KINEMATICS AND UNCERTAINTY RELATIONS OF A QUANTUM TEST PARTICLE IN A CURVED SPACE-TIME

A possible model for quantum kinematics of a test particle in a curved space-time is proposed. Every reasonable neighbourhood V-e of a curve d space-time can be equipped with a nonassociative binary operation ca lled 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 + alpha. In a curved space-time infinitesimal geodesic right t ranslations can be used to define the (geodesic) momentum operators. T he 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 der ived and their physical meaning is discussed. (C) 1998 Elsevier Scienc e B.V.