Gravity from Dirac Eigenvalues

We study a formulation of Euclidean general relativity in which the dynamical variables are given by a sequence of real numbers λn, representing the eigenvalues of the Dirac operator on the curved space–time. 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 λn'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 λn'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.