Oscillating dipole with fractional quantum source in Aharonov-Bohm electrodynamics

We show, in the case of a special dipolar source, that electromagnetic fields in fractional quantum mechanics have an unexpected space dependence: propagating fields may have non-transverse components, and the distinction between near-field zone and wave zone is blurred. We employ an extension of Maxwell theory, Aharonov-Bohm electrodynamics, which is compatible with currents $j^\nu$ conserved globally but not locally, we have derived in another work the field equation $\partial_\mu F^{\mu \nu}=j^\nu+i^\nu$, where $i^\nu$ is a non-local function of $j^\nu$, called"secondary current". Y.\ Wei has recently proved that the probability current in fractional quantum mechanics is in general not locally conserved. We compute this current for a Gaussian wave packet with fractional parameter $a=3/2$ and find that in a suitable limit it can be approximated by our simplified dipolar source. Currents which are not locally conserved may be present also in other quantum systems whose wave functions satisfy non-local equations. The combined electromagnetic effects of such sources and their secondary currents are very interesting both theoretically and for potential applications.


A B S T R A C T
We show, in the case of a special dipolar source, that electromagnetic fields in fractional quantum mechanics have an unexpected space dependence: propagating fields may have non-transverse components, and the distinction between near-field zone and wave zone is blurred. We employ an extension of Maxwell theory, Aharonov-Bohm electrodynamics, which is compatible with currents j ν conserved globally but not locally; we have derived in another work the field equation ∂ µ F µν = j ν + i ν , where i ν is a non-local function of j ν , called "secondary current". Y. Wei has recently proved that the probability current in fractional quantum mechanics is in general not locally conserved. We compute this current for a Gaussian wave packet with fractional parameter a = 3/2 and find that in a suitable limit it can be approximated by our simplified dipolar source. Currents which are not locally conserved may be present also in other quantum systems whose wave functions satisfy non-local equations. The combined electromagnetic effects of such sources and their secondary currents are very interesting both theoretically and for potential applications.
c 2018 Elsevier B. V. All rights reserved.
Aharonov-Bohm electrodynamics [1] is a natural extension of Maxwell theory which allows to couple the electromagnetic field also to a current density j µ that is not locally conserved, i.e., it is such that ∂ µ j µ 0 in some region (µ = 0, 1, 2, 3). A coupling of this kind would of course be inconsistent in the Maxwell theory, since the Maxwell field equations with source are written as ∂ µ F µν = j ν and the tensor F µν is antisymmetric. Aharonov-Bohm electrodynamics has only reduced gauge invariance and one additional degree of freedom, namely the scalar field S = ∂ µ A µ (which is a pure gauge mode in Maxwell theory).
If only locally conserved sources are present, the Aharonov-Bohm theory reduces to Maxwell theory. This happens for all classical sources and for quantum sources which obey a wave equation with locally conserved current, like the Schrödinger equation or Ginzburg-Landau equation. In [2] it was shown, after obtaining a covariant formulation of the Aharonov-Bohm theory and its explicit solution for S , that a censorship property holds: the measurable field strength F µν does not allow to "see" electromagnetically a non-conserved source j ν , because it satisfies the equation ∂ µ F µν = j ν +i ν , where i ν is a non-local function of j ν , called "secondary current" (see eq. (3)), and j ν + i ν is conserved.
Aharonov-Bohm electrodynamics can be applied in principle to systems which exhibit quantum charge anomalies [3] or to superconducting states described by non-local equations [4,5]. Very recently, however, a new possible direct application has emerged. As shown by Wei [6], the Schrödinger equation of fractional quantum mechanics [7] has in general a current which is not locally conserved. This may allow particle teleportation and represents a problem for the compatibility of fractional quantum mechanics with Maxwell electrodynamics, but not for Aharonov-Bohm electrodynamics, where instead it leads to new interesting physical possibilities.
In order to illustrate these features in a paradigmatic, simplified situation, we analyze here the features of the Aharonov-arXiv:1703.05114v1 [physics.gen-ph] 5 Mar 2017 Bohm electromagnetic field generated by a non-conserved current of the form This corresponds to a charge which oscillates periodically between the positions (x − a) and (x + a), without an intermediate current, therefore with a kind of teleportation, as allowed by fractional quantum mechanics. The source (1) can be seen as the limit of a suitably defined wave packet (see below). In order to find the electromagnetic field generated by this source one must solve the Aharonov-Bohm equations, which in covariant form are (with Let us define, with reference to the four-current (1), the quantity θ(t) = ∂ α j α = ∂ 0 j 0 . Eq. (3), which gives the secondary current, can be rewritten as i ν = −∂ ν u ret , where u ret is the mathematical analogue of the retarded electric potential in Lorentz gauge of a charge θ(t), satisfying the equation ∂ 2 u ret = θ. This allows us to compute i ν at any point P (see Fig. 1). Note that the potential u ret will not be equal to the familiar dipole potential of Maxwell theory, because the source (1) is different from a physical oscillating particle.
From eq. (2) and general unicity theorems ( [8]; see also [9]), we deduce that the field F µν R at any point R is the Maxwell field with source j ν + i ν . This must include all the contributions from i ν at each point P, retarded according to the distance RP. Note that such contributions will in general not be transversal with respect to the direction − − → OR. The distinction between near field and wave zone will also be blurred, since the secondary source i ν extends far beyond the localized primary source j ν .
Finally, let us relate our simplified Ansatz (1) to a nonconserved current in fractional quantum mechanics. According to Wei [6], the correct probability continuity equation in fractional quantum mechanics is ∂ t ρ + ∇ · j a = I a , where 1 < a ≤ 2 (a = 2 corresponds to ordinary quantum mechanics) and the extra source term is I a = −iD a a−1 ∇ψ * (−∇ 2 ) a/2−1 ∇ψ − c.c. . As an example, Wei computes I a for a wave function of the form ψ = ψ 1 + ψ 2 , where ψ 1 and ψ 2 are two plane waves with different energies. In order to extend this to a more realistic localized wave function, we define a Gaussian wave packet of the form where k j = K 0 + j, K = K 0 + n 1 , n 1 = 1 + n/2. The fractional extra source term can then be written as As an example, we have computed numerically I a for a = 3/2, at t = 0. With parameters n = 20 (number of wavelets), K 0 = 10 3 (main wave number), = 10 −2 , D 3/2 = 5·10 −3 (which simply define the length and mass scale) we obtain the function in Fig. 2. By tuning the parameters we can obtain a wave packet Electromagnetic field generated by a four-current j ν (t) = ( j 0 (t), 0) which is not locally conserved. The oscillating dipolar source θ(t) in O generates a secondary four-current i ν at any point P in space. i ν is the fourgradient of −u P,ret ; u P,ret is mathematically equal to the retarded electric potential generated in P by an equivalent charge θ(t). The observable field F µν R at any point R outside the dipole is found by solving the Maxwell equations with four-current j ν + i ν at that point (but j ν is zero in R). Therefore F µν R receives "twice retarded" secondary contributions from any P and is not generally transverse to OR. with two peaks more localized in space and a small frequency spread; the result will be approximated well by our Ansatz (1). The full computation of F µν R requires in any case a numerical solution.