Journal of Geometry and Physics最新文献

英文中文

Conformal and contact kinetic dynamics and their geometrization

IF 1.6 3区数学 Q1 MATHEMATICS

Journal of Geometry and Physics

Pub Date : 2024-11-14 DOI: 10.1016/j.geomphys.2024.105369

Oğul Esen , Ayten Gezici , Miroslav Grmela , Hasan Gümral , Michal Pavelka , Serkan Sütlü

We propose a conformal generalization of the reversible Vlasov equation of kinetic plasma dynamics, called conformal kinetic theory. In order to arrive at this formalism, we start with the conformal Hamiltonian dynamics of particles and lift it to the dynamical formulation of the associated kinetic theory. The resulting theory represents a simple example of a geometric pathway from dissipative particle motion to dissipative kinetic motion. We also derive the kinetic equations of a continuum of particles governed by the contact Hamiltonian dynamics, which may be interpreted in the context of relativistic mechanics. Once again we start with the contact Hamiltonian dynamics and lift it to a kinetic theory, called contact kinetic dynamics. Finally, we project the contact kinetic theory to conformal kinetic theory so that they form a geometric hierarchy.

引用次数: 0

Klein-Gordon oscillators and Bergman spaces 克莱因-戈登振荡器和伯格曼空间

IF 1.6 3区数学 Q1 MATHEMATICS

Journal of Geometry and Physics

Pub Date : 2024-11-14 DOI: 10.1016/j.geomphys.2024.105368

Alexander D. Popov

We consider classical and quantum dynamics of relativistic oscillator in Minkowski space

R^{3, 1}

. It is shown that for a non-zero frequency parameter ω the covariant phase space of the classical Klein-Gordon oscillator is a homogeneous Kähler-Einstein manifold

Z_{6} = Ad S_{7} / U (1) = U (3, 1) / U (3) \times U (1)

. In the limit

ω \to 0

, this manifold is deformed into the covariant phase space

T^{⁎} H^{3}

of a free relativistic particle, where

H^{3} = H_{+}^{3} \cup H_{-}^{3}

is a two-sheeted hyperboloid in momentum space. Quantization of this model with

ω \neq 0

leads to the Klein-Gordon oscillator equation which we consider in the Segal-Bargmann representation. It is shown that the general solution of this model is given by functions from the weighted Bergman space of square-integrable holomorphic (for particles) and antiholomorphic (for antiparticles) functions on the Kähler-Einstein manifold

Z_{6}

. This relativistic model is Lorentz covariant, unitary and does not contain non-physical states.

我们考虑了闵科夫斯基空间 R3,1 中相对论振荡器的经典和量子动力学。研究表明，对于非零频率参数ω，经典克莱因-戈登振荡器的协变相空间是一个均质凯勒-爱因斯坦流形 Z6=AdS7/U(1)=U(3,1)/U(3)×U(1)。在极限ω→0 时，这个流形变形为自由相对论粒子的协变相空间 T⁎H3，其中 H3=H+3∪H-3 是动量空间中的两片双曲面。用 ω≠0 对这一模型进行量子化，可以得到克莱因-戈登振荡器方程，我们在西格尔-巴格曼表示法中考虑了这一方程。研究表明，这个模型的一般解是由凯勒-爱因斯坦流形 Z6 上可平方积分全形（粒子）和反全形（反粒子）函数的加权伯格曼空间的函数给出的。这种相对论模型具有洛伦兹协变性、单元性，并且不包含非物理状态。

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引用次数: 0

Hom-actions and class equation for Hom-groups 同域群的同作用和类方程

IF 1.6 3区数学 Q1 MATHEMATICS

Journal of Geometry and Physics

Pub Date : 2024-11-13 DOI: 10.1016/j.geomphys.2024.105371

Zoheir Chebel , Hadjer Adimi , Hassane Bouremel

The notion of Hom-groups is defined as a generalization of a non-associative group. They can be obtained by twisting the associative operation with a compatible bijection mapping. In this article, we provide some constructions by twisting and also discuss properties related to Hom-groups. We introduce different notions of actions concerning Hom-groups. We then present a theorem for a class equation, which is proven. Following that, we illustrate some applications for p-Hom groups.

Hom-群的概念被定义为非关联群的一般化。它们可以通过将关联操作与相容的双射映射进行扭转而得到。在本文中，我们将通过扭转提供一些构造，并讨论与同群组相关的性质。我们介绍了有关 Hom 群的不同作用概念。然后，我们提出了一个类方程定理，并对其进行了证明。随后，我们说明了 p-Hom 群的一些应用。

引用次数: 0

Tau functions of modified CKP hierarchy 修改后的 CKP 层次结构的 Tau 功能

IF 1.6 3区数学 Q1 MATHEMATICS

Journal of Geometry and Physics

Pub Date : 2024-11-13 DOI: 10.1016/j.geomphys.2024.105367

Shen Wang , Wenchuang Guan , Jipeng Cheng

Modified CKP (mCKP) hierarchy is an important integrable hierarchy, that is related to CKP hierarchy through Miura link. It has been proven that there exists a tau pair

(τ_{0}, τ_{1})

for mCKP hierarchy. Further we find that mCKP hierarchy can be fully determined by CKP tau function and corresponding CKP eigenfunction. Based on this, we construct mCKP tau functions by CKP Darboux transformations and also present the vacuum expectation value of free bosons. As a byproduct, determinant formula for

〈 1 | e^{H (t_{o})} β_{0} e^{\frac{β_{n}^{2}}{2}} e^{\frac{β_{n - 1}^{2}}{2}} \dots e^{\frac{β_{1}^{2}}{2}} g | 0 〉

is also derived.

修正 CKP（mCKP）层次结构是一种重要的可积分层次结构，它通过 Miura 链接与 CKP 层次结构相关。研究证明，mCKP 层次结构存在一个 tau 对（τ0,τ1）。我们进一步发现，mCKP 层次结构可以完全由 CKP tau 函数和相应的 CKP 特征函数决定。在此基础上，我们通过 CKP 达布变换构建了 mCKP tau 函数，并给出了自由玻色子的真空期望值。作为副产品，我们还推导出了〈1|eH(to)β0eβn22eβn-122⋯eβ122g|0〉的行列式。

引用次数: 0

On the Kostant-Souriau prequantization of scalar fields with polysymplectic structures 关于具有多折射结构的标量场的科斯坦-索里奥预量化

IF 1.6 3区数学 Q1 MATHEMATICS

Journal of Geometry and Physics

Pub Date : 2024-11-12 DOI: 10.1016/j.geomphys.2024.105365

Tom McClain

In this paper, I present a novel, purely differential geometric approach to the quantization of scalar fields, with a special focus on the familiar case of Minkowski spacetimes. This approach is based on using the natural geometric structures of polysymplectic Hamiltonian field theory to produce an analog of the Kostant-Souriau prequantization map familiar from geometric quantization. I show that while the resulting operators are quite different from those of canonical quantum field theory, the approach is nonetheless able to reproduce a few of canonical quantum field theory's most fundamental results. I finish by elaborating the current limitations of this approach and briefly discussing future prospects.

在这篇论文中，我提出了一种新颖的纯微分几何方法来研究标量场的量子化，并特别关注我们熟悉的闵科夫斯基空间。这种方法的基础是利用多折射哈密顿场论的自然几何结构，生成几何量子化中熟悉的科斯坦-苏里奥预量子化映射。我的研究表明，虽然由此产生的算子与经典量子场论的算子大相径庭，但这种方法仍能重现经典量子场论的一些最基本结果。最后，我阐述了这种方法目前的局限性，并简要讨论了未来的前景。

引用次数: 0

Baryogenesis in Minkowski spacetime 闵科夫斯基时空中的重力发生

IF 1.6 3区数学 Q1 MATHEMATICS

Journal of Geometry and Physics

Pub Date : 2024-11-07 DOI: 10.1016/j.geomphys.2024.105346

Felix Finster, Marco van den Beld-Serrano

Based on a mechanism originally suggested for causal fermion systems, the present paper paves the way for a rigorous treatment of baryogenesis in the language of differential geometry and global analysis. Moreover, a formula for the rate of baryogenesis in Minkowski spacetime is derived.

本文基于最初为因果费米子系统提出的机制，为用微分几何和全局分析语言严格处理重子发生铺平了道路。此外，本文还推导出了闵科夫斯基时空中的重子发生率公式。

引用次数: 0

Transposed Poisson structures on Virasoro-type algebras 维拉索类代数上的反转泊松结构

IF 1.6 3区数学 Q1 MATHEMATICS

Journal of Geometry and Physics

Pub Date : 2024-11-06 DOI: 10.1016/j.geomphys.2024.105356

Ivan Kaygorodov , Abror Khudoyberdiyev , Zarina Shermatova

We compute

\frac{1}{2}

-derivations on the deformed generalized Heisenberg-Virasoro¹ algebras and on not-finitely graded Heisenberg-Virasoro algebras

\hat{W} (G)

\tilde{W} (G)

, and

\tilde{H W} (G)

. We classify all transposed Poisson structures on such algebras.

我们计算了变形广义海森堡-维拉索罗1 代数和非无限分级海森堡-维拉索罗代数 Wˆ(G)、W˜(G) 和 HW˜(G)上的 12 衍生。我们对这些代数的所有转置泊松结构进行了分类。

引用次数: 0

Finite orbits of the braid group actions 辫状群作用的有限轨道

IF 1.6 3区数学 Q1 MATHEMATICS

Journal of Geometry and Physics

Pub Date : 2024-11-05 DOI: 10.1016/j.geomphys.2024.105363

Jialin Zhang

We study the finite orbits of the braid group

B_{n}

action on the space of

n \times n

upper-triangular matrices with 1's along the diagonal. On one hand, we give a necessary condition for a matrix M to be in a finite orbit; on the other hand, we classify and provide lengths of finite orbits in low-dimensional matrices and some other important cases. As the finite orbits on

3 \times 3

matrix were crucial to finding the algebraic solutions of the sixth Painlevé equation, we hope the finite orbits on generic

n \times n

matrices to be useful to finding solutions of higher order Painlevé type differential equations.

我们研究了对角线上有 1 的 n×n 上三角矩阵空间的辫状群 Bn 作用的有限轨道。一方面，我们给出了矩阵 M 处于有限轨道的必要条件；另一方面，我们对低维矩阵和其他一些重要情况下的有限轨道进行了分类，并给出了有限轨道的长度。正如 3×3 矩阵上的有限轨道对找到第六个潘列维方程的代数解至关重要，我们希望通用 n×n 矩阵上的有限轨道对找到高阶潘列维类型微分方程的解有用。

引用次数: 0

On conformal collineation and almost Ricci solitons 关于保角邻接和几乎利玛窦孤子

IF 1.6 3区数学 Q1 MATHEMATICS

Journal of Geometry and Physics

Pub Date : 2024-11-05 DOI: 10.1016/j.geomphys.2024.105354

Adara M. Blaga , Bang-Yen Chen

We provide conditions for a Riemannian manifold with a nontrivial closed affine conformal Killing vector field to be isometric to a Euclidean sphere or to the Euclidean space. Also, we formulate some triviality results for almost Ricci solitons with affine conformal Killing potential vector field.

我们提供了具有非难封闭仿射共形基林向量场的黎曼流形与欧几里得球面或欧几里得空间等距的条件。此外，我们还为具有仿射共形基林势向量场的几乎利玛窦孤子提出了一些三性结果。

引用次数: 0

Quantization in fibering polarizations, Mabuchi rays and geometric Peter–Weyl theorem 纤维偏振中的量子化、马布奇射线和几何彼得-韦尔定理

IF 1.6 3区数学 Q1 MATHEMATICS

Journal of Geometry and Physics

Pub Date : 2024-10-31 DOI: 10.1016/j.geomphys.2024.105355

Thomas Baier , Joachim Hilgert , Oguzhan Kaya , José M. Mourão , João P. Nunes

In this paper we use techniques of geometric quantization to give a geometric interpretation of the Peter–Weyl theorem. We present a novel approach to half-form corrected geometric quantization in a specific type of non-Kähler polarizations and study one important class of examples, namely cotangent bundles of compact connected Lie groups K. Our main results state that this canonically defined polarization occurs in the geodesic boundary of the space of

K \times K

-invariant Kähler polarizations equipped with Mabuchi's metric, and that its half-form corrected quantization is isomorphic to the Kähler case. An important role is played by invariance of the limit polarization under a torus action.

Unitary parallel transport on the bundle of quantum states along a specific Mabuchi geodesic, given by the coherent state transform of Hall, relates the non-commutative Fourier transform for K with the Borel–Weil description of irreducible representations of K.

在本文中，我们利用几何量子化技术对彼得-韦尔定理进行了几何解释。我们提出了一种在特定类型的非凯勒极化中进行半形校正几何量子化的新方法，并研究了一类重要的例子，即紧凑连通李群 K 的共切束。我们的主要结果表明，这种规范定义的极化出现在配备马渊度量的 K×K 不变凯勒极化空间的大地边界中，其半形校正量子化与凯勒情况同构。沿特定马渊测地线的量子态束上的单元平行传输由霍尔的相干态变换给出，它将 K 的非交换傅里叶变换与 K 的不可还原表征的伯尔-韦尔描述联系起来。

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引用次数: 0

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