Semiclassical analysis of quantum asymptotic fields in the Yukawa theory

TL;DR

This paper rigorously connects the scattering theory of the Yukawa quantum field model to the nonlinear Schrödinger-Klein-Gordon equation in the semiclassical limit, revealing properties of asymptotic states and ground states.

Contribution

It establishes a rigorous link between Yukawa model asymptotic fields and the Schrödinger-Klein-Gordon equation, including phase space concentration and ground state characterization.

Findings

Transition amplitudes converge to wave operators of the S-KG equation.

Asymptotic vacuum states concentrate around classical solutions.

Unique minimizer of S-KG energy identified with Yukawa ground states.

Abstract

In this article, we study the asymptotic fields of the Yukawa particle-field model of quantum physics, in the semiclassical regime $\hslash\to 0$, with an interaction subject to an ultraviolet cutoff. We show that the transition amplitudes between final (respectively initial) states converge towards explicit quantities involving the outgoing (respectively incoming) wave operators of the nonlinear Schr\"odinger-Klein-Gordon (S-KG) equation. Thus, we rigorously link the scattering theory of the Yukawa model to that of the Schr\"odinger-Klein-Gordon equation. Moreover, we prove that the asymptotic vacuum states of the Yukawa model have a phase space concentration property around classical radiationless solutions. Under further assumptions, we show that the S-KG energy admits a unique minimizer modulo symmetries and identify exactly the semiclassical measure of Yukawa ground states. Some additional consequences of asymptotic completeness are also discussed, and some further open questions are raised.