Skip to main content
SpringerLink
Log in

Moyal Planes are Spectral Triples

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Axioms for nonunital spectral triples, extending those introduced in the unital case by Connes, are proposed. As a guide, and for the sake of their importance in noncommutative quantum field theory, the spaces R2N endowed with Moyal products are intensively investigated. Some physical applications, such as the construction of noncommutative Wick monomials and the computation of the Connes–Lott functional action, are given for these noncommutative hyperplanes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Amiet, J.-P., Huguenin, P.: Mécaniques classique et quantique dans l’espace de phase. Université de Neuchatel, Neuchatel, 1981

  2. Baez, J.C., Segal, I.E., Zhou, Z.: Introduction to Algebraic and Constructive Quantum Field Theory. Princeton, New Jersey: Princeton University Press, 1992

  3. Bahns, D., Doplicher, S., Fredenhagen, K., Piacitelli, G.: On the unitarity problem in space/time noncommutative theories. Phys. Lett. B533, 178–181 (2002)

    Google Scholar 

  4. Bahns, D., Doplicher, S., Fredenhagen, K., Piacitelli, G.: Ultraviolet finite quantum field theory on quantum spacetime. Commun. Math. Phys. 237, 221–241 (2003)

    MATH  Google Scholar 

  5. Banach, S.: Remarques sur les groupes et les corps métriques. Studia Math 10, 178–181 (1948)

    MathSciNet  MATH  Google Scholar 

  6. Bartlett, M.S., Moyal, J.E.: The exact transition probabilities of quantum-mechanical oscillators calculated by the phase-space method. Proc. Cambridge Philos. Soc. 45, 545–553 (1999)

    MATH  Google Scholar 

  7. Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization II: Physical applications. Ann. Phys. (NY) 111, 111–151 (1978)

    MATH  Google Scholar 

  8. Birman, M.Sh., Karadzhov, G.E., Solomyak, M.Z.: Boundedness conditions and spectrum estimates for the operators b(X)a(D) and their analogs. Adv. in Soviet Math. 7, 85–106 (1991)

    MATH  Google Scholar 

  9. Brouder, C., Fauser, B., Frabetti, A., Oeckl, R.: Quantum groups and quantum field theory: I. The free scalar field. Paris, 2001, hep-th/0311253

  10. Carey, A., Phillips, J., Sukochev, F.: Spectral flow and Dixmier traces. Adv. Math. 173, 68–113 (2003)

    Article  MATH  Google Scholar 

  11. Chakraborty, P.S., Goswami, D., Sinha, K.B.: Probability and geometry on some noncommutative manifolds. J. Oper. Theory 49, 187–203 (2003)

    Google Scholar 

  12. Chamseddine, A.H.: Noncommutative gravity. Ann. Henri Poincaré 4 Suppl. 2, S881–S887 (2003)

  13. Chamseddine, A.H., Connes, A.: Universal formula for noncommutative geometry actions: Unification of gravity and the Standard Model. Phys. Rev. Lett. 177, 4868–4871 (1996)

    Article  MathSciNet  Google Scholar 

  14. Connes, A.: C*-algèbres et géométrie différentielle. C. R. Acad. Sci. Paris 290, 599–604 (1980)

    MATH  Google Scholar 

  15. Connes, A.: Noncommutative differential geometry. Publ. Math. IHES 39, 257–360 (1985)

    MathSciNet  Google Scholar 

  16. Connes, A.: The action functional in noncommutative geometry. Commun. Math. Phys. 117, 673–683 (1988)

    MathSciNet  MATH  Google Scholar 

  17. Connes, A.: Noncommutative Geometry. London and San Diego: Academic Press, 1994

  18. Connes, A.: Noncommutative geometry and reality. J. Math. Phys. 36, 6194–6231 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  19. Connes, A.: La notion de variété et les axiomes de la géométrie. Course at the Collège de France, January–March 1996

  20. Connes, A.: Gravity coupled with matter and the foundation of noncommutative geometry. Commun. Math. Phys. 182, 155–176 (1996)

    MathSciNet  MATH  Google Scholar 

  21. Connes, A., Lott, J.: Particle models and noncommutative geometry. Nucl. Phys. B (Proc. Suppl.) 18, 29–47 (1990)

    Article  MathSciNet  Google Scholar 

  22. Connes, A., Lott, J.: The metric aspect of noncommutative geometry. Cargèse Summer Conference, J. Fröhlich et al. (eds), London-New York: 1992, pp. 53–93

  23. Connes, A., Moscovici, H.: The local index formula in noncommutative geometry. Geom. Func. Anal. 5, 174–243 (1995)

    MathSciNet  MATH  Google Scholar 

  24. Connes, A., Douglas, M.R., Schwarz, A.: Noncommutative geometry and Matrix theory: Compactification on tori. J. High Energy Phys. 02, 003 (1998)

    Google Scholar 

  25. Connes, A., Landi, G.: Noncommutative manifolds, the instanton algebra and isospectral deformations. Commun. Math. Phys. 221, 141–159 (2001)

    MATH  Google Scholar 

  26. Connes, A., Dubois-Violette, M.: Noncommutative finite-dimensional manifolds. I. Spherical manifolds and related examples. Commun. Math. Phys. 230, 539–579 (2002)

    Article  MATH  Google Scholar 

  27. Cwikel, M.: Weak type estimates for singular values and the number of bound states of Schrödinger operators. Ann. Math. 106, 93–100 (1977)

    MATH  Google Scholar 

  28. Daubechies, I.: Continuity statements and counterintuitive examples in connection with Weyl quantization. J. Math. Phys. 24, 1453–1461 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  29. Dimassi, M., Sjöstrand, J.: Spectral Asymptotics in the Semi-Classical Limit. Cambridge: Cambridge University Press, 1999

  30. Dixmier, J., Malliavin, P.: Factorisations de fonctions et de vecteurs indéfiniment différentiables. Bull. Soc. Math. France 102, 305–330 (1978)

    MATH  Google Scholar 

  31. Dubois-Violette, M., Kriegl, A., Maeda, Y., Michor, P.W.: Smooth *-algebras. Prog. Theor. Phys. Suppl. 144, 54–78 (2001)

    MATH  Google Scholar 

  32. Estrada, R.: Some Tauberian theorems for Schwartz distributions. Publ. Math. Debrecen 61, 1–9 (2002)

    MathSciNet  MATH  Google Scholar 

  33. Estrada, R., Gracia-Bondía, J.M., Várilly, J.C.: On asymptotic expansions of twisted products. J. Math. Phys. 30, 2789–2796 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  34. Estrada, R., Gracia-Bondía, J.M., Várilly, J.C.: On summability of distributions and spectral geometry. Commun. Math. Phys. 191, 219–248 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  35. Figueroa, H.: Function algebras under the twisted product. Bol. Soc. Paranaense Mat. 11, 115–129 (1990)

    MathSciNet  MATH  Google Scholar 

  36. Folland, G.B.: Harmonic Analysis in Phase Space. Princeton, NJ: Princeton University Press, 1989

  37. Gadella, M., Gracia-Bondía, J.M., Nieto, L.M., Várilly, J.C.: Quadratic Hamiltonians in phase space quantum mechanics. J. Phys. A 22, 2709–2738 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  38. Gayral, V.: The action functional for Moyal planes. Marseille, hep-th/0307220, Lett. Math. Phys. 65, 147–157 (2003)

    Google Scholar 

  39. Gomis, J., Mehen, T.: Space-time noncommutative field theories and unitarity. Nucl. Phys. B591, 265–276 (2000)

    Google Scholar 

  40. Gracia-Bondía, J.M.: Generalized Moyal quantization on homogeneous symplectic spaces. In: Deformation Theory and Quantum Groups with Application to Mathematical Physics, J. Stasheff, M. Gerstenhaber, eds., Contemp. Math. 134, 93–114 (1992)

    Google Scholar 

  41. Gracia-Bondía, J.M., Lizzi, F., Marmo, G., Vitale, P.: Infinitely many star products to play with. J. High Energy Phys. 04, 026 (2002)

    Article  Google Scholar 

  42. Gracia-Bondía, J.M., Martín, C.P.: Chiral gauge anomalies on noncommutative ℝ4. Phys. Lett. B479, 321–328 (2000)

    Google Scholar 

  43. Gracia-Bondía, J.M., Várilly, J.C.: Algebras of distributions suitable for phase-space quantum mechanics I. J. Math. Phys. 29, 869–879 (1988)

    Article  MathSciNet  Google Scholar 

  44. Gracia-Bondía, J.M., Várilly, J.C., Figueroa, H.: The dual space of the algebra L b (S). San José, 1989, unpublished

  45. Gracia-Bondía, J.M., Várilly, J.C., Figueroa, H.: Elements of Noncommutative Geometry. Birkhäuser Advanced Texts, Boston: Birkhäuser, 2001

  46. Groenewold, H.J.: On the principles of elementary quantum mechanics. Physica 12, 405–460 (1946)

    MathSciNet  MATH  Google Scholar 

  47. Grossmann, A., Loupias, G., Stein, E.M.: An algebra of pseudodifferential operators and quantum mechanics in phase space. Ann. Inst. Fourier (Grenoble) 18, 343–368 (1968)

    MATH  Google Scholar 

  48. Hansen, F.: Quantum mechanics in phase space. Rep. Math. Phys. 19, 361–381 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  49. Hansen, F.: The Moyal product and spectral theory for a class of infinite dimensional matrices. Publ. RIMS (Kyoto) 26, 885–933 (1990)

    MATH  Google Scholar 

  50. Higson, N.: On the Connes–Moscovici residue cocycle. Preprint, Penn. State University, 2003; Lectures at the Clay Mathematics Institute Spring School on Noncommutative Geometry and Applications, Nashville, May 2003

  51. Higson, N., Roe, J.: Analytic K-Homology. Oxford: Oxford University Press, 2000

  52. Hörmander, L.: The Analysis of Partial Differential Operators III. Berlin: Springer, 1986

  53. Horváth, J.: Topological Vector Spaces and Distributions I. Reading, MA: Addison-Wesley, 1966

  54. Howe, R.: Quantum Mechanics and partial differential equations. J. Funct. Anal. 38, 188–254 (1980)

    MathSciNet  MATH  Google Scholar 

  55. Iochum, B., Schücker, T.: A left-right symmetric model à la Connes–Lott. Lett. Math. Phys. 32, 153–166 (1994)

    MathSciNet  MATH  Google Scholar 

  56. Kammerer, J.-B.: Analysis of the Moyal product in flat space. J. Math. Phys. 27, 529–535 (1986)

    Article  MathSciNet  Google Scholar 

  57. Kastler, D.: The C*-algebras of a free boson field I. Commun. Math. Phys. 1, 14–48 (1965)

    MATH  Google Scholar 

  58. Langmann, E., Mickelsson, J.: Scattering matrix in external field problems. J. Math. Phys. 37, 3933–3953 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  59. Lassner, G., Lassner, G.A.: Qu*-algebras and twisted product. BiBoS preprint 246, Bielefeld, 1987

  60. Lizzi, F., Szabo, R.J., Zampini, A.: Geometry of the gauge algebra in noncommutative Yang–Mills theory. J. High Energy Phys. 08, 032 (2001)

    Google Scholar 

  61. Loday, J.-L.: Cyclic Homology. Berlin: Springer, 1992

  62. Martín, C.P.: The UV and IR origin of non-Abelian chiral gauge anomalies on noncommutative Minkowski spacetime. J. Phys. A: Math. Gen. 34, 9037–9055 (2001)

    Article  MathSciNet  Google Scholar 

  63. Martín, C.P., Gracia-Bondía, J.M., Várilly, J.C.: The Standard Model as a noncommutative geometry: The low energy regime. Phys. Reps. 294, 363–406 (1998)

    Article  Google Scholar 

  64. Melo, S.T., Merklen, M.I.: On a conjectured noncommutative Beals–Cordes-type characterization. Proc. Am. Math. Soc. 130, 1997–2000 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  65. Moyal, J.E.: Quantum mechanics as a statistical theory. Proc. Camb. Philos. Soc. 45, 99–124 (1949)

    MathSciNet  MATH  Google Scholar 

  66. von Neumann, J.: Die Eindeutigkeit der Schrödingerschen Operatoren. Math. Ann. 104, 570–578 (1931)

    MATH  Google Scholar 

  67. Nicola, F.: Trace functionals for a class of pseudo-differential operators in ℝn. Math. Phys. Anal. Geom. 6, 89–105 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  68. Ortner, N., Wagner, P.: Applications of weighted D‘ L p spaces to the convolution of distributions. Bull. Acad. Pol. Sci. Math. 37, 579–595 (1989)

    Google Scholar 

  69. Poincaré, H.: Les méthodes nouvelles de la Mécanique Céleste. Vol. 3, Paris: Gauthier-Villars, 1892

  70. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, II: Fourier Analysis, Self Adjointness. New York: Academic Press, 1975

  71. Rennie, A.: Commutative geometries are spin manifolds. Rev. Math. Phys. 13, 409–464 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  72. Rennie, A.: Poincaré duality and spinc structures for complete noncommutative manifolds. math-ph/0107013, Adelaide, 2001

  73. Rennie, A.: Smoothness and locality for nonunital spectral triples. K-Theory 28, 127–165 (2003)

    Article  MATH  Google Scholar 

  74. Rieffel, M.A.: C*-algebras associated with irrational rotations. Pac. J. Math. 93, 415–429 (1981)

    MathSciNet  MATH  Google Scholar 

  75. Rieffel, M.A.: Deformation Quantization for Actions of ℝ d. Mem. Am. Math. Soc. 506, Providence, RI: AMS, 1993

  76. Rieffel, M.A.: Compact quantum groups associated with toral subgroups. In: Representation Theory of Groups and Algebras, J. Adams et al, eds., Providence, RI: Am. Math. Soc., Contemp. Math. 145, 465–491 (1993)

    Google Scholar 

  77. Schwartz, L.: Théorie des Distributions. Paris: Hermann, 1966

  78. Seiberg, N., Susskind, L., Toumbas, N.: Strings in background electric field, space/time noncommutativity and a new noncritical string theory. J. High Energy Phys. 06, 021 (2000)

    MATH  Google Scholar 

  79. Seiberg, N., Witten, E.: String theory and noncommutative geometry. J. High Energy Phys. 09, 032 (1999)

    MATH  Google Scholar 

  80. Shubin, M.A.: Pseudodifferential Operators and Spectral Theory. Berlin: Springer, 1980

  81. Simon, B.: Trace Ideals and Their Applications. Cambridge: Cambridge University Press, 1979

  82. Sitarz, A.: Rieffel’s deformation quantization and isospectral deformations. Int. J. Theor. Phys. 40, 1693–1696 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  83. Terashima, S.: A note on superfields and noncommutative geometry. Phys. Lett. B 482, 276–282 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  84. Treves, F.: Topological Vector Spaces, Distributions and Kernels. New York: Academic Press, 1967

  85. Tuynman, G.M.: Prequantization is irreducible. Indag. Math. 9, 607–618 (1998)

    MathSciNet  MATH  Google Scholar 

  86. van Hove, L.: Sur certaines représentations unitaires d’un group infini de transformations. Mém. Acad. Roy. Belgique Cl. Sci. 26, 1–102 (1951)

    Google Scholar 

  87. Várilly, J.C.: Quantum symmetry groups of noncommutative spheres. Commun. Math. Phys. 221, 511–523 (2001)

    Article  MathSciNet  Google Scholar 

  88. Várilly, J.C.: Hopf algebras in noncommutative geometry. In: Geometrical and Topological Methods in Quantum Field Theory. A. Cardona, H. Ocampo, S. Paycha, eds., Singapore: World Scientific, 2003, pp. 1–85

  89. Várilly, J.C., Gracia-Bondía, J.M.: Los grupos simplécticos y su representación en la teoría del producto cuántico. I. Sp(2,ℝ). Cienc. Tec. (CR) 11, 65–83 (1987)

    Google Scholar 

  90. Várilly, J.C., Gracia-Bondía, J.M.: Algebras of distributions suitable for phase-space quantum mechanics II: Topologies on the Moyal algebra. J. Math. Phys. 29, 880–887 (1988)

    Article  MathSciNet  Google Scholar 

  91. Várilly, J.C., Gracia-Bondía, J.M.: Connes’ noncommutative differential geometry and the Standard Model. J. Geom. Phys. 12, 223–301 (1993)

    Article  MathSciNet  Google Scholar 

  92. Várilly, J.C., Gracia-Bondía, J.M.: On the ultraviolet behaviour of quantum fields on noncommutative manifolds. Int. J. Mod. Phys. A14, 1305–1323 (1999)

    Google Scholar 

  93. Voros, A.: An algebra of pseudodifferential operators and the asymptotic of quantum mechanics. J. Funct. Anal. 29, 104–132 (1978)

    MATH  Google Scholar 

  94. Weidl, T.: Another look at Cwikel’s inequality. Am. Math. Soc. Transl. 189, 247–254 (1999)

    MATH  Google Scholar 

  95. Wightman, A.S., Gårding, L.: Fields as operator-valued distributions in relativistic quantum theory. Arkiv för Fysik 28, 129–189 (1965)

    MATH  Google Scholar 

  96. Wildberger, N.J.: Characters, bimodules and representations in Lie group harmonic analysis. In: Harmonic Analysis and Hypergroups, K. A. Ross et al, eds., Boston: Birkhäuser, 1998, pp. 227–242

  97. Wodzicki, M.: Excision in cyclic homology and in rational algebraic K-theory. Ann. of Math. 129, 591–639 (1989)

    MathSciNet  MATH  Google Scholar 

  98. Zinn-Justin, J.: Quantum Field Theory and Critical Phenomena. Oxford: Clarendon Press, 2002

Download references

Author information

Authors and Affiliations

  1. Centre de Physique Théorique, CNRS–Luminy, Case 907, 13288, Marseille Cedex 9, France

    V. Gayral, B. Iochum & T. Schücker

  2. Université de Provence, France

    V. Gayral, B. Iochum & T. Schücker

  3. Departamento de Física, Universidad de Costa Rica, 2060, San Pedro, Costa Rica

    J.M. Gracia-Bondía

  4. Departamento de Matemáticas, Universidad de Costa Rica, 2060, San Pedro, Costa Rica

    J.C. Várilly

  5. Regular Associate of the Abdus Salam ICTP, 34014, Trieste, Italy

    J.C. Várilly

Authors
  1. V. Gayral
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J.M. Gracia-Bondía
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. Iochum
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. T. Schücker
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. J.C. Várilly
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. Iochum.

Additional information

Communicated by A. Connes

Unité Propre de Recherche 7061

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gayral, V., Gracia-Bondía, J., Iochum, B. et al. Moyal Planes are Spectral Triples. Commun. Math. Phys. 246, 569–623 (2004). https://doi.org/10.1007/s00220-004-1057-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-004-1057-z

Keywords

Search

Navigation