Letters in Mathematical Physics

, Volume 28, Issue 2, pp 123–137 | Cite as

Stochastic differential calculus, the Moyal *-product, and noncommutative geometry

  • Aristophanes Dimakis
  • Folkert Müller-Hoissen


A reformulation of the Itô calculus of stochastic differentials is presented in terms of a differential calculus in the sense of noncommutative geometry (with an exterior derivative operator d satisfying d2 = 0 and the Leibniz rule). In this calculus, differentials do not commute with functions. The relation between both types of differential calculi is mediated by a generalized Moyal *-product. In contrast to the Itô calculus, the new framework naturally incorporates analogues of higher-order differential forms. A first step is made towards an understanding of their stochastic meaning.

Mathematics Subject Classifications (1991)

17B37 58A10 58G32 81R50 81S20 


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  1. 1.
    Connes, A., Non-commutative differential geometry,Publ. IHES 62, 257 (1986); Dubois-Violette, M., Dérivations et calcul différentiel non commutatif,CR Acad. Sci. Paris Série I 307 403 (1988); Coquereaux, R., Noncommutative geometry and theoretical physics,J. Geom. Phys. 6, 425 (1989); Manin, Yu. I.,Topics in Noncommutative Geometry, Princeton University Press, Princeton, 1991.Google Scholar
  2. 2.
    Jaffe, A., Quantum physics as non-commutative geometry, in K. Schmüdgen (ed),Mathematical Physics X, Springer, Berlin, 1992, p. 281.Google Scholar
  3. 3.
    Bellissard, J., Ordinary quantum Hall effect and non-commutative cohomology, in W. Weller and P. Ziesche (eds),Localization of Disordered Systems, Teubner, Leipzig, 1988.Google Scholar
  4. 4.
    Woronowicz, S. L., Differential calculus on compact matrix pseudogroups (quantum groups),Comm. Math. Phys. 122, 125 (1989).Google Scholar
  5. 5.
    Connes, A. and Lott, J., Particle models and noncommutative geometry,Nuclear Phys. B (Proc. Suppl.)18, 29 (1990); Chamseddine, A. H., Felder, G., and Fröhlich, J., Grand unification in noncommutative geometry, Preprint ETH/TH/92-41; Dubois-Violette, M., Kerner, R., and Madore, J., Noncommutative differential geometry and new models of gauge theory,J. Math. Phys. 31, 323 (1990); Coquereaux, R., Esposito-Farèse, G., and Scheck, F., Noncommutative geometry and graded algebras in electroweak interactions,J. Mod. Phys. A 7, 6555 (1992).Google Scholar
  6. 6.
    Dimakis, A. and Müller-Hoissen, F., Quantum mechanics on a lattice andq-deformations,Phys. Lett. B 295, 242 (1992); Dimakis, A., Müller-Hoissen, F., and Striker, T., From continuum to lattice theory via deformation of the differential calculus,Phys. Lett. B 300, 141 (1993); Noncommutative differential calculus and lattice gauge theory, to appear inJ. Phys. A. Google Scholar
  7. 7.
    Dimakis, A. and Müller-Hoissen, F., Noncommutative differential calculus, gauge theory and gravitation, Preprint GOET-TP 33/92.Google Scholar
  8. 8.
    Arnold, L.,Stochastic Differential Equations, Wiley, New York, 1974.Google Scholar
  9. 9.
    Guerra, F., Structural aspects of stochastic mechanics and stochastic field theory,Phys. Rep. 77, 263 (1981); Damgaard, P. H. and Hüffel, H., Stochastic quantization,Phys. Rep. 152, 227 (1987); Namiki, M.,Stochastic Quantization, Lect. Notes in Phys. m9, Springer, Berlin, 1992.Google Scholar
  10. 10.
    Nelson, E., Derivation of the Schrödinger equation from Newtonian mechanics,Phys. Rev. 150, 1079 (1966);Dynamical Theories of Brownian Motion, Princeton University Press, Princeton, 1967;Quantum Fluctuations, Princeton University Press, Princeton, 1985.Google Scholar
  11. 11.
    Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., and Sternheimer, D., Deformation theory and quantization,Ann. Phys. 111, 61 and 111 (1978).Google Scholar
  12. 12.
    Meyer, P. A., A differential geometric formalism for the Ito calculus, in D. Williams (ed),Stochastic Integrals, Lecture Notes in Mathematics 851, Springer, Berlin, 1981, p. 256; Meyer, P.-A., Qu'est ce qu'une différentielle d'ordren?Exposition Math. 7, 249 (1989).Google Scholar
  13. 13.
    Moyal, J. E., Quantum mechanics as a statistical theory,Camb. Phil. Soc. 45, 99 (1949).Google Scholar
  14. 14.
    Dunne, G. V., Quantum canonical invariance - A Moyal approach,J. Phys. A. 21, 2321 (1988).Google Scholar
  15. 15.
    Fletcher, P., The uniqueness of the Moyal algebra,Phys. Lett. B 248, 323 (1990).Google Scholar
  16. 16.
    Emery, M.,Stochastic Calculus in Manifolds, Springer, Berlin, 1989.Google Scholar
  17. 17.
    Müller-Hoissen, F., Differential calculi on the quantum group GLp,q(2),J. Phys. A 25, 1703 (1992); Müller-Hoissen, F., and Reuten, C., Bicovariant differential calculus on GLp,q(2) and quantum subgroups, to appear inJ. Phys. A. Google Scholar
  18. 18.
    Hudson, R. L. and Robinson, P., Quantum diffusions and the noncommutative torus,Lett. Math. Phys. 15, 47 (1988).Google Scholar

Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • Aristophanes Dimakis
    • 1
  • Folkert Müller-Hoissen
    • 2
  1. 1.Department of MathematicsUniversity of CreteIraklionGreece
  2. 2.Institut für Theoretische PhysikGöttingenGermany

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