Oscillating dipole with fractional quantum source in Aharonov-Bohm electrodynamics

TL;DR

This paper explores how fractional quantum mechanics alters electromagnetic field behavior, revealing non-transverse components and blurred near- and far-field distinctions using Aharonov-Bohm electrodynamics with non-local currents.

Contribution

It demonstrates the impact of non-local, non-conserved quantum currents on electromagnetic fields within an extended Maxwell framework, highlighting novel space dependence effects.

Findings

Propagating fields may have non-transverse components.

Near-field and wave zone distinctions become blurred.

Non-local quantum currents influence electromagnetic field behavior.

Abstract

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.

Figures

Equations

$$j_0=\left[{\delta }^3\left(\mathbf{x}-\mathbf{a}\right)-{\delta }^3\left(\mathbf{x}+\mathbf{a}\right)\right]e^{i\omega t};\mathbf{j}=0.$$

$${\partial }_{\mu }{F}^{\mu \nu }=j^{\nu }+i^{\nu }$$

$$i^{\nu }=-{\partial }^{\nu }{\partial }^{-2}\left({\partial }_{\alpha }{j}^{\alpha }\right)$$

$$\psi =\sum _{j=1}^{n+1}{\psi }_j;{\psi }_j=e^{-4(k_j-K{)}^2{n}_1^{-2}}{e}^{i\hslash k_{j}x}{e}^{-i\hslash {\omega }_{j}t},$$

$$\begin{array}[]{l}I_{a}=-iD_{a}\hbar^{a-1}\sum_{i,j=1}^{n+1}(1-\delta_{ij})\left[k_{i}k_{j}^{a-1}(\psi_{i}^{*}\psi_{j}-c.c.)+\right.\\ \left.\qquad+k_{i}^{a-1}k_{j}(\psi_{j}^{*}\psi_{i}-c.c.)\right].\end{array}$$

Full text

††journal: Results in Physics

Aharonov-Bohm electrodynamics [1] is a natural extension of Maxwell theory which allows to couple the electromagnetic field also to a current density $j^{\mu}$ that is not locally conserved, i.e., it is such that $\partial_{\mu}j^{\mu}\neq 0$ in some region ($\mu=0,1,2,3$). A coupling of this kind would of course be inconsistent in the Maxwell theory, since the Maxwell field equations with source are written as $\partial_{\mu}F^{\mu\nu}=j^{\nu}$ and the tensor $F^{\mu\nu}$ is antisymmetric. Aharonov-Bohm electrodynamics has only reduced gauge invariance and one additional degree of freedom, namely the scalar field $S=\partial_{\mu}A^{\mu}$ (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 $F^{\mu\nu}$ does not allow to “see” electromagnetically a non-conserved source $j^{\nu}$, because it satisfies the equation $\partial_{\mu}F^{\mu\nu}=j^{\nu}+i^{\nu}$, where $i^{\nu}$ is a non-local function of $j^{\nu}$, called “secondary current” (see eq. (3)), and $j^{\nu}+i^{\nu}$ 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-Bohm electromagnetic field generated by a non-conserved current of the form

[TABLE]

This corresponds to a charge which oscillates periodically between the positions $\left({{\bf{x}}-{\bf{a}}}\right)$ and $\left({{\bf{x}}+{\bf{a}}}\right)$, 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 $\partial^{2}=\partial_{\alpha}\partial^{\alpha}=\partial_{t}^{2}-\nabla$)

[TABLE]

[TABLE]

Let us define, with reference to the four-current (1), the quantity $\theta(t)={\partial_{\alpha}}{j^{\alpha}}={\partial_{0}}{j^{0}}$. Eq. (3), which gives the secondary current, can be rewritten as $i^{\nu}=-\partial^{\nu}u_{ret}$, where $u_{ret}$ is the mathematical analogue of the retarded electric potential in Lorentz gauge of a charge $\theta(t)$, satisfying the equation $\partial^{2}u_{ret}=\theta$. This allows us to compute $i^{\nu}$ 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 $F^{\mu\nu}_{R}$ at any point $R$ is the Maxwell field with source $j^{\nu}+i^{\nu}$. This must include all the contributions from $i^{\nu}$ 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 $\overrightarrow{OR}$. The distinction between near field and wave zone will also be blurred, since the secondary source $i^{\nu}$ extends far beyond the localized primary source $j^{\nu}$.

Finally, let us relate our simplified Ansatz (1) to a non-conserved current in fractional quantum mechanics. According to Wei [6], the correct probability continuity equation in fractional quantum mechanics is $\partial_{t}\rho+\nabla\cdot{\bf j}_{a}=I_{a}$, where $1<a\leq 2$ ($a=2$ corresponds to ordinary quantum mechanics) and the extra source term is $I_{a}=-iD_{a}\hbar^{a-1}\left[\nabla\psi^{*}(-\nabla^{2})^{a/2-1}\nabla\psi-c.c.\right]$. As an example, Wei computes $I_{a}$ for a wave function of the form $\psi=\psi_{1}+\psi_{2}$, where $\psi_{1}$ and $\psi_{2}$ 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

[TABLE]

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

[TABLE]

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), $\hbar=10^{-2}$, $D_{3/2}=5\cdot 10^{-3}$ (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 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 $F^{\mu\nu}_{R}$ requires in any case a numerical solution.