Algebraic approach to quantum field theory on a class of noncommutative curved spacetimes

TL;DR

This paper develops an algebraic framework for quantizing a scalar field on noncommutative curved spacetimes using deformation quantization and Drinfel'd twists, providing explicit constructions of deformed operators and observable algebras.

Contribution

It introduces a method to quantize scalar fields on noncommutative curved spacetimes via algebraic deformation, extending previous approaches to more general geometries.

Findings

Deformed fundamental solutions depend on the spacetime deformation parameter.

In the case of Killing deformations, the observable algebra is independent of the deformation.

Explicit construction of deformed equations of motion and Green's operators.

Abstract

In this article we study the quantization of a free real scalar field on a class of noncommutative manifolds, obtained via formal deformation quantization using triangular Drinfel'd twists. We construct deformed quadratic action functionals and compute the corresponding equation of motion operators. The Green's operators and the fundamental solution of the deformed equation of motion are obtained in terms of formal power series. It is shown that, using the deformed fundamental solution, we can define deformed *-algebras of field observables, which in general depend on the spacetime deformation parameter. This dependence is absent in the special case of Killing deformations, which include in particular the Moyal-Weyl deformation of the Minkowski spacetime.