GEOMETRICAL OPTICS AS AN ABELIAN U(1) GAUGE THEORY IN A VACUUM SPACE-TIME
DOI:
10.29303/ipr.v7i1.278Downloads
Abstract
As a consequence that geometrical optics (the eikonal equation) can be derived from Maxwell’s equations and Maxwell’s theory is nothing but an Abelian U(1) gauge theory, we propose that geometrical optics could also be treated as an Abelian U(1) gauge theory. We formulate geometrical optics as an Abelian U(1) gauge theory in a (3+1)-dimensional vacuum space-time as an approximation of the weak field. We show the explicit form of the phase, the gauge potential, and the field strength tensor related to the refractive index. We calculate numerically the refractive index and the magnetic field using the suitable parameters that we choose to mimic the real condition of nature. We obtain (without unit) the values of the refractive index  =1.0001 to represent a vacuum space-time and the amplitude Ï = 0.55853 related to magnitude of the magnetic field  = 0.10452 to represent the weak field.  The view of geometrical optics as gauge theory could be generalized or related to topological field theory where geometrical optics could have a topological structure in the case of the weak field.
Keywords:
Geometrical optics Abelian U(1) gauge theory Eikonal equation Vacuum space-time Weak field TopologyReferences
L. D. Landau , E. M. Lifshitz, “The Classical Theory of Fieldsâ€, Pergamon Press, 1994.
The distance between two nearest points in the plane wave having the same phase is equal to a wavelength. In other words, the planes in the plane waves represents the wavefront, spaced a wavelength apart (see David Halliday, Robert Resnick, “Fundamentals of Physicsâ€, John Wiley & Sons, 1976).
L. D. Landau, E. M. Lifshitz, “Electrodynamics of Continuous Mediaâ€, Pergamon Press, 1984.
A. K. Ghatak, K. Thyagarajan, “Contemporary Opticsâ€, Plenum Press, 1978.
Ramamurti Shankar, “Fundamental of Physics II: Electromagnetism, Optics, and Quantum Mechanicsâ€, Yale University Press, 2016.
Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, “Gravitationâ€, W. H. Freeman and Company, 1973, p.573.
L. D. Landau, E. M. Lifshitz, “Quantum Electrodynamicsâ€, Elsevier Ltd, 1982.
Alexander B. Balakin, Alexei E. Zayats, “Non-minimal Einstein-Maxwell theory: the Fresnel equation and the Petrov classification of a trace-free susceptibility tensorâ€, ArXiV, https://arxiv.org/pdf/1710.08013.pdf, 2018.
A. B. Balakin, A. E. Zayats, “Ray Optics in the Field of a Non- minimal Dirac Monopoleâ€, Gravitation and Cosmology, Vol. 14, No. 1, pp. 86-94, 2008.
Arnold Sommerfeld, “Optics, Lectures on Theoretical Physicsâ€, Volume IVâ€, Academic Press, 1949.
Max Born, Emil Wolf, “Principles of Opticsâ€, Pergamon Press, 1993.
J. L. Synge, “Geometrical Mechanics and de Broglie Wavesâ€, Cambridge University Press, 2010.
Antonio F. Ranada, “A Topological Theory of the Electromagnetic Fieldâ€, Letters in Mathematical Physics 18: 97-106, 1989.
Dimitar Simeonov, “On some properties of the electromagnetic field and its interaction with a charged particleâ€, ArXiV, https://arxiv.org/pdf/2004.09273.pdf, 2020.
Wikipedia.org, “Amplitudeâ€.
H. J. Pain, “The Physics of Vibrations and Wavesâ€, John Wiley & Sons, 1983.
Antonio F Ranada, “Topological electromagnetismâ€, J. Phys. A: Math. Gen. 25 (1992) 1621-1641.
Edward M. Purcell, David J. Morin, “Electricity and Magnetismâ€, Cambridge University Press, 2013. Roald K. Wangsness, “Electromagnetic Fieldsâ€, John Wiley & Sons, 1986.
A. F. Ranada, A. Tiemblo, “A Topological Structure in the Set of Classical Free Radiation Electromagnetic Fieldsâ€, ArXiV, arXiv:1407.8145v1 [physics.class-ph], 2014.
Roman Jackiw, “Lectures on Fluid Dynamicsâ€, Springer, 2002.
J. F. Nye, “Natural Focusing and Fine Structure of Lightâ€, IOP Publishing Ltd, 1999.
The complex eikonal equation in a 3-dimensional space where the eikonal, ψ1, is treated as a complex scalar field is considered (see A. Wereszczynski, Knots, “Braids and Hedgehogs from the Eikonal Equationâ€, ArXiV, https://arxiv.org/pdf/math-ph/0506035v1.pdf, 2018).
Miftachul Hadi, “Knot in geometrical opticsâ€, and all references therein, viXra, https://vixra.org/abs/2207.0114, 2022.
Miftachul Hadi, “Geometrical optics as U (1) local gauge theory in a flat space-timeâ€, OSF, https://osf.io/zpxkg, 2023.
Eugene Hecht, “Opticsâ€, Addison-Wesley, 2002.
Miftachul Hadi, “Knot in weak-field geometrical opticsâ€, and all references therein, OSF, https://osf.io/e7afj/, 2023.
License
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Authors retain the copyright to their work and grant Indonesian Physical Review a non-exclusive right of first publication. All articles are published under the Creative Commons Attribution-ShareAlike 4.0 International License (CC BY-SA 4.0). Authors retain their publishing rights without restrictions and may distribute, deposit, or reuse the published version of their work, provided that its initial publication in Indonesian Physical Review is properly acknowledged. Indonesian Physical Review does not require authors to transfer copyright to the journal or publisher.
