Worldline as a spin chain

The general theoretical ground for models based on compact angle coordinates is presented. It is observed that the proper dependence on compact coordinates has to be through the group elements and is achieved most naturally in a discrete-time formulation of the theory. By the construction, the discrete worldline inlaid by compact coordinates resembles the spin chains of magnetic systems. As examples, the models based on the groups U(1), Z_N and SU(2) are explicitly constructed and their exact energy spectra are obtained. As the consequence of the minima in the spectra, the models exhibit a phase transition of first order. We attempt to fit the dynamics by the U(1) group to the proposed role for monopoles in the dual Meissner effect of the confinement mechanism. (orig.)