A Simple Factor in Canonical Quantization yields Affine Quantization Even for Quantum Gravity

Canonical quantization (CQ) is built around $[Q,P]=i\hbar1\!\!1$, while affine quantization (AQ) is built around $[Q,D]=i\hbar\,Q$, where $D\equiv(PQ+QP)/2$. The basic CQ operators must fit $-\infty<P, Q<\infty$, while the basic AQ operators can fit $-\infty<P<\infty$ and $ 0<Q<\infty$, $-\infty<Q<0$, or even $-\infty<Q\neq0<\infty$. AQ can also be the key to quantum gravity, as our simple outline demonstrates.

Not every set of classical variables lead to valid quantum operators, neither for CQ or for AQ.The correct choice of classical variables to promote to quantum operators for CQ are Cartesian coordinates, which creat a constant vanishing curvature. The classical coordinates that creat a constant negative curvature 1 are the correct coordinates to promote to quantum operators for AQ. 2 Regarding CQ and AQ operators, the principal difference is that CQ operators must span the whole real line, while the AQ operator Q strictly spans all positive coordinates, all negative coordinates, or both, provided Q = 0. 3 Different sets of classical problems require valid CQ or AQ quantizations. For example, a traditional harmonic oscillator, with −∞ < p, q < ∞, employs CQ, while a half-harmonic oscillator, with −∞ < p < ∞, but 0 < q < ∞, employs AQ [3]. Partial-harmonic oscillators, with −∞ < p < ∞, while −b < q < ∞ and 0 < b < ∞, also employ AQ [4].
Examples from scalar field theories show that ϕ 12 3 leads to "free" results for CQ while "non-free" results for AQ [5], and also "free" results for ϕ 4 4 [6,7,8] while AQ leads to a "non-free" result [9]. It becomes clear that CQ is not the only quantization procedure for various classical systems! The author has even suggested that quantum field theory might benefit by adding AQ to the usual procedures [10].
Regarding gravity, an affine result can offer a strictly positive metric, i.e., ds(x) 2 = g ab (x) dx a dx b > 0, provided dx c ≡ / 0. This ability offers a strong contribution to the quantization of gravity, as sketched below.

The Essence of Quantum Gravity
The author has shown that an affine quantization is the natural procedure to quantize Einstein's gravity, e.g., [2,11]. Here we offer an easy and straightforward derivation that features the highlights of a natural quantum gravity. Our procedure is based on the analysis of other quantum procedures that lead toward a conventual Hamiltonian as part of a standard Schrödinger's representation followed by a suitable version of Schrödinger's equation. 1 A constant negative curvature is discussed in [1]. 2 Classical variables for spin variables that form a constant positive curvature are the final set of correct classical variables that promote to valid quantum operators. The constant (+, 0, -1) curvatures are all addressed in [2], Sec. 1.

Classical gravity
The classical Hamiltonian, according to ADM [12], uses the metric g ab (x) = g ba (x) and its determinant g(x) ≡ det[g ab (x)], 4 the momentum π cd (x) = π dc (x), or more useful, the momentric 5 π c d (x) ≡ π ce (x) g de (x), all of which leads to where R(x) is the Ricci scalar [13]. The Poisson brackets for the momentric and metric fields are given by Observe that these Poisson brackets are valid even if we change g ab (x) to −g ab (x), a property that allows us to retain only g ab (x) > 0. This is not possible with the Poisson brackets for canonical variables.

Quantum variables
Passing to operator commutations by a promotion of the set of Poisson brackets to operator commutations leads, as established in [11], to As with the Poisson brackets, these commutators are valid if we changeĝ ab (x) to −ĝ ab (x). For the momentric and metric operators, we again find that we can retain onlyĝ ab (x) > 0. Moreover, with suitable regularization, it follows thatπ a b (x)ĝ(x) −1/2 = 0, which is proved in [11].

Schrödinger's representation and equation
Schrödinger's representation choosesĝ ab (x) = g ab (x) and which is well evaluated using a suitable regularization [11]. Finally, we are led to Schrödinger's equation

Potential wave functions
The expressionπ a b (x) g(x) −1/2 = 0 points to a form of wave function given by Ψ({g}) = W ({g})Π y g(y) −1/2 . The wave functions for the Hamiltonian are continuous in the metrics that make up {g}, which is natural for a Hamiltonian that contains spatial derivatives of the metric fields and enforces continuity within the Ricci scalar, R(x).
The normalization of such wave functions may be given by while a matrix expression for an operator A may be given by It may be noticed that the factors Π y g(y) −1/2 act somewhat like 'coherent states', e.g., [14], in these equations.

Conclusion
The forgoing study is focussed on the gravity Hamiltonian, which is the most difficult aspect of quantum gravity. Additional topics, such as fulfilling constraints, are examined in [11], along with a somewhat different viewpoint in [2]. An earlier paper [15] is especially focussed on establishing a path through the multiple constraints that quantum gravity faces. Contributions by other researchers using affine procedures to quantize gravity are welcome to add to the present story.