On the Lorentz space, particles are axiomatized as unitary projective representations of the Poincaré group (according to Wigner if I recall correctly). It is then possible to specify a (non-charged) particle by its mass and spin (or helicity).
It is said that there is no natural notion of particle in a curved spacetime (eg. here). But then, even without considering a QFT formalism, one can build classical particles of a given spin and mass by considering sections of the suitable fibre bundles constructed from the tangent bundle and some spin structure if need be (see for example Wald's General Relativity for the spinors).
Is there a geometric approach unifying the construction on flat spacetime by studying representations of the isometry group (hence from a quantum perspective), and the construction on curved spacetime that essentially starts from assuming that a particle (or a field) can be specified by a spin (and then a mass)?
