# Use of Lagrange multiplier fields to eliminate multiloop corrections

## TL;DR

This paper introduces a Lagrange multiplier field method in quantum gravity and gauge theories to restrict path integrals to classical solutions, effectively eliminating multiloop corrections beyond one loop and maintaining unitarity.

## Contribution

It presents a novel approach using Lagrange multipliers to reduce loop corrections in quantum field theories, with detailed analysis of gauge invariances and BRST symmetry.

## Key findings

- Reduces multiloop contributions to one loop in quantum gravity and Yang-Mills theories.
- Maintains unitarity through BRST invariance despite loop reduction.
- Provides a framework for implementing background field quantization with Lagrange multipliers.

## Abstract

The problem of eliminating divergences arising in quantum gravity is generally addressed by modifying the classical Einstein-Hilbert action. These modifications might involve the introduction of local supersymmetry, the addition of terms that are higher-order in the curvature to the action, or invoking compactification of superstring theory from ten to four dimensions. An alternative to these approaches is to introduce a Lagrange multiplier field that restricts the path integral to field configurations that satisfy the classical equations of motion; this has the effect of doubling the usual one-loop contributions and of eliminating all effects beyond one loop. We show how this reduction of loop contributions occurs and find the gauge invariances present when such a Lagrange multiplier is introduced into the Yang-Mills and Einstein-Hilbert actions. Moreover, we quantize using the path integral, discuss the renormalization, and then show how Becchi-Rouet-Stora-Tyutin (BRST) invariance can be used to both demonstrate that unitarity is retained and to find BRST relations between Greens functions. In the Appendices, we show how the background field quantization can be implemented, consider the use of a Lagrange multiplier field to restrict higher-order contributions in supersymmetric theories, and derive the BRST equations satisfied by the generating functional.