The problem of quantization of lightcone QCD

TL;DR

This paper discusses the challenges in constructing a quantum algebra for lightcone QCD beyond perturbation theory, highlighting issues with boundary gauge fields, symplectic reduction, and quantization.

Contribution

It identifies key problems in formulating a consistent quantum algebra in non-Abelian lightcone QCD with boundary fields and complex Hamiltonians.

Findings

Boundary gauge fields are essential for classical dynamics.

Symplectic reduction leads to complex Hamiltonians with infinite powers.

Poisson algebra formulation faces issues with canonical relations and boundary degrees.

Abstract

There exists the problem to construct a quantum algebra of observables in lightcone QCD beyond the perturbative regime. It has recently established that the boundary gauge fields are crucial for a consistent construction of the classical dynamic system. If the gauge group is non-Abelian and there are four or more space-time dimensions then the procedure of symplectic reduction gives a classical dynamical system with very complicated Hamiltonian having infinite power over the coupling constant. Then, to quantize the theory one should to construct a Poisson algebra and to quantize it. Careful analysis shows that a Poisson formulation has a problem with: canonical commutation relations, spatial invariance, and the boundary degrees of freedom in the Hamiltonian.