International Journal of Geometric Methods in Modern Physics最新文献

{"title":"Ricci solitons and curvature inheritance on Robinson–Trautman spacetimes","authors":"Absos Ali Shaikh, Biswa Ranjan Datta","doi":"10.1142/s0219887824501639","DOIUrl":"https://doi.org/10.1142/s0219887824501639","url":null,"abstract":"<p>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-<span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>η</mi></math></span><span></span>-Ricci soliton, almost-gradient <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>η</mi></math></span><span></span>-Ricci soliton. As a generalization of curvature inheritance [K. L. Duggal, Curvature inheritance symmetry in Riemannian spaces with applications to fluid space times, <i>J. Math. Phys.</i><b>33</b>(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, <i>J. Math. Phys.</i><b>10</b>(4) (1969) 617–629], in this paper, we introduce the notion of <i>generalized curvature inheritance</i> 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.</p>","PeriodicalId":50320,"journal":{"name":"International Journal of Geometric Methods in Modern Physics","volume":"2 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140209814","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}

{"title":"A flat FLRW dark energy model in f(Q,C)-gravity theory with observational constraints","authors":"Anirudh Pradhan, Archana Dixit, M. Zeyauddin, S. Krishnannair","doi":"10.1142/s0219887824501676","DOIUrl":"https://doi.org/10.1142/s0219887824501676","url":null,"abstract":"&lt;p&gt;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, &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}

{"title":"Some inequalities for bi-slant Riemannian submersions in complex space forms","authors":"Nergiz Poyraz, Yılmaz Gündüzalp, Mehmet Akif Akyol","doi":"10.1142/s0219887824501500","DOIUrl":"https://doi.org/10.1142/s0219887824501500","url":null,"abstract":"<p>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 <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mspace width=\".17em\"></mspace><msub><mrow><mi>φ</mi></mrow><mrow><mo stretchy=\"false\">∗</mo></mrow></msub><mo>.</mo></math></span><span></span> Also, we mention some examples.</p>","PeriodicalId":50320,"journal":{"name":"International Journal of Geometric Methods in Modern Physics","volume":"55 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140188872","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}

{"title":"The Kastler–Kalau–Walze-type theorems about J-Witten deformation","authors":"Siyao Liu, Yong Wang","doi":"10.1142/s0219887824501743","DOIUrl":"https://doi.org/10.1142/s0219887824501743","url":null,"abstract":"<p>In this paper, we obtain a Lichnerowicz-type formula for <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>J</mi></math></span><span></span>-Witten deformation and give the proof of the Kastler–Kalau–Walze-type theorems associated with <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>J</mi></math></span><span></span>-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 <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>J</mi></math></span><span></span>-Witten deformation on four-dimensional manifold with boundary.</p>","PeriodicalId":50320,"journal":{"name":"International Journal of Geometric Methods in Modern Physics","volume":"11 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140188853","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}

{"title":"First-order quantum correction of thermodynamics in a charged accelerating AdS black hole with gauge potential","authors":"Riasat Ali, Rimsha Babar, Houcine Aounallah, Ali Övgün","doi":"10.1142/s0219887824501494","DOIUrl":"https://doi.org/10.1142/s0219887824501494","url":null,"abstract":"<p>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.</p>","PeriodicalId":50320,"journal":{"name":"International Journal of Geometric Methods in Modern Physics","volume":"23 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140124114","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}

{"title":"The Hodge–Dirac operator and Dabrowski–Sitarz–Zalecki-type theorems for manifolds with boundary","authors":"Tong Wu, Yong Wang","doi":"10.1142/s0219887824501627","DOIUrl":"https://doi.org/10.1142/s0219887824501627","url":null,"abstract":"<p>Dabrowski <i>et al.</i> [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 <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>d</mi><mo stretchy=\"false\">+</mo><mi>δ</mi></math></span><span></span> on an oriented even-dimensional Riemannian manifold. In this paper, we generalize the results of Dabrowski <i>et al.</i> 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.</p>","PeriodicalId":50320,"journal":{"name":"International Journal of Geometric Methods in Modern Physics","volume":"4 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140124110","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}

{"title":"A background independent notion of causality","authors":"A. Capolupo, A. Quaranta","doi":"10.1142/s0219887824501597","DOIUrl":"https://doi.org/10.1142/s0219887824501597","url":null,"abstract":"<p>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 <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>σ</mi></math></span><span></span> algebras. We then discuss how a natural notion of measure can be associated to the algebra of causal sets.</p>","PeriodicalId":50320,"journal":{"name":"International Journal of Geometric Methods in Modern Physics","volume":"41 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140124175","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}

{"title":"From the classical Frenet–Serret apparatus to the curvature and torsion of quantum-mechanical evolutions. Part II. Nonstationary Hamiltonians","authors":"Paul M. Alsing, Carlo Cafaro","doi":"10.1142/s0219887824501512","DOIUrl":"https://doi.org/10.1142/s0219887824501512","url":null,"abstract":"&lt;p&gt;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 &lt;span&gt;&lt;math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;〉&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; to the state vector &lt;span&gt;&lt;math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi mathvariant=\"normal\"&gt;Ψ&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;〉&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; 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 &lt;span&gt;&lt;math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;〉&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; to the state vector &lt;span&gt;&lt;math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi mathvariant=\"normal\"&gt;Ψ&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;〉&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;, orthogonal to &lt;span&gt;&lt;math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;〉&lt;/mo&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;mo&gt;|&lt;/mo&gt;&lt;mi mathvariant=\"normal\"&gt;Ψ&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;〉&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; 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","PeriodicalId":50320,"journal":{"name":"International Journal of Geometric Methods in Modern Physics","volume":"14 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140124173","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}

{"title":"Introduction to loop quantum gravity. The Holst’s action and the covariant formalism","authors":"L. Fatibene, A. Orizzonte, A. Albano, S. Coriasco, M. Ferraris, S. Garruto, N. Morandi","doi":"10.1142/s0219887824400164","DOIUrl":"https://doi.org/10.1142/s0219887824400164","url":null,"abstract":"&lt;p&gt;We review Holst formalism and dynamical equivalence with standard GR (in dimension &lt;span&gt;&lt;math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;). 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}

{"title":"Quantum mechanics on a p-adic Hilbert space: Foundations and prospects","authors":"Paolo Aniello, Stefano Mancini, Vincenzo Parisi","doi":"10.1142/s0219887824400176","DOIUrl":"https://doi.org/10.1142/s0219887824400176","url":null,"abstract":"<p>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 <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi></math></span><span></span>-adic numbers. In our approach, we are inspired by the idea — first postulated in [I. V. Volovich, <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi></math></span><span></span>-adic string, <i>Class. Quantum Grav.</i><b>4</b> (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 <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi></math></span><span></span>-adic Hilbert space. Next, after introducing all necessary mathematical tools — in particular, various classes of linear operators in a <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi></math></span><span></span>-adic Hilbert space — we consider an algebraic definition of physical states in <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi></math></span><span></span>-adic quantum mechanics. The corresponding observables, whose definition completes the statistical interpretation of the theory, are introduced as SOVMs, a <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi></math></span><span></span>-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 <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi></math></span><span></span>-adic setting, with an <i>affine</i> geometry; therefore, a symmetry transformation of a <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi></math></span><span></span>-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 <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi></math></span><span></span>-adic quantum system has a richer structure with respect to the case of standard quantum mechanics over the complex numbers.</p>","PeriodicalId":50320,"journal":{"name":"International Journal of Geometric Methods in Modern Physics","volume":"43 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140124174","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}