Dual Magnetic and Electric Dipole Symmetry: Pseudo Angular Momentum in Parity Space and the Electric Landé $g$-Factor
Michael E. Tobar
TL;DR
This work develops a symmetry-based duality between magnetic and electric dipoles, casting induced EDMs as parity-space analogues of Zeeman physics through a pseudo-angular momentum $\hat{\vec{J}}_p$ built from the Runge-Lenz vector. In hydrogen, the Zeeman-like action of $\hat{\vec{J}}$ and the Stark-induced $\hat{\vec{A}}_{\rm sc}$ reveal a common structure: $SO(4)$ degeneracy is reduced to $SO(2)$ by $\vec{B}$ and parabolic mixing by $\vec{E}$ yields a $J_p$-based description with an electric Landé factor $g_E$. The dual Ohanian construction defines an internal polarization $\vec{P}$ whose curl gives a magnetic current $\vec{J}_m$, producing an EDM via $\langle \hat{\vec{d}}_{\rm tot}\rangle$, including a purely orbital piece $d_B$ and, if present, an intrinsic part with $g_E^{e}=2 d_{int}/d_B$. The results show a concrete, quantifiable link between parity mixing and circulating currents, clarifying how induced EDMs arise and providing a unified perspective that extends to polarization phenomena in other quantum systems. Overall, the dual framework highlights EM symmetry as a guiding principle for understanding dipole moments and their responses to external fields.
Abstract
EDMs probe fundamental symmetries and underpin BSM searches. We give a symmetry-based description, analogous to the Zeeman effect, that puts magnetic and electric dipoles on equal footing under EM duality. In hydrogen, $\vec B$ (pseudovector) couples to $\hat{\vec J}$ and reduces $SO(4)$ to $SO(2)$ generated by $\hat J_z$. A static $\vec E$ (polar) couples within a fixed $n$ to a scaled Runge-Lenz operator $\hat{\vec A}_{\rm sc}$, mixes parities, and preserves $SO(2)\times SO(2)$ generated by $\hat J_z$ and $\hat A_{{\rm sc},z}$. This motivates a pseudo-angular momentum $\hat{\vec J}_p$ built from $\hat{\vec A}_{\rm sc}$ and a Landé factor $g_E$, so the orbital dipole is $\hat{\vec d}_{\rm orb}=g_E d_B \hat{\vec J}_p/\hbar$, with $d_B=ea_0=2μ_B/(cα)$. Stark mixing of $2s$ and $2p_{m=0}$ gives $|\langle d_{\rm orb}\rangle|=3d_B$ ($g_E=3$). Following Ohanian's magnetisation formalism, we construct its electric dual: the microscopic polarisation $\vec P$ has nonzero curl, defining a magnetic probability current $\vec J_m=-ε_0^{-1}\nabla\times\vec P$, and the EDM expectation is $\langle \hat{\vec d}_{\rm tot}\rangle=-\frac{ε_0}{2}\int \vec r\times \vec J_m\, d^3r=d_B\big[g_E\langle \hat{\vec J}_p\rangle/\hbar+g_E^{e}\langle \hat{\vec S}\rangle/\hbar\big]$, with $g_E^{e}=2d_{\rm int}/d_B$. Here $\hat{\vec S}$ encodes any intrinsic EDM $d_{\rm int}$, while $\hat{\vec J}_p$ captures the Stark-induced pseudo-angular momentum from Runge-Lenz symmetry. The dual framework shows that induced EDMs arise from circulating magnetic probability currents, mirroring magnetic dipoles from circulating electric probability currents.