Noncommutative Geometry as a Regulator

TL;DR

This paper explores how noncommutative geometry can serve as a natural regulator for quantum field theories, potentially eliminating infinities through the quantization of space-time itself.

Contribution

It introduces a perturbative quantization method for noncommutative space-time with non-central commutators, proposing it as a new way to address renormalization.

Findings

Successfully quantized space-time with non-central commutators.

Reproduced all counter terms for ${}^4$ theory using noncommutative geometry.

Suggested that renormalization is equivalent to space-time quantization.

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 ${\phi}^4$ 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 $QR^4$ . We conjecture therefore that renormalization of quantum field theory is equivalent to the quantization of the underlying space-time $R^4$ .