String theory in a vertex operator representation: a simple model for testing loop quantum gravity

TL;DR

This paper applies loop quantum gravity techniques to the free bosonic string, constructing a novel representation and analyzing different regularizations of the Hamiltonian constraint, revealing connections to known quantum field models.

Contribution

It introduces a loop quantum gravity-inspired representation for the string and proposes two regularizations of the Hamiltonian constraint, one anomaly-free and one with anomaly.

Findings

Anomaly-free regularization resembles a 2D φ^4 model

Another regularization resembles a φ^3 model with anomalies

Discussion of relations to conventional Fock space quantization

Abstract

The loop quantum gravity technique is applied to the free bosonic string. A Hilbert space similar to loop space in loop quantum gravity as well as representations of diffeomorphism and hamiltonian constraints on it are constructed. The string in this representation can be viewed as a set of interacting relativistic particles each carrying a certain momentum. Two different regularizations of the hamiltonian constraint are proposed. The first of them is anomaly-free and give rise to interaction very similar to that of two dimensional $\phi^4$-model. The second version of hamiltonian constraint is similar to $\phi^3$-model and contains an anomaly. A possible relation of these two models to the conventional quantization of the string based on Fock space representation is discussed.