Kinematics and uncertainty relations of a quantum test particle in a curved space-time

Abstract

A possible model for quantum kinematics of a test particle in a curved space-time is proposed. Every reasonable neighbourhood V_e of a curved space-time can be equipped with a nonassociative binary operation called the geodesic multiplication of space-time points. In the case of the Minkowski space-time, left and right translations of the geodesic multiplication coincide and amount to a rigid shift of the space-time x->x+a. In a curved space-time infinitesimal geodesic right translations can be used to define the (geodesic) momentum operators. The commutation relations of position and momentum operators are taken as the quantum kinematic algebra. As an example, detailed calculations are performed for the space-time of a weak plane gravitational wave. The uncertainty relations following from the commutation rules are derived and their physical meaning is discussed.