Abstract
We first exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its infinitesimal length elementds. Its unitary representations correspond to Riemannian metrics and Spin structure whileds is the Dirac propagatords=x−x=D −1, whereD is the Dirac operator. We extend these simple relations to the non-commutative case using Tomita's involutionJ. We then write a spectral action, the trace of a function of the length element, which when applied to the non-commutative geometry of the Standard Model will be shown ([CC]) to give the SM Lagrangian coupled to gravity. The internal fluctuations of the non-commutative geometry are trivial in the commutative case but yield the full bosonic sector of SM with all correct quantum numbers in this slightly non-commutative case. The group of local gauge transformations appears spontaneously as a normal subgroup of the diffeomorphism group.
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[CC] Chamseddine, A., Connes, A.: The spectral action principle. To appear
[C] Connes, A.: Non-commutative geometry and reality. J. Math. Phys.36, no 11 (1995)
[Co] Connes, A.: Non-commutative geometry. New York-London: Academic Press, 1994
[CL] Connes, A., Lott, J.: Particle models and non-commutative geometry. Nucl. Phys. B18B (1990), suppl., 29–47 (1991)
[CM] Connes, A., Moscovici, H.: The local index formula in non-commutative geometry. GAFA
[CR] Connes, A., Rieffel, M.: Yang Mills for non-commutative two tori. In: Operator algebras and mathematical physics. Iowa City, Iowa: Univ. of Iowa Press, 1985, pp. 237–266
[CS] Connes, A., Skandalis, G.: The longitudinal index theorem for foliations. Publ. Res. Inst. Math. Sci. Kyoto20, 1139–1183 (1984)
[G] Gromov, M.: Carnot-Caratheodory spaces seen from within. Preprint IHES/M/94/6
[GKP] Grosse, H., Klimcik, C., Presnajder, P.: On finite 4 dimensional quantum field theory in non-commutative geometry. CERN Preprint TH/96-51 Net Hep-th/9602115
[K] Kastler, D.: The Dirac operator and gravitation. Commun. Math. Phys.166, 633–643 (1995)
[KW] Kalau, W., Walze, M.: Gravity, non-commutative geometry and the Wodzicki residue. J. of Geom. and Phys.16, 327–344 (1995)
[L] Loday, J.L.: Cyclic homology, Berlin: Springer, 1992
[LM] Lawson, B., Michelson, M.L.: Spin Geometry: Princeton, NJ: Princeton University Press, 1989
[M] Manin, Y.: Quantum groups and non-commutative geometry. Centre Recherche Math. Univ. Montréal (1988)
[PV] Pimsner, M., Voiculescu, D.: Exact sequences forK groups and Ext groups of certain crossed productC * algebra. J. Operator Theory4, 93–118 (1980)
[Ri] Rieffel, M.A.:C *-algebras associated with irrational rotations. Pacific J. Math.93, 415–429 (1981)
[S] Sullivan, D.: Geometric periodicity and the invariants of manifolds. Lecture Notes in Math.197, Berlin-Heidelberg-New York: Springer, 1971
[Ta] Takesaki, M.: Tomita's theory of modular Hilbert algebras and its applications. Lecture Notes in Math.128, Berlin-Heidelberg-New York: Springer, 1970
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Communicated by A. Jaffe
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Connes, A. Gravity coupled with matter and the foundation of non-commutative geometry. Commun.Math. Phys. 182, 155–176 (1996). https://doi.org/10.1007/BF02506388
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DOI: https://doi.org/10.1007/BF02506388