Perturbative or Path-Integral Approach versus Operator-Formalism Approach

TL;DR

This paper compares the perturbative/path-integral and operator-formalism approaches to 2D quantum gravity, highlighting differences in anomaly treatment and the implications of using T*-products.

Contribution

It clarifies the fundamental differences between the two approaches, especially regarding anomaly handling and the effects of T*-product usage in path-integral methods.

Findings

Path-integral approach uses T*-product, violating field equations.

Extra one-loop Feynman diagrams arise in path-integral approach.

Explicit demerits of the path-integral approach are demonstrated.

Abstract

In the conformal-gauge two-dimensional quantum gravity, the solution obtained by the perturbative or path-integral approach is compared with the one obtained by the operator-formalism approach. Treatments of the anomaly problem in both approaches are different. This difference is found to be essentially caused by the fact that the perturbative or path-integral approach is based on the T*-product (covariantized T-product), which generally violates field equations. Indeed, this fact induces some extra one-loop Feynman diagrams, which would not exist unless a nonzero contribution arose from a zero field. Some demerits of the path-integral approach are explicitly demonstrated.