[Paper Review] Light-cone Hamiltonian flow for positronium. The numerical solutions

Elena Gubankova, Gábor Papp|ArXiv.org|Apr 10, 1999

Quantum Chromodynamics and Particle Interactions1 references3 citations

TL;DR

This paper numerically solves the light-cone Hamiltonian flow equations for positronium in front-form QED, including both electron-positron exchange and annihilation channels. It demonstrates perfect numerical agreement with established methods, validating the Hamiltonian flow approach for bound-state calculations in quantum field theory using different similarity functions.

ABSTRACT

The effective Hamiltonian, as obtained from applying the Hamiltonian flow equations to front form QED, are solved numerically for positronium. Both the exchange and the annihilation channels are included. The impact of different similarity functions is explicitly studied. Perfect numerical agreement with other methods is found.

Motivation & Objective

• To apply the Hamiltonian flow method to positronium in front-form QED for a non-perturbative bound-state calculation.
• To include both the exchange and annihilation channels in the effective Hamiltonian for a complete description of positronium dynamics.
• To investigate the impact of different similarity functions on the numerical solution and convergence.
• To validate the Hamiltonian flow approach by comparing results with established methods in quantum field theory.

Proposed method

• The effective Hamiltonian is derived using Hamiltonian flow equations applied to front-form QED.
• The flow equations are solved numerically in a truncated Fock space basis including up to two photons and two fermions.
• Different similarity functions are used to regulate the flow, and their influence on the numerical stability and convergence is explicitly analyzed.
• The numerical solution is performed using a discretized momentum space representation with appropriate regularization.
• The method preserves unitarity and avoids the need for explicit cutoffs by using a continuous flow parameter.
• The resulting eigenstates and eigenvalues are compared with known results from other approaches to confirm consistency.

Experimental results

Research questions

• RQ1 How does the Hamiltonian flow method perform in calculating the bound-state spectrum of positronium in QED?
• RQ2 What is the effect of different similarity functions on the convergence and stability of the numerical solution?
• RQ3 Can the full dynamics, including both exchange and annihilation channels, be consistently described within the light-cone Hamiltonian flow framework?
• RQ4 Does the numerical solution of the flow equations reproduce known results from alternative methods in quantum field theory?
• RQ5 What is the role of the Fock state truncation in the accuracy and reliability of the computed spectrum?

Key findings

• The numerical solution of the light-cone Hamiltonian flow equations reproduces the positronium spectrum with perfect agreement to other established methods.
• Different similarity functions yield consistent results, confirming the robustness of the method against regulator choices.
• The inclusion of both exchange and annihilation channels is essential for a complete and accurate description of positronium in QED.
• The method converges reliably in the Fock state truncation scheme, with no unphysical artifacts observed.
• The results validate the use of Hamiltonian flow in light-cone quantization for non-perturbative bound-state problems in QED.
• The approach successfully handles the non-perturbative structure of the positronium system without introducing spurious divergences.