Scaled Affine Quantization of Ultralocal ${\phi }_2^4$ a comparative Path Integral Monte Carlo study with Canonical Quantization

After the success of affine quantization in proving through Monte Carlo analysis that the covariant euclidean scalar field theory, ${\phi }_n^r$, where $r$ denotes the power of the interaction term and $n=s+1$ with $s$ the spatial dimension and $1$ adds imaginary time, such that $r\ge 2n\mathrm{/}(n-2)$ can be acceptably quantized and the resulting theory is nontrivial, unlike what happens using canonical quantization, we show here that the same has to be expected for $r>2$ and any $n$ even for the ultralocal field theory. In particular we consider the ultralocal ${\phi }_2^4$ model and study its renormalized properties for both the scaled canonical quantization version and the scaled affine quantization version through path integral Monte Carlo.