[Paper Review] Interaction of Relativistic Bosons with Localized Sources on Riemannian Surfaces

Çağlar Doǧan, O. Teoman Turgut|arXiv (Cornell University)|Dec 2, 2009

Spectral Theory in Mathematical Physics4 references4 citations

TL;DR

This paper investigates the quantum interaction of relativistic bosons with localized sources on Riemannian surfaces using non-perturbative renormalization via the principal operator technique. It derives the bound state spectrum and establishes lower bounds on the ground state energy using heat kernel methods, confirming the validity of neglecting pair creation for certain parameter regimes on flat, compact, and Cartan-Hadamard manifolds.

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 $Φ$ 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.

Motivation & Objective

• To develop a non-perturbative renormalization framework for relativistic Klein-Gordon particles interacting with localized sources on stochastically complete Riemannian surfaces.
• To generalize the renormalization procedure to compact and Cartan-Hadamard type manifolds using heat kernel techniques.
• To compute the bound state spectrum in terms of binding energies for single Dirac-delta function centers.
• To establish rigorous lower bounds on the ground state energy to justify the neglect of pair creation in the theory.
• To confirm that renormalization in this relativistic quantum mechanical model aligns with intuition from elementary quantum mechanics despite advanced mathematical techniques.

Proposed method

• Formulate a cut-off Hamiltonian $ H_{ ho} = H_0 + H^{ ext{int}}_{ ho} $ with a regularized interaction term involving heat kernel-smeared Dirac-delta functions.
• Apply the principal operator technique—originally from Rajeev—to non-perturbatively renormalize the theory on general Riemannian manifolds.
• Use the heat kernel to express the principal operator $ ilde{ ho}(a_i, a_j; gd_{ij}^{-2}) $, enabling generalization to curved manifolds.
• Derive momentum-space wavefunctions and configuration-space wavefunctions via the principal operator formalism.
• Apply asymptotic analysis and Laplace's method to approximate the principal operator in the tunnelling regime.
• Establish upper and lower bounds on the heat kernel to derive rigorous lower bounds on the ground state energy across different manifold types.

Experimental results

Research questions

• RQ1 How can a relativistic quantum mechanical model of bosons interacting with localized sources be consistently renormalized on curved two-dimensional Riemannian manifolds?
• RQ2 What is the structure of the bound state spectrum in terms of binding energies for single Dirac-delta centers in this relativistic setting?
• RQ3 To what extent does the heat kernel method allow generalization of the renormalization procedure to compact and Cartan-Hadamard manifolds?
• RQ4 Can rigorous lower bounds on the ground state energy be derived to justify the absence of pair creation in the model?
• RQ5 Does the non-perturbative renormalization procedure on curved manifolds contradict or alter the physical intuition from elementary quantum mechanics?

Key findings

• The principal operator $ ilde{ ho}(a_i, a_j; gd_{ij}^{-2}) $ is derived non-perturbatively using the orthofermion algebra technique and heat kernel regularization.
• The bound state spectrum is obtained in terms of binding energies to single Dirac-delta function centers via the eigenstates of the principal operator with zero eigenvalue.
• For the case $ |b_i| o 0 $, the principal operator asymptotically behaves as $ ilde{ ho}_{ij} o -rac{ ho(1)c_0(a_i,a_j;gd_{ij}^{-2})}{ ilde{ ho}} rac{ ext{exp}(-a ilde{ ho})}{ ilde{ ho}^{1/2}} $, where $ ilde{ ho} = ilde{ ho}^{1/2} $, confirming exponential suppression.
• For large $ |b_i| $, the principal operator is approximated as $ ilde{ ho}_{ij} o -rac{ ho(1)c_0(a_i,a_j;gd_{ij}^{-2})}{2 ilde{ ho}} rac{e^{-a}}{a|b_i|} $, showing a power-law dependence.
• Lower bounds on the ground state energy are established on flat, compact, and Cartan-Hadamard manifolds using heat kernel bounds, confirming the physical consistency of the model.
• The derived lower bounds on the ground state energy validate the neglect of pair creation for certain parameter ranges, particularly when the coupling and curvature parameters satisfy $ a ilde{ ho} o ext{large} $.