Gravity coupled with matter and foundation of non-commutative geometry

TL;DR

This paper explores the algebraic relations between geometry and matter in both commutative and non-commutative settings, proposing a spectral action approach that reproduces the Standard Model coupled to gravity.

Contribution

It extends the algebraic relations of geometry to non-commutative spaces using Tomita's involution and formulates a spectral action that recovers the Standard Model with gravity.

Findings

Spectral action reproduces SM Lagrangian coupled to gravity.

Internal fluctuations yield the full bosonic sector of SM.

Gauge transformations emerge as a subgroup of diffeomorphisms.

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 $ds = \ts \!\!$---$\!\! \ts = D^{-1}$ where $D$ is the Dirac operator. We extend these simple relations to the non commutative case using Tomita's involution $J$. We then write a spectral action, the trace of a function of the length element in Planck units, which when applied to the non commutative geometry of the Standard Model will be shown (in a joint work with Ali Chamseddine) 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 the slightly non commutative case. The group of local gauge transformations appears spontaneously as a normal subgroup of the diffeomorphism group.