How to define spin in quantum field theory on curved spacetime

[paweld]

How one can define a spin in Qunatum Filed Theory in curved spacetime. If the
space is flat it's invarainat under Poincare group - so in particular it's invariant under SO(3).
Spin operators are simply generators of SO(3). If the space isn't flat we cannot define
spin in this way. I know that in curved spacetime we should think of fileds rather than particles,
because notion of particle is not always well defined. So maybe better question is
how we define spin in globally hyberbolic and stationary spacetime?

 

[DarMM]

You just form a bundle over the spacetime with the same Fiber as in the Minkowski case.

See Mikio Nakahara "Geometry, Topology and Physics, Second Edition" Section 11.6

It not only tells you how to define it, but what sort of manifolds do not admit a spinor field.