The GNewton to 0 limit of Euclidean quantum gravity

Abstract

Using the Ashtekar formulation, it is shown that the GNewton to 0 limit of Euclidean or complexified general relativity is not a free field theory, but is a theory that describes a linearized self-dual connection propagating on an arbitrary anti-self-dual background. This theory is quantized in the loop representation and, as in the full theory, an infinite dimensional space of exact solutions to the constraints are found. An inner product is also proposed. The path integral is constructed from the Hamiltonian theory and the measure is explicitly computed non-perturbatively, without relaying on a semiclassical expansion. This theory could provide the starting point for a new approach to a perturbation theory in GNewton that does not rely on a background field expansion and in which full diffeomorphism invariance is satisfied at each order.