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Landi G., Rovelli C.
General Relativity in terms of Dirac Eigenvalues
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.