UV/IR mixing as an artifact of non-covariant quantisation

Abstract

We study the path integral quantisation of a scalar field on a generic noncommutative deformation of Minkowski space, built as a quantum homogeneous space of a deformed Poincaré group. We show that the procedure depends on the choice of noncommutative functional derivative, and we isolate two natural choices, distinguished by the space in which the Leibniz rule remains undeformed. The first, which carries the undeformed statistics, reproduces the standard scheme and yields n-point functions that break the deformed Poincaré covariance and exhibit the UV/IR mixing of [8]. The second, which adapts the functional calculus to the braided statistics of the fields in the spirit of [20], yields covariant n-point functions free of this mixing. We trace both the covariance breaking and the mixing to a single source, the intertwining of external and loop momenta in the non-planar contributions, and conclude that, in the models considered, the UV/IR mixing of [8] is an artifact of a quantisation that breaks the deformed symmetry rather than a feature of noncommutativity itself. We further disentangle covariance, fixed by the quantisation scheme, from finiteness, fixed independently by the propagator, and illustrate the formalism on the T-Minkowski models, the Euclidean three-dimensional quantum gravity model, and the quantum two-sphere.