Lorentzian Connes Distance, Spectral Graph Distance and Loop Gravity
  Connes' formula defines a distance in loop quantum gravity, via the spinfoam Dirac operator. A simple notion of spectral distance on a graph can be extended do the discrete Lorentzian context, providing a physically natural example of Lorentzian spectral geometry, with a neat space of Dirac operators. The Hilbert structure of the fermion space is Lorentz covariant rather than invariant. 
Carlo Rovelli
gr-qc math-ph math.MP