General Relativity in Terms of Dirac Eigenvalues

The eigenvalues of the Dirac operator are diffeomorphism-invariant functions of the geometry, namely, “observables” for general relativity. Recent work by Chamseddine and Connes suggests taking them as gravity's dynamical variables. We compute their Poisson brackets, find that these can be expressed in terms of energy momenta $T$ of the eigenspinors, and show that $T$ is the Jacobian matrix of the transformation from metric to eigenvalues. We consider a small modification of the spectral action that gets rid of the cosmological term, and derive its equations of motion. These are solved if $T$ scales linearly. We show that such a scaling law yields Einstein equations.