Quasitemporal structure and symmetries in histories based generalized quantum mechanics in curved space-time

The formalism for histories-based generalized quantum mechanics developed in two earlier papers is applied to the treatment of histories (of particles or fields or more general objects) in curved spacetimes (which need not admit foliation in spacelike hypersurfaces). The construction of the space of temporal supports (a partial semigroup generalizing the space of finite time sequences employed in traditional temporal description of histories) employs spacelike subsets of spacetime having dimensionality less than or equal to three. Definition of symmetry is sharpened by the requirement of continuity of mappings (employing topological partial semigroups). It is shown that with this proviso, a symmetry in our formalism implies a conformal isometry of the spacetime metric.