The Secret to Fixing Incorrect Canonical Quantizations

Covariant scalar field quantization, nicknamed $(φ^r)_n$, where $r$ denotes the power of the interaction term and $n=s+1$ where $s$ is the spatial dimension and $1$ adds time. Models such that $r<2n/(n-2)$ can be treated by canonical quantization, while models such that $r>2n/(n-2)$ are trivial, or, if treated as a unit, emerge as `free theories'. Models such as $r=2n/(n-2)$, e.g., $r=n=4$, again using canonical quantization also become `free theories', which must be considered quantum failures. However, there exists a different approach called affine quantization that promotes a different set of classical variables to become the basic quantum operators and it offers different results. It is well-known that the canonical quantization of $\varphi^4_4$ fails. Here is how to fix it along with many other problems.