1+1 string with quarks at the ends revisited

TL;DR

This paper revisits the classical and quantum descriptions of a string with pointlike masses at its ends in 1+1 dimensions, comparing two gauges and analyzing their quantum equivalence and Poincare covariance.

Contribution

It explicitly constructs the canonical transformation between gauges and demonstrates the non-unitary equivalence of the quantum theories, highlighting gauge-dependent quantum properties.

Findings

Classical theories are equivalent via a canonical transformation.

Quantum theories are not unitarily equivalent due to interaction dependence.

Poincare algebra remains closed in both gauges, preserving covariance.

Abstract

The canonical description is presented for the string with pointlike masses at the ends in 1+1 dimensions in two different gauges: in the proper time gauge and in the light cone one. The classical canonical transformation is written out explicitly, which relates physical variables in both gauges, and equivalence of two classical theories is demonstrated in such a way. Both theories are quantized, and it is shown that quantum theories are not unitary equivalent. It happens due to the fact that the canonical transformation depends on interaction. The quantum Poincare algebra proves to be closed in both cases, so that the requirement of Poincare covariance is not able to distinguish between two versions of the theory.