Landi G., Rovelli C
Gravity from Dirac Eigenvalues
(44K, latex)

ABSTRACT.  This is a new and enlarged version of the paper pubblished in 
Phys Rev Lett 78 (1997) 3051. A section on matter couplings has been added.
Relations between the constraints on the eigenvalues of the Dirac
operators and Connes' axioms for noncommutative geometry are hinted at.
Also, a new derivation of the (modified) gravitational spectral action is
given in term of the heat kernel expansion.
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.