Model of a quantum particle in spacetime

In their paper Doplicher, Fredenhagen and Roberts (DFR) proposed a simple model of a particle in quantum spacetime. We give a new formulation of the model and propose some small changes and additions which improve the physical interpretation. In particular, we show that the internal degrees of freedom e and m of the particle represent external forces acting on the particle. To obtain this result we follow a constructive approach. The model is formulated as a covariance system. It has projective representations in which not only the spacetime coordinates but also the conjugated momenta are two-by-two noncommuting. These momenta are of the form P-mu - (b/c)A(mu), where b is the charge of the particle. The electric and magnetic fields obtained from the vector potential A(mu) coincide with the variables e and m postulated by DFR. Similarly, the spacetime position operators are of the form Q(mu) - (al(2)/hc)Omega (mu), where a is a generalized charge and l a fundamental length, and with vector potentials Omega (mu), which are in some sense dual w.r.t. the A mu.