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Conservation of Gravitational Energy-Momentum and Inner Diffeomorphism Group Gauge Invariance
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作者 Christian Wiesendanger 《Journal of Modern Physics》 2013年第8期37-47,共11页
Viewing gravitational energy momentum as equal by observation, but different in essence from inertial energy-momentum requires two different symmetries to account for their independent conservations—spacetime and inn... Viewing gravitational energy momentum as equal by observation, but different in essence from inertial energy-momentum requires two different symmetries to account for their independent conservations—spacetime and inner translation invariance. Gauging the latter a generalization of non-Abelian gauge theories of compact Lie groups is developed resulting in the gauge theory of the non-compact group of volume-preserving diffeomorphisms of an inner Minkowski space M4. As usual the gauging requires the introduction of a covariant derivative, a gauge field and a field strength operator. An invariant and minimal gauge field Lagrangian is derived. The classical field dynamics and the conservation laws for the new gauge theory are developed. Finally, the theory’s Hamiltonian in the axial gauge is expressed by two times six unconstrained independent canonical variables obeying the usual Poisson brackets and the positivity of the Hamiltonian is related to a condition on the support of the gauge fields. 展开更多
关键词 GAUGE Field Theory volume-preserving diffeomorphism group INNER MINKOWSKI Space
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General Relativity as the Classical Limit of the Renormalizable Gauge Theory of Volume Preserving Diffeomorphisms
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作者 Christian Wiesendanger 《Journal of Modern Physics》 2014年第10期948-958,共11页
The different roles and natures of spacetime appearing in a quantum field theory and in classical physics are analyzed implying that a quantum theory of gravitation is not necessarily a quantum theory of curved spacet... The different roles and natures of spacetime appearing in a quantum field theory and in classical physics are analyzed implying that a quantum theory of gravitation is not necessarily a quantum theory of curved spacetime. Developing an alternative approach to quantum gravity starts with the postulate that inertial energy-momentum and gravitational energy-momentum need not be the same for virtual quantum states. Separating their roles naturally leads to the quantum gauge field theory of volume-preserving diffeomorphisms of an inner four-dimensional space. The classical limit of this theory coupled to a quantized scalar field is derived for an on-shell particle where inertial energy-momentum and gravitational energy-momentum coincide. In that process the symmetry under volume-preserving diffeomorphisms disappears and a new symmetry group emerges: the group of coordinate transformations of four-dimensional spacetime and with it General Relativity coupled to a classical relativistic point particle. 展开更多
关键词 QUANTUM Gravity QUANTUM Gauge Theory of volume-preserving diffeomorphism group GR Emerging AS the Classical LIMIT of Above Different Roles of Inertial and Gravitational Momentum Observability of Spacetime at Microscopic Level
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A Renormalizable Theory of Quantum Gravity: Renormalization Proof of the Gauge Theory of Volume Preserving Diffeomorphisms
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作者 Christian Wiesendanger 《Journal of Modern Physics》 2014年第10期959-983,共25页
Inertial and gravitational mass or energy momentum need not be the same for virtual quantum states. Separating their roles naturally leads to the gauge theory of volume-preserving diffeomorphisms of an inner four-dime... Inertial and gravitational mass or energy momentum need not be the same for virtual quantum states. Separating their roles naturally leads to the gauge theory of volume-preserving diffeomorphisms of an inner four-dimensional space. The gauge-fixed action and the path integral measure occurring in the generating functional for the quantum Green functions of the theory are shown to obey a BRST-type symmetry. The related Zinn-Justin-type equation restricting the corresponding quantum effective action is established. This equation limits the infinite parts of the quantum effective action to have the same form as the gauge-fixed Lagrangian of the theory proving its spacetime renormalizability. The inner space integrals occurring in the quantum effective action which are divergent due to the gauge group’s infinite volume are shown to be regularizable in a way consistent with the symmetries of the theory demonstrating as a byproduct that viable quantum gauge field theories are not limited to finite-dimensional compact gauge groups as is commonly assumed. 展开更多
关键词 RENORMALIZATION PROOF of GAUGE Field THEORY of volume-preserving diffeomorphismS
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Bounding Topology via Geometry,A-Simple Fundamental Groups
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作者 Xiaochun Rong Xuchao Yao 《Communications in Mathematical Research》 CSCD 2020年第4期489-505,共17页
We call a group A-simple,if it has no non-trivial normal abelian subgroup.We will present finiteness results in controlled topology via geometry on manifolds whose fundamental groups are A-simple.
关键词 A-simple fundamental group collapsing with bounded sectional curvature finiteness of fundamental groups and diffeomorphic types.
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A Note on the Stability of Geodesics on Diffeomorphism Groups with One-side Invariant Metric
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作者 Li Juan ZHOU 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2020年第5期620-630,共11页
In this note,we consider the stability of geodesics on volume-preserving diffeomorphism groups with one-side invariant metric.We showed that for non-Beltrami fields on a three-dimensional compact manifold,there does n... In this note,we consider the stability of geodesics on volume-preserving diffeomorphism groups with one-side invariant metric.We showed that for non-Beltrami fields on a three-dimensional compact manifold,there does not exist Eulerian stable flow which is Lagrangian exponential unstable.We noticed that a stationary flow corresponding to the KdV equation can be Eulerian stable while the corresponding motion of the fluid is at most exponentially unstable. 展开更多
关键词 volume-preserving diffeomorphism group Bott-Virasoro group Euler equation GEODESIC Eulerian INSTABILITY Lagrangian INSTABILITY
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数学物理中的变换群理论
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作者 冯承天 《上海师范大学学报(自然科学版)》 2000年第3期43-45,共3页
较系统地阐明了变换群的基本概念。
关键词 变换群 小群 微分间胚 李群 可迁作用 数学物理
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球面以及相关几何(英文)
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作者 王庆 《苏州科技学院学报(自然科学版)》 CAS 2013年第3期5-8,共4页
主要研究球面以及相关几何,特别讨论了球面S3和S7。研究结果表明,特殊正交群SO(4)和SO(8)各自微分同胚于空间S3×SO(3)和S7×SO(7)。
关键词 球面 特殊正交群 微分同胚
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基于Demons的微分同胚非刚性配准研究 被引量:5
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作者 徐挺 刘伟 +1 位作者 李传富 冯焕清 《北京生物医学工程》 2010年第1期49-54,共6页
Demons算法的一个局限是它无法处理大形变,且不能产生微分同胚的变换以满足计算解剖学的形态分析需要。利用李群中的指数映射,把原来Demons形变场相加的更新方式改进为若干次形变场间的复合,同时又保证了较高的运算效率。实验表明,新算... Demons算法的一个局限是它无法处理大形变,且不能产生微分同胚的变换以满足计算解剖学的形态分析需要。利用李群中的指数映射,把原来Demons形变场相加的更新方式改进为若干次形变场间的复合,同时又保证了较高的运算效率。实验表明,新算法能配准大形变问题,且真实颅脑CT的实验结果与Demons算法的结果相近,但产生的形变场光滑可逆,并有相对更小的形变能量。 展开更多
关键词 非刚性配准 微分同胚 指数映射 优化
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基于自适应切空间的MRI图像配准
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作者 刘薇 陈雷霆 《计算机应用》 CSCD 北大核心 2017年第4期1193-1197,共5页
微分同胚是一种光滑可逆的变换,在MRI图像配准中可以保证图像形变后的拓扑结构保持不变,同时避免出现不合理的物理现象。为了在空间变换中获得更合理的同胚映射,高维空间中数据的非线性结构被考虑,基于流形学习方法提出一种自适应切空间... 微分同胚是一种光滑可逆的变换,在MRI图像配准中可以保证图像形变后的拓扑结构保持不变,同时避免出现不合理的物理现象。为了在空间变换中获得更合理的同胚映射,高维空间中数据的非线性结构被考虑,基于流形学习方法提出一种自适应切空间的MRI图像配准算法。首先,把MRI数据构造成对称正定(SPD)的协方差矩阵,然后形成李群;接着,利用样本点邻域的局部切空间来表示李群的几何结构的非线性;接下来,在流形上用自适应邻域选择的方法形成的线性子空间去逼近局部切空间,提高切空间的局部线性化程度,从而最大限度地保留流形的局部非线性结构,得到最优的同胚映射。仿真数据和临床数据的实验结果显示,与传统的非参数微分同胚配准算法相比,该算法在高维稠密形变场上获得更高的拓扑保持度,最终提高图像配准精度。 展开更多
关键词 微分同胚 切空间 李群 邻域选择 MRI图像配准
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A generalized π2-diffeomorphism finiteness theorem 被引量:1
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作者 Xiaochun RONG Xuchao YAO 《Frontiers of Mathematics in China》 SCIE CSCD 2020年第2期399-418,共20页
Theπ2-diffeomorphism finiteness result of F.Fang-X.Rong and A.Petrunin-W.Tuschmann(independently)asserts that the diffeomorphic types of compact n-manifolds M with vanishing first and second homotopy groups can be bo... Theπ2-diffeomorphism finiteness result of F.Fang-X.Rong and A.Petrunin-W.Tuschmann(independently)asserts that the diffeomorphic types of compact n-manifolds M with vanishing first and second homotopy groups can be bounded above in terms of n,and upper bounds on the absolute value of sectional curvature and diameter of M.In this paper,we will generalize thisπ2-diffeomorphism finiteness by removing the condition thatπ1(M)-0 and asserting the diffeomorphism finiteness on the Riemannian universal cover of M. 展开更多
关键词 Collapsing with bounded sectional curvature diffeomorphism finiteness vanishing second homotopy group
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微分同胚群Diff(s)在参数长度不固定情形下的推广
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作者 李秀林 《杭州师范学院学报》 1993年第3期54-58,共5页
当把弦的参数长度从固定的情形推广到不固定的情形时,我们发现必须把微分同胚群Diff(s)推广到某个类群G,此类群的商代数正同构于Diff(s),且Diff(s)是此类群G的一个子群.
关键词 弦的参数长度 微分同胚群 类群 商代数 搭桥元素 微分同胚映射
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Conservation of Gravitational Energy Momentum and Renormalizable Quantum Theory of Gravitation
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作者 Christian Wiesendanger 《Journal of Modern Physics》 2013年第8期133-152,共20页
Viewing gravitational energy-momentum as equal by observation, but different in essence from inertial energymomentum naturally leads to the gauge theory of volume-preserving diffeomorphisms of an inner Minkowski space... Viewing gravitational energy-momentum as equal by observation, but different in essence from inertial energymomentum naturally leads to the gauge theory of volume-preserving diffeomorphisms of an inner Minkowski space which can describe gravitation at the classical level. This theory is quantized in the path integral formalism starting with a non-covariant Hamiltonian formulation with unconstrained canonical field variables and a manifestly positive Hamiltonian. The relevant path integral measure and weight are then brought into a Lorentz- and gauge-covariant form allowing to express correlation functions—applying the De Witt-Faddeev-Popov approach—in any meaningful gauge. Next the Feynman rules are developed and the quantum effective action at one loop in a background field approach is renormalized which results in an asymptotically free theory without presence of other fields and in a theory without asymptotic freedom including the Standard Model (SM) fields. Finally the BRST apparatus is developed as preparation for the renormalizability proof to all orders and a sketch of this proof is given. 展开更多
关键词 Path Integral QUANTIZATION GAUGE Theory volume-preserving diffeomorphismS
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Scattering Cross-Sections in Quantum Gravity—The Case of Matter-Matter Scattering
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作者 Christian Wiesendanger 《Journal of Modern Physics》 2015年第3期273-282,共10页
Viewing gravitational energy-momentum PG<sup style='margin-left:-7px;'>μ as equal by observation, but different in essence from inertial energy-momentum PI<sup style='margin-left:-7px;'>μ... Viewing gravitational energy-momentum PG<sup style='margin-left:-7px;'>μ as equal by observation, but different in essence from inertial energy-momentum PI<sup style='margin-left:-7px;'>μ naturally leads to the gauge theory of volume-preserving diffeomorphisms of a four-dimensional inner space. To analyse scattering in this theory, the gauge field is coupled to two Dirac fields with different masses. Based on a generalized LSZ reduction formula the S-matrix element for scattering of two Dirac particles in the gravitational limit and the corresponding scattering cross-section are calculated to leading order in perturbation theory. Taking the non-relativistic limit for one of the initial particles in the rest frame of the other the Rutherford-like cross-section of a non-relativistic particle scattering off an infinitely heavy scatterer calculated quantum mechanically in Newtonian gravity is recovered. This provides a non-trivial test of the gauge field theory of volume-preserving diffeomorphisms as a quantum theory of gravity. 展开更多
关键词 Renormalizable QUANTUM GRAVITY SCATTERING CROSS-SECTIONS in QUANTUM GRAVITY Gauge Theory of volume-preserving diffeomorphismS
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用对称原理构造参量长度不固定的玻色开弦和闭弦的相互作用量
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作者 李秀林 《杭州师范学院学报》 1992年第3期31-39,共9页
文献[3]利用一些对称原理构造了玻色开弦和闭弦的相互作用量,并研究了弦的参数长度固定的情形。本文则沿着文献[3]的思路,把它推广到弦的参数长度不固定的情形。
关键词 玻色开弦和闭弦 参量长度 微分同胚群 相互作用量
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Longitudinal Image Analysis via Path Regression on the Image Manifold 被引量:1
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作者 Shi-Hui Ying Xiao-Fang Zhang +1 位作者 Ya-Xin Peng Ding-Gang Shen 《Journal of the Operations Research Society of China》 EI CSCD 2019年第4期599-614,共16页
Longitudinal image analysis plays an important role in depicting the development of the brain structure,where image regression and interpolation are two commonly used techniques.In this paper,we develop an efficient m... Longitudinal image analysis plays an important role in depicting the development of the brain structure,where image regression and interpolation are two commonly used techniques.In this paper,we develop an efficient model and approach based on a path regression on the image manifold instead of the geodesic regression to avoid the complexity of the geodesic computation.Concretely,first we model the deformation by diffeomorphism;then,a large deformation is represented by a path on the orbit of the diffeomorphism group action.This path is obtained by compositing several small deformations,which can be well approximated by its linearization.Second,we introduce some intermediate images as constraints to the model,which guides to form the best-fitting path.Thirdly,we propose an approximated quadratic model by local linearization method,where a closed form is deduced for the solution.It actually speeds up the algorithm.Finally,we evaluate the proposed model and algorithm on a synthetic data and a real longitudinal MRI data.The results show that our proposed method outperforms several state-of-the-art methods. 展开更多
关键词 Longitudinal image analysis Path regression diffeomorphism group Image registration Infant brain development
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Univalent Functions and Diff(S^1)/S^1
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作者 Simon DAVIS 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2012年第6期1233-1236,共4页
The relation between Diff(S^1)/S^1 and the space of univalent analytic functions on the disk is elucidated and shown to provide upper bounds for the volumes of exhaustive approximations to an analytic submanifold of... The relation between Diff(S^1)/S^1 and the space of univalent analytic functions on the disk is elucidated and shown to provide upper bounds for the volumes of exhaustive approximations to an analytic submanifold of an infinite-dimensional space. The maximum magnitudes of the coefficients in the series expansions of univalent superanalytic functions on the superdisk are inferred. 展开更多
关键词 UNIVALENT diffeomorphism group superanalytic functions
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Twistor quantization of the space of half-differentiable vector functions on the circle revisited
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作者 SERGEEV Armen 《Science China Mathematics》 SCIE 2009年第12期2714-2729,共16页
We discuss the twistor quantization problem for the classical system (V d ,A d ), represented by the phase space V d , identified with the Sobolev space H 0 1/2 (S 1,? d ) of half-differentiable vector functions on th... We discuss the twistor quantization problem for the classical system (V d ,A d ), represented by the phase space V d , identified with the Sobolev space H 0 1/2 (S 1,? d ) of half-differentiable vector functions on the circle, and the algebra of observables A d , identified with the semi-direct product of the Heisenberg algebra of V d and the algebra Vect(S 1) of tangent vector fields on the circle. 展开更多
关键词 twistor quantization Sobolev space of half-differentiable functions group of diffeomorphisms of the circle 58E20 53C28 32L25
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On 1-connected 8-manifolds with the Same Homology as S^(3)×S^(5)
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作者 Xue Qi WANG 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2021年第6期941-956,共16页
In this article,we classify 1-connected 8-dimensional Poincarécomplexes,topological manifolds and smooth manifolds whose integral homology groups are isomorphic to those of S^(3)×S^(5).A topic related to a p... In this article,we classify 1-connected 8-dimensional Poincarécomplexes,topological manifolds and smooth manifolds whose integral homology groups are isomorphic to those of S^(3)×S^(5).A topic related to a paper of Escher and Ziller is also discussed. 展开更多
关键词 8-manifolds Poincarécomplexes diffeomorphism groups S^(1)-bundles
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On a Lagrangian Formulation of the Incompressible Euler Equation
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作者 INCI Hasan 《Journal of Partial Differential Equations》 CSCD 2016年第4期320-359,共40页
In this paper we show that the incompressible Euler equation on the Sobolev space H^S(R^n), s 〉 n/2+1, can be expressed in Lagrangian coordinates as a geodesic equation on an infinite dimensional manifold. Moreove... In this paper we show that the incompressible Euler equation on the Sobolev space H^S(R^n), s 〉 n/2+1, can be expressed in Lagrangian coordinates as a geodesic equation on an infinite dimensional manifold. Moreover the Christoffel map describing the geodesic equation is real analytic. The dynamics in Lagrangian coordinates is described on the group of volume preserving diffeomorphisms, which is an ana- lytic submanifold of the whole diffeomorphism group. Furthermore it is shown that a Sobolev class vector field integrates to a curve on the diffeomorphism group. 展开更多
关键词 Euler equation diffeomorphism group.
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