Ward Identities for the 2PI effective action in QED

TL;DR

This paper demonstrates that 2PI functional techniques in QED preserve all Ward identities at any approximation order, ensuring gauge symmetry is maintained in various 2PI-derived n-point functions.

Contribution

It derives and proves that all Ward identities are exactly satisfied in the 2PI framework for QED, clarifying the behavior of different n-point functions under approximations.

Findings

2PI-resummed vertex functions satisfy standard Ward identities.

Certain n-point functions derived from the two-point function extremum also satisfy Ward identities.

Gauge-field polarization tensor is not transverse in approximations, but this does not violate Ward identities.

Abstract

We study the issue of symmetries and associated Ward-like identities in the context of two-particle-irreducible (2PI) functional techniques for abelian gauge theories. In the 2PI framework, the $n$-point proper vertices of the theory can be obtained in various different ways which, although equivalent in the exact theory, differ in general at finite approximation order. We derive generalized (2PI) Ward identities for these various $n$-point functions and show that such identities are exactly satisfied at any approximation order in 2PI QED. In particular, we show that 2PI-resummed vertex functions, i.e. field-derivatives of the so-called 2PI-resummed effective action, exactly satisfy standard Ward identities. We identify another set of $n$-point functions in the 2PI framework which exactly satisfy the standard Ward identities at any approximation order. These are obtained as field-derivatives of the two-point function $\bcG^{-1}[\phi]$, which defines the extremum of the 2PI effective action. We point out that the latter is not constrained by the underlying symmetry. As a consequence, the well-known fact that the corresponding gauge-field polarization tensor is not transverse in momentum space for generic approximations does not constitute a violation of (2PI) Ward identities. More generally, our analysis demonstrates that approximation schemes based on 2PI functional techniques respect all the Ward identities associated with the underlying abelian gauge symmetry. Our results apply to arbitrary linearly realized global symmetries as well.