Research Article Open Access

Fractional Quantum Field Theory on Multifractals Sets

El-Nabulsi Ahmad Rami1
  • 1 , Afganistan
American Journal of Engineering and Applied Sciences
Volume 4 No. 1, 2011, 133-141

DOI: https://doi.org/10.3844/ajeassp.2011.133.141

Submitted On: 1 July 2008 Published On: 15 March 2011

How to Cite: Rami, E. A. (2011). Fractional Quantum Field Theory on Multifractals Sets. American Journal of Engineering and Applied Sciences, 4(1), 133-141. https://doi.org/10.3844/ajeassp.2011.133.141

Abstract

Problem statement: This study is a contribution to the general program of describing complex dynamical systems using the tool of fractional calculus of variations. Approach: Following our previous work, fractional quantum field theory based on the fractional actionlike variational approach supported by Saxena-Kumbhat fractional integrals functionals, fractional derivative of order (α, β) and dynamical fractional exponent on multi-fractal sets is considered. Results: In order to build the required theory, we introduce the Saxena-Kumbhat hypergeometric fractional functionals determined on the functions on a multifractal sets. We prove, developing the corresponding fractional calculus of variations, that a hierarchy of differential equations can be developed from the extended fractional Lagrangian formalism. Besides, a generalization of the resulting Hamiltonian and Lagrangian dynamics on the complex plane is addressed. Conclusion: The new complexified dynamics guides to a new dynamics which may differ totally from the classical mechanics cardinally and may bring new appealing consequences. Some additional interesting results are explored and discussed in some details.

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Keywords

  • Fractional action-like variational approach
  • multifractal sets
  • Euler-Lagrange equations
  • Saxena-Kumbhat fractional integral
  • fractional derivative
  • dynamical fractional exponent