1.1. Classical Starting Point
There are different ways to promote classical to quantum expressions that are useful. For the classical, canonical Hamiltonian model, we have
where and . The ingredients in this expression are the classical field and the momentum field . These fields obey the Poisson bracket .
However, we can describe the same Hamiltonian in a different way. Let us choose the affine field , instead of the momentum field, but still keep . However, it is necessary to keep for otherwise if , then means nothing. Let us exam (1), the same classical Hamiltonian, but now in the new coordinates, leading to
This new set of fields leads to the Poisson bracket .
1.2. Quantum Starting Point
Promotion of the fields and , leads to the traditional quantum expression for our Hamiltonian, which is given by
Now, knowing that the classical variables were no longer the canonical choice but rather the affine coordinates, and after the promotion of affine field variables to new quantum field variables, the new quantum Hamiltonian becomes
Do not worry about because we have already insisted that ; hence . In a previous usage, which proved itself by modifying , while served as a safeguard .
It is noteworthy that , which, in Schrödinger’s representation, leads to and . Following a suitable regularization process , this yields the stated result.
1.3. Advantages of an Affine Quantization
Using the results of the previous sections we propose that which exposes our choice for general wave functions as given by . A regularized version, using where of this expression looks like , where is dimensionless and .
We now take a Fourier transformation of the absolute square of our regulated wave function that looks like
Normalization ensures that if all , then , which leads to
Now, at last, we can let to fix the Fourier transformation1
Observe that the affine quantization his led to a Poisson distribution, which is the only other term, besides a Gaussian expression, as dictated by The Central Limit Theorem . Nevertheless, the same expression as in (7) could have arisen when , or even when , asserting that our final result is definitely not a Gaussian! Of significant is the fact that if the coupling g, or even the mass m, are smoothly changed, there are only continuous changes within . Also, the fact that , which is a dramatic change from canonical theory’s equivalent relation, i.e., , makes a big difference; indeed, the factor in (5) is the key to avoiding a Gaussian result. Apparently, this behavior of affine quantization adopts the least final domain at the outset, which overcomes any threat of nonrenormalizability.2
We have obtained a continuous, fully regularized, expression that implicitly involves a large sample of quantum field models. The application of affine quantization, but not canonical quantization, has offered us a treasure of interest that presently rests in the Fourier representation space. Understanding the physics needed to clarify our results requires a second Fourier transformation back into the original space of the classical field, here given by . That issue is purely a mathematical task, and the implications of such an effort are certainly of great interest!
1Any change of due to is left implicit.
2For those who wish to learn more about affine quantization see . For beginners, canonical quantization deals with the harmonic oscillator, but the half-harmonic oscillator requires affine quantization .
 Aizenman, M. (1981) Proof of the Triviality of Field Theory and Some Mean-Field Features of Ising Models for d > 4. Physical Review Letters, 47, 1-4, E-886.
 Fantoni, R. and Klauder, J.R. (2021) Affine Quantization of φ44 Succeeds, While Canonical Quantization Fails. Physical Review D, 103, Article ID: 076013.
 Klauder, J.R. (2020) Using Affine Quantization to Analyze Non-Renormalizable Scalar Fields and the Quantization of Einstein’s Gravity. Journal of High Energy Physics, Gravitation and Cosmology, 6, 802-816.