# Self-Adjoint Extension Approach to Motion of Spin-1/2 Particle in the Presence of External Magnetic Fields in the Spinning Cosmic String Spacetime

^{*}

## Abstract

**:**

**Data Set License:**license under which the data set is made available (CC0, CC-BY, CC-BY-SA, CC-BY-NC, etc.)

## 1. Introduction

## 2. Dirac Equation in the Spinning Cosmic String Spacetime

## 3. Self-Adjoint Extension

## 4. The Energy Spectrum

- (i)
- For the case ${\xi}_{m}=0$:$$\begin{array}{cc}\hfill {\mathcal{E}}_{R,+}^{\left(>\right)}& =\pm \sqrt{{M}^{2}+4nM{\omega}_{c}},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{for}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{J}_{+}>0,\hfill \end{array}$$$$\begin{array}{cc}\hfill {\mathcal{E}}_{R,+}^{\left(<\right)}& =-\frac{2aM{\omega}_{c}}{\alpha}\pm \frac{1}{\alpha}\sqrt{{\left(2aM{\omega}_{c}\right)}^{2}+4{\alpha}^{2}M{\omega}_{c}\left(n-{j}_{+}\right)+{\alpha}^{2}{M}^{2}},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{for}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{J}_{+}<0,\hfill \end{array}$$$$\begin{array}{cc}\hfill {\mathcal{E}}_{R,-}^{\left(>\right)}& =\pm \sqrt{{M}^{2}+4M{\omega}_{c}\left(n+1\right)},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{for}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{J}_{-}>0,\hfill \end{array}$$$$\begin{array}{cc}\hfill {\mathcal{E}}_{R,-}^{\left(<\right)}& =-\frac{2aM{\omega}_{c}}{\alpha}\pm \frac{1}{\alpha}\sqrt{{\left(2aM{\omega}_{c}\right)}^{2}+4{\alpha}^{2}M{\omega}_{c}\left(n-{j}_{-}+1\right)+{\alpha}^{2}{M}^{2}},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{for}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{J}_{-}<0.\hfill \end{array}$$
- (ii)
- For the case ${\xi}_{m}=\infty $:$$\begin{array}{cc}\hfill {\mathcal{E}}_{I,+}^{\left(>\right)}& =-\frac{2aM{\omega}_{c}}{\alpha}\pm \frac{1}{\alpha}\sqrt{{\left(2aM{\omega}_{c}\right)}^{2}+4{\alpha}^{2}M{\omega}_{c}\left(n-{j}_{+}\right)+{\alpha}^{2}{M}^{2}},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{for}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{J}_{+}>0,\hfill \end{array}$$$$\begin{array}{cc}\hfill {\mathcal{E}}_{I,+}^{\left(<\right)}& =\pm \sqrt{{M}^{2}+4nM{\omega}_{c}},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{for}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{J}_{+}<0,\hfill \end{array}$$$$\begin{array}{cc}\hfill {\mathcal{E}}_{I,-}^{\left(>\right)}& =-\frac{2aM{\omega}_{c}}{\alpha}\pm \frac{1}{\alpha}\sqrt{{\left(2aM{\omega}_{c}\right)}^{2}+4{\alpha}^{2}M{\omega}_{c}\left(n-{j}_{-}+1\right)+{\alpha}^{2}{M}^{2}},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{for}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{J}_{-}>0,\hfill \end{array}$$$$\begin{array}{cc}\hfill {\mathcal{E}}_{I,-}^{\left(<\right)}& =\pm \sqrt{{M}^{2}+4M{\omega}_{c}\left(n+1\right)},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{for}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{J}_{-}<0,\hfill \end{array}$$$$\begin{array}{cc}\hfill {j}_{+}& =\frac{1}{\alpha}\left(m-\varphi +\frac{\left(1-\alpha \right)}{2}\right),\hfill \end{array}$$$$\begin{array}{cc}\hfill {j}_{-}& =\frac{1}{\alpha}\left(m+1-\varphi -\frac{\left(1-\alpha \right)}{2}\right).\hfill \end{array}$$

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1. | Do not confuse the Greek letter $\alpha $ used to represent the matrices of Dirac ${\alpha}^{i}$ and the parameter $\alpha $ in the metric (3). |

**Figure 1.**Sketch of the energy levels ${\mathcal{E}}_{R,+}^{\left(<\right)}$ (Equation (71)) as a function of ${\omega}_{c}$. In (

**a**), $a=0.2$, $\alpha =0.8$, $\varphi =0.5$. In (

**b**), $a=0.2$, $\alpha =0.6$, $\varphi =1$.

**Figure 2.**Sketch of the energy levels ${\mathcal{E}}_{R,+}^{\left(<\right)}$ (Equation (71)) as a function of $\alpha $. In (

**a**), $a=0.8$, $\omega =2$ and $\varphi =0.5$. In (

**b**), $a=0.4$, $\omega =0.4$ and $\varphi =0.5$.

**Figure 3.**Sketch of the energy levels ${\mathcal{E}}_{R,+}^{\left(<\right)}$ (Equation (71)) as a function of a. In (

**a**), we use $\alpha =0.5$, $\varphi =2.5$ and $\omega =1$ for $m=0$ and $m=2$. In (

**b**), we use $\omega =1$, $\varphi =2.5$ and $m=1$ for $\alpha =0.4$ and $\alpha =0.9$.

**Figure 4.**Sketch of the energy levels ${\mathcal{E}}_{R,+}^{\left(<\right)}$ (Equation (71)) as a function of $\varphi $. In (

**a**), $a=0.5$, $\alpha =0.5$ and $\omega =1$ and, In (

**b**), $a=0.1$, $\alpha =0.2$ and $\omega =9$.

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M. Cunha, M.; O. Silva, E.
Self-Adjoint Extension Approach to Motion of Spin-1/2 Particle in the Presence of External Magnetic Fields in the Spinning Cosmic String Spacetime. *Universe* **2020**, *6*, 203.
https://doi.org/10.3390/universe6110203

**AMA Style**

M. Cunha M, O. Silva E.
Self-Adjoint Extension Approach to Motion of Spin-1/2 Particle in the Presence of External Magnetic Fields in the Spinning Cosmic String Spacetime. *Universe*. 2020; 6(11):203.
https://doi.org/10.3390/universe6110203

**Chicago/Turabian Style**

M. Cunha, Márcio, and Edilberto O. Silva.
2020. "Self-Adjoint Extension Approach to Motion of Spin-1/2 Particle in the Presence of External Magnetic Fields in the Spinning Cosmic String Spacetime" *Universe* 6, no. 11: 203.
https://doi.org/10.3390/universe6110203