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

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. A Preface

## 2. An Introduction to the Variables

#### 2.1. A Survey of Principal Topics

#### 2.2. A Familiar Example of Classical Variables

#### 2.3. Selected Canonical Topics

## 3. Phase Space, Poisson Brackets, and Constant Curvature Spaces

#### 3.1. A Brief Review of Spin Quantization

#### 3.1.1. Spin Coherent States

#### 3.1.2. A Brief Review of Affine Quantization

#### 3.1.3. Affine Coherent States

#### 3.2. Summarizing Constant Curvatures and Coherent States

## 4. Learning to Quantize Selected Problems

#### 4.1. Choosing a Canonical Quantization

#### 4.1.1. First Canonical Example

#### 4.1.2. Second Canonical Example

**——————————————————————————**

**A Simple Truth**

**Consider A × B = C, as well as A = B/C**

**If $B=C=0$, what is A?**

**If $B=C=\infty $, what is A?**

**To ensure getting A one must require $0<\left|B\right|\phantom{\rule{0.277778em}{0ex}}\&\phantom{\rule{0.277778em}{0ex}}\left|C\right|<\infty $.**

**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 $q\left(t\right)$ is continuous, while $p\left(t\right)$ 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.**

**——————————————————————————**

#### 4.1.3. First Affine Example

#### 4.1.4. Second Affine Example

#### 4.1.5. A Canonical Version of the Half-Harmonic Oscillator

#### 4.1.6. A CQ Attempt to Solve the Half-Harmonic Oscillator

## 5. Using CQ and AQ to Examine ‘The Particle in a Box’

#### 5.1. An Example That Needs More Analysis

#### 5.1.1. Failure of the Canonical Quantization of the Particle in a Box

**For a moment, we take an about face.**

#### 5.1.2. Removing a Single Point

**A Vector Version:**The point we now wish to remove is $\overrightarrow{q}=0$; stated, we want to retain all the variables that obey ${\overrightarrow{q}}^{2}>0$ and all those of ${\overrightarrow{p}}^{2}\ge 0$. In addition, we introduce ${\overrightarrow{P}}^{2}={\Sigma}_{j=1}^{s}\phantom{\rule{0.166667em}{0ex}}{P}_{j}^{2}$ and ${\overrightarrow{Q}}^{2}={\Sigma}_{j=1}^{s}\phantom{\rule{0.166667em}{0ex}}{Q}_{j}^{2}>0$.

#### 5.2. Lessons from Canonical and Affine Quantization Procedures

## 6. Ultralocal Field Models

#### 6.1. Introduction

#### 6.2. What Is the Meaning of Ultralocal

#### 6.3. Classical and Quantum Scalar Field Theories

**——————————————————————————**

**Another Simple Truth**

**Consider $A\left(x\right)\times B\left(x\right)=C\left(x\right)$ as well as $A\left(x\right)=C\left(x\right)/B\left(x\right)$**

**If $B\left(x\right)=C\left(x\right)=0$ what is $A\left(x\right)$?**

**If $B\left(x\right)=C\left(x\right)=\infty $ what is $A\left(x\right)$?**

**To ensure getting $A\left(x\right)$ one must require $0<\left|B\right(x\left)\right|\phantom{\rule{0.277778em}{0ex}}\&\phantom{\rule{0.277778em}{0ex}}\left|C\right(x\left)\right|<\infty $.**

**This is good mathematics, but physics has an opinion as well.**

**Consider $k\left(x\right)=\pi \left(x\right)\phi \left(x\right)$, where $\phi \left(x\right)$ is a chosen physical field, $\pi \left(x\right)$ is its momentum field, and their product is $\kappa \left(x\right)$, which we will call the dilation field. Since $\pi \left(x\right)$ serves as the time derivative of $\phi \left(x\right)$, it can vanish along with $\kappa \left(x\right)$. However, requiring that both plus and minus sides of $\phi \left(x\right)\ne 0$ are acceptable, since the derivative term ensures it will still seem to come from a continuous function. Moreover, if $\phi \left(x\right)=0$ it could be confused with any other field, e.g., $\alpha \left(x\right)=0$ ( if you think dimensions can distinguish two such fields, we can eliminate dimensional features by first introducing $\phi \left(y\right)\ne 0$ and $\alpha \left(z\right)\ne 0$. Now, dimensionless factors lead to $\phi \left(x\right)/\phi \left(y\right)=0=\alpha \left(x\right)/\alpha \left(z\right)$. Thus, omitting points, or streams of them, where $\phi \left(x\right)=0$, do not violate any physics. In fact, it may seem logical to say that $\phi \left(x\right)=0$ never even belonged in physics. It fact, since numbers were used to count physical things, in very early times, zero $=0$, was**banned for 1500 years

**; see [9]).**

**It is good math for finite integrations if there are examples where the fields may reach infinity, e.g., ${\int}_{-1}^{1}{\phi}^{-2/3}\phantom{\rule{0.166667em}{0ex}}d\phi <\infty$. However, such cases are very likely to be bad physics because no item of nature reaches infinity. Accepting $\kappa \left(x\right)\phantom{\rule{0.277778em}{0ex}}(=\pi (x\left)\phantom{\rule{0.166667em}{0ex}}\phi \right(x\left)\right)$ and $\phi \left(x\right)\ne 0$, instead of $\pi \left(x\right)$ and $\phi \left(x\right)$, as the basic variables, will have profound consequences.**

**For example, the classical Hamiltonian expressed as**

**in which $0\le \left|\kappa \right(x\left)\right|<\infty $ and $0<\left|\phi \right(x\left)\right|<\infty $, to well represent $\pi \left(x\right)$, fulfills the remarkable property that $H\left(x\right)<\infty $, where $H=\int \phantom{\rule{-0.166667em}{0ex}}H\left(x\right)\phantom{\rule{0.277778em}{0ex}}{d}^{s}\phantom{\rule{-0.166667em}{0ex}}x$,**as nature requires!

**This fact shows that $\kappa \left(x\right)$ and $\phi \left(x\right)\ne 0$ should be the new variables!**

**We now point to our new ABC-items to remind the reader of their relevance.**

**——————————————————————————**

#### 6.4. Canonical Ultralocal Scalar Fields

#### 6.5. An Affine Ultralocal Scalar Field

## 7. An Ultralocal Gravity Model

**fixed**and

**not**made into any operator.

#### 7.1. An Affine Quantization of Ultralocal Gravity

#### 7.2. A Regularized Affine Ultralocal Quantum Gravity

#### 7.3. The Main Lesson from Ultralocal Gravity

## 8. How to Quantize Relativistic Fields

#### 8.1. Reexamining the Classical Territory

#### 8.1.1. A Simple Way to Avoid Integrable-Infinities

#### 8.1.2. The Absence of Infinities by Using Affine Field Variables

#### 8.2. Affine Quantization of Relativistic Field Models

#### 8.2.1. Affine Classical Variables for Selected Field Theories

#### 8.2.2. An Affine Quantization of Relativistic Fields

#### 8.2.3. Schrödinger’s Representation and Equation

## 9. How to Quantize Einstein’s Gravity

#### 9.1. Gravity and AQ, Using Basic Operators

#### Additional Aspects of Quantum Gravity

#### 9.2. Gravity and AQ, Using Path Integration

#### 9.2.1. Introducing the Favored Classical Variables

#### 9.2.2. The Gravity Coherent States

#### 9.2.3. A Special Measure for the Lagrange Multipliers

#### 9.3. The Affine Gravity Path Integration

## 10. Summary, and Outlook

#### Each Field Problem Needs AQ or CQ, Otherwise, There Can Be Incorrect Results

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Appendix to Section 5

## References

- Wikipedia: “Fubini–Study Metric”. Available online: https://en.wikipedia.org/wiki/Fubini%E2%80%93Study_metric (accessed on 14 September 2023).
- Wikipedia: “Pseudosphere”. Available online: https://en.wikipedia.org/wiki/Pseudosphere (accessed on 14 September 2023).
- Kkauder, J.R. The Benefits of Affine Quantization. J. High Energy Phys. Gravity Cosmol.
**2020**, 6, 175. [Google Scholar] [CrossRef] - Dirac, P.A.M. The Principles of Quantum Mechanics; Claredon Press: Oxford, UK, 1958; p. 114. [Google Scholar]
- Gouba, L. Affine Quantization on the Half Line. J. High Energy Phys. Gravit. Cosmol.
**2021**, 7, 352. [Google Scholar] [CrossRef] - Handy, C. Affine Quantization of the Harmonic Oscillator on the Semi-bounded Domain (−b,∞) for b:0→∞. arXiv
**2021**, arXiv:2111.10700. [Google Scholar] - Ashtekar, A. New Variables for Classical and Quantum Gravity. Phys. Rev. Lett.
**1986**, 57, 2244. [Google Scholar] [CrossRef] [PubMed] - Wikipedia: “The Particle in a Box”. Available online: https://en.wikipedia.org/wiki/Particle_in_a_box (accessed on 14 September 2023).
- Watch: “Why the Number 0 Was Banned for 1500 Years”. Available online: https://www.youtube.com/watch?v=ndmwB8F2kxA (accessed on 14 September 2023).
- Fantoni, R.; Klauder, J.R. Scaled Affine Quantization of Ultralocal ${\varphi}_{2}^{4}$ a comparative Path Integral Monte Carlo study with Scaled Canonical Quantization. Phys. Rev. D
**2022**, 106, 114508. [Google Scholar] [CrossRef] - Arnowitt, R.; Deser, S.; Misner, C. Gravitation: An Introduction to Current Research; Witten, L., Ed.; Wiley & Sons: New York, NY, USA, 1962; p. 227. [Google Scholar]
- Klauder, J.R. Quantization of Constrained Systems. Lect. Notes Phys.
**2001**, 572, 143–182. [Google Scholar]

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Klauder, J.R.; Fantoni, R.
The Magnificent Realm of Affine Quantization: Valid Results for Particles, Fields, and Gravity. *Axioms* **2023**, *12*, 911.
https://doi.org/10.3390/axioms12100911

**AMA Style**

Klauder JR, Fantoni R.
The Magnificent Realm of Affine Quantization: Valid Results for Particles, Fields, and Gravity. *Axioms*. 2023; 12(10):911.
https://doi.org/10.3390/axioms12100911

**Chicago/Turabian Style**

Klauder, John R., and Riccardo Fantoni.
2023. "The Magnificent Realm of Affine Quantization: Valid Results for Particles, Fields, and Gravity" *Axioms* 12, no. 10: 911.
https://doi.org/10.3390/axioms12100911