DIRAC'S LINEARIZATION APPLIED TO THE FUNCTIONAL, WITH MATRIX ASPECT, FOR THE TIME OF FLIGHT OF LIGHT

Abstract

In this paper, we adopt a matrix treatment to solve the variational problem that consists of determining the physical path traveled by light between two points in a medium whose refractive index depends on a spatial coordinate. The considered treatment begins with the trivial repetition of the expression of the value of the considered functional, repetition expressed in the form of a matrix. Next, we adopt the trick (of Dirac) originally used as part of the construction of the dynamic equation of relativistic quantum mechanics, which allows us to rewrite the (now) matrix integrand in the expression of the value of the functional in terms of the sum of two (non-diagonal) matrices brought externally to the problem, which are determined based on some requirements. As a result of this development, we obtain two equivalent versions of Snell's law.

Keywords

• Variational principle in optics;
• Matrix treatment of a functional and Euler
• Dirac

References

1. O.D. Kichmarenko, L.I. Plotnikova, N.V. Skripnik, Variational Calculus (in Russian), Odessa, Astroprint (2009).
2. A.V. Panteleev, Calculus of Variations in examples and problems (in Russian), Ed. MAI (2000).
3. M.L. Krasnov, G.I. Makarenko, A.I. Kiseliov, Calculo Variacional (in Spanish), Ed. Mir-Moscu (1976).
4. J. Chaves, Introduction to Nonimaging Optics, second edition, Taylor & Francis Group (2016).
5. J. Garcia, Como llego Dirac a su ecuacion available on the internet, http://aula141.cat/wp-content/uploads/2015/10/Presentaci%C3%B3n_Dirac.pdf
6. J. Lopez-Bonilla, R. Lopez-Vazquez, S. Vidal-Beltran, Linearization of (second order operator)1=2, World Scientic News, 108 (2018) 224-228.
7. B. Mahon, The forgotten genius of Oliver Heaviside, Prometheus Books (2017).
8. P. A. M. Dirac, The quantum theory of the electron, Proc. R. Soc. Lond. A117 (1928) 610-624.

9. H. S. Kragh, The genesis of Dirac's relativistic theory of electrons, Archive for History of Exact Sciences 24(1) (1981) 31-67.
10. P. Lam-Estrada, J. Lopez-Bonilla, R. Lopez-Vazquez, On the Noether's Theorem, Prespace-time Journal, 6(4) (2015) 322-325 .