Quantum Extremism: Effective Potential and Extremal Paths

TL;DR

This paper investigates the effective potential in quantum field theories, comparing Euclidean and Minkowskian approaches, and analyzes dominant field configurations in the N=1 linear sigma model to address the convexity problem.

Contribution

It extends the study of the effective potential and extremal paths from Euclidean to Minkowskian space-time, clarifying dominant configurations in the path integral.

Findings

Euclidean and Minkowskian treatments yield different effective potentials.

Identifies the dominant field configurations in Minkowskian space-time.

Provides insights into the convexity problem in quantum field theory.

Abstract

The reality and convexity of the effective potential in quantum field theories has been studied extensively in the context of Euclidean space-time. It has been shown that canonical and path-integral approaches may yield different results, thus resolving the `convexity problem'. We discuss the transferral of these treatments to Minkowskian space-time, which also necessitates a careful discussion of precisely which field configurations give the dominant contributions to the path integral. In particular, we study the effective potential for the N=1 linear sigma model.