The Secret to Fixing Incorrect Canonical Quantizations

John Klauder; Riccardo Fantoni

doi:10.20944/preprints202407.1397.v1

Preprint

Brief Report

The Secret to Fixing Incorrect Canonical Quantizations

This version is not peer-reviewed.

This version is not peer-reviewed.

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Submitted:

16 July 2024

Posted:

17 July 2024

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Abstract

It is well-known that the canonical quantization of $\varphi^4_4$ fails. Here is how to fix it along with many other problems.

Keywords:

Canonical quantization

;

affine quantization

;

field theory

Subject:

Physical Sciences - Theoretical Physics

I. Introduction

There have been many models with the same `illness’ as that of

φ_{4}^{4}

[1,2] The secret to a valid canonical quantization (CQ) is remarkably simple. All you need is the addition of a single, fixed potential, which is not seen in the classical Hamiltonian, but it puts things in proper position elsewhere. This single, additional, potential is just

2 ℏ^{2} / φ {(x)}^{2}

. Give it a try-out on the model

φ_{4}^{4}

and see for yourself! You can also give it a new try with

φ_{4}^{8}

and be surprised! That additional `potential’, put just after $\hat{π} {(x)}^{2}$ , is all you need to use.

II. Amazing Results by Just Removing $φ (x) = 0$

The special ℏ-term has arisen from the fact that

φ (x) = 0

has been removed, which then means that the momentum is no longer self-adjoint. The next step leads to introducing

\hat{κ} (x) = [\hat{π} {(x)}^{†} φ (x) + φ (x) π (x)] / 2

, and with scaling can lead to become

π {(x)}^{2} = κ {(x)}^{2} / φ {(x)}^{2} \to \hat{κ} (x) φ {(x)}^{- 2} \hat{κ} (x) \Rightarrow \hat{π} {(x)}^{2} + b ℏ^{2} / φ {(x)}^{2},

(2.1)

and the factor

b = 2

has been chosen to fit our particular problem. Observe that choosing $φ (x) \neq 0$ has permitted introducing the `polynomial’-like term $2 ℏ^{2} / φ {(x)}^{2}$ .

A. Understanding How Scaling Works

Initially

\hat{κ} (x) (φ {(x)}^{- 2}) \hat{κ} (x) = {\hat{π}}^{2} + ℏ^{2} δ {(0)}^{2 s} / φ {(x)}^{2}

, where

δ (0)

is Daric’s special function, where

δ (x) = 0

for all

0 < | x |

, while

\int δ (x) d x = 1

which leads to

δ (0) = \infty

. Now our ∞ is reduced to

b ℏ^{2} W^{2} < \infty

, and W will be sent to ∞ latter on.

This now becomes

\hat{π} {(x)}^{2} + b ℏ^{2} W^{2} / φ {(x)}^{2}

. Next

(\hat{π} {(x)}^{2} & φ {(x)}^{2}) \to W (\hat{π} {(x)}^{2} & φ {(x)}^{2})

. This leads to

W \hat{π} {(x)}^{2} + b ℏ^{2} W^{2} / W φ {(x)}^{2}

, and now a full multiplication by

W^{- 1}

leads to the final result which is

\hat{π} {(x)}^{2} + b ℏ^{2} / φ {(x)}^{2}

. Now W can be sent to infinity.

III. Selected Topics of Affine Quantization

A major feature of CQ is that

- \infty < q & φ (x) < \infty

. It is that fact which affine quantization (AQ) overcomes by introducing a vast variety of parts of incompete space, such as these retained space,

q > 0, | q | > 0, q^{2} < b^{2}, q^{2} > b^{2},

etc. For quantum field theory, the most important change is that

φ (x) \neq 0

and now that equation has been fully removed.

Observe, that CQ requires

0 \leq | φ (x) | < \infty

, while AQ seeks to find missing equations which shows that a specific field value, namely, $φ (x) = 0$ is removed,- or explained differently, now $0 < | φ (x) | < \infty$ .

A. An Introduction to AQ

Only AQ can correctly solve all examples that have missing space regions, and it can do it correctly only with remaining space examples.

There is something else that CQ can surely fail on, namely the example of the “Particle in a Box”, which is an example with missing space, and has been traditionally `solved’ using CQ. However, that very model can, and has, been correctly solved now by using AQ [3].

If you wish to read up on AQ, here are two examples where AQ has been well explained; see [4,5]

References

J. Fröhlich, “On the Triviality of $φ_{d}^{4}$ . Theories and the Approach to the Critical Point in d ≥ 4 Dimensions”, Nuclear Physics B 200, 281-296 (1982).
R. Fantoni and J. Klauder, “Affine Quantization of $φ_{4}^{4}$ Succeeds, while Canonical Quantization Fails”, Phys. Rev. D 103, 076013 (2021); arXiv:2012.09991.
J. Klauder, “Particle in a Box Warrants an Examination”, Journal of High Energy Physics, Gravitation and Cosmology, Vol.8 No.3, 2022; arXiv:2204.07577.
J. Klauder, “The Benefits of Affine Quantization”, Journal of High Energy Physics, Gravitation and Cosmology, 6, 175 (2020); arXiv:1912.08047.
J. Klauder and R. Fantoni, “The Magnificent Realm of Affine Quantization: valid results for particles, fields, and gravity”, Axioms 12, 911 (2023); arXiv:2303.13792.

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© 2024 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 (http://creativecommons.org/licenses/by/4.0/).

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The Secret to Fixing Incorrect Canonical Quantizations

Abstract

Keywords:

Subject:

I. Introduction

II. Amazing Results by Just Removing φ ( x ) = 0

A. Understanding How Scaling Works

III. Selected Topics of Affine Quantization

A. An Introduction to AQ

References

MDPI Initiatives

Important Links

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II. Amazing Results by Just Removing $φ (x) = 0$