Deformation of the Poisson Structure of a Point Particle Due to Gravitational Back Reaction

The dynamics of a massive particle in a frame of a test particle in 3 + 1 spacetime dimensions is considered with gravitational interaction taken into account. The total action (gravity+particles) collapses to a boundary separating the massive particle and the test particle, and is further reduced to a finite dimensional action depending only on relative particle coordinates and momenta. It turns out that the momentum space is a coadjoint orbit of the Lorentz group. The momentum space is thus curved and its curvature falls off with the particle relative distance according to the Newton law. This defines the modified form of the Poisson brackets. At the quantum level, this results in non-commutativity and partial discreteness in coordinate space.