Worldsheet Covariant Path Integral Quantization of Strings

TL;DR

This paper presents a covariant path integral quantization method for bosonic strings that maintains tensorial properties and clarifies the role of anomalies, offering a broader framework applicable to non-Weyl invariant theories.

Contribution

It introduces a covariant functional integral approach that separates anomalies from operator renormalization and constructs background-independent measures for string quantization.

Findings

Covariant regularization of vertex operators achieved.

Path integral measure constructed without gauge fixing.

Critical dimension D=26 derived through novel approach.

Abstract

We discuss a covariant functional integral approach to the quantization of the bosonic string. In contrast to approaches relying on non-covariant operator regularizations, interesting operators here are true tensor objects with classical transformation laws, even on target spaces where the theory has a Weyl anomaly. Since no implicit non-covariant gauge choices are involved in the definition of the operators, the anomaly is clearly separated from the issue of operator renormalization and can be understood in isolation, instead of infecting the latter as in other approaches. Our method is of wider applicability to covariant theories that are not Weyl invariant, but where covariant tensor operators are desired. After constructing covariantly regularized vertex operators, we define a class of background-independent path integral measures suitable for string quantization. We show how gauge invariance of the path integral implies the usual physical state conditions in a very conceptually clean way. We then discuss the construction of the BRST action from first principles, obtaining some interesting caveats relating to its general covariance. In our approach, the expected BRST related anomalies are encoded somewhat differently from other approaches. We conclude with an unusual but amusing derivation of the value $D= 26$ of the critical dimension.