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Physical Review Letters

Röntgen Quantum Phase Shift: A Semiclassical Local Electrodynamical Effect?

S. A. R. Horsley, M. Babiker

Published Jun 29, 2005• 19 citations
Publisher

Abstract

The Röntgen quantum phase shift is exhibited by the interference of point particles endowed with an electric dipole moment due to their motion relative to a source of the magnetic field. Here we show, using arguments involving the classical concepts of force and its impulse, that the Röntgen phase shift arises within a largely classical (semiclassical) theoretical framework. All the subtleties normally associated with the nonlocality of magnetic (Aharonov-Bohm-type) quantum phase phenomena are uncontroversially absent in the classical treatment.

DOI
10.1103/physrevlett.95.010405

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The Röntgen quantum phase shift is exhibited by the interference of point particles endowed with an electric dipole moment due to their motion relative to a source of the magnetic field. Here we show, using arguments involving the classical concepts of force and its impulse, that the Röntgen phase shift arises within a largely classical (semiclassical) theoretical framework. All the subtleties normally associated with the nonlocality of magnetic (Aharonov-Bohm-type) quantum phase phenomena are uncontroversially absent in the classical treatment.
Abstract In the Aharonov-Bohm (AB) effect, interference fringes are observed for a charged particle in the absence of the local overlap with the external electromagnetic field. This notion of the apparent “nonlocality” of the interaction or the significant role of the potential have recently been challenged and are under debate. The quantum electrodynamic approach provides a microscopic picture of the characteristics of the interaction between a charge and an external field. We explicitly show the gauge invariance of the local phase shift in the magnetic AB effect, which is in contrast to the results obtained using the usual semiclassical vector potential. Our study can resolve the issue of the locality in the magnetic AB effect. However, the problem is not solved in the same way in the electric counterpart wherein virtual scalar photons play an essential role.
We provide arguments which demonstrate that the Berry phase effects involving particle interference in the presence of external electric and magnetic fields—namely the Aharonov–Bohm (AB), Aharonov–Casher (AC) and the Röntgen (R) effects—cannot be unambiguously explained (as has been controversially argued) through the use of any classical mechanical force. In fact, in the three scenarios which we consider, the only means to reclaim the standard quantum mechanical results is to use the expressions for the force that fail to satisfy the necessary conditions of gauge invariance. Our arguments explain the foundation of the controversy and confirm the standard idea that the Berry phase effects in question are non-local phenomena which only arise in the application of quantum theory. We lay special emphasis on the use of the concept of a canonical force, this being the quantity which ensures the agreement between the classical and quantum formulations. It is shown that if the Aharonov–Bohm effect is interpreted as being due to a force, through the use of the Bohmian formulation of quantum mechanics, then this force is equal in magnitude to the canonical force.
A new classical electromagnetic analysis is presented suggesting that the Aharonov-Bohm phase shift is overwhelmingly likely to arise from a classical lag effect based upon classical electromagnetic forces. The analysis makes use of several aspects of classical electromagnetic theory which are unfamiliar to most physicists, including the Darwin Lagrangian, acceleration-based electric fields, internal electromagnetic momentum in a magnet, and a magnet model involving at least three mutually-interacting particles. Only when the acceleration-based electric forces acting on the passing charge are included do we find consistency with all the relativistic conservation laws: energy, linear momentum, angular momentum, and constant center-of-mass velocity. The electric forces on the passing charge lead to a lag effect which accounts quantitatively for the Aharonov-Bohm phase shift. Thus the classical analysis strongly suggests that the Aharonov-Bohm phase shift (observed when electrons pass a long solenoid which corresponds to a line of magnetic dipoles) is the analogue of the Matteucci-Pozzi phase shift (observed when electrons pass a line of electric dipoles). The classical electromagnetic analysis suggests experiments to distinguish the proposed classical-based lag effect from the presently accepted view that the Aharonov-Bohm phase shift is a quantum topological effect arising from magnetic fluxes in the absence of classical electromagnetic forces.
Classical electrodynamics foresees that the effective interaction force between a moving charge and a magnetic dipole is modified by the time-varying total momentum of the interaction fields. We derive the equations of motion of the particles from the total stress-energy tensor, assuming the validity of Maxwell’s equations and the total momentum conservation law. Applications to the effects of Aharonov–Bohm type show that the observed phase shift may be due to the relative lag between interfering particles caused by the effective local force.
The Aharonov-Bohm effect is a physical phenomenon in which the quantum state of a charged particle acquires a phase shift that is directly proportional to the magnetic flux, ΦΦΦ, due to a (classical) magnetic field, B{\mathbf B}B, which is confined in a spatial region from which the magnetic field cannot escape. Even though the charged particle is not allowed to interact with the magnetic field, it accumulates a phase shift that affects the interference pattern produced. Not surprisingly, this apparent nonlocality is puzzling and counter intuitive. In this work, we provide an explanation that explains the physics underlying this apparent nonlocality. We find that the role of the confined magnetic field is to impart a puncture in the configuration space, R2\mathbb{R}^2R2, of the charge. Therefore, the quantum state corresponding to the charged quantum particle acquires the phase shift due to its response to the modified topology of the configuration space, R2−{0}\mathbb{R}^2-\{0\}R2−{0}, corresponding to the charge.