2 research outputs found
General Relativity in terms of Dirac Eigenvalues
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.Comment: 6 pages, RevTe
Observables of the Euclidean Supergravity
The set of constraints under which the eigenvalues of the Dirac operator can
play the role of the dynamical variables for Euclidean supergravity is derived.
These constraints arise when the gauge invariance of the eigenvalues of the
Dirac operator is imposed. They impose conditions which restrict the
eigenspinors of the Dirac operator.Comment: Revised version, some misprints in the ecuations (11), (13) and (17)
corrected. The errors in the published version will appear cortected in a
future erratu