Noncommutative geometry as a regulator

We give a perturbative quantization of $R^4$ space-time in the case where the commutators $C^{\mu \nu }=\left[X^{\mu }{,X}^{\nu }\right]$ 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 ${\phi }^4$ theory, that by choosing appropriately the commutators $C^{\mu \nu }$ we can remove all the infinities by reproducing all the counterterms. In other words, the renormalized action on $R^4$ plus the counterterms can be rewritten as only a renormalized action on the quantum space-time ${\mathrm{QR}}^4.$ We conjecture therefore that the renormalization of quantum field theory is equivalent to the quantization of the underlying space-time $R^4.$