Journal of Applied Mathematics

Relativistic wave equations with fractional derivatives and pseudodifferential operators

Petr Závada

Full-text: Open access


We study the class of the free relativistic covariant equations generated by the fractional powers of the d′Alembertian operator (1/n). The equations corresponding to n=1 and 2 (Klein-Gordon and Dirac equations) are local in their nature, but the multicomponent equations for arbitrary n>2 are nonlocal. We show the representation of the generalized algebra of Pauli and Dirac matrices and how these matrices are related to the algebra of SU(n) group. The corresponding representations of the Poincaré group and further symmetry transformations on the obtained equations are discussed. The construction of the related Green functions is suggested.

Article information

J. Appl. Math., Volume 2, Number 4 (2002), 163-197.

First available in Project Euclid: 30 March 2003

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 81R20: Covariant wave equations 15A66: Clifford algebras, spinors
Secondary: 47G30: Pseudodifferential operators [See also 35Sxx, 58Jxx] 26A33: Fractional derivatives and integrals 34B27: Green functions


Závada, Petr. Relativistic wave equations with fractional derivatives and pseudodifferential operators. J. Appl. Math. 2 (2002), no. 4, 163--197. doi:10.1155/S1110757X02110102.

Export citation


  • D. G. Barci, C. G. Bollini, L. E. Oxman, and M. C. Rocca, Lorentz-invariant pseudo-differential wave equations, Internat. J. Theoret. Phys. 37 (1998), no. 12, 3015–3030.
  • N. Fleury and M. Rausch de Traubenberg, Linearization of polynomials, J. Math. Phys. 33 (1992), no. 10, 3356–3366.
  • ––––, Finite-dimensional representations of Clifford algebras of polynomials, Adv. Appl. Clifford Algebras 4 (1994), no. 2, 123–130.
  • L. L. Foldy, Synthesis of covariant particle equations, Phys. Rev. (2) 102 (1956), 568–581.
  • W. I. Fushchich and A. G. Nikitin, Symmetries of čommentAuthor please check the correctness of the title in [5,11a,11b,15,7,3,13,8,9,4,2,6,14.?] čommentPlease provide the date you accessed [12?]. čommentConcerning [17?], please provide the range of pages and the date you accessed this cite if possible. Equations of Quantum Mechanics, Allerton Press, New York, 1994.
  • M. S. Joshi, Introduction to pseudo-differential operators, lecture notes, June 1999.
  • R. Kerner, ${Z}\sb 3$-grading and the cubic root of the Dirac equation, Classical Quantum Gravity 9 (1992), 137–146.
  • C. Lämmerzahl, The pseudodifferential operator square root of the Klein-Gordon equation, J. Math. Phys. 34 (1993), no. 9, 3918–3932.
  • J. Patera and H. Zassenhaus, The Pauli matrices in $n$ dimensions and finest gradings of simple Lie algebras of type ${A}\sb {n-1}$, J. Math. Phys. 29 (1988), no. 3, 665–673.
  • M. S. Plyushchay and M. Rausch de Traubenberg, Cubic root of Klein-Gordon equation, Phys. Lett. B 477 (2000), no. 1-3, 276–284.
  • A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Vol. 1, Gordon & Breach Science Publishers, New York, 1986.
  • A. Ramakrishnan, $L$-Matrix Theory or the Grammar of Dirac Matrices, Tata McGraw-Hill Publishing, New Delhi, 1972.
  • A. Raspini, Dirac equation with fractional derivatives of order 2/3, Fizika B 9 (2000), no. 2, 49–54.
  • M. Rausch de Traubenberg, Clifford algebras, supersymmetry and $Z_N$ symmetries: application in field theory, Habilitation thesis, Université Louis Pasteur, Laboratoire de Physique Théorique, France, February 1998,.
  • J. R. Smith, Second quantization of the square root Klein-Gordon operator, microscopic causality, propagators, and interactions, Tech. report, Phys. Dept., Univ. of California, Davis, 1993.
  • J. Sucher, Relativistic invariance and the square-root Klein-Gordon equation, J. Mathematical Phys. 4 (1963), no. 1, 17–23.
  • J. Szwed, The “square root” of the Dirac equation within supersymmetry, Phys. Lett. B 181 (1986), no. 3-4, 305–307.
  • F. Trèves, Introduction to Pseudodifferential and Fourier Integral Operators. Vol. 1, Plenum Press, New York, 1980.
  • ––––, Introduction to Pseudodifferential and Fourier Integral Operators. Vol. 2, Plenum Press, New York, 1980.
  • M. W. Wong, An Introduction to Pseudo-Differential Operators, 2nd ed., World Scientific Publishing, New Jersey, 1999.
  • P. Závada, Operator of fractional derivative in the complex plane, Comm. Math. Phys. 192 (1998), no. 2, 261–285.