Quantum Field Theory on Curved Noncommutative Spacetimes

TL;DR

This paper develops a framework for quantum field theory on noncommutative curved spacetimes using Drinfel'd twist deformation, establishing isomorphisms with undeformed theories and analyzing their implications.

Contribution

It introduces a method to define and quantize scalar fields on noncommutative curved spacetimes, proving symplectic and algebraic isomorphisms with classical theories.

Findings

Existence of unique deformed Green's operators under support conditions

Construction of the deformed solution space with symplectic structure

Symplectic and *-algebra isomorphisms between deformed and undeformed theories

Abstract

We summarize our recently proposed approach to quantum field theory on noncommutative curved spacetimes. We make use of the Drinfel'd twist deformed differential geometry of Julius Wess and his group in order to define an action functional for a real scalar field on a twist-deformed time-oriented, connected and globally hyperbolic Lorentzian manifold. The corresponding deformed wave operator admits unique deformed retarded and advanced Green's operators, provided we pose a support condition on the deformation. The solution space of the deformed wave equation is constructed explicitly and can be canonically equipped with a (weak) symplectic structure. The quantization of the solution space of the deformed wave equation is performed using *-algebras over the ring C[[\lambda]]. As a new result we add a proof that there exist symplectic isomorphisms between the deformed and the undeformed symplectic R[[\lambda]]-modules. This immediately leads to *-algebra isomorphisms between the deformed and the formal power series extension of the undeformed quantum field theory. The consequences of these isomorphisms are discussed.