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

TL;DR

This paper extends histories-based generalized quantum mechanics to curved spacetimes, introducing a formalism that relates symmetries to conformal isometries of the spacetime metric.

Contribution

It develops a formalism for histories-based quantum mechanics in curved spacetime, defining a space of temporal supports and characterizing symmetries as conformal isometries.

Findings

Symmetries imply conformal isometries of spacetime.

Construction of temporal support space using spacelike subsets.

Formalism applies to particles, fields, and general objects in curved spacetime.

Abstract

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.