Landi G., Rovelli C.
General Relativity in terms of Dirac Eigenvalues
(31K, revtex)

ABSTRACT.  The eigenvalues of the Dirac operator on a curved spacetime are 
diffeomorphism-invariant functions of the geometry.  They 
form an infinite set of ``observables'' for general relativity.  
Recent work of Chamseddine and Connes suggests that they can be taken 
as variables for an invariant description of the gravitational 
field's dynamics.  We compute the Poisson brackets of these eigenvalues 
and find them in terms of the energy-momentum of the eigenspinors and the 
propagator of the linearized Einstein equations.  We show that the 
eigenspinors' energy-momentum is the Jacobian matrix of the change of 
coordinates from metric to eigenvalues.  We also consider a minor 
modification of the spectral action, which eliminates the disturbing 
huge cosmological term and derive its equations of motion. These are  
satisfied if the energy momentum of the trans Planckian 
eigenspinors scale linearly with the eigenvalue; we argue that this 
requirement approximates the Einstein equations.