Lagrangian for Interacting Electric Charge and Magnetic Dipole. Derivation of Interaction Forces in Quantum Effects of the Aharonov-Bohm Type

We construct the classical interaction Lagrangian for an electric charge q and a magnetic dipole m in relative motion. In the rest frame of m the resulting force acting on q is \(\textbf{f}_{q}=q\textbf{E}+c^{-1} \mathbf {v\times B}+c^{-1}q(\mathbf {v\cdot \nabla })\textbf{A}\). Application to the Aharonov-Bohm (AB), and the equivalent Spavieri effect, indicates that the observed AB phase shift is due to the classical lag effect between interfering particles caused by the local force \(c^{-1}q(\mathbf {v\cdot \nabla })\textbf{A}=(\mathbf {v\cdot \nabla })\textbf{Q}_{em}\) with nonvanishing longitudinal component in the direction of motion and with \(\textbf{Q}_{em}\) representing the gauge-invariant electromagnetic momentum. Our results confirm the validity of the same expression for \(\textbf{f}_{q}\) derived in literature with an approach based on the stress-energy tensor \(T^{\mu \nu }\), Maxwell’s equations, and the momentum conservation law. Similar results apply to the force \(\textbf{f}_{m}=-\textbf{f}_{q}\) acting on m, indicating conservation of the action and reaction principle in the effects of AB type, which can be interpreted classically in terms of the lag effect caused by a local force.