A Functional Integral Equation for the Complete Effective Action in Quantum Field Theory

TL;DR

This paper proposes a novel functional integral equation for the complete effective action in quantum field theory, offering a new perspective on gauge theories and nonperturbative phenomena, with applications to QED and implications for vacuum energy and gravity.

Contribution

It introduces a fixed point functional integral equation for the effective action, challenging traditional distinctions and providing a new framework for interacting gauge theories like QED.

Findings

Derived a fixed point equation for the effective action in QFT.

Applied the approach to approximate the QED coupling constant.

Discussed implications for vacuum energy and induced gravity.

Abstract

Based on a methodological analysis of the effective action approach certain conceptual foundations of quantum field theory are reconsidered to establish a quest for an equation for the effective action. Relying on the functional integral formulation of Lagrangian quantum field theory a functional integral equation for the complete effective action is proposed which can be understood as a certain fixed point condition. This is motivated by a critical attitude towards the distinction artificial from an experimental point of view between classical and effective action. While for free field theories nothing new is accomplished, for interacting theories the concept differs from the established paradigm. The analysis of this new concept is concentrated on gauge field theories treating QED as the prototype model. An approximative approach to the functional integral equation for the complete effective action of QED is exploited to obtain certain nonperturbative information about the quadratic kernels of the action. As particular application the approximative calculation of the QED coupling constant $\alpha$ is explicitly studied. It is understood as one of the characteristics of a fixed point given as a solution of the functional integral equation proposed. Finally, within the present approach the vacuum energy problem is considered and possible implications on the induced gravity concept are contemplated.