Gravity from Dirac Eigenvalues

Abstract

We study a formulation of euclidean general relativity in which the dynamical variables are given by a sequence of real numbers ${\lambda }_n$, representing the eigenvalues of the Dirac operator on the curved spacetime. These quantities are diffeomorphism-invariant functions of the metric and they form an infinite set of ``physical observables'' for general relativity. Recent work of Connes and Chamseddine suggests that they can be taken as natural variables for an invariant description of the dynamics of gravity. We compute the Poisson brackets of the ${\lambda }_n$'s, and find that these can be expressed in terms of the propagator of the linearized Einstein equations and the energy-momentum of the eigenspinors. We show that the eigenspinors' energy-momentum is the Jacobian matrix of the change of coordinates from the metric to the ${\lambda }_n$'s. We study a variant of the Connes-Chamseddine spectral action which eliminates a disturbing large cosmological term. We analyze the corresponding equations of motion and find that these are solved if the energy momenta of the eigenspinors scale linearly with the mass. Surprisingly, this scaling law codes Einstein's equations. Finally we study the coupling to a physical fermion field.

An enlarged and improved version which will be pubblished in Mod. Phys. Lett. A

References in corpus (2)

• General Relativity in terms of Dirac Eigenvalues
• Observables of the Euclidean Supergravity

Cited by in corpus (13)

• Quantum Gravity: a Progress Report
• Forces from Connes' geometry
• Noncommutative AdS Supergravity in three Dimensions
• Towards Spectral Geometry for Causal Sets
• Towards a background independent approach to M theory
• Noncommutative Geometry, Strings and Duality
• On Local Constraints of D=4 Supergravity in Terms of Dirac Eigenvalues
• Constraints on spacetime manifold in Euclidean supergravity in terms of Dirac eigenvalues
• Quantum Topology Change and Large N Gauge Theories
• Duality of Coordinates and Matter Fields in Curved Spacetime
• Strings, Noncommutative Geometry and the Size of the Target Space
• On The Symplectic Two-Form of Gravity in Terms of Dirac Eigenvalues
• Toward Supergravity Spectral Action