolution.
In the chosen phase convention, the deeper-level phase is
\[\tag{12} \chi(a)=\frac{\pi a}{2}, \] Its rate of change is determined by the internal dynamics, rather than the charge label:
\[\tag{13} \boxed{\frac{d\chi}{dt}=\frac{\pi}{2}\frac{da}{dt}.} \] 4. The Full Electron and Positron Operator
In the reduced description considered here, represent the full particle operator by independent dynamical, closure, and charge factors:
\[\tag{14} \boxed{\mathcal J_{\sigma,\eta}(a,b)=J_\sigma(a,b)Q_\Gamma(a)J_{q,\eta},\qquad\sigma=1,2,\quad\eta=\pm1.} \] Here, \(J_1(a,b)=J(a,b)\) and \(J_2(a,b)=J(b,a)\); in both realizations, \(a\) retains its meaning as the internal phase parameter, and \(b\) as the external motion parameter. The factor \(Q_\Gamma\) specifies two-sheeted closure; \(J_{q,\eta}\) specifies a separate charge sector. The independent factors act on different spaces and are multiplied after extending them by identity operators on the remaining factors. States of different particles are combined by a tensor product.
The electron and positron may share the same dynamical realization and closure factor, while differing in their charge sectors:
\[\tag{15} \boxed{\begin{aligned} \mathcal J_{e^-,\sigma}(a,b)&=J_\sigma(a,b)Q_\Gamma(a)J_{q,+},\\ \mathcal J_{e^+,\sigma}(a,b)&=J_\sigma(a,b)Q_\Gamma(a)J_{q,-}. \end{aligned}} \] Associate the measured charge with the label \(\eta\) by convention, where \(e>0\):
\[\tag{16} \boxed{ q_{\eta}=-e\eta. } \] The label \(\eta\) does not denote spin-up or spin-down. Both realizations \(\sigma=1,2\) are allowed for the electron and the positron. Exchanging the primary channels does not change charge, and inverting \(Q_\Gamma\) does not automatically specify a transition between charge sectors. A possible physical connection between the charge sign and the orientation of the internal contour requires a separate law; it is not derived in this article.
5. Phase Correlation of the Closure Factors
First, consider only the phase factors describing closure for two particles:
\[\tag{17} Q_{\Gamma,k}=\pmp+\pme e^{i\chi_k},\qquad\chi_k=\frac{\pi a_k}{2}. \] The product involving conjugation contains the difference of the deeper-level phases:
\[\tag{18} Q_{\Gamma,1}\overline{Q_{\Gamma,2}}=\pmp+\pme e^{i(\chi_1-\chi_2)}. \] The direct product contains their sum:
\[\tag{19} Q_{\Gamma,1}Q_{\Gamma,2}=\pmp+\pme e^{i(\chi_1+\chi_2)}. \] Use the identity
\[\tag{20} \cos(x-y)-\cos(x+y) = 2\sin x\sin y. \] The difference between the real parts of the conjugated and direct products isolates the product of two sine components. Denote the resulting quantity as the closure phase correlation:
\[\tag{21} \boxed{\begin{aligned} \mathcal C^{(\Gamma)}_{12}&=\frac12\operatorname{Re}\left(Q_{\Gamma,1}\overline{Q_{\Gamma,2}}-Q_{\Gamma,1}Q_{\Gamma,2}\right)\\ &=\pme\sin\chi_1\sin\chi_2. \end{aligned}} \] Equation (21) is an exact trigonometric result, but it does not determine the sign of electric charge. For \(\chi_2=\chi_1+\delta\), uniform averaging over a period gives \(2\langle\sin\chi_1\sin\chi_2\rangle=\cos\delta\). The result depends on the relative phase and can change sign or vanish. Without an additional physical constraint, this correlation therefore cannot be identified with the universal sign of an electrostatic interaction.
6. Independent Charge Correlation
Introduce the charge-label observable \(\widehat\eta\) with eigenvalues \(\pm1\) and the charge operator \(\widehat q=-e\widehat\eta\). For normalized charge states \(|\eta_k\rangle\), assume \(\widehat\eta|\eta_k\rangle=\eta_k|\eta_k\rangle\). This is a separate representation of a physical attribute, rather than a derivation of charge from \(Q_\Gamma\). For definite charges, the normalized charge correlation is
\[\tag{22} C_{q,12}=\frac{q_1q_2}{e^2}=\eta_1\eta_2, \] For two particles with definite charges, set \(|\Psi_{12}^{(q)}\rangle=|\eta_1\rangle\otimes|\eta_2\rangle\). The same correlation can then be expressed in terms of the charge sector:
\[\tag{23} \boxed{C_{q,12}=\langle\Psi_{12}^{(q)}|\widehat\eta\otimes\widehat\eta|\Psi_{12}^{(q)}\rangle=\eta_1\eta_2.} \] Matching charge labels give a positive correlation; opposite labels give a negative one. After adopting the rule connecting the correlation to potential energy in Section 7, this corresponds to
\[\tag{24} \boxed{ \begin{aligned} \eta_1=\eta_2 &\quad\Longrightarrow\quad C_{q,12}=+1 \quad\Longrightarrow\quad \text{repulsion},\\ \eta_1=-\eta_2 &\quad\Longrightarrow\quad C_{q,12}=-1 \quad\Longrightarrow\quad \text{attraction}. \end{aligned} } \] Since \(q_k=-e\eta_k\), the product of charges has the same relative sign:
\[\tag{25} q_1q_2 = e^2\eta_1\eta_2, \qquad \boxed{ \operatorname{sgn}(q_1q_2) = \eta_1\eta_2. } \] The relative sign of the charge product is independent of internal phases and of the choice of their phase origins. Assigning \(\eta=+1\) to the electron with charge \(-e\) and \(\eta=-1\) to the positron with charge \(+e\) is a convention. This convention does not replace the physical law relating the charge product to interaction energy.
7. Combining the Correlation Sign with the Geometric Force Magnitude
In a separate article, the difference between projections of nearby internal branches yields the potential kernel \(1/\ell\) in the far-field region once an energy mapping is adopted. Here, the positive kernel \(U_0\) and its gradient are considered. The relation \(U_{12}=C_{q,12}U_0\) is adopted as a separate physical interaction rule; it does not yet follow from idempotent algebra alone.
\[\tag{26} U_0(\ell) \approx E_*\frac{\Delta r}{\ell}, \qquad F_0(\ell) = -\frac{dU_0}{d\ell} \approx E_*\frac{\Delta r}{\ell^2}. \] Under this rule, the pair’s potential energy and radial force in the far-field region have the form
\[\tag{27} \boxed{ U_{12}(\ell) \approx \eta_1\eta_2 E_*\frac{\Delta r}{\ell}, } \] \[\tag{28} \boxed{ F_{E,\ell} = -\frac{dU_{12}}{d\ell} \approx \eta_1\eta_2 E_*\frac{\Delta r}{\ell^2}. } \] The direction of increasing separation \(\ell\) is chosen as positive. Thus, \(F_{E,\ell}>0\) means repulsion, while \(F_{E,\ell}<0\) means attraction.
For the electron and positron realizations, use the adopted energy scale and the condition on the effective branch separation:
\[\tag{29} E_*=m_ec^2, \qquad \Delta r_{\mathrm{eff}}=r_e, \qquad m_ec^2r_e = \alpha_{\mathrm{fs}}\hbar c = \frac{e^2}{4\pi\varepsilon_0}. \] The electric force is consequently written as
\[\tag{30} \boxed{ F_{E,\ell} \approx \eta_1\eta_2 \frac{\alpha_{\mathrm{fs}}\hbar c}{\ell^2} = \eta_1\eta_2 \frac{e^2}{4\pi\varepsilon_0\ell^2} = \frac{q_1q_2}{4\pi\varepsilon_0\ell^2}. } \] Thus, in the far-field region, the chosen geometry and energy mapping yield the force magnitude, while the separate rule \(U_{12}=C_{q,12}U_0\) relates its sign to the charge correlation. The condition \(\Delta r_{\mathrm{eff}}=r_e\) provides the Coulomb normalization and requires independent justification.
8. A Moving Charge and the Direction of Current
The carrier velocities below refer to a single chosen reference frame. In each realization \(\sigma\), the external mapping reads the channel containing \(b\), rather than a fixed idempotent regardless of the channel exchange.
The external motion of a particle is specified 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 current direction is determined by charge transport, rather than velocity alone. For an elementary particle,
\[\tag{32} q_kv_k = -e\eta_kc\beta_k. \] Define a dimensionless charge-transport quantity. This is not a current in amperes, but a normalized product of charge and velocity:
\[\tag{33} \boxed{ \iota_k = \frac{q_kv_k}{ec} = -\eta_k\beta_k. } \] For the electron, \(\eta=+1\), so negative-charge transport, expressed in the conventional-current convention, is opposite to the carrier velocity. For the positron, \(\eta=-1\), and these directions coincide. The product \(\beta_1\beta_2\) alone is insufficient if the flows contain carriers with different charge signs. All velocities and signed currents are compared using consistent orientations of the longitudinal axes.
9. Carrier Flow and Macroscopic Current
Let \(n_k\) be the linear number density of identical carriers. The signed linear charge density is then
\[\tag{34} \Lambda_k = n_kq_k = -en_k\eta_k. \] The current they produce has the form
\[\tag{35} \boxed{ I_k = \Lambda_kv_k = -en_kc\eta_k\beta_k. } \] If several carrier species contribute to the current, their contributions add:
\[\tag{36} \boxed{ I = \sum_s n_sq_sv_s. } \] In an electrically neutral conductor, the static contributions of positive and negative charges cancel on average. Moving carriers can nevertheless produce a nonzero directed current, so the magnetic interaction between conductors remains.
10. Current Correlation and Agreement with Ampère’s Force Law
For two flows of elementary carriers with equal charge magnitudes, the product of currents is
\[\tag{37} I_1I_2 = e^2n_1n_2c^2 \eta_1\eta_2\beta_1\beta_2. \] The current correlation combines independent charge labels and velocities. Denote \(C_{v,12}=\beta_1\beta_2\) and define
\[\tag{38} \boxed{C_{I,12}=\iota_1\iota_2=\eta_1\eta_2\beta_1\beta_2=C_{q,12}C_{v,12}.} \] For long parallel conductors carrying steady currents in vacuum, in the magnetostatic approximation and far from their ends, the standard magnetic force per unit length has the form
\[\tag{39} \boxed{ \frac{F_{A,\ell}}{L} = -\frac{\mu_0I_1I_2}{2\pi\ell}. } \] The minus sign follows from the adopted magnetic force law and the spatial orientation of the field and force, rather than from the product of charge labels alone. In this article, Ampère’s force formula is used to make the current correlation consistent with the standard magnetic response:
\[\tag{40} \boxed{ \begin{aligned} \eta_1\eta_2\beta_1\beta_2>0 &\quad\Longrightarrow\quad I_1I_2>0 \quad\Longrightarrow\quad \text{attraction},\\ \eta_1\eta_2\beta_1\beta_2<0 &\quad\Longrightarrow\quad I_1I_2<0 \quad\Longrightarrow\quad \text{repulsion}. \end{aligned} } \] For electron and positron flows moving in the same direction, \(\beta_1\beta_2>0\), but \(\eta_1\eta_2=-1\), so their signed currents are opposite. If their motion is in opposite directions, the currents point in the same direction. This illustrates the sign rule for the magnetic component of parallel-flow interactions; it does not determine the total force between individual charged particles, which also includes electric interaction.
Equations (39)–(40) describe the magnetic component of the interaction between long parallel currents. For neutral conductors, all charge and current contributions must be summed. In general, the total force may include electric fields, end effects, and other influences.
11. Two Correlations of the Full Multilevel State
The charge correlation and transport velocity cannot be extracted from a single sign of the deeper-level phase. In the adopted formulation, charge belongs to \(J_{q,\eta}\), closure to \(Q_\Gamma\), and external velocity to the parameter \(b\) of the dynamical operator:
\[\tag{41} \boxed{\begin{aligned} J_{q,\eta}&\longrightarrow\text{charge label }\eta,\\ Q_\Gamma(a)&\longrightarrow\text{two-sheeted closure},\\ b&\longrightarrow\beta=\sin(\pi b),\\ (q,\beta)&\longrightarrow\iota=(q/e)\beta. \end{aligned}} \] The principal practical rule is
\[\tag{42} \boxed{ \begin{aligned} \text{stationary charges:} &\qquad \operatorname{sgn}F_{E,\ell} = \operatorname{sgn}(\eta_1\eta_2),\\ \text{parallel currents:} &\qquad \operatorname{sgn}F_{A,\ell} = -\operatorname{sgn} \left( \eta_1\eta_2\beta_1\beta_2 \right). \end{aligned} } \] The first case uses definite charge labels and the adopted electric energy rule. The second uses signed currents and the adopted magnetic force law. Neither rule requires synchronization of the carriers’ internal phases.
12. What Follows from Mathematics and What Remains a Physical Mapping
The mathematics of the second level implies: the form of \(Q_\Gamma\), its invertibility and algebraic unitarity; the deeper-level phase \(\pi a/2\); periodicity in \(a\); and identity (21) for the phase correlation. These properties do not determine the sign of electric charge or prove the force sign.
Specifying an independent charge sector yields: \(q_\eta=-e\eta\); the correlation of definite charges \(C_{q,12}=\eta_1\eta_2\); normalized transport \(\iota_k=-\eta_k\beta_k\); and the current correlation \(C_{I,12}=\eta_1\eta_2\beta_1\beta_2\). These are independent of the phase of \(Q_\Gamma\).
Additional physical conditions remain: the mapping of deeper-level channels onto spatial branches; their effective separation \(\Delta r_{\mathrm{eff}}=r_e\); the specific structure of the charge sector; the energy rule \(U_{12}=C_{q,12}U_0\); and a complete derivation of the magnetic field and force response from unified dynamics. Here, the magnetic force sign is made consistent with the standard Ampère force law, rather than independently derived from the phase correlation.
The electric-force dependence \(1/\ell^2\) is not derived again here. It is recovered in the far-field region from the gradient of the chosen energy projection in a separate work. This article defines an independent charge correlation and explicitly distinguishes it from the closure phase correlation and from the physical interaction law.
Conclusion
The multilevel description separates the dynamical operator, two-sheeted closure, and the charge sector. The electron and positron may have identical \(J_\sigma(a,b)\) and \(Q_\Gamma(a)\) while differing in \(J_{q,+}\) and \(J_{q,-}\). Neither the spin realization nor the initial internal phase replaces the charge label.
The direct and conjugated products of closure factors yield the phase correlation (21). The electric interaction sign is determined by a different object: the correlation of independent charge sectors:
\[\tag{43} \boxed{ C_{q,12} = \eta_1\eta_2. } \] After the energy rule is adopted, this quantity multiplies the geometric force magnitude in the far-field region:
\[\tag{44} \boxed{ F_{E,\ell} \approx \eta_1\eta_2 \frac{\alpha_{\mathrm{fs}}\hbar c}{\ell^2} = \frac{q_1q_2}{4\pi\varepsilon_0\ell^2}. } \] For a moving charge, the transport velocity supplements the charge label. The current correlation contains the product \(\eta_1\eta_2\beta_1\beta_2\). In the stated magnetostatic regime, the standard magnetic law gives
\[\tag{45} \boxed{ \frac{F_{A,\ell}}{L} = -\frac{\mu_0I_1I_2}{2\pi\ell}. } \] Charge correlation, closure phase correlation, and current correlation have different definitions. Separating them brings the article into agreement with the independent charge and closure sectors. Attraction and repulsion are reproduced after specifying the electric energy rule and the magnetic force law; deriving these rules from the model’s general dynamics remains the next task.