[Paper Review] Flow equations for the quantum electrodynamics on the light-front

Elena Gubankova|ArXiv.org|Feb 16, 1999

Quantum Chromodynamics and Particle Interactions4 references3 citations

TL;DR

This paper applies flow equations to quantum electrodynamics on the light-front, transforming the Hamiltonian to eliminate particle-number-violating terms and obtain an effective electron-positron Hamiltonian. The method yields accurate results for the positronium spectrum, including the Bohr energy levels and hyperfine splitting, with high agreement to experiment, while preserving rotational invariance and avoiding infrared divergences except for longitudinal ones, and performs ultraviolet renormalization without longitudinal infrared issues via coupling coherence and normal ordering.

ABSTRACT

The method of flow equations is applied to QED on the light front. Requiring that the particle number conserving terms in the Hamiltonian are considered to be diagonal and the other terms off-diagonal an effective Hamiltonian is obtained which reduces the positronium problem to a two-particle problem, since the particle number violating contributions are eliminated. Using an effective electron-positron Hamiltonian, obtained in the second order in coupling, we analyze the positronium bound state problem analytically and numerically. The results obtained for Bohr spectrum and hyperfine splitting coincide to a high accuracy with experimental values. The rotational invariance, that is not manifest symmetry on the light-front, is recovered for positronium mass spectrum. Except for the longitudinal infrared divergences, that are special for the light-front gauge calculations, no infrared divergences appear. The ultraviolet renormalization in the second order in coupling constant is performed simultaneously. To preserve boost invariance we take into account the diagrams arising from the normal ordering of instantaneous interactions. Using flow equations and coupling coherence we obtain the counterterms for electron and photon masses, which are free from longitudinal infrared divergences.

Motivation & Objective

• To develop a systematic method for handling particle-number-changing interactions in light-front QED using flow equations.
• To derive an effective two-body Hamiltonian for positronium by eliminating particle-number-violating terms.
• To achieve accurate predictions for the positronium energy spectrum, including hyperfine splitting, matching experimental values.
• To preserve rotational invariance in the mass spectrum despite its non-manifest nature on the light-front.
• To perform ultraviolet renormalization free of longitudinal infrared divergences through coupling coherence and normal ordering of instantaneous interactions.

Proposed method

• Utilizes flow equations to continuously deform the light-front Hamiltonian, diagonalizing particle-number-conserving terms and off-diagonalizing particle-number-violating terms.
• Constructs an effective Hamiltonian in second order in the coupling constant, focusing on electron-positron interactions.
• Applies normal ordering to include contributions from instantaneous photon exchange, preserving boost invariance.
• Uses coupling coherence to systematically derive counterterms for electron and photon masses.
• Performs ultraviolet renormalization simultaneously with the flow equation transformation, avoiding longitudinal infrared divergences.
• Solves the bound state problem analytically and numerically using the effective Hamiltonian.

Experimental results

Research questions

• RQ1 Can flow equations effectively decouple particle-number-violating sectors in light-front QED to yield a reliable two-body effective Hamiltonian?
• RQ2 To what extent can the positronium spectrum, including hyperfine splitting, be reproduced accurately using this effective Hamiltonian?
• RQ3 How is rotational invariance recovered in the mass spectrum despite its non-manifest form on the light-front?
• RQ4 Are longitudinal infrared divergences the only source of infrared problems in this light-front approach?
• RQ5 Can ultraviolet counterterms be derived consistently without introducing longitudinal infrared divergences?

Key findings

• The effective Hamiltonian successfully reduces the positronium problem to a two-particle system by eliminating particle-number-violating terms.
• The calculated Bohr spectrum and hyperfine splitting agree with experimental values to high accuracy.
• Rotational invariance is recovered in the positronium mass spectrum, despite the non-manifest nature of the symmetry on the light-front.
• No infrared divergences appear except for longitudinal ones, which are inherent to light-front gauge calculations.
• Ultraviolet renormalization is performed successfully, with counterterms for electron and photon masses derived free of longitudinal infrared divergences.
• The method preserves boost invariance through inclusion of normal-ordered instantaneous interaction diagrams.