Using a Toy Model to Improve the Quantization of Gravity and Field Theories

A half-harmonic oscillator, which gets its name because the coordinate is strictly positive, has been quantized and determined that it was a physically correct quantization. This positive result was found using affine quantization (AQ). The main purpose of this paper is to compare results of this new quantization procedure with those of canonical quantization (CQ). Using Ashtekar-like classical variables and CQ, we quantize the same toy model. While these two quantizations lead to different results, they both would reduce to the same classical Hamiltonian if $\hbar\rightarrow0$. Since these two quantizations have differing results, only one of the quantizations can be physically correct. Two brief sections illustrate how AQ can correctly help quantum gravity and the quantization of most field theory problems.