A Comparison of the Proper Time Equation and the Renormalization Group $\beta$-Function in String Theory

TL;DR

This paper compares the proper time equation with the renormalization group beta-function in string theory, illustrating their relationship through calculations of corrections to Maxwell and Yang-Mills equations, emphasizing the proper time method's full equations of motion.

Contribution

It clarifies the connection between the proper time equation and the beta-function, demonstrating the proper time method's ability to produce full equations of motion with derivative corrections.

Findings

Calculated cubic order corrections to Maxwell's equations.

Tested gauge covariance with higher derivative corrections to Yang-Mills.

Showed the proper time equation encapsulates the full equations of motion.

Abstract

It is known that there is a proportionality factor relating the $\beta$-function and the equations of motion viz. the Zamolodchikov metric. Usually this factor has to be obtained by other methods. The proper time equation, on the other hand, is the full equation of motion. We explain the reasons for this and illustrate it by calculating corrections to Maxwell's equation. The corrections are calculated to cubic order in the field strength, but are exact to all orders in derivatives. We also test the gauge covariance of the proper time method by calculating higher (covariant) derivative corrections to the Yang-Mills equation.