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Conventional quantization of covariant scalar field models $\phi^4_n$, forspacetime dimensions $n\ge5$ are trivial, and this may also be true for $n=4$as well. However, an alternative ${\cal O}(\hbar)$ counterterm leads tonontrivial results for all $n\ge4$, and provides a different quantization for$n=2,3$ as well. In this article we determine the counterterm that providesthese desirable properties as simply and directly as possible. The samecounterterm also resolves models such as $\phi^p_n$ for all even $p$, includingthose where $p>2n/(n-2)$, which are traditionally regarded asnonrenormalizable.

Nontrivial Quantization of $\phi^4_n$, $n\ge2$