# The Magnificent Realm of Affine Quantization: Valid Results for Particles, Fields, and Gravity

**Round 1**

*Reviewer 1 Report
*

It is an interesting review. The manuscript is well written, and I think I can recommend it to be published in its current form.

*Author Response*

The Referee recommends publication of the paper in its current form.

*Reviewer 2 Report
*

Dear Editor,

I wish to recommend that this paper will be accepted for publication after some issues will be addressed by the authors.

This paper reviews a relatively novel set of tools that can help to build quantum theories constrained by boundaries, wherein the canonical formalism may fail. The authors then show how these new tools could be used to build a quantum theory of gravity.

I find the paper interesting, but I have some comments about the claims of the paper.

A most fundamental guiding principle in building a quantum theory is the notion of symmetry. For example, quantum electrodynamics may be defined as the quantum field theory defined by its symmetry through local U(1) times the Poincare group, rather than the set of rules through which the final Hamiltonian is reached. The preservation of this symmetry is also a fundamental constraint in the formulation of the renormalization group equations that allow to build a meaningful quantum field theory. Lorentz covariant quantum field theories are defined by their Hamiltonian together with a set of renormalization group equations that state how the parameters of the theory (or running coupling constants) depend on the energies involved in an interaction.

In this paper the authors announce to have found a path towards a quantum theory of gravity, that is, a formal Hamiltonian together with a (set of) renormalization group equation for the gravitational constant that are consistent with the required group of local (gauge) Lorentz invariance. However, this aim is certainly not achieved in the paper. In fact, the authors do not even address the issue of how to handle the gauge symmetries of the theory.

I am not trying to play down the achievements of the paper, which may be many, and I fully understand that not every issue must be addressed in a single paper. Nonetheless, I would find more appropriate if these unresolved difficulties were stated already at the beginning of the paper. In its current form, a reader that starts reading the paper expects from the authors' statements all the way almost to the end of the paper to find a consistently formulated quantum theory of gravity, but this is quite far from being the case.

*Author Response*

Thank you for your report. In any position to get from a to b you might find a different

way along the path but both paths end in the proper position. Now I use that argument

to say you might have your way to quantize gravity items you want to use but if there is

another path using standard procedures about quantization in general such as if you

can collect the proper operators and put it in the proper Hamiltonian you can go forward.

Maybe we have taken slightly different paths to achieve our goal. In particular we have

no idea what your use of symmetry has in the world of quantization. Perhaps is q and p can

be exchanged but in affine quantization the things are not of that kind. Moreover

it is our choice to seek the rules and the road to quantizing fields and gravity. We have

deliberately constructed a proper and satisfactory road to quantized gravity and perhaps

alternative paths which include other parts of the story but you may have a belief that that

is true and perhaps your article proposal would have been slightly different. On the other

hand we are convinced that the road we have taken is perfectly adequate to reach quantum gravity.

Thank you for your considerations and suggestions however.

*Reviewer 3 Report
*

The paper addresses the fundamental problem of quantization and reviews the so called affine quantization, which is a generalization of the canonical quantization. It is claimed that the affine quantization works in some cases, where the canonical quantisation leads to divergences, discontinuities, etc.

Though I cannot say that I have understood everything well, I believe that these thoughts and proposals can be published. May be they could be of interest for people who understand the ideas of this manuscript better. To improve readability, I recommend the authors to take into account the following comments (mainly about clarity of explanations):

1. P. 5, the first paragraph ("A brief example of the affine procedures..." I cannot understand what is the problem in this example. The authors write that H can be finite even if pi(x) is infinite at some point, but, physically, pi(x) should be always finite. So, what's the problem? If pi(x) is a bounded function, then H is well-defined.

2. Which object do we define in Eq. (3.5)? What is d sigma(theta, phi)?

3. A general question about Section III.A: What is meant by "spin quantization" if we already start with the quantum mechanical description (the last paragraph on p.7) and Eq. (3.4)? So, the last but one paragraph in this section "These classical variables can not lead to a physically correct canonical quantization.." is completely unclear. What are classical variables to be quantized here?

4. P. 8, line 4 from bottom. Why do we choose the quantization P^dag Q+QP, not QP^dag+PQ? The same question is about P.13, the third paragraph, the definition of D, where Q is replaced by Q+b.

5. P. 13, the last sentence of Section 3: "Evidently, an affine quantization fails to quantize a full harmonic oscillator". Why? In the previous paragraph, it has been shown that the canonical quantization is restored from the affine quantization (4.3) if b goes to infinity. So, what's the problem?

4. P. 13, Section 4: "Second affine example". What is the difference between this section and section 3? In the 2nd line of P.14, we see the same Hamiltonian H'(D,Q) as in Section 3 if we set b=0.

5. Eq. (4.5). If we consider the Schrodinger equation on the half-line x>0, then the boundary conditions at x=0 are required. What are the boundary conditions?

6. In the second half of p.16, the authors claim that the canonical quantization of the particle in a box has problems. Again, I cannot understand what the problems are. Why the authors write that the derivative of the wave function is discontinuous? (pi/2b)sin(pi x/2b) is a continuous function, isn't it? Why different derivatives in the ends of the box (+b and -b) is a problem?

From the other side, according to the standard treatment, the wave function is zero outside the box and then the derivatives are indeed discontinuous. But the textbooks on quantum mechanics do not see a problem here.

Typos:

1. P. 4, line 5 from bottom: Comma in the end of the paragraph.

2. P. 5, line 5: it seems that k should be replaced by kappa.

3. P. 16, the formulas in the 4th line of Section 1: cos'(x)=-sin(x), not -(pi/2b)sin(pi x/2b).

4. P. 16, the second formula in the 4th line of Section 1: it is better to replace -\pm by \mp.

*Author Response*

1. Yes in the classical regime we need both Hamiltonian and its density

having only finite behavior but when you quantize such as in path integration

the path inside the formulation will reach infinity for example take the

example where you integrate from minus 1 to plus 1 and have dx divided by

absolute x to the 1/2 that means the answer is finite of the integral but at x=0

the tiny little field is zero meaning it has reached infinity. That is why is there

because of the potential possibility of quantization of your situation.

2. d sigma(theta, phi) is the Fubini-Study metric. See reference Wik-1.

3. Well yes. Traditionally q runs from minus to plus infinity like a straight line

however there is the situation where the position runs around in a circle and stops

at the edge end of that circle so that's why theta can be interested

in these kind of things. Indeed the quantization has discrete momentum because it moves

the figure around and that's how it tells correct momenta. Momenta would be, I moved the

circle here. Well, the circle moves into a circle but that would be zero. However, if I make

one wave that goes around the circle outside and come back to the beginning point. It comes

up and back 180 degrees later and then comes back to 360. So that has two points on the

circle. So that tells you that the momentum can have two values. And then it can do it for

any number of points of contact. That's the momentum. The position is theta from 0 to 360.

That's not canonical quantization so that's the SU(1) kind of quantization. Analogously

you proceed for a sphere.

4. The basic rule is if I have A and B both real and I consider (A+iB)dagger=A-iB that is

the rule. Now if I have a non real story perhaps it could be different. Take

(AdaggerB)dagger=BdaggerA in other words it flips its arguments and flips the daggers.

That is the rule. So (PdaggerQ+QP)dagger=PdaggerQ+QP. It's the way these kind of variables

has to shuffle and repeat.

5. The range of canonical quantization the q there classical or quantum runs from minus

infinity to plus infinity. For the affine quantization the q runs between -b and +b when b

is a finite number. The fact that works for affine quantization require b being finite. But

if you choose you can push b to infinity. That makes it the kind of problem q of infinity

that is canonical quantization. In other words if you can complete the entire coordinates of

q from plus to minus infinity you have entered canonical quantization. However if there is ANY

space removed from that then canonical quantization cannot solve the problem and affine can

certainly solve the problem. Even running from 0 to infinity positively like q>0 one of the

tails can go to infinity or many many more if you take these all on different islands but the

only one that is canonical is the one completely q from minus to plus infinity while any space

removed must have affine quantization.

6. It is true that sin is a continuous function but the wavefunction is not simply a cos but

is a cos inside the well and zero outside. And canonical quantization REQUIRES considering the

whole line from minus to plus infinity.

We corrected all the Typos pointed out by the Referee.