Communications in Mathematical Physics

, Volume 246, Issue 3, pp 569–623 | Cite as

Moyal Planes are Spectral Triples

  • V. Gayral
  • J.M. Gracia-Bondía
  • B. Iochum
  • T. Schücker
  • J.C. Várilly


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.


Field Theory Quantum Field Theory Physical Application Functional Action Spectral Triple 
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  1. 1.
    Amiet, J.-P., Huguenin, P.: Mécaniques classique et quantique dans l’espace de phase. Université de Neuchatel, Neuchatel, 1981Google Scholar
  2. 2.
    Baez, J.C., Segal, I.E., Zhou, Z.: Introduction to Algebraic and Constructive Quantum Field Theory. Princeton, New Jersey: Princeton University Press, 1992Google Scholar
  3. 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. 4.
    Bahns, D., Doplicher, S., Fredenhagen, K., Piacitelli, G.: Ultraviolet finite quantum field theory on quantum spacetime. Commun. Math. Phys. 237, 221–241 (2003)MATHGoogle Scholar
  5. 5.
    Banach, S.: Remarques sur les groupes et les corps métriques. Studia Math 10, 178–181 (1948)MathSciNetMATHGoogle Scholar
  6. 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)MATHGoogle Scholar
  7. 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)MATHGoogle Scholar
  8. 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)MATHGoogle Scholar
  9. 9.
    Brouder, C., Fauser, B., Frabetti, A., Oeckl, R.: Quantum groups and quantum field theory: I. The free scalar field. Paris, 2001, hep-th/0311253Google Scholar
  10. 10.
    Carey, A., Phillips, J., Sukochev, F.: Spectral flow and Dixmier traces. Adv. Math. 173, 68–113 (2003)CrossRefMATHGoogle Scholar
  11. 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. 12.
    Chamseddine, A.H.: Noncommutative gravity. Ann. Henri Poincaré 4 Suppl. 2, S881–S887 (2003)Google Scholar
  13. 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)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Connes, A.: C*-algèbres et géométrie différentielle. C. R. Acad. Sci. Paris 290, 599–604 (1980)MATHGoogle Scholar
  15. 15.
    Connes, A.: Noncommutative differential geometry. Publ. Math. IHES 39, 257–360 (1985)MathSciNetGoogle Scholar
  16. 16.
    Connes, A.: The action functional in noncommutative geometry. Commun. Math. Phys. 117, 673–683 (1988)MathSciNetMATHGoogle Scholar
  17. 17.
    Connes, A.: Noncommutative Geometry. London and San Diego: Academic Press, 1994Google Scholar
  18. 18.
    Connes, A.: Noncommutative geometry and reality. J. Math. Phys. 36, 6194–6231 (1995)CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Connes, A.: La notion de variété et les axiomes de la géométrie. Course at the Collège de France, January–March 1996Google Scholar
  20. 20.
    Connes, A.: Gravity coupled with matter and the foundation of noncommutative geometry. Commun. Math. Phys. 182, 155–176 (1996)MathSciNetMATHGoogle Scholar
  21. 21.
    Connes, A., Lott, J.: Particle models and noncommutative geometry. Nucl. Phys. B (Proc. Suppl.) 18, 29–47 (1990)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Connes, A., Lott, J.: The metric aspect of noncommutative geometry. Cargèse Summer Conference, J. Fröhlich et al. (eds), London-New York: 1992, pp. 53–93Google Scholar
  23. 23.
    Connes, A., Moscovici, H.: The local index formula in noncommutative geometry. Geom. Func. Anal. 5, 174–243 (1995)MathSciNetMATHGoogle Scholar
  24. 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. 25.
    Connes, A., Landi, G.: Noncommutative manifolds, the instanton algebra and isospectral deformations. Commun. Math. Phys. 221, 141–159 (2001)MATHGoogle Scholar
  26. 26.
    Connes, A., Dubois-Violette, M.: Noncommutative finite-dimensional manifolds. I. Spherical manifolds and related examples. Commun. Math. Phys. 230, 539–579 (2002)CrossRefMATHGoogle Scholar
  27. 27.
    Cwikel, M.: Weak type estimates for singular values and the number of bound states of Schrödinger operators. Ann. Math. 106, 93–100 (1977)MATHGoogle Scholar
  28. 28.
    Daubechies, I.: Continuity statements and counterintuitive examples in connection with Weyl quantization. J. Math. Phys. 24, 1453–1461 (1983)CrossRefMathSciNetMATHGoogle Scholar
  29. 29.
    Dimassi, M., Sjöstrand, J.: Spectral Asymptotics in the Semi-Classical Limit. Cambridge: Cambridge University Press, 1999Google Scholar
  30. 30.
    Dixmier, J., Malliavin, P.: Factorisations de fonctions et de vecteurs indéfiniment différentiables. Bull. Soc. Math. France 102, 305–330 (1978)MATHGoogle Scholar
  31. 31.
    Dubois-Violette, M., Kriegl, A., Maeda, Y., Michor, P.W.: Smooth *-algebras. Prog. Theor. Phys. Suppl. 144, 54–78 (2001)MATHGoogle Scholar
  32. 32.
    Estrada, R.: Some Tauberian theorems for Schwartz distributions. Publ. Math. Debrecen 61, 1–9 (2002)MathSciNetMATHGoogle Scholar
  33. 33.
    Estrada, R., Gracia-Bondía, J.M., Várilly, J.C.: On asymptotic expansions of twisted products. J. Math. Phys. 30, 2789–2796 (1989)CrossRefMathSciNetMATHGoogle Scholar
  34. 34.
    Estrada, R., Gracia-Bondía, J.M., Várilly, J.C.: On summability of distributions and spectral geometry. Commun. Math. Phys. 191, 219–248 (1998)CrossRefMathSciNetMATHGoogle Scholar
  35. 35.
    Figueroa, H.: Function algebras under the twisted product. Bol. Soc. Paranaense Mat. 11, 115–129 (1990)MathSciNetMATHGoogle Scholar
  36. 36.
    Folland, G.B.: Harmonic Analysis in Phase Space. Princeton, NJ: Princeton University Press, 1989Google Scholar
  37. 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)CrossRefMathSciNetMATHGoogle Scholar
  38. 38.
    Gayral, V.: The action functional for Moyal planes. Marseille, hep-th/0307220, Lett. Math. Phys. 65, 147–157 (2003)Google Scholar
  39. 39.
    Gomis, J., Mehen, T.: Space-time noncommutative field theories and unitarity. Nucl. Phys. B591, 265–276 (2000)Google Scholar
  40. 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. 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)CrossRefGoogle Scholar
  42. 42.
    Gracia-Bondía, J.M., Martín, C.P.: Chiral gauge anomalies on noncommutative ℝ4. Phys. Lett. B479, 321–328 (2000)Google Scholar
  43. 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)CrossRefMathSciNetGoogle Scholar
  44. 44.
    Gracia-Bondía, J.M., Várilly, J.C., Figueroa, H.: The dual space of the algebra L b(S). San José, 1989, unpublishedGoogle Scholar
  45. 45.
    Gracia-Bondía, J.M., Várilly, J.C., Figueroa, H.: Elements of Noncommutative Geometry. Birkhäuser Advanced Texts, Boston: Birkhäuser, 2001Google Scholar
  46. 46.
    Groenewold, H.J.: On the principles of elementary quantum mechanics. Physica 12, 405–460 (1946)MathSciNetMATHGoogle Scholar
  47. 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)MATHGoogle Scholar
  48. 48.
    Hansen, F.: Quantum mechanics in phase space. Rep. Math. Phys. 19, 361–381 (1984)CrossRefMathSciNetMATHGoogle Scholar
  49. 49.
    Hansen, F.: The Moyal product and spectral theory for a class of infinite dimensional matrices. Publ. RIMS (Kyoto) 26, 885–933 (1990)MATHGoogle Scholar
  50. 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 2003Google Scholar
  51. 51.
    Higson, N., Roe, J.: Analytic K-Homology. Oxford: Oxford University Press, 2000Google Scholar
  52. 52.
    Hörmander, L.: The Analysis of Partial Differential Operators III. Berlin: Springer, 1986Google Scholar
  53. 53.
    Horváth, J.: Topological Vector Spaces and Distributions I. Reading, MA: Addison-Wesley, 1966Google Scholar
  54. 54.
    Howe, R.: Quantum Mechanics and partial differential equations. J. Funct. Anal. 38, 188–254 (1980)MathSciNetMATHGoogle Scholar
  55. 55.
    Iochum, B., Schücker, T.: A left-right symmetric model à la Connes–Lott. Lett. Math. Phys. 32, 153–166 (1994)MathSciNetMATHGoogle Scholar
  56. 56.
    Kammerer, J.-B.: Analysis of the Moyal product in flat space. J. Math. Phys. 27, 529–535 (1986)CrossRefMathSciNetGoogle Scholar
  57. 57.
    Kastler, D.: The C*-algebras of a free boson field I. Commun. Math. Phys. 1, 14–48 (1965)MATHGoogle Scholar
  58. 58.
    Langmann, E., Mickelsson, J.: Scattering matrix in external field problems. J. Math. Phys. 37, 3933–3953 (1996)CrossRefMathSciNetMATHGoogle Scholar
  59. 59.
    Lassner, G., Lassner, G.A.: Qu*-algebras and twisted product. BiBoS preprint 246, Bielefeld, 1987Google Scholar
  60. 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. 61.
    Loday, J.-L.: Cyclic Homology. Berlin: Springer, 1992Google Scholar
  62. 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)CrossRefMathSciNetGoogle Scholar
  63. 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)CrossRefGoogle Scholar
  64. 64.
    Melo, S.T., Merklen, M.I.: On a conjectured noncommutative Beals–Cordes-type characterization. Proc. Am. Math. Soc. 130, 1997–2000 (2002)CrossRefMathSciNetMATHGoogle Scholar
  65. 65.
    Moyal, J.E.: Quantum mechanics as a statistical theory. Proc. Camb. Philos. Soc. 45, 99–124 (1949)MathSciNetMATHGoogle Scholar
  66. 66.
    von Neumann, J.: Die Eindeutigkeit der Schrödingerschen Operatoren. Math. Ann. 104, 570–578 (1931)MATHGoogle Scholar
  67. 67.
    Nicola, F.: Trace functionals for a class of pseudo-differential operators in ℝn. Math. Phys. Anal. Geom. 6, 89–105 (2003)CrossRefMathSciNetMATHGoogle Scholar
  68. 68.
    Ortner, N., Wagner, P.: Applications of weighted DLp spaces to the convolution of distributions. Bull. Acad. Pol. Sci. Math. 37, 579–595 (1989)Google Scholar
  69. 69.
    Poincaré, H.: Les méthodes nouvelles de la Mécanique Céleste. Vol. 3, Paris: Gauthier-Villars, 1892Google Scholar
  70. 70.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, II: Fourier Analysis, Self Adjointness. New York: Academic Press, 1975Google Scholar
  71. 71.
    Rennie, A.: Commutative geometries are spin manifolds. Rev. Math. Phys. 13, 409–464 (2001)CrossRefMathSciNetMATHGoogle Scholar
  72. 72.
    Rennie, A.: Poincaré duality and spinc structures for complete noncommutative manifolds. math-ph/0107013, Adelaide, 2001Google Scholar
  73. 73.
    Rennie, A.: Smoothness and locality for nonunital spectral triples. K-Theory 28, 127–165 (2003)CrossRefMATHGoogle Scholar
  74. 74.
    Rieffel, M.A.: C*-algebras associated with irrational rotations. Pac. J. Math. 93, 415–429 (1981)MathSciNetMATHGoogle Scholar
  75. 75.
    Rieffel, M.A.: Deformation Quantization for Actions of ℝ d. Mem. Am. Math. Soc. 506, Providence, RI: AMS, 1993Google Scholar
  76. 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. 77.
    Schwartz, L.: Théorie des Distributions. Paris: Hermann, 1966Google Scholar
  78. 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)MATHGoogle Scholar
  79. 79.
    Seiberg, N., Witten, E.: String theory and noncommutative geometry. J. High Energy Phys. 09, 032 (1999)MATHGoogle Scholar
  80. 80.
    Shubin, M.A.: Pseudodifferential Operators and Spectral Theory. Berlin: Springer, 1980Google Scholar
  81. 81.
    Simon, B.: Trace Ideals and Their Applications. Cambridge: Cambridge University Press, 1979Google Scholar
  82. 82.
    Sitarz, A.: Rieffel’s deformation quantization and isospectral deformations. Int. J. Theor. Phys. 40, 1693–1696 (2001)CrossRefMathSciNetMATHGoogle Scholar
  83. 83.
    Terashima, S.: A note on superfields and noncommutative geometry. Phys. Lett. B 482, 276–282 (2000)CrossRefMathSciNetMATHGoogle Scholar
  84. 84.
    Treves, F.: Topological Vector Spaces, Distributions and Kernels. New York: Academic Press, 1967Google Scholar
  85. 85.
    Tuynman, G.M.: Prequantization is irreducible. Indag. Math. 9, 607–618 (1998)MathSciNetMATHGoogle Scholar
  86. 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. 87.
    Várilly, J.C.: Quantum symmetry groups of noncommutative spheres. Commun. Math. Phys. 221, 511–523 (2001)CrossRefMathSciNetGoogle Scholar
  88. 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. 1–85Google Scholar
  89. 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. 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)CrossRefMathSciNetGoogle Scholar
  91. 91.
    Várilly, J.C., Gracia-Bondía, J.M.: Connes’ noncommutative differential geometry and the Standard Model. J. Geom. Phys. 12, 223–301 (1993)CrossRefMathSciNetGoogle Scholar
  92. 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. 93.
    Voros, A.: An algebra of pseudodifferential operators and the asymptotic of quantum mechanics. J. Funct. Anal. 29, 104–132 (1978)MATHGoogle Scholar
  94. 94.
    Weidl, T.: Another look at Cwikel’s inequality. Am. Math. Soc. Transl. 189, 247–254 (1999)MATHGoogle Scholar
  95. 95.
    Wightman, A.S., Gårding, L.: Fields as operator-valued distributions in relativistic quantum theory. Arkiv för Fysik 28, 129–189 (1965)MATHGoogle Scholar
  96. 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. 227–242Google Scholar
  97. 97.
    Wodzicki, M.: Excision in cyclic homology and in rational algebraic K-theory. Ann. of Math. 129, 591–639 (1989)MathSciNetMATHGoogle Scholar
  98. 98.
    Zinn-Justin, J.: Quantum Field Theory and Critical Phenomena. Oxford: Clarendon Press, 2002Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • V. Gayral
    • 1
    • 2
  • J.M. Gracia-Bondía
    • 3
  • B. Iochum
    • 1
    • 2
  • T. Schücker
    • 1
    • 2
  • J.C. Várilly
    • 4
    • 5
  1. 1.Centre de Physique ThéoriqueCNRS–LuminyMarseille Cedex 9France
  2. 2.Université de ProvenceFrance
  3. 3.Departamento de FísicaUniversidad de Costa RicaSan PedroCosta Rica
  4. 4.Departamento de MatemáticasUniversidad de Costa RicaSan PedroCosta Rica
  5. 5.Regular Associate of the Abdus Salam ICTPTriesteItaly

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