Yang–Mills theory on a cylinder coupled to point particles

Abstract

A model of quantum Yang–Mills theory with a finite number of gauge invariant degrees of freedom is studied. The gauge field has only a finite number of degrees of freedom since it is assumed that space–time is a two-dimensional cylinder. The gauge field is coupled to matter, modeled by either one or two nonrelativistic point particles. These problems can be solved without any gauge fixing, by generalizing the canonical quantization methods of S. G. Rajeev [Phys. Lett. B 212, 203 (1988)] to the case including matter. For this, the geometry of the space of connections is used, which has the structure of a principal fiber bundle with an infinite-dimensional fiber. Both problems are reduced to finite-dimensional, exactly solvable, quantum mechanics problems. In the case of one particle, it is found 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’’).