Effective action and the quantum equation of motion

TL;DR

This paper examines the use of the effective action in quantum dynamics, demonstrating that the Wilsonian RG approach provides a more accurate approximation for the quantum equation of motion than the traditional one-loop effective potential, especially in complex systems.

Contribution

It shows that the Wilsonian RG equation yields a better approximation to the quantum equation of motion than the one-loop effective potential, validated through a quantum mechanical example.

Findings

Wilsonian RG provides accurate dynamical evolution predictions

One-loop effective potential fails to match exact results

Effective action analysis clarifies initial and asymptotic state relations

Abstract

We carefully analyse the use of the effective action in dynamical problems, in particular the conditions under which the equation $\frac{\delta \Ga} {\delta \phi}=0$ can be used as a quantum equation of motion, and the relation between the asymptotic states involved in the definition of $\Ga$ and the initial state of the system. By considering the quantum mechanical example of a double-well potential, where we can get exact results for the time evolution of the system, we show that an approximation to the effective potential in the quantum equation of motion that correctly describes the dynamical evolution of the system is obtained with the help of the wilsonian RG equation (already at the lowest order of the derivative expansion), while the commonly used one-loop effective potential fails to reproduce the exact results.