Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 040, 10 pages      arXiv:0804.1060      http://dx.doi.org/10.3842/SIGMA.2008.040
Contribution to the Special Issue on Deformation Quantization

Quantum Dynamics on the Worldvolume from Classical su(n) Cohomology

José M. Isidro and Pedro Fernández de Córdoba
Grupo de Modelización Interdisciplinar Intertech, Instituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, Valencia 46022, Spain

Received February 05, 2008, in final form March 28, 2008; Published online April 15, 2008

Abstract
A key symmetry of classical p-branes is invariance under worldvolume diffeomorphisms. Under the assumption that the worldvolume, at fixed values of the time, is a compact, quantisable Kähler manifold, we prove that the Lie algebra of volume-preserving diffeomorphisms of the worldvolume can be approximated by su(n), for n → ∞. We also prove, under the same assumptions regarding the worldvolume at fixed time, that classical Nambu brackets on the worldvolume are quantised by the multibrackets corresponding to cocycles in the cohomology of the Lie algebra su(n).

Key words: branes; Nambu brackets; Lie-algebra cohomology.

pdf (239 kb)   ps (166 kb)   tex (14 kb)

References

  1. Wigner E., On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40 (1932), 749-759.
  2. Groenewold H., On the principles of elementary quantum mechanics, Physica 12 (1946), 405-460.
  3. Moyal J., Quantum mechanics as a statistical theory, Proc. Camb. Phil. Soc. 45 (1949), 99-124.
  4. Bayen F., Flato M., Fronsdal C., Lichnerowicz A., Sternheimer D., Deformation theory and quantization. I. Deformations of symplectic structures, Ann. Physics 111 (1978), 61-110.
    Bayen F., Flato M., Fronsdal C., Lichnerowicz A., Sternheimer D., Deformations of symplectic structures. II. Physical applications, Ann. Physics 111 (1978), 111-151.
  5. Kontsevich M., Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), 157-216, q-alg/9709040.
  6. Cattaneo A., Keller B., Torossian C., Bruguières A., Déformation, quantification, théorie de Lie, Panoramas et Synthèses, Vol. 20, Société Mathématique de France, Paris, 2005.
  7. Dito G., Sternheimer D., Deformation quantization: genesis, developments and metamorphoses, math.QA/0201168.
  8. Nambu Y., Generalized Hamiltonian dynamics, Phys. Rev. D 7 (1973), 2405-2412.
  9. Curtright T., Zachos C., Classical and quantum Nambu mechanics, Phys. Rev. D 68 (2003), 085001, 29 pages, hep-th/0212267.
  10. Knapp A., Lie groups, Lie algebras and cohomology, Princeton University Press, Princeton, 1988.
  11. Goldberg S., Curvature and homology, Dover, New York, 1982.
  12. Hoppe J., Quantum theory of a massless relativistic surface and a two dimensional bound state problem, Ph.D. Thesis, MIT, 1982, http://www.aei.mpg.de/~hoppe.
    de Wit B., Hoppe J., Nicolai H., On the quantum mechanics of supermembranes, Nuclear Phys. B 305 (1988), 545-581.
    de Witt B., Supermembranes and supermatrix models, in Quantum Aspects of Gauge Theories, Supersymmetry and Unification, Editors A. Ceresole, C. Kounnas, D. Lüst and S. Theisen, Springer, Berlin, 1999, 45-69.
  13. Hoppe J., On M-algebras, the quantisation of Nambu mechanics, and volume preserving diffeomorphisms, Helv. Phys. Acta 70 (1997), 302-317, hep-th/9602020.
    Hoppe J., Membranes and matrix models, hep-th/0206192.
  14. Fairlie D., Fletcher P., Zachos C., Trigonometric structure constants for new infinite algebras, Phys. Lett. B 218 (1989), 203-206.
    Hoppe J., DiffAT2, and the curvature of some infinite dimensional manifolds, Phys. Lett. B 215 (1988), 706-710.
  15. Bordemann M., Hoppe J., Schaller P., Schlichenmaier M., gl(∞) and geometric quantization, Comm. Math. Phys. 138 (1991), 209-244.
    Bordemann M., Meinrenken E., Schlichenmaier M., Toeplitz quantization of Kähler manifolds and gl(N), N → ∞, limits, Comm. Math. Phys. 165 (1994), 281-296, hep-th/9309134.
    Hoppe J., Olshanetsky M., Theisen S., Dynamical systems on quantum tori algebras, Comm. Math. Phys. 155 (1993), 429-448.
  16. Carter R., Segal G., Macdonald I., Lectures on Lie groups and Lie algebras, London Mathematical Society Student Texts, Vol. 32, Cambridge University Press, Cambridge, 1995.
  17. Witten E., Bound states of strings and p-branes, Nuclear Phys. B 460 (1996), 335-350, hep-th/9510135.
  18. Banks T., Fischler W., Shenker S., Susskind L., M-theory as a matrix model: a conjecture, Phys. Rev. D 55 (1997), 5112-5128, hep-th/9610043.
  19. Galaviz I., García-Compeán H., Przanowski M., Turrubiates F., Weyl-Wigner-Moyal formalism for Fermi classical systems, Ann. Physics 323 (2008), 267-290, hep-th/0612245.
    Galaviz I., García-Compeán H., Przanowski M., Turrubiates F., Deformation quantization of Fermi fields, hep-th/0703125.
    Sánchez L., Galaviz I., García-Compeán H., Deformation quantization of relativistic particles in electromagnetic fields, arXiv:0705.2259.
  20. Isidro J.M., Fernández de Córdoba P., Dirichlet branes and a cohomological definition of time flow, arXiv:0712.1961.

Previous article   Next article   Contents of Volume 4 (2008)