Reduced dynamics with Poincar\'e symmetry in an open quantum system

We consider how the reduced dynamics of an open quantum system coupled to an environment admits the Poincar\'e symmetry. The reduced dynamics is described by a dynamical map, which is given by tracing out the environment from the total unitary evolution without initial correlations. We investigate the dynamical map which is invariant under the Poincar\'e group. Based on the unitary representation theory of the Poincar\'e group, we develop a systematic way to give such a dynamical map. Using this way, we derive the dynamical map of a massive particle with a finite spin and a massless particle with a finite spin and a nonzero momentum. The dynamical map of a spinless massive particle is exemplified and the conservation of the Poincar\'e generators is discussed. We then find the map with the Poincar\'e invariance and the four-momentum conservation. Further, we show that the conservation of the angular momentum and the boost operator makes the map of a spinless massive particle unitary