Relativistic Lee Model on Riemannian Manifolds

TL;DR

This paper constructs and analyzes the relativistic Lee model on static Riemannian manifolds nonperturbatively, demonstrating consistent renormalization across coordinate systems and dimensions, and providing bounds on ground state energy.

Contribution

It introduces a nonperturbative construction of the relativistic Lee model on Riemannian manifolds using heat kernel techniques and establishes renormalization procedures consistent with flat space.

Findings

Renormalization procedures are consistent across coordinate systems and dimensions.

Ground state energy is positive in the large boson number limit.

In 2+1 dimensions, only mass renormalization is needed, with rigorous bounds on ground state energy.

Abstract

We study the relativistic Lee model on static Riemannian manifolds. The model is constructed nonperturbatively through its resolvent, which is based on the so-called principal operator and the heat kernel techniques. It is shown that making the principal operator well-defined dictates how to renormalize the parameters of the model. The renormalization of the parameters are the same in the light front coordinates as in the instant form. Moreover, the renormalization of the model on Riemannian manifolds agrees with the flat case. The asymptotic behavior of the renormalized principal operator in the large number of bosons limit implies that the ground state energy is positive. In 2+1 dimensions, the model requires only a mass renormalization. We obtain rigorous bounds on the ground state energy for the n-particle sector of 2+1 dimensional model.