Bohr's correspondence principle in quantum field theory and classical renormalization scheme: the Nelson model

TL;DR

This paper proves that the classical limit of Nelson's quantum field theory aligns with a Schrödinger-Klein-Gordon system, clarifying the role of renormalization as a normalization strategy for classical Hamiltonian PDEs.

Contribution

It establishes Bohr's correspondence principle for Nelson's model and interprets the renormalization method as a normalization process for classical Hamiltonian PDEs.

Findings

Quantum dynamics converge to Schrödinger-Klein-Gordon system in classical limit

Renormalization acts as a normalization strategy for classical Hamiltonian PDEs

Clarifies the classical-quantum transition in Nelson's model

Abstract

In the mid Sixties Edward Nelson proved the existence of a consistent quantum field theory that describes the Yukawa-like interaction of a non-relativistic nucleon field with a relativistic meson field. Since then it is thought, despite the renormalization procedure involved in the construction, that the quantum dynamics should be governed in the classical limit by a Schr\"odinger-Klein-Gordon system with Yukawa coupling. In the present paper we prove this fact in the form of a Bohr correspondence principle. Besides, our result enlighten the nature of the renormalization method employed in this model which we interpret as a strategy that allows to put the related classical Hamiltonian PDE in a normal form suitable for a canonical quantization.