Bohr's Correspondence Principle for the Renormalized Nelson Model

This paper studies the derivation of the nonlinear system of Schrödinger--Klein--Gordon (S-KG) equations, coupled by a Yukawa-type interaction, from a microscopic quantum field model of nonrelativistic particles interacting with a relativistic scalar field introduced by Edward Nelson in the mid 1960s. In particular, we prove that the quantum states evolved by the microscopic dynamics converge, in the classical limit, to their Wigner measures pushed forward by the S-KG flow. To define the microscopic dynamics it is not sufficient to quantize the classical energy, since the system requires a self-energy renormalization; it is therefore noteworthy, as well as one of the main technical difficulties of the analysis, that the classical limit is not affected by such renormalization. This last fact is proved with the aid of a classical dressing transformation.