The Weyl equation under an external electromagnetic field in the cosmic string space–time

Abstract

In this paper we have considered a massless spinor Dirac particle in the presence of an external electromagnetic field in the cosmic string space–time. To study the Weyl equation in the cosmic string framework using the general definition of Laplacian in the curved space, elements of covariant derivative have been constructed and the Weyl equation has been rewritten in the considered framework. Then we have obtained the equation of energy eigenvalues by using the Nikiforov–Uvarov (NU) method. The wave function has been obtained in terms of Laguerre polynomials. An important result obtained is that the degeneracy of the Minkowski space spectral is broken in the transition from Minkowski to cosmic string space.

Appendix A. The NU method

We consider the following differential equation [37]:

$$\begin{aligned} \frac{\mathrm{d}^2{\Psi (r)}}{\mathrm{d}{r^2}}&+ \frac{{\alpha _1}-{\alpha _2}r}{r\left( 1-{\alpha _3}r\right) } \frac{\mathrm{d}{\Psi (r)}}{\mathrm{d}{r}}+\frac{1}{{r^2}\left( 1-{\alpha _3}r\right) ^2} \nonumber \\&\times \{- {\xi _1}r^2+ {\xi _2}r -{\xi _3}\} \Psi (r)= 0. \end{aligned}$$

(A1)

According to the NU method, the equations of energy eigenvalues and eigenfunctions, respectively, are obtained from

$$\begin{aligned}&{\alpha _2}n-(2n+1){\alpha _5}+(2n+1)(\sqrt{\alpha _9}+{\alpha _3}\sqrt{\alpha _8}) \nonumber \\&\quad + n(n-1){\alpha _3}+ {\alpha _7}+ 2{\alpha _3}{\alpha _8}+ 2 \sqrt{{\alpha _8}{\alpha _9}} = 0, \end{aligned}$$

(A2)

$$\begin{aligned} \Psi (r)= & {} r^{\alpha _{12}}\left( 1-{\alpha _3}r\right) ^{{- \alpha _{12}}-\left( {\alpha _{13}}/{\alpha _3}\right) } \nonumber \\&\times P_n^{\left( {\alpha _{10}}-1,\left( {\alpha _{11}}/{\alpha _3}\right) -{\alpha _{10}}-1\right) } \left( 1-2{\alpha _3}r\right) , \end{aligned}$$

(A3)

where

$$\begin{aligned}&{\alpha _4} = \frac{1}{2} \left( {1 - {\alpha _1}} \right) , \quad {\alpha _5} = \frac{1}{2} \left( {{\alpha _2} - 2 {\alpha _3}} \right) , \nonumber \\&{\alpha _6} = \alpha _5^2 + {\xi _1}, \quad {\alpha _7} = 2\,{\alpha _4}\,{\alpha _5}\, - {\xi _2}, \nonumber \\&{\alpha _8} = \alpha _4^2 + {\xi _3}, \quad {\alpha _9} = {\alpha _3}\,{\alpha _7}\, + \,\alpha _3^2\,{\alpha _8}\, + \,{\alpha _6}, \nonumber \\&{\alpha _{10}} = {\alpha _1}\, + \,2\,{\alpha _4} + 2\sqrt{{\alpha _8}}, \nonumber \\&{\alpha _{11}} = {\alpha _2}\, - \,2\,{\alpha _5} + 2( {\sqrt{{\alpha _9}} \, + \,{\alpha _3}\,\sqrt{{\alpha _8}} } ), \nonumber \\&{\alpha _{12}} = {\alpha _4} + \sqrt{{\alpha _8}}, \quad {\alpha _{13}} = \,{\alpha _5}\, - \,( {\sqrt{{\alpha _9}} \, + \,\,{\alpha _3}\,\sqrt{{\alpha _8}} } ). \nonumber \\ \end{aligned}$$

(A4)

In the special case of ${\alpha _3} = 0$ [38]

$$\begin{aligned}&\mathop {\lim }\limits _{{\alpha _3} \rightarrow 0} P_n^{({\alpha _{10}-1, ({\alpha _{11}}/{\alpha _3})}-{\alpha _{10}}-1)} \left( 1-{\alpha _3} r\right) \nonumber \\&\quad = L_n^{\alpha _{10}-1} \left( \alpha _{11} r\right) , \end{aligned}$$

(A5)

$$\begin{aligned}&\mathop {\lim }\limits _{{\alpha _3} \rightarrow 0} \left( 1-{\alpha _3} r\right) ^{-\alpha _{12}} - \frac{\alpha _{13}}{\alpha _3} = \mathrm {e}^{\alpha _{13}r} \end{aligned}$$

(A6)

and

$$\begin{aligned} {\Psi _n}(r) = r^{\alpha _{12}} \mathrm {e}^{\alpha _{13}r} {L_n}^{\alpha _{10}-1} \left( {\alpha _{11}r}\right) . \end{aligned}$$

(A7)