Narrow Results

Following a modest comparison between canonical and affine quantization, which points to positive features in the affine procedures. We prove through Monte Carlo analysis that the covariant Euclidean scalar field theory, ${\phi }_n^r$, where $r$ denotes the power of the interaction term and $n=s+1$ where $s$ is the spatial dimension and 1 adds imaginary time, such that $r=n=4$ can be acceptably quantized using scaled affine quantization and the resulting theory is nontrivial, unlike what happens using canonical quantization.

Affine quantization, which is a parallel procedure with canonical quantization, needs to use its principal quantum operators, most simply $D=(PQ+QP)/2$ and $Q\ne 0$, to represent appropriate kinetic factors, normally $P^2$, which involve only one canonical quantum operator. The need for this requirement stems from the quantization of selected problems that require affine quantization to achieve valid Monte Carlo results. This task is resolved for introductory examples as well as examples that involve scalar quantum field theories.