Conformal Field Theory on R x S^3 from Quantized Gravity

Conformal algebra on R x S^3 derived from quantized gravitational fields is examined. The model we study is a renormalizable quantum theory of gravity in four dimensions described by a combined system of the Weyl action for the traceless tensor mode and the induced Wess-Zumino action managing non-perturbative dynamics of the conformal factor in the metric field. It is shown that the residual diffeomorphism invariance in the radiation^+ gauge is equal to the conformal symmetry, and the conformal transformation preserving the gauge-fixing condition that forms a closed algebra quantum mechanically is given by a combination of naive conformal transformation and a certain field-dependent gauge transformation. The unitarity issue of gravity is discussed in the context of conformal field theory. We construct physical states by solving the conformal invariance condition and calculate their scaling dimensions. It is shown that the conformal symmetry mixes the positive-metric and the negative-metric modes and thus the negative-metric mode does not appear independently as a gauge invariant state at all.Comment: 42 page

Resummation and Higher Order Renormalization in 4D Quantum Gravity

Higher order renormalization in 4D quantum gravity is carried out using dimensional regularization with great care concerning the conformal-mode dependence. In this regularization, resummation can be automatically carried out without making an assumption like that of David, Distler and Kawai. In this paper we consider a model of 4D quantum gravity coupled to QED. Resummation inevitably implies a four-derivative quantum gravity. The renormalizability is directly checked up to $O(e_r^6)$ and $O(t_r^2)$, where $e_r$ and $t_r$ are the running coupling constants of QED and the traceless gravitational mode. There is no other running coupling constant in our model. The conformal mode is treated exactly, which means it is unrenormalized. It is found that Hathrell's results are included in our results. As a by-product, it is found that a higher-order gravitational correction to the beta function of QED is negative. An advantage of our model is that in the very high-energy regime, it closely resembles exactly solvable 2D quantum gravity. Thus, we can study physical states of 4D quantum gravity in this regime in parallel to those of 2D quantum gravity, which can be described with diffeomorphism invariant composite fields.Comment: 39 pages, minor typo error correcte

A Candidate for Renormalizable and Diffeomorphism Invariant 4D Quantum Theory of Gravity

We present evidence that there is a 4D model that satisfies the conditions of renormalizability and diffeomorphism invariance simultaneously at the 2-loop level. The traceless mode is treated perturbatively, while the conformal mode can be managed exactly. The two conditions constrain the theory strongly and determine the measure of the gravitational field uniquely. Quantum corrections of the cosmological constant are computed in part to 3-loop diagrams. The method to remove the negative-metric states is also discussed from the viewpoint of diffeomorphism invariance in analogy to the $R_{\xi}$ gauge in spontaneously broken gauge theory. The model may be a candidate for a continuum version of 4D simplicial quantum geometry realized in recent numerical simulations.Comment: 33 pages, revised and extended version, to appear in Pog. Theor. Phy

RG Analysis for Quantum Gravity with A Single Dimensionless Coupling

We study the quantum conformal gravity whose dynamics is governed by a single dimensionless gravitational coupling with negative beta function. Since the Euler term is not dynamical classically, the constant in front of it is not an independent coupling. Quantum mechanically, however, it induces the Riegert conformal-factor dynamics with BRST conformal symmetry representing background free nature. In this paper, we propose how to handle the Euler term systematically incorporating such dynamics on the basis of renormalization group analysis using dimensional regularization. As a non-trivial test of renormalization, we explicitly calculate the three-loop anomalous dimension of the cosmological constant operator and show that it agrees with the exact expression derived using the BRST conformal symmetry. The physical significance to inflation and CMB is also discussed.Comment: 18 pages, 2 figures, added many sentences and paragraphs to explain the model more in detail, and corrected an error in the tex

Determination of Gravitational Counterterms Near Four Dimensions from RG Equations

The finiteness condition of renormalization gives a restriction on the form of the gravitational action. By reconsidering the Hathrell's RG equations for massless QED in curved space, we determine the gravitational counterterms and the conformal anomalies as well near four dimensions. As conjectured for conformal couplings in 1970s, we show that at all orders of the perturbation they can be combined into two forms only: the square of the Weyl tensor in $D$ dimensions and $E_D=G_4 +(D-4)\chi(D)H^2 -4\chi(D) \nabla^2 H$, where $G_4$ is the usual Euler density, $H=R/(D-1)$ is the rescaled scalar curvature and $\chi(D)$ is a finite function of $D$ only. The number of the dimensionless gravitational couplings is also reduced to two. $\chi(D)$ can be determined order by order in series of $D-4$, whose first several coefficients are calculated. It has a universal value of $1/2$ at $D=4$. The familiar ambiguous $\nabla^2 R$ term is fixed. At the $D \to 4$ limit, the conformal anomaly $E_D$ just yields the combination $E_4=G_4-2\nabla^2 R/3$, which induces Riegert's effective action.Comment: 29 pages, minor corrections, a reference added, to appear in Phys.Rev.