The fail of determinism and the limits of Bohm hidden variable theory

We give a counter example to show that determinism as such is in contradiction to quantum mechanics. More precisely, we consider a simple quantum system and its environment, including the measurement device, and make the assumption that the time evolution of the global system, the quantum system plus this environment, is deterministic, i. e., the time evolution is given by a dynamical trajectory from some initial conditions. From this, we prove a type of Bell inequalities which are violated by quantum mechanics, reaching the conclusion that our quantum system evolves in a non deterministic way. In order to seize the interest of this conclusion, one must realize that it cannot be reached from the present experimental violation of Bell inequalities predicted by quantum mechanics, since this violation is compatible with non-local realism.