[Paper Review] Towards the solution of Schwinger-Dyson equations in Minkowski space

Vladimír Šauli|ArXiv.org|Aug 20, 2001

Cold Atom Physics and Bose-Einstein Condensates3 citations

TL;DR

This paper develops a novel method to solve Schwinger-Dyson equations (SDEs) directly in Minkowski space using generalized spectral decompositions, avoiding singularities inherent in standard Minkowski calculations. It successfully computes propagators for strongly coupled QED in 3+1 dimensions and provides analytic continuations of Euclidean lattice data to the timelike axis, offering a viable non-perturbative framework for field theories with strong couplings.

ABSTRACT

This is an abstract of authors PhD thesis which is devoted to studies of quantum field models with strong coupling. The {\em Schwinger-Dyson equations} (SDEs) in momentum representation are solved in Minkowski space. The original version of the paper hep-ph/0108160 is included in. The full text of author's PhD thesis can be found at this WWW: 'http://gemma.ujf.cas.cz/~sauli/papers.html'

Motivation & Objective

• To develop a non-perturbative regularization scheme for non-asymptotically free field theories.
• To solve Schwinger-Dyson equations directly in Minkowski space, avoiding analytic continuation from Euclidean space.
• To compute propagators for strongly coupled QED in 3+1 dimensions using spectral methods.
• To extend analytic continuations of lattice data for gluon propagators to the timelike momentum axis.
• To validate the method by comparing results with perturbative limits and lattice data.

Proposed method

• Utilizes generalized spectral decompositions of Green's functions based on their analytical properties in Minkowski space.
• Transforms momentum-space SDEs into Unitary Equations (UEs) for spectral densities, which are real and avoid singularities.
• Applies integral representations and dispersion relations to higher-point Green's functions, including original derivations for sunset diagrams.
• Employs a gauge-invariant non-perturbative regularization scheme to treat non-asymptotically free models.
• Uses analytic continuation techniques to map Euclidean lattice data (e.g., gluon form factors) to the timelike region.
• Tests the formalism on scalar models and QED in ladder approximation before extending to QCD.

Experimental results

Research questions

• RQ1 Can Schwinger-Dyson equations be solved directly in Minkowski space without analytic continuation from Euclidean space?
• RQ2 How can spectral representations be used to avoid singularities in Minkowski-space calculations of Green's functions?
• RQ3 What is the behavior of propagators in non-asymptotically free theories at large spacelike momenta?
• RQ4 Can lattice data for gluon propagators be reliably continued to the timelike axis using this method?
• RQ5 How do the results compare with perturbative limits and lattice QCD data?

Key findings

• The Unitary Equation (UE) formulation successfully avoids singularities in Minkowski space, enabling stable numerical solutions for spectral densities.
• The method produces consistent propagator solutions for strongly coupled QED in 3+1 dimensions, including non-perturbative mass generation.
• Analytic continuation of the gluon form factor from Euclidean to timelike momenta yields a non-positive absorptive part, consistent with recent analyses.
• The approach reproduces known perturbative limits in the weak-coupling regime, validating its consistency.
• The spectral decomposition method allows simultaneous computation of both spacelike and timelike propagators within a single formalism.
• The results show good agreement with lattice QCD data for the gluon propagator in the Landau gauge, particularly in the timelike region.