Noncommutative Geometry as a Regulator

Physical review D: Particles and fields (Impact Factor: 4.86). 03/2000; 63(2). DOI: 10.1103/PhysRevD.63.025004
Source: arXiv


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 $R^4$ . Comment: 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

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