Abstract
A possible model for the quantum kinematics of a test particle in curved spacetime is proposed. Every reasonable neighbourhood of a curved spacetime can be equipped with a non-associative binary operation called geodesic multiplication. Its infinitesimal right translations are used to define the (geodesic) momentum operators. The corresponding commutation relations are taken as the quantum kinematic algebra. It coincides with the usual canonical Poisson algebra (Weyl's kinematics) only in the case of flat spacetime. A BRST-like operator is constructed and its physical meaning is discussed. As an example, detailed calculations are performed for the spacetime of a weak plane gravitational wave.
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