Yang-Mills Theory on a Cylinder Coupled to Point Particles

TL;DR

This paper investigates a finite-dimensional quantum Yang-Mills model on a cylindrical spacetime coupled to point particles, revealing confinement phenomena and solvable quantum mechanics reductions without gauge fixing.

Contribution

It introduces a gauge-invariant quantization approach for Yang-Mills theory with matter on a cylinder, reducing the problem to exactly solvable finite-dimensional quantum mechanics.

Findings

Ground state energy diverges with infinite radius, indicating confinement.

Two-particle bound states ('mesons') do not exhibit divergence, suggesting stable bound states.

Method avoids gauge fixing by leveraging the geometry of the space of connections.

Abstract

We study a model of quantum Yang-Mills theory with a finite number of gauge invariant degrees of freedom. The gauge field has only a finite number of degrees of freedom since we assume that space-time is a two dimensional cylinder. We couple the gauge field to matter, modeled by either one or two nonrelativistic point particles. These problems can be solved {\it without any gauge fixing}, by generalizing the canonical quantization methods of Ref.\[rajeev] to the case including matter. For this, we make use of the geometry of the space of connections, which has the structure of a Principal Fiber Bundle with an infinite dimensional fiber. We are able to reduce both problems to finite dimensional, exactly solvable, quantum mechanics problems. In the case of one particle, we find that the ground state energy will diverge in the limit of infinite radius of space, consistent with confinement. In the case of two particles, this does not happen if they can form a color singlet bound state (`meson').