  In this paper we analyze the classical limit of the Nelson model with cut
off, when both non-relativistic and relativistic particles number goes to
infinity. We prove convergence of quantum observables to the solutions of
classical equations, and find the evolution of quantum fluctuations around the
classical solution. Furthermore we analyze the convergence of transition
amplitudes of normal ordered products of creation and annihilation operators
between different types of initial states. In particular the limit of normal
ordered products between states with a fixed number of both relativistic and
non-relativistic particles yields an unexpected quantum residue: instead of the
product of classical solutions we obtain an average of the product of solutions
corresponding to varying initial conditions.
