On The BRST Formulation of Diffeomorphism Invariant 4D Quantum Gravity

In this note we give some remarks on the BRST formulation of a renormalizable and diffemorphism invariant 4D quantum gravity recently proposed by the author, which satisfies the integrability condition by Riegard, Fradkin and Tseytlin at the 2-loop level. Diffeomorphism invariance requires an addition of the Wess-Zumino action, from which the Weyl action can be induced by expanding around a vacuum expectation value of the conformal mode. This fact suggests the theory has in itself a mechanism to remove extra negative-metric states dynamically. E-mail address : [email protected] Diffeomorphism invariance requires that 4D quantum gravity becomes 4th order derivative theory for gravity sector. We recently showed [1, 2] that 4th order actions, including the Wess-Zumino (WZ) action [3, 4], are uniquely determined by diffeomorphism invariance. Then, the theory also becomes renormalizable [1]. Especially, our model satisfies the integrability condition on the WZ action discussed by Riegard, Fradkin and Tseytlin [3], which is generalized by the author to the form that can be applied to higher loops. A problem in 4th-order theories is that there are extra negative-metric states. Thus, the unitarity becomes obscure [5]. In this paper we shall see that there is a posibility that diffeomorphism invariance also ensures the unitarity. Here, we briefly explain how to realize diffeomorphism invariance. The details of the argument were discussed in our previous papers [1, 2]. Perturbation theory is defined by replacing the invariant measure with the measure defined on the background-metric. As a lesson from 2D quantum gravity [6]– [11], in order to preserve background-metric independence, or diffeomorphism invariance, we must add an action, S, which satisfies the WZ condition [12], as Z = ∫ [dφ]ĝ[ede]ĝ[df ]ḡ vol(diff.) exp[−S(φ, ḡ)− I(f, g)] , (1) where f is a matter field and I is an invariant action. The metric is now decomposed as gμν = eḡμν and ḡμν = (ĝe)μν , where tr(h) = 0 [11]. The measures of the metric fields are defined on the background-metric by the norms: < dφ, dφ >ĝ= ∫