Higher Derivative Quantum Gravity on a Simplicial Lattice

A major difficulty with conventional formulations of euclidean quantum gravity is the fact that the Einstein action I, can become arbitrarily negative [l]. This means that the path integral of exp( -In) does not converge. The problem persists for lattice formulations of gravity [2] and provides an obstacle to progress for calculations of anything other than the weak field limit [3]. A possible solution to the problem has been described by Hawking [l], who suggests performing the integration in a conformal gauge in which the Einstein action is non-negative, and then integrating over all conformal factors. A second possibility is to add to the Einstein action extra terms, including higher derivative ones [4] like R*, in a carefully chosen combination which makes the total action non-negative. This paper is based on the description of gravity known as Regge calculus [2,5] in which the Einstein theory is expressed in terms of simplicial decompositions of space-time manifolds. Its use in quantum gravity is prompted by the desire to make use of techniques developed in lattice gauge theories, but with a lattice which reflects the structure of space-time rather than just providing a flat passive background. The difficulty of defining conformal transformations for the simplicial lattice leads us to explore the second of the two possible solutions mentioned above. In particular, the