# A Non-Local Action for Electrodynamics: Duality Symmetry and the Aharonov-Bohm Effect, Revisited

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## Abstract

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## 1. Introduction

## 2. The Free Non-Local (Duality Invariant) Action

#### 2.1. Non-Local Formulation for the Electromagnetic Field in Terms of $\mathbf{E}$ and $\mathbf{B}$

#### 2.2. Electric-Magnetic Duality Symmetry

## 3. The Non-Local Action with Matter

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Interaction Term in the A.B. Effect

- The expression for the magnetic field is $\mathbf{B}\left(\mathbf{x}\right)=\widehat{\mathbf{z}}\Theta ({R}^{2}-{x}^{2}-{y}^{2})$, where $\Theta $ is the Heaviside step function and $\mathbf{x}=(x,y,z)$.
- The volume region is a cylinder C of radius R, with a length ${L}_{1}$ and ${L}_{2}$ over and under the xy plane respectively. Furthermore, it will be assumed that the cylinder is long i.e., ${L}_{1}^{2},{L}_{2}^{2}\gg {R}^{2},{x}^{2}+{y}^{2}$.

## References

- DeWitt, B. Quantum theory without electromagnetic potentials. Phys. Rev.
**1962**, 125, 2189. [Google Scholar] [CrossRef] - Coleman, S. Aspects of Symmetry; Cambridge University Press: Cambridge, UK, 1985. [Google Scholar]
- Faddeev, L.; Jackiw, R. Hamiltonian reduction of unconstrained and constrained systems. Phys. Rev. Lett.
**1988**, 60, 1692. [Google Scholar] [CrossRef] [PubMed] - Henneaux, M.; Teitelboim, C. Quantization of Gauge Systems; Princeton University: Princeton, NJ, USA, 1992. [Google Scholar]
- Barbero, G.F.J.; Díaz, B.; Margalef-Bentabol, J.; Villaseñor, E.J.S. Dirac’s algorithm in the presence of boundaries: A practical guide to a geometric approach. arXiv
**2019**, arXiv:1904.11790. [Google Scholar] - Jackiw, R. Constrained quantization without tears. In Constraint Theory and Quantization Methods: From Relativistic Particles to Field Theory and General Relativity; Colomo, F., Lusanna, L., Marmo, G., Eds.; World Scientific: River Edge, NJ, USA, 1994; p. 448. [Google Scholar]
- Jackson, J.D. Classical Electrodynamics; Wiley: New York, NY, USA, 1998. [Google Scholar]
- Peskin, M.E.; Schroeder, D.V. An Introduction to Quantum Field Theory; Westview Press: Boulder, CO, USA, 1995. [Google Scholar]
- Feynman, R.P.; Hibbs, A.R. Quantum Mechanics and Path Integrals; McGraw-Hill: New York, NY, USA, 1965. [Google Scholar]
- Aharonov, Y.; Bohm, D. Significance of electromagnetic potentials in the quantum theory. Phys. Rev.
**1959**, 115, 485. [Google Scholar] [CrossRef] - Aharonov, Y.; Bohm, D. Further considerations on electromagnetic potentials in the quantum theory. Phys. Rev.
**1961**, 123, 1511. [Google Scholar] [CrossRef] - Feynman, R.P.; Leighton, R.B.; Sands, M. The Feynman Lectures on Physics Vol. 2: Mainly Electromagnetism and Matter; Addison-Wesley: Boston, MA, USA, 1979. [Google Scholar]
- Peshkin, M.; Tonomura, A. The Aharonov-Bohm Effect; Springer: Berlin, Germany, 1989. [Google Scholar]
- Sakurai, J.J.; Napolitano, J. Modern Quantum Mechanics; Cambridge University Press: Cambridge, UK, 2017. [Google Scholar]
- Griffiths, D.J. Introduction to Electrodynamics; Prentice Hall: Upper Saddle River, NJ, USA, 1981. [Google Scholar]
- Bjorken, J.D.; Drell, S.D. Relativistic Quantum Fields; McGraw-Hill: New York, NY, USA, 1965. [Google Scholar]
- Vogel, W.; Welsch, D.G. Quantum Optics; John Wiley and Sons: Hoboken, NJ, USA, 2006. [Google Scholar]
- Weinberg, S. Lectures on Quantum Mechanics; Cambridge University Press: Cambridge, UK, 2015. [Google Scholar]
- Calkin, M.G. An invariance property of the free electromagnetic field. Am. J. Phys.
**1965**, 33, 958–960. [Google Scholar] [CrossRef] - Deser, S.; Teitelboim, C. Duality transformations of Abelian and non-Abelian gauge fields. Phys. Rev. D
**1976**, 13, 1592. [Google Scholar] [CrossRef] - Deser, S. Off-shell electromagnetic duality invariance. J. Phys. A
**1982**, 15, 1053. [Google Scholar] [CrossRef] - Agullo, I.; del Rio, A.; Navarro-Salas, J. Electromagnetic duality anomaly in curved spacetimes. Phys. Rev. Lett.
**2017**, 118, 111301. [Google Scholar] [CrossRef] - Agullo, I.; del Rio, A.; Navarro-Salas, J. Gravity and handedness of photons. Int. J. Mod. Phys. D
**2017**, 26, 1742001. [Google Scholar] [CrossRef] - Agullo, I.; del Rio, A.; Navarro-Salas, J. Classical and quantum aspects of electric-magnetic duality rotations in curved spacetimes. Phys. Rev. D
**2018**, 98, 125001. [Google Scholar] [CrossRef] [Green Version] - Agullo, I.; del Rio, A.; Navarro-Salas, J. On the Electric-Magnetic Duality Symmetry: Quantum Anomaly, Optical Helicity, and Particle Creation. Symmetry
**2018**, 10, 763. [Google Scholar] [CrossRef] - Stewart, A.M. Role of nonlocality of the vector potential in the Aharonov-Bohm effect. Can. J. Phys.
**2013**, 91, 373. [Google Scholar] [CrossRef] - Majumdar, P.; Ray, A. Maxwell Electrodynamics in Terms of Physical Potentials. Symmetry
**2019**, 11, 915. [Google Scholar] [CrossRef]

**Figure 1.**Experimental setup we will consider to analyse the magnetic AB effect. A source of electrons is located at the point ${\mathbf{x}}_{1}$, from which one is emitted at a time ${t}_{1}$. Between the source and a screen on the other side of the setup there is a wall, containing two slits A and B, and a long impenetrable cylinder behind it. Inside the cylinder, oriented parallel to the z-axis, there is a magnetic field $\mathbf{B}=\widehat{\mathbf{z}}{B}_{0}$, while outside $\mathbf{B}=0$. The electrons can trace two types of $\mathit{deterministic}$ paths to reach the point ${\mathbf{x}}_{\mathrm{f}}$ on the screen at a time ${t}_{\mathrm{f}}$, either above (e.g., ${\gamma}_{1}$) or below (e.g., ${\gamma}_{2}$) the cylinder.

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**MDPI and ACS Style**

Bernabeu, J.; Navarro-Salas, J.
A Non-Local Action for Electrodynamics: Duality Symmetry and the Aharonov-Bohm Effect, Revisited. *Symmetry* **2019**, *11*, 1191.
https://doi.org/10.3390/sym11101191

**AMA Style**

Bernabeu J, Navarro-Salas J.
A Non-Local Action for Electrodynamics: Duality Symmetry and the Aharonov-Bohm Effect, Revisited. *Symmetry*. 2019; 11(10):1191.
https://doi.org/10.3390/sym11101191

**Chicago/Turabian Style**

Bernabeu, Joan, and Jose Navarro-Salas.
2019. "A Non-Local Action for Electrodynamics: Duality Symmetry and the Aharonov-Bohm Effect, Revisited" *Symmetry* 11, no. 10: 1191.
https://doi.org/10.3390/sym11101191