Can classical electrodynamics predict nonlocal effects?

Abstract

Classical electrodynamics is a local theory describing local interactions between charges and electromagnetic fields and therefore one would not expect that this theory could predict nonlocal effects. But this perception implicitly assumes that the electromagnetic configurations lie in simply connected regions. In this paper, we consider an electromagnetic configuration lying in a non-simply connected region, which consists of a charged particle encircling an infinitely long solenoid enclosing a uniform magnetic flux, and show that the electromagnetic angular momentum of this configuration describes a nonlocal interaction between the encircling charge outside the solenoid and the magnetic flux confined inside the solenoid. We argue that the nonlocality of this interaction is of topological nature by showing that the electromagnetic angular momentum of the configuration is proportional to a winding number. The magnitude of this electromagnetic angular momentum may be interpreted as the classical counterpart of the Aharonov–Bohm phase.

Topology, nonlocality and duality in classical electrodynamics

“…The proof that minus the curl of (8) yields the electric field in ( 6) is entirely similar to that given in Appendix B of Ref. 1 with the dual changes A → C and Φ m → −Φ e . From (8) it follows that…”

“…The proof that the Laplacian of ( 8) yields ( 7) is entirely similar to that given in Appendix A of Ref. 1 with the changes A → C and Φ m → −Φ e . The potential C is not defined at ρ = R but one can regularise it to obtain C(R) = −Φ e φ /(2πR) which indicates that C is continuous at ρ = R. Equation ( 8) satisfies the Coulomb gauge:…”

“…In this section and in the two subsequent sections we closely follow the discussion given in Ref. 1 by making appropriate dual changes and putting more emphasis on relevant results than on formal details which can be found in Ref. 1 .…”

“…1 by making appropriate dual changes and putting more emphasis on relevant results than on formal details which can be found in Ref. 1 .…”

“…In a recent paper 1 we have pointed out that classical electrodynamics can predict nonlocal effects when considering electromagnetic configurations lying in non-simply connected regions. We have shown that the configuration formed by an electric charge q encircling an infinitely-long magnetic solenoid enclosing a uniform magnetic flux Φ m accumulates the electromagnetic angular momentum…”