Bound States in Gauge Theories as the Poincare Group Representations

TL;DR

This paper constructs a gauge theory bound state generating functional using the Dirac Hamiltonian approach, classifies states via the Poincaré group, and links perturbative and nonperturbative QCD through derived relations and condensate analysis.

Contribution

It introduces a novel bound state generating functional in gauge theories based on the Dirac Hamiltonian and Poincaré classification, connecting perturbative and nonperturbative regimes.

Findings

Derived the Gell-Mann-Oakes-Renner relation from SD and BS equations.

Established relations between perturbative QCD and low-energy models.

Calculated constituent quark masses from a self-consistent equation.

Abstract

The bound state generating functional is constructed in gauge theories. This construction is based on the Dirac Hamiltonian approach to gauge theories, the Poincar\'e group classification of fields and their nonlocal bound states, and the Markov-Yukawa constraint of irreducibility. The generating functional contains additional anomalous creations of pseudoscalar bound states: para-positronium in QED and mesons in QCD in the two gamma processes of the type of \gamma + \gamma = \pi_0+para-positronium. The functional allows us to establish physically clear and transparent relations between the perturbative QCD to its nonperturbative low energy model by means of normal ordering and the quark and gluon condensates. In the limit of small current quark masses, the Gell-Mann-Oakes-Renner relation is derived from the Schwinger-Dyson (SD) and Bethe-Salpeter (BS) equations. The constituent quark masses can be calculated from a self-consistent non-linear equation.