Nontrivial Quantization of ${\varphi }_n^4$, $n\ge 2$

Abstract

Conventional quantization of covariant scalar field models ${\varphi }_n^4$, for spacetime dimensions $n\ge 5$ are trivial, and this may also be true for $n=4$ as well. However, an alternative $\mathcal{O}(\hslash )$ counterterm leads to nontrivial results for all $n\ge 4$, and provides a different quantization for $n=2,3$ as well. In this article we determine the counterterm that provides these desirable properties as simply and directly as possible. The same counterterm also resolves models such as ${\varphi }_n^p$ for all even $p$, including those where $p>2n/(n-2)$, which are traditionally regarded as nonrenormalizable.

10 pages. arXiv admin note: substantial text overlap with arXiv:1112.0803, updated text, added dedication