Noncommutative Geometry as a Regulator

Abstract

We give a perturbative quantization of space-time $R^4$ 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-times can be used as regulators for quantum field theories . In particular we show in the case of the $ϕ^4$ theory that by choosing appropriately the commutators $C^{μν}$ we can remove all the infinities by reproducing all the counter terms . In other words the renormalized action on $R^4$ plus the counter terms can be rewritten as only a renormalized action on the quantum space-time $QR^4$ . We conjecture therefore that renormalization of quantum field theory is equivalent to the quantization of the underlying space-time $R^4$ .