First Quantized Noncritical Relativistic Polyakov String

TL;DR

This paper develops a consistent first quantization framework for the noncritical relativistic Polyakov string in dimensions 2 to 24, explicitly calculating the propagator and physical state space without relying on conformal invariance.

Contribution

It introduces a path integral quantization approach for noncritical strings that avoids conformal invariance, explicitly constructs the propagator and physical states, and links to the massive string model.

Findings

Exact propagator for arbitrary string configurations

Explicit construction of the physical state Hilbert space

Equivalence to the Fairlie-Chodos-Thorn massive string model

Abstract

The first quantization of the relativistic Brink-DiVecchia-Howe-Polyakov (BDHP) string in the range $1<d<25$ is considered. It is shown that using the Polyakov sum over bordered surfaces in the Feynman path integral quantization scheme one gets a consistent quantum mechanics of relativistic 1-dim extended objects in the range $1<d<25$. In particular the BDHP string propagator is exactly calculated for arbitrary initial and final string configurations and the Hilbert space of physical states of noncritical BDHP string is explicitly constructed. The resulting theory is equivalent to the Fairlie-Chodos-Thorn massive string model. In contrast to the conventional conformal field theory approach to noncritical string and random surfaces in the Euclidean target space the path integral formulation of the Fairlie-Chodos-Thorn string obtained in this paper does not rely on the principle of conformal invariance. Some consequences of this feature for constructing a consistent relativistic string theory based on the "splitting-joining" interaction are discussed.}