String Representation of Quantum Loops

TL;DR

This paper presents a comprehensive string-based framework for representing quantum states of relativistic loops, integrating holographic, bulk, and boundary dynamics with renormalization considerations.

Contribution

It introduces a unified string representation of quantum loops, deriving the Polyakov action from a covariant phase space approach, and discusses a covariant matrix coordinate formulation.

Findings

Derived the Polyakov string action from a phase space covariant Schild action.

Showed the boundary quantum dynamics are induced by bulk quantum fluctuations.

Proposed a covariant matrix coordinate approach for points, loops, and strings.

Abstract

We recover a general representation for the quantum state of a relativistic closed line (loop) in terms of string degrees of freedom.The general form of the loop functional splits into the product of the Eguchi functional, encoding the holographic quantum dynamics, times the Polyakov path integral, taking into account the full Bulk dynamics, times a loop effective action, which is needed to renormalize boundary ultraviolet divergences. The Polyakov string action is derived as an effective actionfrom a phase space,covariant,Schild action, by functionally integrating out the world-sheet coordinates.The area coordinates description of the boundary quantum dynamics, is shown to be induced by the ``zero mode'' of the bulk quantum fluctuations. Finally, we briefly comment about a ``unified, fully covariant'' description of points, loops and strings in terms of Matrix Coordinates.