Noncommutative geometry as a regulator

We give a perturbative quantization of R⁴ space-time in the case where the commutators C^μν=[X^μ,X^ν] of the underlying algebra generators are not central. We argue that this kind of quantum space-time can be used as a regulator for quantum field theories. In particular we show, in the case of φ⁴ theory, that by choosing appropriately the commutators C^μν we can remove all the infinities by reproducing all the counterterms. In other words, the renormalized action on R⁴ plus the counterterms can be rewritten as only a renormalized action on the quantum space-time QR⁴. We conjecture therefore that the renormalization of quantum field theory is equivalent to the quantization of the underlying space-time R⁴.