Dual Magnetic and Electric Dipole Symmetry: Pseudo Angular Momentum in Parity Space and the Electric Land\'e $g$-Factor

Abstract

Electric dipole moments (EDMs) are sensitive probes of fundamental symmetries and central to searches for physics beyond the Standard Model. We present a symmetry-based, Zeeman-analogue operator framework that places magnetic and electric dipole physics on parallel footing under electromagnetic duality, and introduce a polar-sector pseudo-angular-momentum degree of freedom in parity space together with an associated electric Land\'e factor that organizes induced orbital dipoles. Following Ohanian's effective-current formulation of the Zeeman effect, we construct its electric dual: the wavefunction's microscopic polarization admits an equivalent effective magnetic probability-current representation, providing a field-equivalence description of parity-mixed charge displacement. In this notation the total EDM expectation takes the unified form $\langle \hat{\vec d}_{\rm tot}\rangle= d_B(g_E\,\frac{\langle \hat{\vec J}_p\rangle}{\hbar}+ g_E^{e}\,\frac{\langle \hat{\vec S}\rangle}{\hbar})$, with $g_E^{e}=\frac{2d_{\rm int}}{d_B}$, where $\hat{\vec J}_p$ captures Stark-induced pseudo-angular momentum and $\hat{\vec S}$ encodes any intrinsic (spin-aligned) EDM $d_{\rm int}$ from symmetry-violating interactions. We define a natural electric dipole unit (the ``Bohr EDM'') as $d_B \equiv e a_0=\frac{2\mu_B}{c\alpha}$ ($a_0$ the Bohr radius and $\mu_B$ the Bohr magneton). As a canonical analytic benchmark, we show in the hydrogenic problem that a static electric field couples within a fixed $n$ manifold through the scaled Runge--Lenz structure, yielding a compact Land\'e-like description and reproducing the Stark doublet (e.g.\ $|\langle d_{\rm orb}\rangle|=3d_B$ for the $2s$--$2p_{m=0}$ mixing).