$N$-cutoff regularization for fields on hyperbolic space

Abstract

We apply a novel background independent regularization scheme, the $N$-cutoffs, to self-consistently quantize scalar and metric fluctuations on the maximally symmetric but non-compact hyperbolic space. For quantum matter fields on a classical background or full Quantum Einstein Gravity (regarded here as an effective field theory) treated in the background field formalism, the $N$-cutoff is an ultraviolet regularization of the fields' mode content that is independent of the background hyperbolic space metric. For each $N > 0$, the regularized system backreacts on the geometry to dynamically determine the self-consistent background metric. The limit in which the regularization is removed then automatically yields the 'physically correct' spacetime on which the resulting quantum field theory lives. When self-consistently quantized with the $N$-cutoff, we find that without any fine-tuning of parameters, the vacuum fluctuations of scalar and (linearized) graviton fields do not lead to the usual cosmological constant problem of a curvature singularity. Instead, the presence of increasingly many field modes tends to reduce the negative curvature of hyperbolic space, leading to vanishing values in the limit of removing the cutoff.