Lagrangian constraints and renormalization of 4D gravity

TL;DR

This paper explores how 4D Einstein gravity can effectively reduce to 3D through Lagrangian constraints, transitioning from classical to path integral formulation, and investigates the two-loop renormalization of the resulting 3D action.

Contribution

It advances the understanding of gravity's renormalization by explicitly analyzing the two-loop renormalization of a 3D reduced gravity model derived from 4D Einstein gravity.

Findings

Reduction of 4D gravity to 3D via Lagrangian constraints.

Path integral formulation of the reduced gravity.

Two-loop renormalization of the 3D gravity action.

Abstract

It has been proposed in \cite{Park:2014tia} that 4D Einstein gravity becomes effectively reduced to 3D after solving the Lagrangian analogues of the Hamiltonian and momentum constraints of the Hamiltonian quantization. The analysis in \cite{Park:2014tia} was carried out at the classical/operator level. We review the proposal and make a transition to the path integral account. We then set the stage for explicitly carrying out the two-loop renormalization procedure of the resulting 3D action. We also address a potentially subtle issue in the gravity context concerning whether renormalizability does not depend on the background around which the original action is expanded.