Inverse-dimensional expansion in quantum gravity

Abstract

Quantum gravity is investigated in the limit of large numbers of dimensions. We develop a gauge-invariant expansion in $\frac{1}{D}$, the inverse number of dimensions. We consider both the case where the extra dimensions are noncompact and the case where they are highly compactified, as in theories utilizing dimensional reduction. The leading term in $\frac{1}{D}$ of any Green's function is shown to obey a version of the Slavnov-Taylor identities that is more restrictive than the usual one. These identities reflect general coordinate invariance, rather than just gravitational Becchi-Rouet-Stora invariance. It is shown that they completely fix the tensor structure of any Green's function. This structure is such that we are able to cancel the infinities arising from the momentum integration with a generally covariant counterterm, unlike in the usual weak-coupling expansion. It is observed that there is an order-by-order (in $\frac{1}{D}$) cancellation of large numbers of graphs. The origin of this cancellation remains a puzzle. However, it enables us to find the leading term of any Green's function by finite computation. It is given by the set of "disjoint bubble graphs." We compute the propagator explicitly. It has a physical pole near the Planck mass, a region where the validity of the expansion is questionable. At $D=\ensuremath{\infty}$, phase-space factors suppress the Feynman integrations and the theory is finite.