Abstract
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 of the eigenspinors, and show that 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 scales linearly. We show that such a scaling law yields Einstein equations.
- Received 10 December 1996
DOI:https://doi.org/10.1103/PhysRevLett.78.3051
©1997 American Physical Society