Covariant non-commutative space-time

We introduce a covariant non-commutative deformation of 3 + 1-dimensional conformal field theory. The deformation introduces a short-distance scale l(p), and thus breaks scale invariance, but preserves all space time isometries. The non-commutative algebra is defined on space times with non-zero constant curvature, i.e. dS(4) or AdS(4). The construction makes essential use of the representation of CFT tensor operators as polynomials in an auxiliary polarization tensor. The polarization tensor takes active part in the non-commutative algebra, which for dS(4) takes the form of so(5, 1), while for AdS(4) it assembles into so (4, 2). The structure of the non-commutative correlation functions hints that the deformed theory contains gravitational interactions and a Regge-like trajectory of higher spin excitations. (C) 2015 The Authors