Dirac Operators on Noncommutative Curved Spacetimes

Symmetry Integrability and Geometry-Methods and Applications

article

Alexander Schenkel¹  Christoph F. Uhlemann² 

[1] Fachbereich C - Physik, Bergische Universität Wuppertal;Department of Physics, University of Washington

关键词: Dirac operators;    Dirac fields;    Drinfeld twists;    deformation quantization;    noncommutative quantum field theory;    quantum field theory on curved spacetimes;   

DOI  :  10.3842/SIGMA.2013.080

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown