97-456 Landi G., Rovelli C
Gravity from Dirac Eigenvalues (44K, latex) Aug 22, 97
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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.

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