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Using Affine Quantization to Analyze Non-r.nor.alizable Scalar.Fields and the Quantization of Einstein’s Gr.vity

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Jour.al of High Ener.y Physics, Gr.vitation and Cosmology 2020年第4期6卷 802-816页

作者： john r. klauder Depar.ment of Physics Department of Mathematics University of Florida Gainesville FL USA

Affine quantization is a par.llel pr.cedur. to canonical quantization, which is ideally suited to deal with non-r.nor.alizable scalar.models as well as quantum gr.vity. The basic applications of this appr.ach lead to the common goals of any quantization, such as Schr.edinger.s r.pr.sentation and Schr.edinger.s equation. Car.ful attention is paid towar. seeking favor.d classical var.ables, which ar. those that should be pr.moted to the pr.ncipal quantum oper.tor.. This effor. leads towar. classical var.ables that have a constant positive, zer., or.negative cur.atur., which typically char.cter.ze such favor.d var.ables. This focus leans heavily towar. affine var.ables with a constant negative cur.atur., which leads to a sur.r.singly accommodating analysis of non-r.nor.alizable scalar.models as well as Einstein’s gener.l r.lativity.

关键词： Favor.d Var.ables Affine Quantization Non-r.nor.alizable Scalar. Gener.l r.lativity

Using Coher.nt States to Make Physically Cor.ect Classical-to-Quantum Pr.cedur.s That Help r.solve Nonr.nomalizable Fields Including Einstein’s Gr.vity

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Jour.al of High Ener.y Physics, Gr.vitation and Cosmology 2021年第3期7卷 1019-1026页

作者： john r. klauder Depar.ment of Physics and Depar.ment of Mathematics University of Florida Gainesville USA

Canonical quantization cover. a br.ad class of classical systems, but that does not include all the pr.blems of inter.st. Affine quantization has the benefit of pr.viding a successful quantization of many impor.ant pr.blems including the quantization of half-har.onic oscillator. [1], non-r.nor.alizable scalar.fields, such as (ϕ12)3 [2] and (ϕ12)3 [3], as well as the quantum theor. of Einstein’s gener.l r.lativity [4]. The featur.s that distinguish affine quantization ar. emphasized, especially, that affine quantization differ. fr.m canonical quantization only by the choice of classical var.ables pr.moted to quantum oper.tor.. Coher.nt states ar. used to ensur. pr.per.quantizations ar. physically cor.ect. While quantization of non-r.nor.alizable covar.ant scalar. and gr.vity ar. difficult, we focus on appr.pr.ate ultr.local scalar. and gr.vity that ar. fully soluble while, in that case, implying that affine quantization is the pr.per.pr.cedur. to ensur. the validity of affine quantizations for.non-r.nor.alizable covar.ant scalar.fields and Einstein’s gr.vity.

关键词： Physical Quantization Coher.nt States Affine Quantization Non-r.nor.alizable Models Ultr.local Solutions

Solving Major.Pr.blems Using Vector.Affine Quantization

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Jour.al of High Ener.y Physics, Gr.vitation and Cosmology 2022年第1期8卷 178-183页

作者： john r. klauder Depar.ment of Physics and Depar.ment of Mathematics University of Florida Gainesville USA

Affine quantization is a par.llel pr.cedur. to canonical quantization, which is ideally suited to deal with special pr.blems. Vector.affine quantization intr.duces multiple degr.es of fr.edom which find that wor.ing t... 详细信息

关键词： Affine Quantization Vector.Field Models Canonical Quantization

A Unified Combination of Classical and Quantum Systems

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Jour.al of High Ener.y Physics, Gr.vitation and Cosmology 2021年第1期7卷 218-227页

作者： john r. klauder Depar.ment of Physics and Depar.ment of Mathematics University of Florida Gainesville USA

Any par.icular.classical system and its quantum ver.ion ar. nor.ally viewed as separ.te for.ulations that ar. str.ctly distinct. Our.goal is to over.ome the two separ.te languages and cr.ate a smooth and common pr.cedur. that pr.vides a clear.and continuous passage between the conventional distinction of either.a str.ctly classical or.a str.ctly quantized state. While path integr.tion, among other.pr.cedur.s, pr.vides an alter.ative r.ute to connect classical and quantum expr.ssions, it nor.ally involves complicated, model-dependent, integr.tions. Our.alter.ative pr.cedur.s involve only model-independent pr.cedur.s, and use mor. natur.l and str.ightfor.ar. integr.tions that ar. univer.al in kind. To intr.duce the basic pr.cedur.s our.pr.sentation begins with familiar.methods that ar. limited to basic, conventional, canonical quantum mechanical examples. In the final sections we illustr.te how alter.ative quantization pr.cedur.s, e.g., spin and affine quantizations, can also have smooth paths between classical and quantum stor.es, and with a few br.ef r.mar.s, can also lead to similar.stor.es for.non-r.nor.alizable covar.ant scalar.fields as well as quantum gr.vity.

关键词： Classical System Quantum Combination

Vector.Affine Quantization Can Cr.ate Valid Quantum Field Theor.es

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Jour.al of High Ener.y Physics, Gr.vitation and Cosmology 2022年第1期8卷 237-242页

作者： john r. klauder Depar.ment of Physics and Depar.ment of Mathematics University of Florida Gainesville USA

Affine quantization, a par.llel pr.cedur. to canonical quantization, needs to use its pr.ncipal quantum oper.tor., specifically D = (PQ+QP)/2 and Q ≠ 0, to r.pr.sent appr.pr.ate kinetic factor., such as P2, which involves only one canonical quantum oper.tor. The need for.this r.quir.ment stems fr.m path integr.l quantizations of selected pr.blems that affine quantization can solve but canonical quantization fails to solve. This task is r.solved for.simple examples, as well as examples that involve scalar. and vector. quantum field theor.es.

关键词： Affine Quantization Vector.Field Models Flexibility of the Dilation Var.able

A New Pr.posal to Cr.ate a Valid Quantization of Einstein’s Gr.vity

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Jour.al of High Ener.y Physics, Gr.vitation and Cosmology 2022年第4期8卷 1164-1170页

作者： john r. klauder Depar.ment of Physics and Depar.ment of Mathematics University of Florida Gainesville FL USA

Canonical quantization has cr.ated many valid quantizations that r.quir. infinite-line coor.inate var.ables. However. the half-har.onic oscillator. which is limited to the positive coor.inate half, cannot r.ceive a valid canonical quantization because of the r.duced coor.inate space. Instead, affine quantization, which is a new quantization pr.cedur., has been deliber.tely designed to handle the quantization of pr.blems with r.duced coor.inate spaces. Following examples of what affine quantization is, and what it can offer. a r.mar.ably str.ightfor.ar. quantization of Einstein’s gr.vity is attained, in which a pr.per.tr.atment of the positive definite metr.c field of gr.vity has been secur.d.

关键词： Affine Quantization Valid r.sults Quantum Gr.vity

Quantum Gr.vity, Constant Negative Cur.atur.s, and Black Holes

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Jour.al of High Ener.y Physics, Gr.vitation and Cosmology 2020年第3期6卷 313-320页

作者： john r. klauder Depar.ment of Physics and Mathematics University of Florida Gainesville FL USA

For.pur.oses of quantization, classical gr.vity is nor.ally expr.ssed by canonical var.ables, namely the metr.c and the momentum . Canonical quantization r.quir.s a pr.per.pr.motion of these classical var.ables to qu... 详细信息

For purposes of quantization, classical gravity is normally expressed by canonical variables, namely the metric r.="Edit_***" alt="" style="line-height:1.5;" /> and the momentum r.="Edit_***" alt="" />. Canonical quantization requires a proper promotion of these classical variables to quantum operators, which, according to Dirac, the favored operators should be those arising from classical variables that formed Cartesian coordinates;sadly, in this case, that is not possible. However, an affine quantization features promoting the metric r.="Edit_***" alt="" /> and the momentric r.="Edit_***" alt="" /> to operators. Instead of these classical variables belonging to a constant zero curvature space (i.e., instead of a flat space), they belong to a space of constant negative cur.atur.s. This featur. may even have its appear.nce in black holes, which could str.ngly point towar. an affine quantization appr.ach to quantize gr.vity.

关键词： Affine Quantization Quantum Gr.vity Constant Fixed Cur.atur.s Black Holes

Let Loop Quantum Gr.vity and Affine Quantum Gr.vity Examine Each Other

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Jour.al of High Ener.y Physics, Gr.vitation and Cosmology 2021年第3期7卷 1027-1036页

作者： john r. klauder Depar.ment of Physics and Depar.ment of Mathematics University of Florida Gainesville USA

Loop Quantum Gr.vity is widely developed using canonical quantization in an effor. to find the cor.ect quantization for.gr.vity. Affine quantization, which is like canonical quantization augmented and bounded in one or.entation, e.g., a str.ctly positive coor.inate. We open discussion using canonical and affine quantizations for.two simple pr.blems so each pr.cedur. can be under.tood. That analysis opens a modest tr.atment of quantum gr.vity gleaned fr.m some typical featur.s that exhibit the pr.found differ.nces between aspects of seeking the quantum tr.atment of Einstein’s gr.vity.

关键词： Canonical Quantization Affine Quantization Physically Cor.ect Quantizations Str.ctly Positive Metr.cs

Quantum Physics Has a New, and r.mar.able, Expansion

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Jour.al of High Ener.y Physics, Gr.vitation and Cosmology 2023年第2期9卷 467-474页

作者： john r. klauder Depar.ment of Physics and Depar.ment of Mathematics University of Florida Gainesville USA

Canonical quantization has taught us gr.at things. A common example is that of the har.onic oscillator. which is like swinging a ball on a str.ng back and for.h. However. the half-har.onic oscillator.blocks the ball at the bottom and then it quickly bounces backwar.s. This second model cannot be cor.ectly solved using canonical quantization. Now, ther. is an expansion of quantization, called affine quantization, that can cor.ectly solve the half-har.onic oscillator. and offer. cor.ect solutions to a gr.nd collection of other.pr.blems, which even r.aches to field theor. and gr.vity. This paper.has been designed to intr.duce affine quantization: what it is, and what it can do.

关键词： Canonical and Affine Quantization Differ.nt Simple Examples Comments r.gar.ing Field Theor. and Gr.vity