Affine quantization is a parallel procedure to canonical quantization, which is ideally suited to deal with non-renormalizable scalar models as well as quantum gravity. The basic applications of this approach lead to ...Affine quantization is a parallel procedure to canonical quantization, which is ideally suited to deal with non-renormalizable scalar models as well as quantum gravity. The basic applications of this approach lead to the common goals of any quantization, such as Schroedinger’s representation and Schroedinger’s equation. Careful attention is paid toward seeking favored classical variables, which are those that should be promoted to the principal quantum operators. This effort leads toward classical variables that have a constant positive, zero, or negative curvature, which typically characterize such favored variables. This focus leans heavily toward affine variables with a constant negative curvature, which leads to a surprisingly accommodating analysis of non-renormalizable scalar models as well as Einstein’s general relativity.展开更多
The similarity between classical and quantum physics is large enough to make an investigation of quantization methods a worthwhile endeavour. As history has shown, Dirac's canonical quantization method works reaso...The similarity between classical and quantum physics is large enough to make an investigation of quantization methods a worthwhile endeavour. As history has shown, Dirac's canonical quantization method works reasonably well in the case of conventional quantum mechanics over R<sup>n</sup> but it may fail in non-trivial phase spaces and also suffer from ordering problems. Affine quantization is an alternative method, similar to the canonical quantization, that may offer a positive result in situations for which canonical quantization fails. In this paper we revisit the affine quantization method on the half-line. We formulate and solve some simple models, the free particle and the harmonic oscillator.展开更多
Canonical quantization (CQ) is built around [<i>Q</i>, <i>P</i>] = <i>iħ</i>1l , while affine quantization (AQ) is built around [<i>Q</i>,<i>D</i>...Canonical quantization (CQ) is built around [<i>Q</i>, <i>P</i>] = <i>iħ</i>1l , while affine quantization (AQ) is built around [<i>Q</i>,<i>D</i>] = <i>iħQ</i>, where <i>D</i> ≡ (<i>PQ</i> +<i>QP</i>) / 2 . The basic CQ operators must fit -∞ < <i>P</i>, <i>Q</i> < ∞ , while the basic AQ operators can fit -∞ < <i>P</i> < ∞ and 0 < <i>Q</i> < ∞ , -∞ < <i>Q</i> < 0 , or even -∞ < <i>Q</i> ≠ 0 < ∞ . AQ can also be the key to quantum gravity, as our simple outline demonstrates.展开更多
Affine quantization, a parallel procedure to canonical quantization, needs to use its principal quantum operators, specifically <i>D</i> = (<i>PQ</i>+<i>QP</i>)/2 and <i>Q<...Affine quantization, a parallel procedure to canonical quantization, needs to use its principal quantum operators, specifically <i>D</i> = (<i>PQ</i>+<i>QP</i>)/2 and <i>Q</i> ≠ 0, to represent appropriate kinetic factors, such as <i>P</i><sup>2</sup>, which involves only one canonical quantum operator. The need for this requirement stems from path integral quantizations of selected problems that affine quantization can solve but canonical quantization fails to solve. This task is resolved for simple examples, as well as examples that involve scalar, and vector, quantum field theories.展开更多
A half-harmonic oscillator, which gets its name because the position coordinate is strictly positive, has been quantized and determined that it was a physically correct quantization. This positive result was found usi...A half-harmonic oscillator, which gets its name because the position coordinate is strictly positive, has been quantized and determined that it was a physically correct quantization. This positive result was found using affine quantization (AQ). The main purpose of this paper is to compare results of this new quantization procedure with those of canonical quantization (CQ). Using Ashtekar-like classical variables and CQ, we quantize the same toy model. While these two quantizations lead to different results, they both would reduce to the same classical Hamiltonian if ħ→ 0. Since these two quantizations have differing results, only one of the quantizations can be physically correct. Two brief sections also illustrate how AQ can correctly help quantum gravity and the quantization of most field theory problems.展开更多
Canonical quantization has created many valid quantizations that require infinite-line coordinate variables. However, the half-harmonic oscillator, which is limited to the positive coordinate half, cannot receive a va...Canonical quantization has created many valid quantizations that require infinite-line coordinate variables. However, the half-harmonic oscillator, which is limited to the positive coordinate half, cannot receive a valid canonical quantization because of the reduced coordinate space. Instead, affine quantization, which is a new quantization procedure, has been deliberately designed to handle the quantization of problems with reduced coordinate spaces. Following examples of what affine quantization is, and what it can offer, a remarkably straightforward quantization of Einstein’s gravity is attained, in which a proper treatment of the positive definite metric field of gravity has been secured.展开更多
Canonical quantization is a wonderful procedure for selected problems, but there are many problems for which it fails. Affine quantization is a different procedure that has shown that it can solve many problems that c...Canonical quantization is a wonderful procedure for selected problems, but there are many problems for which it fails. Affine quantization is a different procedure that has shown that it can solve many problems that canonical quantization cannot. Here, words like succeed and fail to refer to whether the quantization results are correct or incorrect. This paper offers two simple examples that serve to introduce affine quantization, and compare studies of two different quantization procedures. Brief comments about field theory and gravity problems undergoing quantization by affine procedures complete the paper.展开更多
Loop Quantum Gravity is widely developed using canonical quantization in an effort to find the correct quantization for gravity. Affine quantization, which is like canonical quantization augmented and bounded in one o...Loop Quantum Gravity is widely developed using canonical quantization in an effort to find the correct quantization for gravity. Affine quantization, which is like canonical quantization augmented and bounded in one orientation, e.g., a strictly positive coordinate. We open discussion using canonical and affine quantizations for two simple problems so each procedure can be understood. That analysis opens a modest treatment of quantum gravity gleaned from some typical features that exhibit the profound differences between aspects of seeking the quantum treatment of Einstein’s gravity.展开更多
Classical phase-space variables are normally chosen to promote to quantum operators in order to quantize a given classical system. While classical variables can exploit coordinate transformations to address the same p...Classical phase-space variables are normally chosen to promote to quantum operators in order to quantize a given classical system. While classical variables can exploit coordinate transformations to address the same problem, only one set of quantum operators to address the same problem can give the correct analysis. Such a choice leads to the need to find the favored classical variables in order to achieve a valid quantization. This article addresses the task of how such favored variables are found that can be used to properly solve a given quantum system. Examples, such as non-renormalizable scalar fields and gravity, have profited by initially changing which classical variables to promote to quantum operators.展开更多
<span style="line-height:1.5;">For purposes of quantization, classical gravity is normally expressed by canonical variables, namely the metric </span><img src="Edit_7bad0ce2-ecaa-4318-b3c...<span style="line-height:1.5;">For purposes of quantization, classical gravity is normally expressed by canonical variables, namely the metric </span><img src="Edit_7bad0ce2-ecaa-4318-b3c9-5bbcfa7c087e.png" alt="" style="line-height:1.5;" /><span style="line-height:1.5;"></span><span "="" style="line-height:1.5;"><span> and the momentum </span><img src="Edit_c86b710a-9b65-4220-a4e2-cff8eeab9642.png" alt="" /></span><span style="line-height:1.5;"></span><span style="line-height:1.5;">. 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 feature</span><span style="line-height:1.5;">s</span><span "="" style="line-height:1.5;"><span> promoting the metric </span><img src="Edit_d0035f64-c366-4510-9cc7-d1053f755369.png" alt="" /></span><span "="" style="line-height:1.5;"><span> and the momentric </span><img src="Edit_60c18bb8-525b-4896-ae8f-2cd6456eb6f7.png" alt="" /></span><span "="" style="line-height:1.5;"><span> to operators. Instead of these classical variables belonging to a constant zero curvature space (</span><i><span>i.e.</span></i><span>, instead of a flat space), they belong to a space of constant negative curvatures. This feature may even have its appearance in black holes, which could strongly point toward an affine quantization approach to quantize gravity.展开更多
Canonical quantization covers a broad class of classical systems, but that does not include all the problems of interest. Affine quantization has the benefit of providing a successful quantization of many important pr...Canonical quantization covers a broad class of classical systems, but that does not include all the problems of interest. Affine quantization has the benefit of providing a successful quantization of many important problems including the quantization of half-harmonic oscillators [1], non-renormalizable scalar fields, such as (<i>ϕ</i><sup>12</sup>)<sub>3</sub> [2] and (<i>ϕ</i><sup>12</sup>)<sub>3</sub> [3], as well as the quantum theory of Einstein’s general relativity [4]. The features that distinguish affine quantization are emphasized, especially, that affine quantization differs from canonical quantization only by the choice of classical variables promoted to quantum operators. Coherent states are used to ensure proper quantizations are physically correct. While quantization of non-renormalizable covariant scalars and gravity are difficult, we focus on appropriate ultralocal scalars and gravity that are fully soluble while, in that case, implying that affine quantization is the proper procedure to ensure the validity of affine quantizations for non-renormalizable covariant scalar fields and Einstein’s gravity.展开更多
The particle in a box is a simple model that has a classical Hamiltonian H = p<sup>2</sup> (using 2m = 1), with a limited coordinate space, -b q b, where 0 b < ∞. Using canonical quantization, this exa...The particle in a box is a simple model that has a classical Hamiltonian H = p<sup>2</sup> (using 2m = 1), with a limited coordinate space, -b q b, where 0 b < ∞. Using canonical quantization, this example has been fully studied thanks to its simplicity, and it is a common example for beginners to understand. Despite its repeated analysis, there is a feature that puts the past results into question. In addition to pointing out the quantization issue, the procedures of affine quantization can lead to a proper quantization that necessarily points toward more complicated eigenfunctions and eigenvalues, which deserve to be solved.展开更多
Present day Quantum Field Theory (QFT) is founded on canonical quantization, which has served quite well but also has led to several issues. The free field describing a free particle (with no interaction term) can sud...Present day Quantum Field Theory (QFT) is founded on canonical quantization, which has served quite well but also has led to several issues. The free field describing a free particle (with no interaction term) can suddenly become nonrenormalizable the instant a suitable interaction term appears. For example, using canonical quantization <img src="Edit_9f6ab3f7-9277-4093-adcc-cdccf32c2c7c.png" width="15" height="15" alt="" /><sup?style="margin-left:-7px;">, has been deemed a “free” theory with no difference from a truly free field [1] [2]. Using the same model, affine quantization has led to a truly interacting theory [3]. This fact alone asserts that canonical and affine tools of quantization deserve to be open to their procedures together as a significant enlargement of QFT.</sup?style="margin-left:-7px;">展开更多
The understanding of what a black hole is like is not easy and may not yet be well understood. The introduction of canonical quantization into the issue has not been significant to our understanding. However, introduc...The understanding of what a black hole is like is not easy and may not yet be well understood. The introduction of canonical quantization into the issue has not been significant to our understanding. However, introducing affine quantization, a new procedure, offers a very unusual expression that seems to be plausible, and quite profound as well.展开更多
Canonical quantization has taught us great things. A common example is that of the harmonic oscillator, which is like swinging a ball on a string back and forth. However, the half-harmonic oscillator blocks the ball a...Canonical quantization has taught us great things. A common example is that of the harmonic oscillator, which is like swinging a ball on a string back and forth. However, the half-harmonic oscillator blocks the ball at the bottom and then it quickly bounces backwards. This second model cannot be correctly solved using canonical quantization. Now, there is an expansion of quantization, called affine quantization, that can correctly solve the half-harmonic oscillator, and offers correct solutions to a grand collection of other problems, which even reaches to field theory and gravity. This paper has been designed to introduce affine quantization: what it is, and what it can do.展开更多
Today's quantum field theory (QFT) relies heavenly on canonical quantization (CQ), which fails for φ<sub>4</sub>4</sup> leading only to a “free” result. Affine quantization (AQ), an alternativ...Today's quantum field theory (QFT) relies heavenly on canonical quantization (CQ), which fails for φ<sub>4</sub>4</sup> leading only to a “free” result. Affine quantization (AQ), an alternative quantization procedure, leads to a “non-free” result for the same model. Perhaps adding AQ to CQ can improve the quantization of a wide class of problems in QFT.展开更多
文摘Affine quantization is a parallel procedure to canonical quantization, which is ideally suited to deal with non-renormalizable scalar models as well as quantum gravity. The basic applications of this approach lead to the common goals of any quantization, such as Schroedinger’s representation and Schroedinger’s equation. Careful attention is paid toward seeking favored classical variables, which are those that should be promoted to the principal quantum operators. This effort leads toward classical variables that have a constant positive, zero, or negative curvature, which typically characterize such favored variables. This focus leans heavily toward affine variables with a constant negative curvature, which leads to a surprisingly accommodating analysis of non-renormalizable scalar models as well as Einstein’s general relativity.
文摘The similarity between classical and quantum physics is large enough to make an investigation of quantization methods a worthwhile endeavour. As history has shown, Dirac's canonical quantization method works reasonably well in the case of conventional quantum mechanics over R<sup>n</sup> but it may fail in non-trivial phase spaces and also suffer from ordering problems. Affine quantization is an alternative method, similar to the canonical quantization, that may offer a positive result in situations for which canonical quantization fails. In this paper we revisit the affine quantization method on the half-line. We formulate and solve some simple models, the free particle and the harmonic oscillator.
文摘Canonical quantization (CQ) is built around [<i>Q</i>, <i>P</i>] = <i>iħ</i>1l , while affine quantization (AQ) is built around [<i>Q</i>,<i>D</i>] = <i>iħQ</i>, where <i>D</i> ≡ (<i>PQ</i> +<i>QP</i>) / 2 . The basic CQ operators must fit -∞ < <i>P</i>, <i>Q</i> < ∞ , while the basic AQ operators can fit -∞ < <i>P</i> < ∞ and 0 < <i>Q</i> < ∞ , -∞ < <i>Q</i> < 0 , or even -∞ < <i>Q</i> ≠ 0 < ∞ . AQ can also be the key to quantum gravity, as our simple outline demonstrates.
文摘Affine quantization, a parallel procedure to canonical quantization, needs to use its principal quantum operators, specifically <i>D</i> = (<i>PQ</i>+<i>QP</i>)/2 and <i>Q</i> ≠ 0, to represent appropriate kinetic factors, such as <i>P</i><sup>2</sup>, which involves only one canonical quantum operator. The need for this requirement stems from path integral quantizations of selected problems that affine quantization can solve but canonical quantization fails to solve. This task is resolved for simple examples, as well as examples that involve scalar, and vector, quantum field theories.
文摘A half-harmonic oscillator, which gets its name because the position coordinate is strictly positive, has been quantized and determined that it was a physically correct quantization. This positive result was found using affine quantization (AQ). The main purpose of this paper is to compare results of this new quantization procedure with those of canonical quantization (CQ). Using Ashtekar-like classical variables and CQ, we quantize the same toy model. While these two quantizations lead to different results, they both would reduce to the same classical Hamiltonian if ħ→ 0. Since these two quantizations have differing results, only one of the quantizations can be physically correct. Two brief sections also illustrate how AQ can correctly help quantum gravity and the quantization of most field theory problems.
文摘Canonical quantization has created many valid quantizations that require infinite-line coordinate variables. However, the half-harmonic oscillator, which is limited to the positive coordinate half, cannot receive a valid canonical quantization because of the reduced coordinate space. Instead, affine quantization, which is a new quantization procedure, has been deliberately designed to handle the quantization of problems with reduced coordinate spaces. Following examples of what affine quantization is, and what it can offer, a remarkably straightforward quantization of Einstein’s gravity is attained, in which a proper treatment of the positive definite metric field of gravity has been secured.
文摘Canonical quantization is a wonderful procedure for selected problems, but there are many problems for which it fails. Affine quantization is a different procedure that has shown that it can solve many problems that canonical quantization cannot. Here, words like succeed and fail to refer to whether the quantization results are correct or incorrect. This paper offers two simple examples that serve to introduce affine quantization, and compare studies of two different quantization procedures. Brief comments about field theory and gravity problems undergoing quantization by affine procedures complete the paper.
文摘Loop Quantum Gravity is widely developed using canonical quantization in an effort to find the correct quantization for gravity. Affine quantization, which is like canonical quantization augmented and bounded in one orientation, e.g., a strictly positive coordinate. We open discussion using canonical and affine quantizations for two simple problems so each procedure can be understood. That analysis opens a modest treatment of quantum gravity gleaned from some typical features that exhibit the profound differences between aspects of seeking the quantum treatment of Einstein’s gravity.
文摘Classical phase-space variables are normally chosen to promote to quantum operators in order to quantize a given classical system. While classical variables can exploit coordinate transformations to address the same problem, only one set of quantum operators to address the same problem can give the correct analysis. Such a choice leads to the need to find the favored classical variables in order to achieve a valid quantization. This article addresses the task of how such favored variables are found that can be used to properly solve a given quantum system. Examples, such as non-renormalizable scalar fields and gravity, have profited by initially changing which classical variables to promote to quantum operators.
文摘<span style="line-height:1.5;">For purposes of quantization, classical gravity is normally expressed by canonical variables, namely the metric </span><img src="Edit_7bad0ce2-ecaa-4318-b3c9-5bbcfa7c087e.png" alt="" style="line-height:1.5;" /><span style="line-height:1.5;"></span><span "="" style="line-height:1.5;"><span> and the momentum </span><img src="Edit_c86b710a-9b65-4220-a4e2-cff8eeab9642.png" alt="" /></span><span style="line-height:1.5;"></span><span style="line-height:1.5;">. 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 feature</span><span style="line-height:1.5;">s</span><span "="" style="line-height:1.5;"><span> promoting the metric </span><img src="Edit_d0035f64-c366-4510-9cc7-d1053f755369.png" alt="" /></span><span "="" style="line-height:1.5;"><span> and the momentric </span><img src="Edit_60c18bb8-525b-4896-ae8f-2cd6456eb6f7.png" alt="" /></span><span "="" style="line-height:1.5;"><span> to operators. Instead of these classical variables belonging to a constant zero curvature space (</span><i><span>i.e.</span></i><span>, instead of a flat space), they belong to a space of constant negative curvatures. This feature may even have its appearance in black holes, which could strongly point toward an affine quantization approach to quantize gravity.
文摘Canonical quantization covers a broad class of classical systems, but that does not include all the problems of interest. Affine quantization has the benefit of providing a successful quantization of many important problems including the quantization of half-harmonic oscillators [1], non-renormalizable scalar fields, such as (<i>ϕ</i><sup>12</sup>)<sub>3</sub> [2] and (<i>ϕ</i><sup>12</sup>)<sub>3</sub> [3], as well as the quantum theory of Einstein’s general relativity [4]. The features that distinguish affine quantization are emphasized, especially, that affine quantization differs from canonical quantization only by the choice of classical variables promoted to quantum operators. Coherent states are used to ensure proper quantizations are physically correct. While quantization of non-renormalizable covariant scalars and gravity are difficult, we focus on appropriate ultralocal scalars and gravity that are fully soluble while, in that case, implying that affine quantization is the proper procedure to ensure the validity of affine quantizations for non-renormalizable covariant scalar fields and Einstein’s gravity.
文摘The particle in a box is a simple model that has a classical Hamiltonian H = p<sup>2</sup> (using 2m = 1), with a limited coordinate space, -b q b, where 0 b < ∞. Using canonical quantization, this example has been fully studied thanks to its simplicity, and it is a common example for beginners to understand. Despite its repeated analysis, there is a feature that puts the past results into question. In addition to pointing out the quantization issue, the procedures of affine quantization can lead to a proper quantization that necessarily points toward more complicated eigenfunctions and eigenvalues, which deserve to be solved.
文摘Present day Quantum Field Theory (QFT) is founded on canonical quantization, which has served quite well but also has led to several issues. The free field describing a free particle (with no interaction term) can suddenly become nonrenormalizable the instant a suitable interaction term appears. For example, using canonical quantization <img src="Edit_9f6ab3f7-9277-4093-adcc-cdccf32c2c7c.png" width="15" height="15" alt="" /><sup?style="margin-left:-7px;">, has been deemed a “free” theory with no difference from a truly free field [1] [2]. Using the same model, affine quantization has led to a truly interacting theory [3]. This fact alone asserts that canonical and affine tools of quantization deserve to be open to their procedures together as a significant enlargement of QFT.</sup?style="margin-left:-7px;">
文摘The understanding of what a black hole is like is not easy and may not yet be well understood. The introduction of canonical quantization into the issue has not been significant to our understanding. However, introducing affine quantization, a new procedure, offers a very unusual expression that seems to be plausible, and quite profound as well.
文摘Canonical quantization has taught us great things. A common example is that of the harmonic oscillator, which is like swinging a ball on a string back and forth. However, the half-harmonic oscillator blocks the ball at the bottom and then it quickly bounces backwards. This second model cannot be correctly solved using canonical quantization. Now, there is an expansion of quantization, called affine quantization, that can correctly solve the half-harmonic oscillator, and offers correct solutions to a grand collection of other problems, which even reaches to field theory and gravity. This paper has been designed to introduce affine quantization: what it is, and what it can do.
文摘Today's quantum field theory (QFT) relies heavenly on canonical quantization (CQ), which fails for φ<sub>4</sub>4</sup> leading only to a “free” result. Affine quantization (AQ), an alternative quantization procedure, leads to a “non-free” result for the same model. Perhaps adding AQ to CQ can improve the quantization of a wide class of problems in QFT.
