The Proper Time Equation and the Zamolodchikov Metric

TL;DR

This paper explores the relationship between the proper time equation and the Zamolodchikov metric, demonstrating their connection through the beta function and two-point functions, and applies this to open string vectors in electromagnetic fields.

Contribution

It establishes the equivalence between the proper time equation and the Zamolodchikov metric, and develops a systematic, all-orders perturbative method for gauge invariant equations of motion.

Findings

Reproduces the Born-Infeld equation in the uniform field limit.

Provides an exact derivative expansion for the proper time equation.

Validates the method through consistency with known results.

Abstract

The connection between the proper time equation and the Zamolodchikov metric is discussed. The connection is two-fold: First, as already known, the proper time equation is the product of the Zamolodchikov metric and the renormalization group beta function. Second, the condition that the two-point function is the Zamolodchikov metric, implies the proper time equation. We study the massless vector of the open string in detail. In the exactly calculable case of a uniform electromgnetic field strength we recover the Born-Infeld equation. We describe the systematics of the perturbative evaluation of the gauge invariant proper time equation for the massless vector field. The method is valid for non-uniform fields and gives results that are exact to all orders in derivatives. As a non trivial check, we show that in the limit of uniform fields it reproduces the lowest order Born-Infeld equation.