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On Noncommutative Geometric Regularisation

Studies in string theory and in quantum gravity suggest the existence of a finite lower bound to the possible resolution of lengths which, quantum theoretically, takes the form of a minimal uncertainty in positions $Δx_0$. A finite minimal uncertainty in momenta $Δp_0$ has been motivated from the absence of plane waves on generic curved spaces. Both effects can be described as small noncommutative geometric features of space-time. In a path integral approach to the formulation of field theories on noncommutative geometries, we can now generally prove IR regularisation for the case of noncommutative geometries which imply minimal uncertainties $Δp_0$ in momenta.