Equivalence of first and second order formulations of the Einstein-Hilbert theory

TL;DR

The work addresses the question of quantum equivalence between the first-order (with independent connection) and second-order Einstein–Hilbert formulations within a background-field quantization framework. It derives a general relation between generating functionals of connected Green's functions and proves a basic identity between the corresponding background effective actions, showing that $\bar{\Gamma}^{\text{I}}[0,0;\bar{\mathfrak{h}},\mathcal{G}(\bar{\mathfrak{h}})]=\bar{\Gamma}^{\text{II}}[0;\bar{\mathfrak{h}}]$ when the background connection is on-shell, with explicit one-loop verification in a general background gauge. The one-loop calculation yields a divergent self-energy structure in terms of five tensor bases and a counterterm Lagrangian of the form $\mathcal{L}_{CT}^{\text{I}}|_{\bar{G}=\mathcal{G}} = (\sqrt{-\bar{g}})/(16\pi^{2}\epsilon) [ a(\xi) \bar{R}^{2} + b(\xi) \bar{R}_{\mu\nu} \bar{R}^{\mu\nu} ]$, which reduces to the known Goldberg result for $\xi=1$ and matches the second-order theory on-shell. Collectively, these findings reinforce the quantum consistency of using either first- or second-order variables in EH gravity with background fields and pave the way for extensions to theories like Einstein–Cartan or supergravity.

Abstract

We derive a general relation between the background effective actions, which directly proves that the two formulations of the Einstein-Hilbert theory with background fields are equivalent at the quantum level. This basic result has been substantiated in a general background gauge, by explicit calculations at one-loop order of the corresponding counterterm Lagrangians.

Figures (2)

• Figure 1: Diagrams that contributes to the self-energies $\bar{\mathfrak{h}} \bar{\mathfrak{h}}$ (a), $\bar{G} \bar{\mathfrak{h}}$ (b) and $\bar{G} \bar{G}$ (c) in the first order formulation of the EH theory. Wavy and solid lines represents respectively the background fields $\bar{\mathfrak{h} }$ and ${\bar{G}}$. The quantum fields $\mathfrak{h}$ and $\mathfrak{G}$ are represented by springy and double solid lines. Momenta in the loops flow clockwise, so that, $q = p+k$.
• Figure 2: Divergent part of the background self-energies in the first-order Yang-Mills theory. The factor $d_{N}$ is equal to $iN g^{2} \delta^{ab} /16 \pi^{2} \epsilon$.