If the universe were curved and noncommutative

TL;DR

This paper explores quantum field theory in a curved, noncommutative space, deriving the classical action, analyzing divergences, and studying the renormalization group flow, revealing potential asymptotic freedom or cosmological constant sign change.

Contribution

It provides a perturbative formulation of scalar field theory in Snyder--de Sitter space and analyzes its renormalization group behavior in various dimensions.

Findings

Explicit first-order classical action in small-deformation regime

Computed one-loop divergences and beta functions

Identified conditions for asymptotic freedom and cosmological constant sign flip

Abstract

We review recent discussions concerning the definition of a quantum field theory in a curved and noncommutative space, the Snyder--de Sitter space. For a quartic self-interacting scalar field in a spacetime of arbitrary dimension, we show how to perturbatively define the classical action in the small-deformation regime and give its explicit first-order expression. Afterwards, we compute the divergences of the one-loop effective action for the two- and four-dimensional cases. These are employed to calculate the beta functions of the couplings and to numerically analyze the corresponding renormalization group flow. Depending on the initial conditions of the couplings, in four dimensions it is possible to obtain an asymptotic-free theory or a flip in the sign of the cosmological constant as a consequence of the running.