Worldline as a Spin Chain

TL;DR

This paper develops a theoretical framework linking worldlines with compact coordinates to spin chains, explicitly constructing models for U(1), Z_N, and SU(2), revealing phase transitions and potential implications for confinement in gauge theories.

Contribution

It introduces a discrete-time formulation of worldline models with compact coordinates, connecting them to spin chains and analyzing their spectra and phase transitions.

Findings

Models exhibit first-order phase transitions.

Exact energy spectra are obtained for U(1), Z_N, and SU(2).

U(1) dynamics may relate to monopole roles in confinement.

Abstract

The general theoretical ground for the models based on the 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), $\mathbb{Z}_N$ and SU(2) are explicitly constructed and their exact energy spectra are obtained. As the consequence of minima in the spectra, the models exhibit a phase transition of first-order. The dynamics by U(1) group is attempted to be fitted to the proposed role for monopoles in the dual Meissner effect of confinement mechanism.