Note on Quantum Diffeomorphism Invariance, Physical States and Unitarity

TL;DR

This paper develops a four-dimensional quantum gravity model with conformal and traceless modes, exploring its invariance, physical states, and unitarity, revealing complex state structures and potential ghost issues.

Contribution

It introduces a novel 4D quantum geometry framework with exact conformal mode treatment and a unique gravitational coupling satisfying renormalizability and asymptotic freedom.

Findings

Physical states are non-trivial and resemble composite fields.

The theory exhibits asymptotic freedom.

Potential ghost states may be hidden within the physical state conditions.

Abstract

Recently, using a local action satisfying the Wess-Zumino condition as a kinetic term of the conformal mode, we formulated a four-dimensional quantum geometry (4DQG). The conformal mode can be treated exactly, and it was shown that the part of the effective action related to this mode is given by the scale-invariant non-local Riegert action. As for the traceless mode, we introduce dimensionless coupling, which is a unique gravitational coupling of this theory satisfying the conditions of renormalizability and asymptotic freedom. Although this theory is asymptotically free, the physical states are non-trivial, which should be described as composite fields, like the spectrum of 2DQG. The possibility that the physical state conditions representing background-metric independence conceal ghosts is pointed out. The usual graviton state would be realized when the physical state condition breaks down dynamically.