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

SIGMA 9 (2013), 080, 19 pages      arXiv:1308.1929
Contribution to the Special Issue on Deformations of Space-Time and its Symmetries

Dirac Operators on Noncommutative Curved Spacetimes

Alexander Schenkel a and Christoph F. Uhlemann b
a) Fachgruppe Mathematik, Bergische Universität Wuppertal, Gaußstraße 20, 42119 Wuppertal, Germany
b) Department of Physics, University of Washington, Seattle, WA 98195-1560, USA

Received August 09, 2013, in final form December 11, 2013; Published online December 15, 2013

We study the notion of a Dirac operator in the framework of twist-deformed noncommutative geometry. We provide a number of well-motivated candidate constructions and propose a minimal set of axioms that a noncommutative Dirac operator should satisfy. These criteria turn out to be restrictive, but they do not fix a unique construction: two of our operators generally satisfy the axioms, and we provide an explicit example where they are inequivalent. For highly symmetric spacetimes with Drinfeld twists constructed from sufficiently many Killing vector fields, all of our operators coincide. For general noncommutative curved spacetimes we find that demanding formal self-adjointness as an additional condition singles out a preferred choice among our candidates. Based on this noncommutative Dirac operator we construct a quantum field theory of Dirac fields. In the last part we study noncommutative Dirac operators on deformed Minkowski and AdS spacetimes as explicit examples.

Key words: Dirac operators; Dirac fields; Drinfeld twists; deformation quantization; noncommutative quantum field theory; quantum field theory on curved spacetimes.

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