Gravity coupled with matter and the foundation of noncommutative geometry
Abstract
We first exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its infinitesimal length element ds. Its unitary representations correspond to Riemannian metrics and Spin structure while ds is the Dirac propagator ds=xx= D ^{1}, where D is the Dirac operator. We extend these simple relations to the noncommutative case using Tomita's involution J. We then write a spectral action, the trace of a function of the length element, which when applied to the noncommutative geometry of the Standard Model will be shown ([CC]) to give the SM Lagrangian coupled to gravity. The internal fluctuations of the noncommutative geometry are trivial in the commutative case but yield the full bosonic sector of SM with all correct quantum numbers in this slightly noncommutative case. The group of local gauge transformations appears spontaneously as a normal subgroup of the diffeomorphism group.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 December 1996
 DOI:
 10.1007/BF02506388
 arXiv:
 arXiv:hepth/9603053
 Bibcode:
 1996CMaPh.182..155C
 Keywords:

 High Energy Physics  Theory;
 Mathematics  Quantum Algebra
 EPrint:
 30 pages, Plain TeX