Abstract
In this paper it is presented a manifestly covariant formulation of the Aharonov-Bohm (AB) phase difference for the magnetic AB effect. This covariant AB phase is written in terms of the Faraday 2-form F and using the decomposition of F in terms of the electric and magnetic fields as four-dimensional (4D) geometric quantities. It is shown that there is a static electric field outside a stationary solenoid with resistive conductor carrying steady current, which causes that the AB phase difference in the magnetic AB effect may be determined by the electric part of the covariant expression, i.e., by the local influence of the 4D electric field and not, as generally accepted, in terms of nonzero vector potential.
Highlights
In a recent paper [1] the covariant generalizations of the Aharonov-Bohm (AB) effect [2] are considered
It is shown that there is a static electric field outside a stationary solenoid with resistive conductor carrying steady current, which causes that the AB phase difference in the magnetic AB effect may be determined by the electric part of the covariant expression, i.e., by the local influence of the 4D electric field and not, as generally accepted, in terms of nonzero vector potential
As can be seen from [4,5,6,7,8,9], in the four-dimensional (4D) spacetime, in contrast to the usual transformations (UT) of the 3-vectors E and B, Eq (2) here, or Eq (11.148) in [10], according to which the transformed E is expressed by the mixture of the 3-vectors E and B, the mathematically correct Lorentz transformations (LT) always transform the 4D algebraic object representing the electric field only to the electric field; there is no mixing with the magnetic field, Eq (4) here or Eqs. (42) and (43) in [3]
Summary
In a recent paper [1] the covariant generalizations of the Aharonov-Bohm (AB) effect [2] are considered. The mentioned results for the existence of the 4D external electric fields may give the possibility to explain the experimentally observed fringe shift for the magnetic AB effect even in Tonomura’s experiments [14], Sec. 10 in [3]: “in terms of forces, which so far have been overlooked.”. The existence of the overlooked 4D external electric fields is one of the reasons why we do not consider the covariant AB phase in terms of the four-potentials, δαEB = (e/ ) Aμdxμ, Eq (5) in [1] Another reason is that in [15] an axiomatic formulation of the electromagnetism is presented in which only the field equation for F is postulated, Eq (4) in [15], i.e., Eq (20) in [3]. There, [16], the field equations are written in terms of F and the generalized magnetization-polarization bivector M and not, as usual, in terms of F and the electromagnetic excitation tensor H
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