Polymer quantum mechanics on a compact configuration space

"Polymer quantum mechanics" is a quantization scheme inspired by loop quantum gravity in which the configuration space of the theory is chosen to have a discrete topology. Polymer quantization yields a representation of the canonical commutation relations that is genuinely distinct from the conventional "Schrödinger" representation. We summarize the main features of polymer quantum mechanics and investigate the mathematical structure of polymer quantization of systems with compact configuration spaces. We show that the construction of polymer states leads to a Hilbert space of states defined on a finite graph of points. By way of example, we find the exact energy eigenvalues and eigenfunctions for a particle on a ring and discuss similarities and differences from standard Schrödinger quantum mechanics. We also explore the continuum limit of states in this system, and show how the exact eigenfunctions in the position representation approach their continuum counterparts.