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

The conventional quantization of covariant $\phi^4_n$ scalar field models for space–time dimensions $n\ge5$ is trivial, and this may also be true for $n=4$. But an alternative $O(\hbar)$ counterterm leads to nontrivial results for all $n\ge4$ and also provides a different quantization for $n=2,3$. We determine the counterterm that provides these desirable properties as simply and directly as possible. The same counterterm also resolves models such as $\phi^p_n$ for all even $p$ including those where $p>2n/(n-2)$, which are traditionally regarded as nonrenormalizable.