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Ricci solitons and curvature inheritance on Robinson–Trautman spacetimes 罗宾逊-特劳特曼时空中的利玛窦孤子和曲率继承
IF 1.8 3区 物理与天体物理 Q2 PHYSICS, MATHEMATICAL Pub Date : 2024-03-22 DOI: 10.1142/s0219887824501639
Absos Ali Shaikh, Biswa Ranjan Datta

The purpose of this paper is to investigate the existence of Ricci solitons and the nature of curvature inheritance as well as collineations on the Robinson–Trautman (briefly, RT) spacetime. It is shown that under certain conditions RT spacetime admits almost-Ricci soliton, almost-η-Ricci soliton, almost-gradient η-Ricci soliton. As a generalization of curvature inheritance [K. L. Duggal, Curvature inheritance symmetry in Riemannian spaces with applications to fluid space times, J. Math. Phys.33(9) (1992) 2989–2997] and curvature collineation [G. H. Katzin, J. Livine and W. R. Davis, Curvature collineations: A fundamental symmetry property of the space-times of general relativity defined by the vanishing Lie derivative of the Riemann curvature tensor, J. Math. Phys.10(4) (1969) 617–629], in this paper, we introduce the notion of generalized curvature inheritance and examine if RT spacetime admits such a notion. It is shown that RT spacetime also realizes the generalized curvature (resp., Ricci, Weyl conformal, concircular, conharmonic, Weyl projective) inheritance. Finally, several conditions are obtained, under which RT spacetime possesses curvature (resp., Ricci, conharmonic, Weyl projective) inheritance as well as curvature (resp., Ricci, Weyl conformal, concircular, conharmonic, Weyl projective) collineation, and we have also introduced the concept of generalized Lie inheritance and showed that RT spacetime realizes such a notion.

本文旨在研究罗宾逊-特劳特曼时空(简称 RT 时空)上利玛窦孤子的存在、曲率继承的性质以及领线。研究表明,在某些条件下,RT 时空存在近似利玛窦孤子、近似η-利玛窦孤子、近似梯度η-利玛窦孤子。作为曲率继承的一般化 [K. L. Duggal, Curvature inheritance] 。L. Duggal, Curvature inheritance symmetry in Riemannian spaces with applications to fluid space times, J. Math. Phys.33(9) (1992).Phys.33(9)(1992)2989-2997] 和曲率邻接[G. H. Katzin, J. Livine and W. R. Davis, Curvature collineations:G. H. Katzin, J. Livine and W. R. Davis, Curvature collineations: A fundamental symmetry property of the space-times of general relativity defined by the vanishing Lie derivative of the Riemann curvature tensor, J. Math. Phys.10(4) (1969).Phys.10(4)(1969)617-629],在本文中,我们引入了广义曲率继承的概念,并研究了 RT 时空是否存在这种概念。结果表明,RT 时空也实现了广义曲率继承(Ricci、Weyl 保角、共圆、共谐、Weyl 投影)。最后,我们得到了几个条件,在这些条件下,RT 时空具有曲率(Ricci、Weyl 保角、协和、Weyl 投射)继承以及曲率(Ricci、Weyl 保角、协圆、协和、Weyl 投射)勾连,我们还引入了广义烈继承的概念,并证明 RT 时空实现了这样一个概念。
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引用次数: 0
A flat FLRW dark energy model in f(Q,C)-gravity theory with observational constraints 具有观测约束的 f(Q,C)引力理论中的平面 FLRW 暗能量模型
IF 1.8 3区 物理与天体物理 Q2 PHYSICS, MATHEMATICAL Pub Date : 2024-03-21 DOI: 10.1142/s0219887824501676
Anirudh Pradhan, Archana Dixit, M. Zeyauddin, S. Krishnannair
<p>In the recently suggested modified non-metricity gravity theory with boundary term in a flat FLRW spacetime universe, dark energy scenarios of cosmological models are examined in this study. An arbitrary function, <span><math altimg="eq-00002.gif" display="inline" overflow="scroll"><mi>f</mi><mo stretchy="false">(</mo><mi>Q</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Q</mi><mo stretchy="false">+</mo><mi>α</mi><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span><span></span>, has been taken into consideration, where <span><math altimg="eq-00003.gif" display="inline" overflow="scroll"><mi>Q</mi></math></span><span></span> is the non-metricity scalar, <span><math altimg="eq-00004.gif" display="inline" overflow="scroll"><mi>C</mi></math></span><span></span> is the boundary term denoted by <span><math altimg="eq-00005.gif" display="inline" overflow="scroll"><mi>C</mi><mo>=</mo><mover accent="true"><mrow><mi>R</mi></mrow><mo>̈</mo></mover><mo stretchy="false">−</mo><mi>Q</mi></math></span><span></span>, and <span><math altimg="eq-00006.gif" display="inline" overflow="scroll"><mi>α</mi></math></span><span></span> is the model parameter, for the action that is quadratic in <span><math altimg="eq-00007.gif" display="inline" overflow="scroll"><mi>C</mi></math></span><span></span>. The Hubble function <span><math altimg="eq-00008.gif" display="inline" overflow="scroll"><mi>H</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow></msub><msup><mrow><mo stretchy="false">[</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">+</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><mrow><mi>n</mi></mrow></msup><mo stretchy="false">+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy="false">]</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></math></span><span></span>, where <span><math altimg="eq-00009.gif" display="inline" overflow="scroll"><msub><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span><span></span> is the current value of the Hubble constant and <span><math altimg="eq-00010.gif" display="inline" overflow="scroll"><mi>n</mi><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span><span></span> and <span><math altimg="eq-00011.gif" display="inline" overflow="scroll"><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span><span></span> are arbitrary parameters with <span><math altimg="eq-00012.gif" display="inline" overflow="scroll"><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><mo stretchy="false">+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mn>1</mn></math></span><span></span>, has been used to examine the dark energy characteristics of the model. We discovered a transit phase expanding uni
本研究考察了最近提出的带边界项的修正非度量引力理论在平坦的 FLRW 时空宇宙中的宇宙学模型暗能量情景。其中 Q 是非度量标量,C 是边界项,用 C=R̈-Q 表示,α 是模型参数。哈勃函数 H(z)=H0[c1(1+z)n+c2]12,其中 H0 是哈勃常数的当前值,n、c1 和 c2 是 c1+c2=1 的任意参数,被用来检验模型的暗能量特征。我们发现了一个过去减速、现在加速的过境相膨胀宇宙模型,并发现暗能量状态方程(EoS)ω(de)表现为(-1≤ω(de)<2)。Om 诊断分析揭示了现在宇宙的五元行为和晚期宇宙的宇宙常数情景。最后,我们计算了宇宙当前的年龄,发现它与最近的数据非常相似。
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An arbitrary function, &lt;span&gt;&lt;math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;+&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;, has been taken into consideration, where &lt;span&gt;&lt;math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; is the non-metricity scalar, &lt;span&gt;&lt;math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; is the boundary term denoted by &lt;span&gt;&lt;math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mover accent=\"true\"&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;̈&lt;/mo&gt;&lt;/mover&gt;&lt;mo stretchy=\"false\"&gt;−&lt;/mo&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;, and &lt;span&gt;&lt;math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; is the model parameter, for the action that is quadratic in &lt;span&gt;&lt;math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;. The Hubble function &lt;span&gt;&lt;math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo stretchy=\"false\"&gt;[&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=\"false\"&gt;+&lt;/mo&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo stretchy=\"false\"&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; is the current value of the Hubble constant and &lt;span&gt;&lt;math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; and &lt;span&gt;&lt;math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; are arbitrary parameters with &lt;span&gt;&lt;math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo stretchy=\"false\"&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;, has been used to examine the dark energy characteristics of the model. We discovered a transit phase expanding uni","PeriodicalId":50320,"journal":{"name":"International Journal of Geometric Methods in Modern Physics","volume":"45 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140188951","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Some inequalities for bi-slant Riemannian submersions in complex space forms 复空间形式中双斜面黎曼潜影的一些不等式
IF 1.8 3区 物理与天体物理 Q2 PHYSICS, MATHEMATICAL Pub Date : 2024-03-21 DOI: 10.1142/s0219887824501500
Nergiz Poyraz, Yılmaz Gündüzalp, Mehmet Akif Akyol

The goal of this paper is to analyze sharp-type inequalities including the scalar and Ricci curvatures of bi-slant Riemannian submersions in complex space forms. Then, for bi-slant Riemannian submersion between a complex space form and a Riemannian manifold, we give inequalities involving the Casorati curvature of the space ker φ. Also, we mention some examples.

本文旨在分析复空间形式中双斜面黎曼潜影的锐型不等式,包括标量曲率和黎奇曲率。然后,对于复空间形式与黎曼流形之间的双斜黎曼潜影,我们给出了涉及空间 ker φ∗ 的卡索拉提曲率的不等式。此外,我们还提到了一些例子。
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引用次数: 0
The Kastler–Kalau–Walze-type theorems about J-Witten deformation 关于 J-Witten 变形的 Kastler-Kalau-Walze 型定理
IF 1.8 3区 物理与天体物理 Q2 PHYSICS, MATHEMATICAL Pub Date : 2024-03-21 DOI: 10.1142/s0219887824501743
Siyao Liu, Yong Wang

In this paper, we obtain a Lichnerowicz-type formula for J-Witten deformation and give the proof of the Kastler–Kalau–Walze-type theorems associated with J-Witten deformation on four-dimensional and six-dimensional almost product Riemannian manifold with (respectively, without) boundary. We give an explanation of the Einstein–Hilbert action for J-Witten deformation on four-dimensional manifold with boundary.

在本文中,我们得到了 J-Witten 变形的 Lichnerowicz 型公式,并给出了在有边界(分别为无边界)的四维和六维几乎积黎曼流形上与 J-Witten 变形相关的 Kastler-Kalau-Walze 型定理的证明。我们给出了有边界四维流形上 J-Witten 变形的爱因斯坦-希尔伯特作用的解释。
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引用次数: 0
First-order quantum correction of thermodynamics in a charged accelerating AdS black hole with gauge potential 带有规势的带电加速 AdS 黑洞中热力学的一阶量子修正
IF 1.8 3区 物理与天体物理 Q2 PHYSICS, MATHEMATICAL Pub Date : 2024-03-13 DOI: 10.1142/s0219887824501494
Riasat Ali, Rimsha Babar, Houcine Aounallah, Ali Övgün

In this paper, we study the tunneling radiation from a charged-accelerating AdS black hole with gauge potential under the impact of quantum gravity. Using the semi-classical phenomenon known as the Hamilton–Jacobi ansatz, it is studied that tunneling radiation occurs via the horizon of a black hole and also employs the Lagrangian equation using the generalized uncertainty principle. Furthermore, we investigate the impact of charge, gauge potential, and first order correction parameters on the temperature as well as the stable and unstable states of the black hole. We also compute thermodynamic properties such as entropy, internal energy, Helmholtz free energy, enthalpy, specific heat, and Gibbs free energy under the impact of the correction parameter for the black hole. We calculate the logarithmic modification terms for entropy around the equilibrium state to analyze the impacts of logarithmic correction. In the presence of the correction terms, we also check the validity of the thermodynamics. It examines the graphical representation of the influence of logarithmic correction on the thermodynamic properties of black hole stability as well as charged, accelerating, and gauge potential parameters.

本文研究了在量子引力作用下,带电加速AdS黑洞的隧穿辐射。利用被称为汉密尔顿-贾科比方差的半经典现象,研究了隧道辐射是通过黑洞的视界发生的,还利用广义不确定性原理运用了拉格朗日方程。此外,我们还研究了电荷、规势和一阶修正参数对黑洞温度以及稳定和不稳定状态的影响。我们还计算了黑洞修正参数影响下的热力学性质,如熵、内能、亥姆霍兹自由能、焓、比热和吉布斯自由能。我们计算了平衡态附近熵的对数修正项,以分析对数修正的影响。在存在修正项的情况下,我们还检查了热力学的有效性。它研究了对数修正对黑洞稳定性的热力学性质以及带电、加速和规势参数的影响的图示。
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引用次数: 0
The Hodge–Dirac operator and Dabrowski–Sitarz–Zalecki-type theorems for manifolds with boundary 有边界流形的霍奇-狄拉克算子和达布罗夫斯基-西塔尔兹-扎莱基类型定理
IF 1.8 3区 物理与天体物理 Q2 PHYSICS, MATHEMATICAL Pub Date : 2024-03-13 DOI: 10.1142/s0219887824501627
Tong Wu, Yong Wang

Dabrowski et al. [Spectral metric and Einstein functionals for Hodge–Dirac operator, preprint (2023), arXiv:2307.14877] gave spectral Einstein bilinear functionals of differential forms for the Hodge–Dirac operator d+δ on an oriented even-dimensional Riemannian manifold. In this paper, we generalize the results of Dabrowski et al. to the cases of 4-dimensional oriented Riemannian manifolds with boundary. Furthermore, we give the proof of Dabrowski–Sitarz–Zalecki-type theorems associated with the Hodge–Dirac operator for manifolds with boundary.

Dabrowski 等人[霍奇-狄拉克算子的谱度量和爱因斯坦函数,预印本(2023 年),arXiv:2307.14877]给出了定向偶维黎曼流形上霍奇-狄拉克算子 d+δ 的微分形式的谱爱因斯坦双线性函数。在本文中,我们将 Dabrowski 等人的结果推广到有边界的 4 维定向黎曼流形的情况。此外,我们还给出了与有边界流形的霍奇-狄拉克算子相关的 Dabrowski-Sitarz-Zalecki- 型定理的证明。
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引用次数: 0
A background independent notion of causality 与背景无关的因果关系概念
IF 1.8 3区 物理与天体物理 Q2 PHYSICS, MATHEMATICAL Pub Date : 2024-03-13 DOI: 10.1142/s0219887824501597
A. Capolupo, A. Quaranta

We develop a notion of causal order on a generic manifold as independent of the underlying differential and topological structure. We show that sufficiently regular causal orders can be recovered from a distinguished algebra of sets, which plays a role analogous to that of topologies and σ algebras. We then discuss how a natural notion of measure can be associated to the algebra of causal sets.

我们在一般流形上建立了一个因果阶的概念,它与底层微分和拓扑结构无关。我们证明,足够规则的因果阶可以从一个有区别的集合代数中恢复,它的作用类似于拓扑和σ代数。然后,我们将讨论如何将自然的度量概念与因果集合代数联系起来。
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引用次数: 0
From the classical Frenet–Serret apparatus to the curvature and torsion of quantum-mechanical evolutions. Part II. Nonstationary Hamiltonians 从经典的 Frenet-Serret 装置到量子力学演化的曲率和扭转。第二部分.非稳态哈密顿
IF 1.8 3区 物理与天体物理 Q2 PHYSICS, MATHEMATICAL Pub Date : 2024-03-13 DOI: 10.1142/s0219887824501512
Paul M. Alsing, Carlo Cafaro
<p>In this paper, we present a geometric perspective on how to quantify the bending and the twisting of quantum curves traced by state vectors evolving under nonstationary Hamiltonians. Specifically, relying on the existing geometric viewpoint for stationary Hamiltonians, we discuss the generalization of our theoretical construct to time-dependent quantum-mechanical scenarios where both time-varying curvature and torsion coefficients play a key role. Specifically, we present a quantum version of the Frenet–Serret apparatus for a quantum trajectory in projective Hilbert space traced out by a parallel-transported pure quantum state evolving unitarily under a time-dependent Hamiltonian specifying the Schrödinger evolution equation. The time-varying curvature coefficient is specified by the magnitude squared of the covariant derivative of the tangent vector <span><math altimg="eq-00001.gif" display="inline" overflow="scroll"><mo>|</mo><mi>T</mi><mo stretchy="false">〉</mo></math></span><span></span> to the state vector <span><math altimg="eq-00002.gif" display="inline" overflow="scroll"><mo>|</mo><mi mathvariant="normal">Ψ</mi><mo stretchy="false">〉</mo></math></span><span></span> and measures the bending of the quantum curve. The time-varying torsion coefficient, instead, is given by the magnitude squared of the projection of the covariant derivative of the tangent vector <span><math altimg="eq-00003.gif" display="inline" overflow="scroll"><mo>|</mo><mi>T</mi><mo stretchy="false">〉</mo></math></span><span></span> to the state vector <span><math altimg="eq-00004.gif" display="inline" overflow="scroll"><mo>|</mo><mi mathvariant="normal">Ψ</mi><mo stretchy="false">〉</mo></math></span><span></span>, orthogonal to <span><math altimg="eq-00005.gif" display="inline" overflow="scroll"><mo>|</mo><mi>T</mi><mo stretchy="false">〉</mo></math></span><span></span> and <span><math altimg="eq-00006.gif" display="inline" overflow="scroll"><mo>|</mo><mi mathvariant="normal">Ψ</mi><mo stretchy="false">〉</mo></math></span><span></span> and, in addition, measures the twisting of the quantum curve. We find that the time-varying setting exhibits a richer structure from a statistical standpoint. For instance, unlike the time-independent configuration, we find that the notion of generalized variance enters nontrivially in the definition of the torsion of a curve traced out by a quantum state evolving under a nonstationary Hamiltonian. To physically illustrate the significance of our construct, we apply it to an exactly soluble time-dependent two-state Rabi problem specified by a sinusoidal oscillating time-dependent potential. In this context, we show that the analytical expressions for the curvature and torsion coefficients are completely described by only two real three-dimensional vectors, the Bloch vector that specifies the quantum system and the externally applied time-varying magnetic field. Although we show that the torsion is identically zero for an arbitrary time-de
在本文中,我们从几何学的角度阐述了如何量化在非稳态哈密顿力下演化的状态向量所描绘的量子曲线的弯曲和扭曲。具体地说,基于现有的静态哈密顿几何观点,我们讨论了将我们的理论构造推广到时变曲率和扭转系数都起关键作用的时变量子力学场景。具体地说,我们提出了一种量子版的 Frenet-Serret 装置,该装置适用于投影希尔伯特空间中的量子轨迹,该轨迹是由平行传输的纯量子态在指定薛定谔演化方程的时变哈密顿下单元演化出来的。时变曲率系数由切线向量|T〉到状态向量|Ψ〉的协变导数的平方的大小确定,并测量量子曲线的弯曲程度。时变扭转系数则由切线矢量|T〉到状态矢量|Ψ〉的协变导数投影的大小平方给出,与|T〉和|Ψ〉正交,此外还测量量子曲线的扭转。我们发现,从统计学的角度来看,时变设置呈现出更丰富的结构。例如,与不依赖于时间的构型不同,我们发现广义方差的概念不可逆转地进入了量子态在非稳态汉密尔顿下演化出的曲线扭转的定义中。为了从物理上说明我们的构造的意义,我们将其应用于一个由正弦振荡时变势指定的精确可解时变双态拉比问题。在这种情况下,我们证明曲率系数和扭转系数的分析表达式完全可以用两个实三维矢量来描述,即指定量子系统的布洛赫矢量和外部施加的时变磁场。尽管我们证明了扭转在任意随时间变化的单量子位哈密顿演化过程中同等于零,但我们还是研究了曲率系数在不同动力学情况下的时间行为,包括非共振和共振状态,以及强驱动和弱驱动构型。虽然我们的形式主义适用于任意维度的纯量子态,但随着维度的增加,相关曲率的分析推导和轨道模拟会变得相当复杂。因此,最后我们简要评述了将我们的几何形式主义应用于在一般非稳态哈密顿下单元演化的高维量子系统的可能性。
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引用次数: 0
Introduction to loop quantum gravity. The Holst’s action and the covariant formalism 环量子引力入门。霍尔斯特作用和协变形式主义
IF 1.8 3区 物理与天体物理 Q2 PHYSICS, MATHEMATICAL Pub Date : 2024-03-13 DOI: 10.1142/s0219887824400164
L. Fatibene, A. Orizzonte, A. Albano, S. Coriasco, M. Ferraris, S. Garruto, N. Morandi
<p>We review Holst formalism and dynamical equivalence with standard GR (in dimension <span><math altimg="eq-00001.gif" display="inline" overflow="scroll"><mn>4</mn></math></span><span></span>). Holst formalism is written for a spin coframe field <span><math altimg="eq-00002.gif" display="inline" overflow="scroll"><msubsup><mrow><mi>e</mi></mrow><mrow><mi>μ</mi></mrow><mrow><mi>I</mi></mrow></msubsup></math></span><span></span> and a Spin(3,1)-connection <span><math altimg="eq-00003.gif" display="inline" overflow="scroll"><msubsup><mrow><mi>ω</mi></mrow><mrow><mi>μ</mi></mrow><mrow><mi>I</mi><mi>J</mi></mrow></msubsup></math></span><span></span> on spacetime <span><math altimg="eq-00004.gif" display="inline" overflow="scroll"><mi>M</mi></math></span><span></span> and it depends on the <i>Holst parameter</i><span><math altimg="eq-00005.gif" display="inline" overflow="scroll"><mi>γ</mi><mo>∈</mo><mi>ℝ</mi><mo stretchy="false">−</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></math></span><span></span>. We show the model is dynamically equivalent to standard GR, in the sense that up to a pointwise Spin(3,1)-gauge transformation acting on (uppercase Latin) frame indices, solutions of the two models are in one-to-one correspondence. Hence, the two models are classically equivalent. One can also introduce new variables by splitting the spin connection into a pair of a Spin(3)-connection <span><math altimg="eq-00006.gif" display="inline" overflow="scroll"><msubsup><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow><mrow><mi>i</mi></mrow></msubsup></math></span><span></span> and a Spin(3)-valued 1-form <span><math altimg="eq-00007.gif" display="inline" overflow="scroll"><msubsup><mrow><mi>k</mi></mrow><mrow><mi>μ</mi></mrow><mrow><mi>i</mi></mrow></msubsup></math></span><span></span>. The construction of these new variables relies on a particular algebraic structure, called a <i>reductive splitting</i>. A weaker structure than requiring that the gauge group splits as the products of two sub-groups, as it happens in Euclidean signature in the selfdual formulation originally introduced in this context by Ashtekar, and it still allows to deal with the Lorentzian signature without resorting to complexifications. The reductive splitting of <span><math altimg="eq-00008.gif" display="inline" overflow="scroll"><mstyle><mtext>SL</mtext></mstyle><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></math></span><span></span> is not unique and it is parameterized by a real parameter <span><math altimg="eq-00009.gif" display="inline" overflow="scroll"><mi>β</mi></math></span><span></span> which is called the <i>Immirzi parameter</i>. The splitting is here done <i>on spacetime</i>, not on space as it is usually done in the literature, to obtain a Spin(3)-connection <span><math altimg="eq-00010.gif" display="inline" overflow="scroll"><msubsup><mrow><mi>A</mi></mrow><mrow><mi>μ</mi></mrow><mrow><mi>i</mi></mrow></msubsup><
我们回顾了霍尔斯特形式主义以及它与标准 GR(4 维)的动力学等价性。霍尔斯特形式主义是为时空M上的自旋共框场eμI和自旋(3,1)连接ωμIJ而写的,它取决于霍尔斯特参数γ∈ℝ-{0}。我们证明了这个模型在动力学上等同于标准GR,即在作用于(大写拉丁文)框架指数的点向Spin(3,1)-gauge变换之前,两个模型的解是一一对应的。因此,这两个模型在经典上是等价的。我们还可以通过将自旋连接拆分为一对自旋(3)连接 Aμi 和自旋(3)值 1-form kμi 来引入新变量。这些新变量的构建依赖于一种特殊的代数结构,即还原分裂。这种结构比要求轨距群分裂为两个子群的乘积要弱,因为在欧几里得签名中,阿什特卡尔(Ashtekar)最初在这种情况下提出了自偶公式。SL(2,ℂ)的还原分裂并不是唯一的,它是由一个实数参数β参数化的,这个参数被称为伊米尔兹参数。这里的分裂是在时空中进行的,而不是像文献中通常那样在空间上进行,从而得到一个 Spin(3)-connection Aμi,它被称为时空中的巴贝罗-伊米尔兹(Barbero-Immirzi)连接。我们可以得到一个取决于场(eμI,Aμi,kμi)的协变量模型,它在动力学上等同于标准 GR。为了简单起见,文献中通常设定 β=γ 。在这里,我们将霍尔斯特参数和伊米尔兹参数区分开来,以说明最终只有 β 会在边界场方程中存在。
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Holst formalism is written for a spin coframe field &lt;span&gt;&lt;math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; and a Spin(3,1)-connection &lt;span&gt;&lt;math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;ω&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; on spacetime &lt;span&gt;&lt;math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; and it depends on the &lt;i&gt;Holst parameter&lt;/i&gt;&lt;span&gt;&lt;math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;ℝ&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;−&lt;/mo&gt;&lt;mo stretchy=\"false\"&gt;{&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo stretchy=\"false\"&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;. We show the model is dynamically equivalent to standard GR, in the sense that up to a pointwise Spin(3,1)-gauge transformation acting on (uppercase Latin) frame indices, solutions of the two models are in one-to-one correspondence. Hence, the two models are classically equivalent. One can also introduce new variables by splitting the spin connection into a pair of a Spin(3)-connection &lt;span&gt;&lt;math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; and a Spin(3)-valued 1-form &lt;span&gt;&lt;math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;. The construction of these new variables relies on a particular algebraic structure, called a &lt;i&gt;reductive splitting&lt;/i&gt;. A weaker structure than requiring that the gauge group splits as the products of two sub-groups, as it happens in Euclidean signature in the selfdual formulation originally introduced in this context by Ashtekar, and it still allows to deal with the Lorentzian signature without resorting to complexifications. The reductive splitting of &lt;span&gt;&lt;math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mstyle&gt;&lt;mtext&gt;SL&lt;/mtext&gt;&lt;/mstyle&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;ℂ&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; is not unique and it is parameterized by a real parameter &lt;span&gt;&lt;math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; which is called the &lt;i&gt;Immirzi parameter&lt;/i&gt;. The splitting is here done &lt;i&gt;on spacetime&lt;/i&gt;, not on space as it is usually done in the literature, to obtain a Spin(3)-connection &lt;span&gt;&lt;math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;","PeriodicalId":50320,"journal":{"name":"International Journal of Geometric Methods in Modern Physics","volume":"8 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140124146","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Quantum mechanics on a p-adic Hilbert space: Foundations and prospects p-adic Hilbert 空间上的量子力学:基础与前景
IF 1.8 3区 物理与天体物理 Q2 PHYSICS, MATHEMATICAL Pub Date : 2024-03-11 DOI: 10.1142/s0219887824400176
Paolo Aniello, Stefano Mancini, Vincenzo Parisi

We review some recent results on the mathematical foundations of a quantum theory over a scalar field that is a quadratic extension of the non-Archimedean field of p-adic numbers. In our approach, we are inspired by the idea — first postulated in [I. V. Volovich, p-adic string, Class. Quantum Grav.4 (1987) L83–L87] — that space, below a suitably small scale, does not behave as a continuum and, accordingly, should be modeled as a totally disconnected metrizable topological space, ruled by a metric satisfying the strong triangle inequality. The first step of our construction is a suitable definition of a p-adic Hilbert space. Next, after introducing all necessary mathematical tools — in particular, various classes of linear operators in a p-adic Hilbert space — we consider an algebraic definition of physical states in p-adic quantum mechanics. The corresponding observables, whose definition completes the statistical interpretation of the theory, are introduced as SOVMs, a p-adic counterpart of the POVMs associated with a standard quantum system over the complex numbers. Interestingly, it turns out that the typical convex geometry of the space of states of a standard quantum system is replaced, in the p-adic setting, with an affine geometry; therefore, a symmetry transformation of a p-adic quantum system may be defined as a map preserving this affine geometry. We argue that, as a consequence, the group of all symmetry transformations of a p-adic quantum system has a richer structure with respect to the case of standard quantum mechanics over the complex numbers.

我们回顾了关于标量场量子理论数学基础的一些最新成果,标量场是 p-adic 数的非阿基米德场的二次扩展。在我们的方法中,我们受到了[I. V. Volovich, p-adic string, Class. Quantum Grav.4(1987)L83-L87]一文中首次提出的观点的启发,即空间在适当小的尺度以下不表现为连续体,因此,应该被建模为完全断开的可元拓扑空间,由满足强三角不等式的度量统治。我们构建模型的第一步是给 p-adic Hilbert 空间下一个合适的定义。接下来,在引入所有必要的数学工具--特别是 p-adic Hilbert 空间中的各类线性算子--之后,我们将考虑 p-adic 量子力学中物理状态的代数定义。相应的观测值(其定义完成了理论的统计解释)被引入为 SOVMs,即与复数标准量子系统相关的 POVMs 的 p-adic 对应物。有趣的是,在 p-adic 环境中,标准量子系统状态空间的典型凸几何被仿射几何所取代;因此,p-adic 量子系统的对称变换可以定义为保留这种仿射几何的映射。我们认为,与复数上的标准量子力学相比,p-adic 量子系统的所有对称变换群具有更丰富的结构。
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引用次数: 0
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International Journal of Geometric Methods in Modern Physics
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