Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 061, 19 pages      arXiv:1003.3190      https://doi.org/10.3842/SIGMA.2010.061
Contribution to the Special Issue “Noncommutative Spaces and Fields”

Field Theory on Curved Noncommutative Spacetimes

Alexander Schenkel and Christoph F. Uhlemann
Institut für Theoretische Physik und Astrophysik, Universität Würzburg, Am Hubland, 97074 Würzburg, Germany

Received March 17, 2010, in final form July 14, 2010; Published online August 03, 2010

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.

Key words: noncommutative field theory; Drinfel'd twists; deformation quantization; field theory on curved spacetimes.

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