Abstract
In this paper, a fractional Hamiltonian formulation for Duffin–Kemmer–Petiau’ (DKP) fields is presented and, as done in the framework of the Lagrangian formalism, the fractional DKP equation is deduced. The space-time fractional DKP equation is then solved for both scalar and vectorial cases. The wave functions obtained are expressed in terms of Mittag–Leffler function.
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S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives Theory and Applications (Gordon and Breach Science Publishers, Philadelphia, 1993)
K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (A Wiley-Interscience Publication, New York, 1993)
I. Podlubny, Fractional Differential Equations. Mathematics in Science and Engineering, vol. 198 (Academic Press, New York, 1999)
K.B. Oldham, J. Spanier, The Fractional Calculus (Academic Press, Inc., New York, 1974)
S.I. Muslih, Solutions of a particle with fractional \(\delta \)-potential in a fractional dimensional space. Int. J. Theor. Phys. 49, 2095 (2010)
D. Baleanu, S.I. Muslih, About fractional supersymmetric quantum mechanics. Czech J. Phys. 55, 1063 (2005)
N. Laskin, Fractional quantum mechanics and Levy path integrals. Phys. Lett. A. 268, 298 (2000)
M.I. Bhati, L. Debnath, On fractional Schrodinger and Dirac equations. Int. J. Pure Appl. Math. 15, 1 (2004)
S.I. Muslih, O.P. Agrawal, D. Baleanu, A fractional Dirac equation and its solution. J. Phys. A Math. Theor. 43, 055203 (2010)
A. Raspini, Simple solutions of the fractional Dirac equation of order 2/3. Phys. Scr. 64, 20 (2001)
V.E. Tarasov, Fractional dynamics of relativistic particle. Int. J. Theor. Phys. 49, 293 (2010)
N. Bouzid, M. Merad, D. Baleanu, On fractional Duffin–Kemmer–Petiau equation. Few. Body Syst. 57, 265 (2016)
J.F. Gómez-Aguilara, H. Yépez-Martínezb, R.F. Escobar-Jiméneza, C.M. Astorga-Zaragozaa, L.J. Morales-Mendozac, M. González-Leec, Universal character of the fractional space–time electromagnetic waves in dielectric media. J. Electromagn. Waves Appl. 29, 727 (2015)
V.E. Tarasov, Fractional integro-differential equations for electromagnetic waves in dielectric media. Theor. Math. Phys. 158, 355 (2009)
M. Zubair, M.J. Mughal, Q.A. Naqvi, A.A. Rizvi, Differential electromagnetic equations in fractional space. Progr. Electromagn. Res. 114, 255 (2011)
G. Petiau, Contribution a la theorie des equations d’ondes corpusculaires. University of Paris’ thesis. Acad. Roy. de Belg., A. Sci. Mem. Collect. 16(2), 1–115 (1936)
R.Y. Duffin, On the characteristic matrices of covariant systems. Phys. Rev. 54, 1114 (1938)
N. Kummer, The particle aspect of meson theory. Proc. R. Soc A 173, 91 (1939)
O.P. Agrawal, Formulation of Euler–Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272, 368 (2002)
R. Gorenflo, A. Kilbas, F. Mainardi, S. Rogosin, Mittag–Leffler Functions, Related Topics and Applications (Springer, Berlin, 2014)
Y. Nedjadi, R.C. Barrett, Solution of the central field problem for a Duffin–Kemmer–Petiau vector boson. J. Math. Phys. 19, 87 (1994)
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Bouzid, N., Merad, M. Space-Time Fractional DKP Equation and Its Solution. Few-Body Syst 58, 131 (2017). https://doi.org/10.1007/s00601-017-1295-1
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DOI: https://doi.org/10.1007/s00601-017-1295-1