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 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=ea₀=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=-ε₀^-1nablatimesvec P, and the EDM expectation is langle hat{vec d}_{rm tot}rangle=-frac{ε₀}{2}int vec rtimes vec J_m d^3r=d_Bbig\[g_Elangle hat{vec J}_prangle/hbar+g_E^elangle hat{vec S}rangle/hbarbig\], 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 may be described by circulating magnetic probability currents, mirroring magnetic dipoles from circulating electric probability currents.