Classical limit of the Nelson model with cutoff

In this paper we analyze the classical limit of the Nelson model with cutoff, 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. (C) 2013 American Institute of Physics. [http://dx.doi.org/10.1063/1.4775716