GRAVITY FROM DIRAC EIGENVALUES

We study a formulation of Euclidean general relativity in which the dy namical variables are given by a sequence of real numbers lambda(n), r epresenting the eigenvalues of the Dirac operator on the curved space- time. These quantities are diffeomorphism-invariant functions of the m etric and they form an infinite set of ''physical observables'' for ge neral relativity. Recent work of Comes and Chamseddine suggests that t hey 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 e igenspinors. We show that the eigenspinors' energy-momentum is the Jac obian matrix of the change of coordinates from the metric to the lambd a(n)'s. We study a variant of the Connes-Chamseddine spectral action w hich eliminates a disturbing large cosmological term. We analyze the c orresponding equations of motion and find that these are solved if the energy momenta of the eigenspinors scale linearly with the mass. Surp risingly, this scaling law codes Einstein's equations. Finally we stud y the coupling to a physical fermion field.