Dual Magnetic and Electric Dipole Symmetry: Pseudo Angular Momentum in Parity Space and the Electric Landé $g$-Factor

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 $\widehat{\vec{J}}$ and reduces $SO(4)$ to $SO(2)$ generated by ${\widehat{J}}_z$. A static $\vec{E}$ (polar) couples within a fixed $n$ to a scaled Runge-Lenz operator ${\widehat{\vec{A}}}_{\mathrm{s}\mathrm{c}}$, mixes parities, and preserves $SO(2)\times SO(2)$ generated by ${\widehat{J}}_z$ and ${\widehat{A}}_{\mathrm{s}\mathrm{c},z}$. This motivates a pseudo-angular momentum ${\widehat{\vec{J}}}_p$ built from ${\widehat{\vec{A}}}_{\mathrm{s}\mathrm{c}}$ and a Landé factor $g_E$, so the orbital dipole is ${\widehat{\vec{d}}}_{\mathrm{o}\mathrm{r}\mathrm{b}}=g_E{d}_B{\widehat{\vec{J}}}_p\mathrm{/}\mathrm{\hslash }$, with $d_B=e{a}_0=2{\mu }_B\mathrm{/}(c\alpha )$. Stark mixing of $2s$ and $2p_{m=0}$ gives $\mathrm{\mid }⟨d_{\mathrm{o}\mathrm{r}\mathrm{b}}⟩\mathrm{\mid }=3d_B$ ($g_E=3$). Following Ohanian’s magnetisation formalism, we construct its electric dual: the microscopic polarisation $\vec{P}$ has nonzero curl, defining a magnetic probability current ${\vec{J}}_m=-{\epsilon }_0^{-1}\mathrm{\nabla }\times \vec{P}$, and the EDM expectation is $⟨{\widehat{\vec{d}}}_{\mathrm{t}\mathrm{o}\mathrm{t}}⟩=-\frac{{\epsilon }_0}{2}\int \vec{r}\times {\vec{J}}_m{d}^{3}r=d_B[g_E⟨{\widehat{\vec{J}}}_p⟩\mathrm{/}\mathrm{\hslash }+g_E^e⟨\widehat{\vec{S}}⟩\mathrm{/}\mathrm{\hslash }]$, with $g_E^e=2d_{\mathrm{i}\mathrm{n}\mathrm{t}}\mathrm{/}{d}_B$. Here $\widehat{\vec{S}}$ encodes any intrinsic EDM $d_{\mathrm{i}\mathrm{n}\mathrm{t}}$, while ${\widehat{\vec{J}}}_p$ captures the Stark-induced pseudo-angular momentum from Runge-Lenz symmetry. The dual framework shows that induced EDMs may be described by circulating magnetic probability currents, mirroring magnetic dipoles from circulating electric probability currents.