Effective Actions for Regge State-Sum Models of Quantum Gravity

We study the semiclassical expansion of the effective action for a Regge state-sum model and its dependence on the choice of the path-integral measure and the spectrum of the edge lengths. If the positivity of the edge lengths is imposed in the effective action equation, we find that the semiclassical expansion is not possible for the power-law measures, while the exponential measures allow the semiclassical expansion. Furthermore, a slightly generalized exponential measure can generate the cosmological term in the effective action as a quantum correction, with a naturally small value of the cosmological constant. We also find that in the case of a discrete length spectrum, the semiclassical expansion is allowed only if the spectrum gap is much smaller than the Planck length.