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Canonical quantization for fractional complex scalar and vector field. (English) Zbl 1431.35228

Summary: The fractional form of the complex scalar and vector Lagrangian densities is presented using the Caputo’s fractional derivative. Agrawal procedure is employed to obtain equations of motion for both fields in Caputo’s definition. The Hamiltonian density resulting from Lagrangian densities is obtained in fractional form. Then we quantize the Lagrangian density for both standard and vector fields by constructing the fractional creation and annihilation operators and fractional canonical commutation relations (CCRs).

MSC:

35R11 Fractional partial differential equations
35A15 Variational methods applied to PDEs
81S05 Commutation relations and statistics as related to quantum mechanics (general)
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[1] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. · Zbl 0924.34008
[2] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974. · Zbl 0292.26011
[3] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. · Zbl 0998.26002
[4] J. A. Tenreiro-Machado, Special Issue of Fractional Order Calculus and its Applications, Nonlinear Dynamics, Springer, Berlin, 2002.
[5] O. P. Agrawal, Generalized Euler-Lagrange equations and transversality 100 Emad K. Jaradat conditions for FVPs in terms of the Caputo derivative, J. Vib. Control 13 (2007), 1217-1237. · Zbl 1158.49006
[6] E. K. Jaradat, R. S. Hijjawi and J. M. Khalifeh, Maxwell’s equations and electromagnetic Lagrangian density in fractional form, J. Math. Phys. 53 (2012), 033505. · Zbl 1274.78012
[7] Emad K. Jaradat, Canonical quantization for fractional Schrödinger Lagrangian density in Caputo definition, Jordan Journal of Physics 6(2) (2013), 55-63.
[8] Eyad Hasan, Fractional quantization of holonomic constrained systems using fractional WKB approximation, Advanced Studies in Theoretical Physics 10(5) (2016), 223-234.
[9] Emad K. Jaradat, Proca equations of a massive vector boson field in fractional form, Mu’tah Lil-Buhoth wad-Dirasat 26(2) (2011).
[10] O. P. Agrawal, Fractional variational calculus in terms of Riesz fractional Derivatives, J. Phys. A: Math. Theor. 40 (2007), 6287-6303. · Zbl 1125.26007
[11] D. Baleanu and S. I. Muslih, Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives, Phys. Scripta 72(2-3) (2005), 119-121. · Zbl 1122.70360
[12] S. N. Gupta, Prog. Phys. Soc. A 63 (1952), 681.
[13] S. N. Gupta, Prog. Phys. Soc. A 65 (1952), 426.
[14] P. A. M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Science, Yehiva University, Academic Press, New York, 1964.
[15] Madhat Sadallah, Sami I. Muslih and Dumitru Baleanu, Equations of motion for Einstein’s field in non-integer dimensional space, Czechoslovak Journal of Physics 56(4) (2006), 323-328. · Zbl 1129.83341
[16] David J. Griffiths, Introduction to Quantum Mechanics, 2nd ed., Pearson Prentice Hall, 2004. · Zbl 0818.00001
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