Noncommutative Geometry as a Regulator

Abstract

We give a perturbative quantization of space-time $R^4$ in the case where the commutators $C^{\mu \nu }=[X^{\mu },X^{\nu }]$ 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 ${\varphi }^4$ theory that by choosing appropriately the commutators $C^{\mu \nu }$ 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 $Q{R}^4$ . We conjecture therefore that renormalization of quantum field theory is equivalent to the quantization of the underlying space-time $R^4$ .

Comments: Latex, 30 pages, no figures,typos corrected,references added . Substantial amount of rewriting of the last section . Final interesting remarks added at the end of the paper