Axiomatic quantum electrodynamics: from causality to convexity of effective action

TL;DR

This paper explores the axiomatic foundations of quantum electrodynamics, linking causality constraints to the geometric properties of the effective action, particularly its convexity and curvature in the local limit.

Contribution

It establishes a connection between causality conditions and the convexity and geometric structure of the effective action in quantum electrodynamics.

Findings

Causality imposes constraints on the effective action's geometry.

The local limit of the effective action has a surface with positive Gaussian curvature.

The nonlinear Lagrangian as a function of invariants exhibits specific geometric properties.

Abstract

Theory of electromagnetic field, specified by an effective action functional, is considered. The causality condition is imposed in the form of a requirement that the group velocities of propagation of small and soft disturbances over the background of an external constant field should not exceed the speed of light in vacuum. It is shown that these conditions lead, in particular, to a very definite conclusion about the geometry of the local limit of the effective action. Namely, the surface, which is specified in the local limit by the nonlinear Lagrangian, considered as a function of two invariants of the field has positive Gaussian curvature.