Rami El-Nabulsi


In 2000, Andrea Raspini introduced a fractional Dirac equation of order by taking the cubic root of the Klein-Gordon equation and by proving that a 3-fold factorization of the Klein-Gordon equation leads to a fractional Dirac equation characterized by an elliptic operator of order and where the resulting γ - matrices obey an extended Clifford algebra. In this short communication, we show that an elliptic operator of order may be constructed from the generalized fractional Glaeske-Kilbas-Saigo integral which will be useful to define non-integer dimensional deformations of the canonical spectral triples in noncommutative geometry. We show that the fractional order corresponds to the highest pole of the fractional dimension spectrum of a spectral triple in the case of a four-dimensional torus as formulated by Connes-Moscovici. In this work, we give only a concise exposition of the topic whereas a complete description will be presented elsewhere.

Full Text:



M. Abramowitz, I. A. Stegun, eds. (1965), "Chapter 13", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, pp. 504, ISBN 978-0486612720, MR0167642.

G. E. Andrews, R. Askey, R. Roy, (2000). Special functions. Cambridge University Press.

M. Arzano, G. Calcagni, D. Oriti, M. Scalisi, Fractional and noncommutative spacetimes, Phys. Rev. D84, (2011) 125002-125017.

M. Atiyah, Eigenvalues of the Dirac operator, Lect. Notes in Math. Vol. 1111, (1985) 251-260.

B. Bonila et al, Modified Bessel-type function and solution of differential and integral equations, Indian J. Pure Appl. Math. 31(1), (2000) 93-109.

G. Calcagni, Geometry and field theory in multi-fractional spacetime, JHEP01, (2012) 065-147.

G. Calcagni, Quantum field theory, gravity and cosmology in a fractal universe, JHEP1003, (2010) 120-162.

G. Calcagni, Fractal universe and quantum gravity, Phys. Rev. Lett. 104, 251301-251305.

J. Collins, Renormalization, Cambridge Univ. Press (1984).

A. Connes, Noncommutative Geometry, Academic Press, London and San Diego (1994).

R. A. El-Nabulsi, G.-c. Wu, Fractional complexified field theory from Saxena-Kumbhat fractional integral, fractional derivative of order and dynamical fractional integral exponent, Afric. Disp. J. Math. 13, No. 2 (2012) 45-61.

R. A. El-Nabulsi, Glaeske-Kilbas-Saigo fractional integration and fractional Dixmier trace, Acta Math. Viet. 37, 2, (2012) 149-160.

S. El-Showk, M. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, A. Vichi, Conformal field theories in fractional dimensions, arXiv: 1309.5089.

E. Goldfain, Complexity in quantum field theory and physics beyond the Standard Model, Chaos Solitons Fractals 28 (2009), 913-922.

E. Goldfain, Fractional dynamics and the standard model for particle physics, Comm. Nonlinear Sci. Numer. Simul. 13 (2008) 1397-1404.

E. Goldfain, Fractional dynamics, Cantorian spacetime and the gauge hierarchy problem, Chaos Solitons Fractal. 22 (2004) 513-520.

E. Goldfain, Fractional dynamics and the TeV regime of field theory particle, Comm. Nonlinear Sci. Numer. Simul. 13 (2008) 666-676.

R. Herrmann, Gauge invariance in fractional field theories, Phys. Lett. A372, (2008) 5515-5522.

R. Herrmann, Fractional Calculus-An Introduction for Physicists, World Scientific, Singapore (2011), ISBN: 9789814340243.

R. Hilfer, Application of fractional calculus in physics, World Scientific, Singapore, (2000).

T. Ikebe, Eigenfunction expansions associated with the Schrödinger operators and their application to scattering theory, Arch. Ration. Mech. Anal. 5, (1960) 1-34.

A. A. Kilbas, J. J. Trujillo, Computation of fractional integrals via functions of hypergeometric and Bessel type, J. Comp. Appl. Math. 118 (2000) 223-239.

H. Kleinert, Fractional quantum field theory, path integral, and stochastic differential equation for strongly interacting many-particle systems, Europ. Phys. Letts. 100, (2012) 10001-10005.

S. N. Lakaev, Sh. M. Tilavova, Merging of eigenvalues and resonances of a two-particle Schrödinger operator, Theor. Math. Phys. 101, 2 (1994) 1320-1331.

B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman and Company, ISBN 0-7167-1186-9, San-Francisco (1982).

S. I. Muslih, O. P. Agrawal, D. Baleanu, Solutions of a fractional Dirac equation, Proceedings of the ASME (2009) International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2009 August 30 - September 2, 2009, San Diego, California, USA.

S. I. Muslih, O. P. Agrawal, D. Baleanu, A fractional Dirac equation and its solution, J. Phys. A: Math. Theor. 45, (2010) 055203-055213.

A. Raspini, Dirac equation with fractional derivatives of order 2/3, Fiz. B9, No. 2, (2000) 49-54.

A. Raspini., Simple solution of the fractional Dirac equation of order 2/3. Phys. Scrip. 64, (2001) 20-22.

R. Trinchero, Scalar field on non-integer dimensional spaces, arXiv: 1201.4365.

J. Sabatier, O. P. Agrawal, J. A. Tenreiro Machado, Advance in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, (2007).

K. G. Wilson, M. E. Fisher, Critical exponents in 3.99 dimensions, Phys. Rev. Lett. 28, (1972) 240-243.

K. G. Wilson, Quantum field theory models in less than four-dimensions, Phys. Rev. D7 (1973), 2911-2926.


  • There are currently no refbacks.