The Conformal Constraint in Canonical Quantum Gravity

TL;DR

This paper explores a conformal invariance approach in perturbative canonical quantum gravity coupled with matter, proposing a new constraint linking beta function zeros to fixed, computable physical parameters.

Contribution

It introduces a novel constraint in quantum gravity that relates conformal invariance and beta function zeros to fixed physical constants.

Findings

Conformal invariance constrains matter theories to have vanishing beta functions.

The set of consistent quantum matter theories is discrete or finite.

Physical constants like masses and the cosmological constant are determined by this framework.

Abstract

Perturbative canonical quantum gravity is considered, when coupled to a renormalizable model for matter fields. It is proposed that the functional integral over the dilaton field should be disentangled from the other integrations over the metric fields. This should generate a conformally invariant theory as an intermediate result, where the conformal anomalies must be constrained to cancel out. When the residual metric is treated as a background, and if this background is taken to be flat, this leads to a novel constraint: in combination with the dilaton contributions, the matter lagrangian should have a vanishing beta function. The zeros of this beta function are isolated points in the landscape of quantum field theories, and so we arrive at a denumerable, or perhaps even finite, set of quantum theories for matter, where not only the coupling constants, but also the masses and the cosmological constant are all fixed, and computable, in terms of the Planck units.