Annihilation of Quantum Magnetic Fluxes

Abstract

After introducing the concepts associated with the Aharonov and Bohm effect and with the existence of a quantum of magnetic flux (QMF), we briefly discuss the Ginzburg-Landau theory that explains its origin and fundamental consequences. Also relevant observations of QMFs obtained in the laboratory using superconducting systems (vortices) are mentioned. Next, we describe processes related with the interaction of QMFs with opposite directions in terms of the gauge field geometry related to the vector potential. Then, we discuss the use of a Lagrangian density for a scalar field theory involving radiation in order to describe the annihilation of QMFs, claimed to be responsible for the emission of photons with energies corresponding to that of the annihilated magnetic fields. Finally, a possible application of these concepts to the observed variable dynamics of neutron stars is briefly mentioned.

Appendices

Appendix I

Although the proper way to study the radiation emitted by annihilating QMFs is using tools from quantum field theory, as those followed by Hecht and DeGrand (1990) and by Gleiser and Thorarinson (2007) and briefly mentioned in Sect. 13.6, an indication of the presence of radiation in the interaction of QMFs can already be visualized from the Treumann et al. (2012) approach, as outlined below. A heuristic argument about the force between the flux tubes of Figs. 13.6 and 13.7 can be put forward as follows. The gauge field $\nabla \varLambda = \mathbf{A}$ causes an electric potential:

$$\displaystyle{U = -\partial \varLambda /\partial t.}$$

(cf., Jackson 1975, pp. 220–223) being of pure gauge nature. It is clear that the gauge field around an isolated flux tube is stationary, and U = 0. In the presence of another flux tube, however, information is exchanged between the flux tubes, requiring time. The gauge field becomes nonstationary, acquiring time dependence; the equivalent induced electrostatic potential is non-zero. The time dependence of the gauge field in the Lorentz gauge is taken care of by the wave equation for $\varLambda$:

$$\displaystyle{\nabla ^{2}\varLambda - \frac{1} {c^{2}} \frac{\partial ^{2}\varLambda } {\partial ^{2}t} = 0}$$

of which the solution $\varLambda$ is subject to the boundary conditions on the surfaces of the two flux tubes. These prescribe that $\nabla \varLambda = \mathbf{A}$ on both surfaces.

Formally, the potential caused by the gauge field gives rise to a gauge-Coulomb force

$$\displaystyle{\mathbf{F} = e\nabla U = -e\frac{\partial \nabla \varLambda } {\partial t} = -\frac{2\pi c\hslash } {\phi _{0}} \frac{\partial \nabla \varLambda (\theta,r,t)} {\partial t},}$$

which in the presence of another flux tube evolves a radial component, the sign of which depends on the mutual orientation of the flux tubes. Formally, QMFs behave like electric charges of value $2\pi c\hslash /\phi _{0}$. However, there are no massive charged particles involved on which the force could act in the empty space between the flux tubes. Hence the force must be experienced by the flux tubes only, where for tow flux tubes $\varLambda (\theta,r)$ is given by expression (13.3) for a given time interval. This force causes acceleration and displacement of a flux tube in the presence of another one at distance d.

Appendix II

Since the topic of this Book is Magnetic Reconnection, there is in my opinion an apparently interesting related issue in the material discussed in this chapter, which remains to be explored.

It refers to a possible reconnection scenario associated with the classical magnetic fluxes pervading the space outside the superconducting medium, such as that illustrated in Fig 13.4, when that space is (somehow) filled with plasma.

We assume that externally created magnetic fluxes with opposite directions meet at the middle of the superconducting slab forming vortices and antivortices that can annihilate, as described by Harada et al. (1996). Thus, the region where the annihilation occurs would resemble a “diffusion” region of reconnection, whereas the “cutted” classical magnetic field existing outside the superconductor could be regarded as a “reconnected” field. Therefore, one may tentatively say hat such “reconnected fluxes” would propagate externally as Alven waves carrying magnetic energy, which eventually could be measured in the laboratory.

Thus, the radiation emitted by the annihilating QMFs from the superconductor, together with the classical magnetic energy carried by the Alfven waves outside, would energetically represent the main outcome from the combined annihilation/reconnection system.

As mentioned in this Chapter, the quantum character of the annihilating magnetic fluxes in the superconductor extends outside only over a few London penetration lengths, and further than that one expects to have a classical magnetic flux, as that illustrated in Figure 13.4.

If this scenario turns out to be confirmed by further research, one could apply our knowledge on magnetic reconnection to possible related superconducting/plasma systems, as those associated with the superconducting core and the external magnetized plasma of neutron stars, in which the superconducting core is expected to have a large volume of annihilating QMFs, as described in the last section of this chapter. Thus, from the discussion given above, one may expect that the external reconnected field would propagate in the magnetosphere of the neutron star carrying substantial amount of magnetic energy and perhaps leading into some interesting magnetospheric dynamics that could be studied.