The Magnificent Realm of Affine Quantization: valid results for particles, fields, and gravity

Affine quantization is a relatively new procedure, and it can solve many new problems. This essay reviews this new, and novel, procedure for particle problems, as well as those of fields and gravity. New quantization tools, which are extremely close to, and even constructed from, the tools of canonical quantization, are able to fully solve selected problems that using the standard canonical quantization would fail. In particular, improvements can even be found with an affine quantization of fields, as well as gravity.

The scope of quantum physics has expanded remarkably, as will be clear in this presentation.Many problems have new and novel results.
The basic rules of quantization were largely set around mid- (1920), and have changed very little thereafter.There are many problems that using those rules can lead to acceptable results, but there are many more problems that those rules are inadequate.As an example, the traditional harmonic oscillator, which is set on the whole real line, can be fully solved with the original rules.However, if the harmonic oscillator is set only on the positive real line, it can not be solved with the old quantization rules despite the fact that it can be solved classically.Many problems that can be solved classically cannot be solved with old rules known as canonical quantization (CQ).Those procedures fail on nonrenormalizable examples which include certain relativistic scalar fields and Einstein's gravity.
A new quantization procedure, called affine quantization (AQ), has now been added to the old rules.This procedure is now about 30 years old.AQ is not well known and it deserves to be as strongly known as CQ.While CQ chooses the momentum, e.g., p, and the coordinate, e.g., q, to promote to quantum operators, AQ chooses what we call the dilation, We start slowly with simple models to appreciate what AQ is able to accomplish.Already, using Monte Carlo methods, several nonrenormalizable relativistic scalar models have confirmed what AQ can do for them.Einstein's gravity is more complicated, but the rules of AQ offer considerably positive results.

II. AN INTRODUCTION TO THE VARIABLES
Quantum operators are promoted from classical variables that can play an important role and need to be presented here because it is poorly covered.Our story involves three sets of classical variables that will, later, find their importance when they are promoted to basic quantum variables.

A. A Survey of Principal Topics
The common examination of quantum topics starts with a classical review, and we shall do the same.Our focus features three different classical versions.These three have some similar features as well as their differences, but they all play a role in the quantum story.
The three versions of quantum theory, which develop from the three classical versions, have important and distinct roles to play.After studying the procedures, we will apply them to specific problems.It follows that the various procedures fit specific sets of problems, and fail when the wrong procedures are applied to any wrong set of problems.In particular, problems that are nonrenormalizable quantum problems, and which have been unsolved for decades, can, in fact, be properly solved by using the correct quantum procedures instead of the wrong procedures.While they may have been favored, they also may have been the incorrect procedure for decades!In later chapters, we will solve nonrenormalizable covariant scalar fields as well as Einstein's gravity.

B. A Familiar Example of Classical Variables
The everyday behavior of most objects consists of its position, abbreviated by q, and its momentum, namely, its mass multiplied by its velocity, like p = mv.These objects also change place and/or motion, which is represented by q(t) and p(t), with t serving as time.
In an ideal universe, there would be no friction to slow motion down, while instead energy is typically considered to be a constant.The important Hamilton expression, H(p, q), and the equations of motion are given by q(t) = ∂H(p, q)/∂p(t) , ṗ(t) = −∂H(p, q)/∂q(t) . (2.1) A common example is the harmonic oscillator, for which, like all systems, the energy is contained in the Hamiltonian, This leads to the equations of motion given by q(t) = p(t)/m, while ṗ(t) = −ω 2 m q(t).
These equations lead to q(t) = −ω 2 q(t) and p(t) = −ω 2 p(t), with solutions given by q(t) = A cos(ωt) + B sin(ωt) (2.3) The action functional is an important expression that also leads to the same equations that we dealt with in the section above, e.g., and which leads to tiny variations in the variables, δq(t) and δp(t), and now δq(T ) = δq(0) = 0 as well as δp(T ) = δp(0) = 0.The variations lead to which leads to the correct equations of motion being recovered when arbitrary variations are implied.

SPACES
Phase space consists of a collection of general, continuous, functions p(t) and q(t).These functions can be turned into different functions, such as The family of functions is chosen to observe the integral The Poisson brackets for these variables is given by Poisson brackets play a reducing lever putting multiple expressions into fixed sets.For example, {q, p} = 1 and {q 3 /3, p/q 2 } = 1, and also as {q, pq} = q.
The pair of functions, p(t) & q(t), also has a geometric role to play.Let us assume we choose to create a flat, two-dimensional surface, by using the following expression, where ω is a positive constant that does not depend on p(t) or q(t) in any way.A common name for this case is 'Cartesian variables'.It is noteworthy that this two-dimensional surface is completely identical if you move to any other location.That property may be called a 'constant zero curvature'.
Moreover, such a mathematical plane is infinitely big, meaning that −∞ < p & q < ∞.
Observe that this property of p & q is complete, which means every point in IR 2 is included.
There is no case where q = 17, for example, is excluded from the rest of −∞ < q < ∞.
Instead, they offer a distinct quantization procedure that applies to different problems.
However, Eq. (3.6) makes it clear that if s → ∞, in which case both p and q span the real line, we are led to 'Cartesian coordinates', a basic property of canonical quantization.

A brief review of affine quantization
Consider a classical system for which −∞ < p < ∞, but 0 < q < ∞, that does not lead to a self-adjoint quantum operator P , i.e.P † = P .Perhaps we can do better if we change classical variables.For example, 0 < q < ∞ -or it may arise instead that −∞ < q < 0. To capture these possibilities for q -and thus also for This expression happens to be like the Lie algebra of the "affine group" [Wik-1], and, incidentally, that name has been adopted by "affine quantization".Again, it is useful to choose dimensions such that q & Q are dimensionless while d & D have the dimensions of h.

Affine coherent states
The affine coherent states involve the quantum operators D and Q, where now Q > 0.
It follows that the Fubini-Study metric, for q > 0, becomes This expression leads to a surface that has a 'constant negative curvature' [Sch-1] of magnitude −2/βh, which, like the other curvatures, has been 'created'. 2The set of classical variables can not lead to a physically correct canonical quantization.Instead, they offer a distinct quantization procedure that applies to different problems.Any use of classical variables that do not form a 'constant negative curvature' subject to an affine quantization is very likely not a physically correct quantization.
The inner product of two affine coherent state vectors is given by while |p; q p; q|(1 − 1/2β) dp dq/2πh = 1 1, provided that β > 1/2.While the variable change for all p, q → cp, q/c leaves a Cartesian metric still Cartesian, while it can be seen that 1 The semicolon in |p; q distinguishes the affine ket from the canonical ket |p, q .If −∞ < q < 0, change ln(q) to ln(|q|), but keep q → Q < 0 so that |q|Q = q|Q|. 2As noted, while constant zero and positive curvatures can be seen in our three spatial dimensions, a visualization of a complete constant negative curvature is not possible.A glance of one would be a single point on a saddle, namely, the highest point from the rider's feet direction, and the lowest point from the horse's head direction.
there is no change whatsoever in (3.9), illustrating the significance of the affine Fubini-Study metric.

B. Summarizing Constant Curvatures and Coherent States
These three stories complete our family of 'constant curvature' spaces, specifically, constant positive, zero, and negative curvatures.Additionally, the various coherent states can be used to build "bridges" in each case that enable one to pass from the classical realm to the quantum realm or pass in the other direction.Two relative articles, for different systems, can be found in [Kla-1], [Kla-2].

IV. LEARNING TO QUANTIZE SELECTED PROBLEMS
We begin with two different quantization procedures, and two simple, but distinct, problems one of which is successful and the other one is a failure in trying to use both of the quantization procedures on each example.This exercise serves as a prelude to a valid and straightforward clarification of the fact that affine quantization and canonical quantization solve completely different sets of problems.This fact will help us when we turn to the quantization of field theories and of gravity in later chapters.

A. Choosing a Canonical Quantization
The classical variables, p & q, that are elements of a constant zero curvature, better known as Cartesian variables, such as those featured by Dirac [Dir-1], are promoted to selfadjoint quantum operators P (= P † ) and Q (= Q † ), ranged so that −∞ < P & Q < ∞, and scaled so that [Q, P ] = ih1 1. 3

First canonical example
Our example is just the familiar harmonic oscillator, for which −∞ < p & q < ∞ and a Poisson bracket {q, p} = 1, also a classical Hamiltonian, with the common factors m = ω = 1, given by H(p, q) = (p 2 + q 2 )/2.The quantum Hamiltonian is H(P, Q) = (P 2 + Q 2 )/2, and Schrödinger's representation is given by P Finally, for our example, Schrödinger's equation is given by Solutions to Eq. (3.1) for our example are well known.In particular, for the harmonic oscillator, the eigenvalues are given by E n = h(n + 1/2) for n = 0, 1, 2, , ..., and the eigenfunctions (with h = 1) are given by ψ n (x) = N n H n (x) e −x 2 /2 with n = 0, 1, 2, .... Here, N n serves to enforce normalization, and the remainder is This model is one of the most well understood of all examples.

Second canonical example
For our next example we keep the same classical Hamiltonian, and we retain −∞ < p < ∞, but now we restrict 0 < q < ∞.This new model is called the 'half-harmonic oscillator'.It follows that the operator P † = P , which leads to a different behavior to that when P is self adjoint, i.e., P † = P .In particular, this can lead to infinitely many different self-adjoint Hamiltonians each of which passes to the same classical Hamiltonian that would be (p 2 + q 2 )/2 in this case.Just two of the different quantum Hamiltonians could be H 0 (P, Q) = (P In particular, in [Dir-1], the mid-page of 114, Dirac wrote "However, if the system does have a classical analogue, its connexion with classical mechanics is specially close and one can usually assume that the Hamiltonian is the same function of the canonical coordinates and momenta in the quantum theory as in the classical theory † Footnote †: This assumption is found in practice to be successful only when applied with the dynamical coordinates and momenta referring to a Cartesian system of axes and not to more general curvilinear coordinates." of these quantum Hamiltonians lead to the same classical Hamiltonian, namely (p 2 + q 2 )/2, when h → 0.4 This judgement renders the canonical quantization of the half-harmonic oscillator to be an invalid quantization.
We interrupt our present story to bring the reader an important message. - This is good mathematics, but physics has an opinion as well.
Consider mv = p.If the velocity v = 0, then the momentum p = 0, which makes good sense.However, if the mass m = 0 and the velocity v = 9, then the momentum, p = 0, makes bad physics.However, if any of them are infinite, that is certainly bad math as well as bad physics.
We will especially use this topic for the dilation variable d = pq, where q is the coordinate of a position and p denotes its time derivative (times its mass too).The position q(t) is continuous, while p(t) is traditionally continuous, but it can change sign, like bouncing a ball off a wall.
We may point to an ABC-item to remind the reader of its relevance. - This important notification is finished.

First affine example
The traditional classical affine variables are d ≡ pq and q > 0 (ABC), and they have a Poisson bracket given by {q, d} = q.In addition, we can choose a different dilation variable, The classical affine variables now are −∞ < d ≡ p(q + b) < ∞ and 0 < (q + b) < ∞, while the classical harmonic oscillator Hamiltonian is given by H while, in a proper limit, an affine quantization becomes a canonical quantization when the Evidently, an affine quantization fails to quantize a full harmonic oscillator.

Second affine example
The common canonical operator expression, [Q To confirm this affine expression, let us multiply Canonical quantization implies affine quantization, but adds a limitation, for classical as well as quantum, on the coordinates.
Regarding our problem, now with b = 0, and so the classical affine variables are d ≡ pq and q > 0, which lead to the half-harmonic oscillator H ′ (d, q) = (d 2 /q 2 + q 2 )/2.The basic affine quantum operators are D and Q, where , and Schrödinger's representation is given by Q → x > 0 and Finally, Schrödinger's equation is given by We note that kinetic factors, such as P and D, can annihilate separate features.Adopting Schrödinger's representation, it follows thar P 1 = 0 while Dx −1/2 = 0. We will exploit this simple fact in later chapters.
Solutions of (4.5) have been provided by L. Gouba [Gou-1].Her solutions for the halfharmonic oscillator contain eigenvalues that are equally spaced as are the eigenvalues of the full-harmonic oscillator, although the spacing itself differs in the two cases.The relevant differential equation in (4.5) is known as a 'spiked harmonic oscillator', and its solutions are based on confluent hypergeometric functions.It is noteworthy that every eigenfunction, , which applies for all n = 0, 1, 2, ....The leading factor of the eigenfunctions, i.e., x 3/2 , provides a continuous result after the first derivative, but the second derivative could lead to an x −1/2 behavior, except that [−d 2 /dx 2 +(3/4)/x 2 ] x 3/2 = 0.
This zero ensures that after two derivatives, the wave function is still finite, continuous, and belongs in a Hilbert space.5 It is interesting to consider an increase in the coordinate space by choosing x + b > 0.
This leads to a related Schrödinger's equation, given by which has been shown to also have equally spaced eigenvalues that become narrower as b becomes larger.Moreover, if b → ∞, then the h-term disappears and the full-harmonic oscillator, with its canonical quantization features, is fully recovered .In this fashion, we observe that AQ can pass to CQ, but the reverse is, apparently, impossible.
Finally, we can assert that an affine quantization of the half-harmonic oscillator can be considered to be a correctly solved problem.

A canonical version of the half-harmonic oscillator
We start again with the classical Hamiltonian for the half-harmonic oscillator which is still H = (p 2 +q 2 )/2 and q > 0, but this time we will use different coordinates.To let our new coordinate variables span the whole real line, which makes them 'Ashtekar-like' [Ash-1], we choose q = s 2 , where −∞ < s < ∞.Thus, s is the new coordinate.For the new momentum, r , we choose p = r/2s.We choose it because the Poisson bracket {s, r} = { √ q, 2p √ q} = 1.6 The classical Hamiltonian now becomes H = (p 2 + q 2 )/2 = (r 2 /4s 2 + s 4 )/2.

A CQ attempt to solve the half-harmonic oscillator
For quantization, the new variables use canonical quantum operators, r → R and s → S, with [S, R] = ih1 1.Following the CQ rules, this leads to This quantum operator, using canonical operators where It is self-evident that these two canonical quantum Hamiltonian operators, H AQ and H CQ , have different eigenfunctions and eigenvalues.Does it matter that H AQ < ∞ while H CQ ≤ ∞, due to S = 0 while R = 0? It is clear that answers to these questions are "No".
Trying to quantize the half-harmonic oscillator, using CQ variables, has led to physically incorrect results.
Now we examine a very different model using both CQ and AQ.

V. USING CQ AND AQ TO EXAMINE 'THE PARTICLE IN A BOX'
A. An Example that Needs More Analysis This model has often been used in teaching and it is introduced early in the process as an easy example to solve.The classical Hamiltonian for this model is simply H = p 2 , allowing, for simplicity, that 2m = 1.Now the coordinate space is −b < q < b, where 0 < b < ∞ (which also may be chosen as 0 < q < 2b ≡ L < ∞).To accommodate the CQ operators, we assume that outside the box there are infinte potentials that force any wave functions to be zero in the entire outside region where |x| ≥ b.Inside the box we have the quantum equation It was remarked in Wikipedia's discussion of the particle in a box [Wik-2] that the first derivative was not continuous as it should have been, but effectively, ignoring it afterwards.
In summary, we conclude that by using CQ, the standard treatment and results for the particle in a box are incorrect.
The reduced coordinate space now requires a newly named dilation variable, , along with accepting only −b < q < b.Using affine variables, the classical Hamiltonian now becomes H ′ = d ′2 /(b 2 − q 2 ) 2 .Following the affine quantization rules, means that the and the quantum Hamiltonian is (5. 2) The new h-expression is unravelled later in the Appendix to Chapter V.
When comparing the different h-terms, we find, with using , which mimics the (3/4)-factor for the half-harmonic oscillator.This implies that the x term in eigenfunctions, extremely close to either ±b, should be like ψ(x) ≃ (b 2 − x 2 ) 3/2 (remainder).
For a moment, we take an about face A very different use of (5.2) is to accept the outside space, |x| > b, and reject |x| < b, which then becomes an 'anti-box'.
Note that this system has a similarity to a toy 'black hole'.It could happen that particles would pile up close to an 'end of space', while having been attracted there by a simple, "gravity-like", pull of a potential, such as V (x) = W 2 x 4 .If you choose AQ, then the barracked, h-like term, in (5.2), would prevent the particles from falling 'out of space' [Kla-3], while the shores exhibit light from the fires of trapped trash.

Removing a single point
Assuming that we still have chosen the outside, |x| > b, coordinates, it is noteworthy that if we focus on the region where b → 0, while insisting that |x| > 0. In this case, the h-term becomes 2h 2 /x 2 .However, the previous eigenfunction behavior of (x 2 − b 2 ) 3/2 , now with x 2 > b 2 , implies that any eigenstates (again, having potentials, like V (x) = |x| r , for r ≥ 2, that reach infinity) must start like ψ n (x) ≃ x 3 (remainder n ).This offers effective continuity for the eigenfunction and its first two derivatives, even though x = 0 can permit a more different behavior on either side of x = 0. This, then, is the 'cost' to remove a single point in the usual coordinate space, e.g., in this case, removing just the single point at q = 0.This result has been made possible using AQ and not using CQ, which requires including all x, i.e., −∞ < x < ∞.
A Vector Version: The point we now wish to remove is − → q = 0; stated, we want to retain all the variables that obey − → q 2 > 0 and all those of − → p 2 ≥ 0. In addition, we introduce , that equation also unfolds, in a fashion similar to that shown in the Appendix to Chapter V, below, and leads to the quantum Hamiltonian Just by sending b 2 → 0, we achieve the situation where only the single point, i.e., − → q 2 = 0 → − → Q 2 = 0 is removed from our s-dimensional space.The quantum Hamiltonian in this case is (5.4) To offer a justification that this relation holds for all − → Q including just the case where s = 1, i.e., just Q 2 .To do so, let us introduce the wave function ψ(x) = U(x) W B (Bx j ), by introducing a partial expectation of the Hamiltonian given by in which we have integrated all x j except x = x 1 .Now, for all, but x 1 , we let x j → x j /B, which changes the previous equation to become (5.6) The purpose of this exercise is to show that the original quantum Hamiltonian (5.4) for s many dimensions holds the equation for a final quantum Hamiltonian (5.6) as B → ∞ for a single dimension.
Briefly stated, an (s − 1)-dimensional reduction may be arranged that can force all of those coordinates to become zero.This leaves behind just one of the coordinates, which is part of a proper equation, and is already waiting to fulfill its duty.7

B. Lessons from Canonical and Affine Quantization Procedures
An important lesson from the foregoing set of examples is that canonical quantization requires special classical variables, i.e., −∞ < p & q < ∞, that create a flat surface, to be promoted to valid quantum operators that satisfy −∞ < P & Q < ∞.However, an affine quantization requires different classical variables, e.g.,−∞ < d b = p(b + q) < ∞ and −b < q < ∞, chosen so that 0 < b < ∞, to be promoted to valid affine quantum operators, quantization to show that these non-renormalizable theories can be correctly quantized by affine quantizations; the story of such scalar models is introduced in this chapter.The present chapter will also show that ultralocal gravity can be successfully quantized by affine quantization.
The purpose of this study is to show that a successful affine quantization of any ultralocal field problem would imply that, with properly restored spacial derivatives, the classical theory can, in principle, be guaranteed a successful quantization result using either a canonical quantization in some cases or an affine quantization in different cases.
In particular, Einstein's gravity requires an affine quantization, and it will be successful, as we will find out in a following chapter.

C. Classical and Quantum Scalar Field Theories
The purpose of this section is to review a modest summary of the results of canonical quantization when it has been used to study a variety of covariant scalar field models.
We interrupt our present story to up grade 'A Simple Truth' to prepare the reader for its use with fields, To ensure getting A(x) one must require 0 This is good mathematics, but physics has an opinion as well.
Consider k(x) = π(x)ϕ(x), where ϕ(x) is a chosen physical field, π(x) is its momentum field, and their product is κ(x), which we will call the dilation field.Since π(x) serves as the time derivative of ϕ(x), it can vanish along with κ(x).However, requiring that both plus and minus sides of ϕ(x) = 0 are acceptable, since the derivative term ensures it will still seem to come from a continuous function.Moreover, if ϕ(x) = 0 it could be confused with any other field, e.g., α(x) = 0. a It is good math for finite integrations if there are examples where the fields may reach infinity, e.g., 1 −1 ϕ −2/3 dϕ < ∞.However, such cases are very likely to be bad physics because no item of nature reaches infinity.Accepting κ(x) (= π(x) ϕ(x)) and ϕ(x) = 0, instead of π(x) and ϕ(x), as the basic variables, will have profound consequences.
For example, the classical Hamiltonian expressed as Thus omitting points, or streams of them, where ϕ(x) = 0, do not violate any physics.
In fact, it may seem logical to say that ϕ(x) = 0 never even belonged in physics.It fact, since numbers were used to count physical things, in very early times, zero = 0, was banned for 1,500 years ; see [Zero].

D. Canonical Ultralocal Scalar Fields
These models have a classical (labelled by c) Hamiltonian given by with p = 4, 6, 8, ... and s = 1, 2, 3, ....With n = s + 1 spacetime dimensions, and first using canonical quantization, we examine these models.In preparation for a possible path integration, the domain of H c consists of all, momentum functions π(x) and scalar fields ϕ(x), for which 0 ≤ H c < ∞.
Since all derivatives have now been removed even stronger issues can be expected by path integrations being swamped by integrable-infinities of the field, or by vast numbers of almost integrable-infinities.However, effectively, that strong behavior fails to contribute to the path integration results, e.g., for p ≥ 4, while the middle range contributions have the most influence on the final result.
To confirm that view, Monte Carlo computations have shown an effectively free-like behavior for analogous CQ models [F-K-1].
The basic quantum operators are φ(x) = 0 and κ(x), and their commutator is given by [ φ(x), κ(x ′ )] = ihδ s (x − x ′ ) φ(x).The quantum, ultralocal, affine Hamiltonian, is now given by Clearly this is a formal equation for the Hamiltonian operator, etc.Such expressions deserve a regularization and rescaling of these equations.
It is noteworthy that Monte Carlo computations have shown a reasonable, active behavior, for analogous AQ models [F-K-1], [F-K-6].

VII. AN ULTRALOCAL GRAVITY MODEL
An affine formulation would use the classical metric g ab (x), which, as before, has a positivity requirement, while the momentum field will be replaced by the dilation field, π a b (x) [≡ π ac (x) g bc (x)], summed by c.These basic affine variables are promoted to quantum operators, both of which can be self-adjoint, while the metric operator is also positive as required.
The principle of using ultralocal rules, as before, is that spacial derivatives must be eliminated.To satisfy that rule, we drop the factor (3) R(x), the Ricci scalar field composed of the metric field and its spacial derivatives, and replace it with a new function, Λ(x), which will be called a 'Cosmological Function' to imitate the standard constant factor, Λ, known as the 'Cosmological Constant'.This new function is independent of the dilation and metric functions, and is simply used as a continuous function that obeys 0 < Λ(x) < ∞, or otherwise.

With this substitution, the ultralocal classical Hamiltonian is now given by
Since there are no spatial derivatives, we are given another example that every spatial point x labels a pair of distinct variables, namely π a b (x) and g cd (x).Once again, we find a quantum wave function, using the Schrödinger representation for the metric field g ab (x), that is a product of independent spacial values of the form Ψ({g}) = Π x W (x), where {g} denotes g ab (•) for all x.
When this Hamitonian is quantized, the only variables that are promoted to quantum operators are the metric field, g ab (x), and the dilation (or, sometime known as 'momentric' to include momentum and metric) field, π a b (x) = π ac (x) g bc (x), and the field Λ(x) is fixed and not made into any operator.

A. An Affine Quantization of Ultralocal Gravity
The quantum operators are ĝab (x) and πc d (x), and their Schrödinger representations are given by ĝab (x) = g ab (x) and πa b (x) = −i 1 2 h[g bc (x) (δ/δg ac (x)) + (δ/δg ac (x))g bc (x)].The Schrödinger equation for the ultralocal Hamiltonian is then given by where, as noted, the symbol {g} denotes the full metric matrix.Solutions of (7.2) are governed by the Central Limit Theorem.

B. A Regularized Affine Ultralocal Quantum Gravity
Much like the regularization of the ultralocal scalar fields, we introduce a discrete version of the underlying space such as x → ka, where k ∈ {..., −1, 0, 1, 2, 3, ...} 3 and a > 0 is the spacing between rungs in which, for the Schrödinger representation, g ab (x) → g ab k and πc d (x) → πc d k .It can be helpful by assuming that the metric has been diagonalized so that g ab k → {g 11 k , g 22 k , g 33 k }, as it becomes Take note that πa b k g −1/2 k = 0, where g k = det(g ab k ).We will exploit such an expression one more time.
The regularized Schrödinger equation is now given by ih ∂ψ(g, t)/∂t (7.4) Observe that g k = det(g ab k ) is now the only representative of the metric g ab k .
A normalized, stationary solution to this equation may be given, by some Y (g k ), which The Characteristic Function for such expressions is given by where the scalar g k → g(x) > 0 and Y accommodates any change due to a → 0. The final result is a (generalized) Poisson distribution, which obeys the Central Limit Theorem.
The formulation of Characteristic Functions for gravity establishes the suitability of an affine quantization as claimed.Although this analysis was only for an ultralocal model, it nevertheless points to the existence of proper quantum solutions for Einstein's general relativity.

C. The Main Lesson from Ultralocal Gravity
Just like the success of quantizing ultralocal scalar models, we have also showed that ultralocal gravity can be quantized using affine quantization.The purpose of solving ultralocal scalar models was to ensure that non-renormalizable covariant fields can be solved using affine quantization.Likewise, the purpose of quantizing an ultralocal version of Einstein's gravity shows that we should, in principle, and using affine quantization, be able to quantize the genuine version of Einstein's gravity using affine quantization; see arXiv:2203.15141.
The analysis of certain gravity models with significant symmetry may provide examples that can be completely solved using the tools of affine quantization.

VIII. HOW TO QUANTIZE RELATIVISTIC FIELDS
If the reader thinks that canonical quantization is the best way to quantize relativistic field theories, the reader should read this chapter carefully.

A. Reexamining the Classical Territory
We now turn from ultralocal models to those that are relativistic.These are models that really can represent nature, and they are clearly the most important examples.The principal example of a covariant scalar field theory is the usual one that we focus on, namely This example is meant to deal with fields that obey the rule that |π(x That is a very reasonable restriction, however a path integration can violate that rule.We have in mind integrable-infinities, such as π(x) 2 = 1/|x| 2s/3 , where s is the number of spatial coordinates, i.e., s , which from a classical viewpoint seem unlikely, but from a path integration point of view it seems very likely.
Such integrable-infinities encountered here in the classical analysis lead to nonrenormalizable behavior in which the domain of the variables for a free model, i.e., g = 0, becomes reduced then, when g > 0, and p ≥ 2n/(n − 2), with n = s + 1.Since the domain of the classical variables becomes reduced, it remains that way when the coupling constant is reduced to zero using g → 0. With such behavior for the classical analysis, there is every reason to expect considerable difficulties in using canonical quantization.
To make that statement clear, it is a fact that Monte Carlo calculations for the scalar fields ϕ 12 3 and ϕ 4 4 apparently led to free results, using CQ, as if the coupling constant g = 0 when that was not the case, but offered reasonable results using AQ [Fan-1], [F-K-4].Clearly, integrable-infinities are not welcome!This section will draw on Chapter V to a large extent, although it has been somewhat changed by the introduction of the gradient term.That may lead to some repeats of certain topics.

A simple way to avoid integrable-infinities
Let us, again, introduce a new field, κ(x) ≡ π(x) ϕ(x), as a featured variable rather than π(x), to accompany ϕ(x) = 0 (ABC).We really don't 'change any variable', but just give the usual ones 'a new role'.Some care is needed in choosing κ(x) and ϕ(x) as the new pair of variables, and physics can be a good guide.
Let us recall the simple analog, namely p = mv.If the velocity v = 0, then physics agrees that the momentum p = 0.However, if the mass m = 0 and v = 6, then having p = 0, along with any term being infinity, is very bad physics.Instead, physics requires that 0 ≤ |v| & |p| < ∞ and 0 < m < ∞ makes good physics.This story can apply to other variables, and as has often been noted, we point to such items as (ABC).
In our case, we assume ϕ(x) is a physical field, π(x) is its time derivative, and κ(x) ≡ π(x) ϕ(x), their product, which will be called the 'dilation field', serves as a kind of momentum.Now, using a similar argument as above, we accept the assertion that 0 ≤

The absence of infinities by using affine field variables
Now, let us use κ(x) and ϕ(x) = 0 as the new variables to be used in the classical Hamiltonian (8.1), which then becomes Now, things are different.To represent π(x), then κ(x) and ϕ(x), must serve their role.Hence we require that 0 < |ϕ(x)| < ∞, which implies that 0 < |ϕ(x)| p < ∞ for all 0 < p < ∞ and all s.In addition, we require that |κ(x)| < ∞ for a similar reason.The gradient term, which arises in the spacial derivative ( creates another kind of (ABC) issue that leads to |( − → ∇ϕ(x))| < ∞.The Hamiltonian density, H(x), is now finite everywhere!It follows that the Hamiltonian, H = H(x) d s x, will be finite if it is confined to any finite spacial region, or if the field values taper off sufficiently, as is customary.
Although we have pointed out some difficulties that might arise in a canonical quantization, we follow a careful road to see how far we can get.
The usual continuum limit of the canonical quantum Hamiltonian leads to (8.3) but now there is some confusion.
The confusion arises in comparing [Q k , P l ] = ihδ kl 1 1 with [ φ(x), π(y)] = ihδ(x − y)1 1.As with the ultralocal case, it seems that we have a big difference in scale when p ≥ 2n/(n − 2) and the domain reduction appears when the interaction term is active compered with if it is not active.The same issue applied to the ultralocal case, which the p-value happened even earlier due to the absence of the gradient term, which, then is p > 2. From a path integration viewpoint, fields like |ϕ(x)| > > 1 are less likely to help their contribution.That can also apply to |ϕ(x)| < < 1 about the fields.Indeed, having both π(x) and ϕ(x) fields in 'the middle' tends to make them more prominent features in a path integration.
B. Affine Quantization of Relativistic Field Models

Affine classical variables for selected field theories
We first reexamine the features of a classical Hamiltonian once again, now with the affine variables κ(x) and ϕ(x) = 0, which becomes (8.4)In this case, we need 0 ).This requirement leads to the Hamiltonian density, H(x), which will entirely be 0 ≤ H(x) < ∞, for all x, signally that integrable-infinities may be excluded.That is true, and it must be obeyed, also in a path integration.This rule, regarding quantization, already distinguishes AQ from CQ.
If new variables can calm down the classical Hamiltonian, is it possible that they might also calm down the quantum Hamiltonian?Let's see how we can do just that!
Now is the time to introduce some scaling.Such a feature can adopt π κ → a −s/2 P κ and . Now we re-examine the kinetic factor for which κκ ( φ−2 κ )κ κ = a −s P 2 κ + 2a −2s h2 /a −s Q 2 κ .This regularization now leads to a regularized quantum Hamiltonian (8.6) provided that g a −s(p−2)/2 → g a −s by properly changing g.

Schrödinger's representation and equation
We are now in position to suggest the important affine quantization of standard classical models such as followed by the usual rules leading to As like other Characteristic Functions, such as were used for the ultralocal models, we note that any normalized wave function, such as Π x W (ϕ(x))/ϕ(x) 1/2 in the Hilbert space, a Fourier transformation leads to (8.9)

IX. HOW TO QUANTIZE EINSTEIN'S GRAVITY
If the reader thinks that canonical quantization is the best way to quantize Einstein's gravity, the reader should read this chapter carefully.

A. Gravity and AQ, Using Basic Operators
In order to quantize gravity it is important to render a valid quantization of the Arnowitt, Deser, and Misner classical Hamiltonian [ADM].We first choose our new classical variables which include what we also call the dilation field π a b (x) ≡ π ac (x) g bc (x) (summed on c) along with the metric field g ab (x).We don't need to impose any restriction on the metric field because physics already requires that ds(x) 2 = g ab (x) dx a dx b > 0 provided that Σ 3 a=1 (dx a ) 2 > 0. The metric can also be diagonalized by non-physical, orthogonal matrices, and then it includes only g 11 (x), g 22 (x), & g 33 (x), each of which must be strictly positive as required by physics.8 Next we present the ADM classical Hamiltonian in our chosen affine variables, which, introducing g(x) ≡ det[g ab (x)] > 0, leads to where Finally, we introduce the dilation gravity operator πa b (x) = [π ac (x) † ĝbc (x)+ĝ bc (x) πac (x)]/2 along with ĝab (x) > 0, and adopting Schrödinger's representation and equation, we are led to And now, as before, we close with Schrödinger's equation ih ∂ Ψ(g, t)/∂t = H ′ (π, g) Ψ(g, t) , (9.3) which offers the necessary ingredients for the foundation of a valid quantization of the classical Hamiltonian, which is an important part of the full story.
As before, it may be necessary to introduce some version of regularization for these equations, but these same equations point the way to proceed.In that effort, note that although πac (x) † = πac (x) it can be helpful to know that πac (x) † g bc (x) = πac (x) g bc (x).
A full quantization of gravity must deal with first and likely second order constraints, which are designed to reduce the overall Hilbert space to secure a final quantization.This project is not the proper place to finalize a quantization of gravity, but several of the author's articles have been designed to go further toward the final steps [Kla-4] -[Kla-9].

Additional aspects of quantum gravity
This section is relevant to follow sections which lead toward a path integration.These topics involve constraints required in the ADM approach.The present story, told just above, follows in the pattern of establishing a Schrödinger equation using his representation, has been the rule in discussing prior examples, e.g., the half-harmonic oscillator, quantum field theories over multiple powers of the interaction term, ultralocal examples of fields and gravity, and covariant field theories.Now, in the forthcoming section, we offer a careful treatment of constraints and their analysis, which is prominent in gravity and needs its own analysis.

B. Gravity and AQ, Using Path Integration
We first recall the Arnowitt, Deser, and Misner version of the classical Hamiltonian, seen in [ADM], as originally expressed in the standard classical variables, namely the momentum, π ab (x), the metric, g cd (x), the metric determent, g(x) = det[g ab (x)], and (3) R(x), which is the Ricci scalar for 3 spatial variables.Now the ADM classical Hamiltonian is essentially given by .

Introducing the favored classical variables
The ingredients in providing a path integration of gravity include proper coherent states, the Fubini-Study metric which turns out to be affine in nature, and affine-like Wiener measures are used for the quantizing of the classical Hamiltonian.While that effort is only part of the story, it is an important portion to ensure that the quantum Hamiltonian is a bonafide self-adjoint operator.
According to the ADM classical Hamiltonian, it can also be expressed in affine-like variables, as we did in the previous chapter, namely by introducing, in some papers of this author, the 'momentric' (a name that is the combination of momentum and metric) and, instead, this item is now called the 'dilation variable' becoming π a b (x) (≡ π ac (x) g bc (x)), along with the metric g ab (x).The essential physical requirement is that g ab (x) > 0, which means that ds(x) 2 = g ab (x) dx a dx b > 0, provided that Σ a (dx a ) 2 > 0. which is seen to imitate an affine metric, leading to a constant negative curvature, as well, and that will provide a genuine Wiener-like measure for a path integration.In no way could we transform this metric into a proper Cartesian form, as was done for the half-harmonic oscillator.That is because there is no physically proper Cartesian metric for the variables π ab (x) and g cd (x).

A special measure for the Lagrange multipliers
To ensure a proper treatment of the operator constraints, we choose a special measure of the Lagrange multipliers, R(N a , N), guided by the following procedures.
The first step is to unite the several classical constraints by using e i(y a Ha(x)+yH(x)) W (u, y a , y, g ab (x)) Π a dy a dy = e −iu[Ha(x)g ab (x)H b (x)+H(x) 2 ] ≡ e −iuHv(x) 2 (9.11) with a suitable measure W .
An elementary Fourier transformation9 given by M δ 2 −δ 2 e iǫτ uy dy/2 = sin(uǫτ δ 2 )/u, using a suitable M, which then ensures that the inverse Fourier transformation, where ǫ represents a tiny spatial interval and τ represents a tiny time interval, as part of a fully regularized integration in space and time, and u is another part of the Lagrange multipliers, N a (nǫ) and is the Meaning of Ultralocal C. Classical and Quantum Scalar Field Theories D. Canonical Ultralocal Scalar Fields E. An Affine Ultralocal Scalar Field VII.An Ultralocal Gravity Model A. An Affine Quantization of Ultralocal Gravity B. A Regularized Affine Ultralocal Quantum Gravity C. The Main Lesson from Ultralocal Gravity VIII.How to Quantize Relativistic Fields A. Reexamining the Classical Territory 1.A simple way to avoid integrable-infinities 2. The absence of infinities by using affine field variables B. Affine Quantization of Relativistic Field Models 1. Affine classical variables for selected field theories 2.An affine quantization of relativistic fields 3. Schrödinger's representation and equation IX.How to Quantize Einstein's Gravity A. Gravity and AQ, Using Basic Operators 1Field Problem Needs AQ or CQ, Otherwise, There Can Be Incorrect Results 35 I.A PREFACE generally, with b > 0. For very large b we can approximate a full-line harmonic oscillator and even see what happens if we choose b → ∞ to mimic the full-line story.

a
.1) in which 0 ≤ |κ(x)| < ∞ and 0 < |ϕ(x)| < ∞, to well represent π(x), fulfills the remarkable property that H(x) < ∞, where H = H(x) d s x, as nature requires!This fact shows that κ(x) and ϕ(x) = 0 should be the new variables!We now point to our new ABC-items to remind the reader of their relevance.If you think dimensions can distinguish two such fields, we can eliminate dimensional features by first introducing ϕ(y) = 0 and α(z) = 0. Now dimensionless factors lead to ϕ(x)/ϕ(y) = 0 = α(x)/α(z).
3) R(x) is the Ricci scalar for three spatial coordinates and which contains all of the derivatives of the metric field.Already this version of the classical Hamiltonian contains reasons that restrict g(x) to 0 < g(x) < ∞, 0 ≤ |π a b (x)| < ∞, and 0 ≤ | (3) R(x)| < ∞, which, like the previous field theory examples, and lead to no integral-infinities for the gravity story.