Field Theory on Curved Noncommutative Spacetimes

TL;DR

This paper develops a framework for scalar field theories on noncommutative curved spacetimes using Drinfel'd twists, deriving deformed wave equations and Green's functions with explicit examples in various geometries.

Contribution

It introduces a general method for formulating scalar fields on noncommutative curved spacetimes with position-dependent noncommutativity, extending previous approaches.

Findings

Explicit deformed Klein-Gordon operators for key spacetimes

Construction of deformed Green's functions and perturbative methods

Calculation of leading noncommutative corrections to Green's functions

Abstract

We study classical scalar field theories on noncommutative curved spacetimes. Following the approach of Wess et al. [Classical Quantum Gravity 22 (2005), 3511 and Classical Quantum Gravity 23 (2006), 1883], we describe noncommutative spacetimes by using (Abelian) Drinfel'd twists and the associated *-products and *-differential geometry. In particular, we allow for position dependent noncommutativity and do not restrict ourselves to the Moyal-Weyl deformation. We construct action functionals for real scalar fields on noncommutative curved spacetimes, and derive the corresponding deformed wave equations. We provide explicit examples of deformed Klein-Gordon operators for noncommutative Minkowski, de Sitter, Schwarzschild and Randall-Sundrum spacetimes, which solve the noncommutative Einstein equations. We study the construction of deformed Green's functions and provide a diagrammatic approach for their perturbative calculation. The leading noncommutative corrections to the Green's functions for our examples are derived.