BRST Conformal Symmetry as A Background-Free Nature of Quantum Gravity

TL;DR

This paper reviews quantum gravity theories that are background-free due to BRST conformal symmetry, emphasizing their implications for physical states, unphysical modes, and the emergence of classical spacetime.

Contribution

It introduces recent developments in BRST conformal algebra and physical states in 4D quantum gravity, highlighting the background-independent formulation and its consequences.

Findings

BRST conformal invariance renders all physical states real and scalar.

Negative-metric modes are unphysical due to symmetry.

The work discusses the emergence of classical spacetime and minimal length without discretization.

Abstract

Quantum gravity that describes the world beyond the Planck scale should be formulated in a background-metric independent manner. Such a background-free nature can be represented as a gauge equivalency under conformal transformations, called the BRST conformal symmetry. In this review, we present quantum field theories of gravity with such symmetry. Since we can choose any background owing to this symmetry as far as it is conformally flat, we here employ the cylindrical background. First, we briefly review the famous BRST Liouville-Virasoro algebra in 2D quantum gravity on $R \times S^1$. We then present recent developments of the BRST conformal algebra and physical states of 4D quantum gravity on $R \times S^3$ whose conformal-factor dynamics is ruled by the Riegert's Wess-Zumino action, which arises in the UV limit of the renormalizable quantum conformal gravity with the "asymptotic background freedom". We find that the BRST conformal invariance makes all physical states real and scalar as well as all negative-metric modes unphysical. We also briefly discuss the dynamics of how the conformal invariance breaks down and how our classical spacetime emerges at low energies, in which a novel interpretation of the "minimal length" we can measure comes out without discretizing spacetime.