On Restricting to One Loop Order the Radiative Effects in Quantum Gravity

TL;DR

This paper demonstrates that using Lagrange multiplier fields in quantum gravity can restrict radiative effects to one loop order, potentially leading to a renormalizable theory including matter fields.

Contribution

It introduces a method employing Lagrange multipliers to eliminate non-renormalizable divergences in quantum gravity and related matter-coupled models.

Findings

Radiative effects can be restricted to one loop order in quantum gravity.

The method renders a scalar field coupled to gravity renormalizable.

Extension to spinor and vector fields suggests a renormalizable Standard Model with dynamical metric.

Abstract

The dimensionful nature of the coupling in the Einstein-Hilbert action in four dimensions implies that the theory is non-renormalizable; explicit calculation shows that beginning at two loop order, divergences arise that cannot be removed by renormalization without introducing new terms in the classical action. It has been shown that, by use of a Lagrange multiplier field to ensure that the classical equation of motion is satisfied in the path integral, radiative effects can be restricted to one loop order. We show that by use of such Lagrange multiplier fields, the Einstein-Hilbert action can be quantized without the occurrence of non-renormalizable divergences. We then apply this mechanism to a model in which there is in addition to the Einstein-Hilbert action, a fully covariant action for a self-interacting scalar field coupled to the metric. It proves possible to restrict loop diagrams involving internal lines involving the metric to one-loop order; diagrams in which the scalar field propagates occur at arbitrary high order in the loop expansion. This model also can be shown to be renormalizable. Incorporating spinor and vector fields in the same way as scalar fields is feasible, and so a fully covariant Standard Model with a dynamical metric field can also be shown to be renormalizable