Interaction of Relativistic Bosons with Localized Sources on Riemannian Surfaces

TL;DR

This paper investigates the quantum interactions of Klein-Gordon bosons with localized sources on Riemannian surfaces, employing non-perturbative renormalization and heat kernel techniques to analyze bound states and energy bounds.

Contribution

It introduces a non-perturbative renormalization method using orthofermion algebra for Klein-Gordon particles on Riemannian surfaces and generalizes the bound state analysis to various manifold types.

Findings

Derived the bound state spectrum in terms of binding energies.

Established lower bounds on ground state energy using heat kernel estimates.

Validated neglect of pair creation within certain parameter ranges.

Abstract

We study the interaction of mutually non-interacting Klein-Gordon particles with localized sources on stochastically complete Riemannian surfaces. This asymptotically free theory requires regularization and coupling constant renormalization. Renormalization is performed non-perturbatively using the orthofermion algebra technique and the principal operator $\Phi$ is found. The principal operator is then used to obtain the bound state spectrum, in terms of binding energies to single Dirac-delta function centers. The heat kernel method allows us to generalize this procedure to compact and Cartan-Hadamard type Riemannian manifolds. We make use of upper and lower bounds on the heat kernel to constrain the ground state energy from below thus confirming that our neglect of pair creation is justified for certain ranges of parameters in the problem.