Dirac operators on Lorentzian manifolds

[Kasper J. Larsen]

Hi all,


I'm a bit confused. Can one define a Dirac operator D on a Lorentzian
manifold in the same way as one defines D on a Riemannian manifold? If
yes, how do the definitions differ? (I am already acquainted with the
Riemannian manifold definition.)

Moreover, assuming that a Lorentzian Dirac operator D can be defined,
what is its physical significance? If you substitute this D for the
"flat space-time" D in the ordinary Dirac equation, do you get a
"curved space-time" equation describing spin-½-particles?


Thanks in advance!
Kasper Jens Larsen