The quantum metric tensor was introduced for defining the distance in the parameter space of a system. However, it is also useful for other purposes, like predicting quantum phase transitions. Due to the physical info...The quantum metric tensor was introduced for defining the distance in the parameter space of a system. However, it is also useful for other purposes, like predicting quantum phase transitions. Due to the physical information this tensor provides, its gauge independence sounds reasonable. Moreover, its original construction was made by looking for this gauge independence. The aim of this paper, however, is to prove that the quantum metric tensor does depend on the gauge. In addition, a real gauge invariant quantum metric tensor is introduced. A related concept is the quantum fidelity, which is also shown to depend on the gauge in this paper. The gauge dependences are explicitly shown by computing the quantum metric tensor and the quantum fidelity of the Landau problem in different gauges. Then, a real gauge independent metric tensor is proposed and computed for the same Landau problem. Since the gauge dependences have not been observed before, the results of this paper might lead to a new study of topics that are believed to be completely understood.展开更多
In the general theory of relativity, the fundamental metric tensor plays a special role, which has its physical basis in the peculiar aspects of gravitation. The fundamental property of gravitational fields provides t...In the general theory of relativity, the fundamental metric tensor plays a special role, which has its physical basis in the peculiar aspects of gravitation. The fundamental property of gravitational fields provides the possibility of establishing an analogy between the motion in a gravitational field and the motion in any external field considered as a noninertial system of reference. Thus, the properties of the motion in a noninertial frame are the same as those in an inertial system in the presence of a gravitational field. In other words, a noninertial frame of reference is equivalent to a certain gravitational field. This is known as the principle of equivalence. From the mathematical viewpoint, the same special role can be played by the small deformation strain tensor, which describes the geometrical properties of any region deformed because of the effect of some external agent. It can be proved that, from that tensor, all the mathematical structures needed in the general theory of relativity can be constructed.展开更多
Majorana's stellar representation provides an intuitive picture in which quantum states in highdimensional Hilbert space can be observed using the trajectory of Majorana stars.We consider the Majorana's stella...Majorana's stellar representation provides an intuitive picture in which quantum states in highdimensional Hilbert space can be observed using the trajectory of Majorana stars.We consider the Majorana's stellar representation of the quantum geometric tensor for a spin state up to spin-3/2.The real and imaginary parts of the quantum geometric tensor,corresponding to the quantum metric tensor and Berry curvature,are therefore obtained in terms of the Majorana stars.Moreover,we work out the expressions of quantum geometric tensor for arbitrary spin in some important cases.Our results will benefit the comprehension of the quantum geometric tensor and provide interesting relations between the quantum geometric tensor and Majorana's stars.展开更多
The authors discuss the existence and uniqueness up to isometries of Enof immersions φ : Ω R^n→ E^n with prescribed metric tensor field(g ij) : Ω→ S^n>, and discuss the continuity of the mapping(gij) →φ d...The authors discuss the existence and uniqueness up to isometries of Enof immersions φ : Ω R^n→ E^n with prescribed metric tensor field(g ij) : Ω→ S^n>, and discuss the continuity of the mapping(gij) →φ defined in this fashion with respect to various topologies. In particular, the case where the function spaces have little regularity is considered. How, in some cases, the continuity of the mapping(gij) →φ can be obtained by means of nonlinear Korn inequalities is shown.展开更多
Let x : M → R n be an umbilical free hypersurface with non-zero principal curvatures, then x is associated with a Laguerre metric g, a Laguerre tensor L, a Laguerre form C, a Laguerre second fundamental form B, which...Let x : M → R n be an umbilical free hypersurface with non-zero principal curvatures, then x is associated with a Laguerre metric g, a Laguerre tensor L, a Laguerre form C, a Laguerre second fundamental form B, which are invariants of x under Laguerre transformation group. A classical theorem of Laguerre geometry states that M(n > 3) is characterized by g and B up to Laguerre equivalence. A Laguerre isopararmetric hypersurface is defined by satisfying the conditions that C = 0 and all the eigenvalues of B with respect to g are constant. It is easy to see that all Laguerre isopararmetric hypersurfaces are Dupin hypersurfaces. In this paper, we established a complete classification for all Laguerre isopararmetric hypersurfaces with three distinct principal curvatures in R7.展开更多
This note provides an explicit proof of the equivalence of Belinfante's energy-momentum tensor and metric energy-momentum tensor for general mixed tensor-spinor fields.
We have obtained exact static plane-symmetric solutions to the spinor field equations with nonlinear terms which are arbitrary functions of invariant , taking into account their own gravitational field. It is shown th...We have obtained exact static plane-symmetric solutions to the spinor field equations with nonlinear terms which are arbitrary functions of invariant , taking into account their own gravitational field. It is shown that the initial set of the Einstein and spinor field equations with a power-law nonlinearity have regular solutions with a localized energy density of the spinor field only if m=0 (m is the mass parameter in the spinor field equations). Equations with power and polynomial nonlinearities are studied in detail. In this case, a soliton-like configuration has negative energy. We have also obtained exact static plane-symmetric solutions to the above spinor field equations in flat space-time. It is proved that in this case soliton-like solutions are absent.展开更多
By studying a fully nonlinear flow deforming conformal metrics on a compact and connected manifold, we prove the long time existence and the exponential convergence of the solutions of the flow for any initial metric ...By studying a fully nonlinear flow deforming conformal metrics on a compact and connected manifold, we prove the long time existence and the exponential convergence of the solutions of the flow for any initial metric g0 with the Schouten tensor Ag0 ∈ Γk.展开更多
Einstein’s General relativity theory and Quantum physics are the main pillars for explaining most modern physics. Obtaining these theories relation between them remains a theoretical physics main question. In the las...Einstein’s General relativity theory and Quantum physics are the main pillars for explaining most modern physics. Obtaining these theories relation between them remains a theoretical physics main question. In the last most decades, works are leading to new physical ideas and mathematical tools broad range. In recent years TOUGMA’s equation is established and solved, and one of these solutions, mostly a real solution is studied in our last article. In this work, complex TOUGMA’s metric is studied, such as the physics concepts implied by this metric, mainly material bodies geodesics orbits. We studied the fact material bodies’ orbits and their limits. This study of the underlying principles and various phenomena in universe are interconnected logic leading to new technologies development such as news engines and telecommunication networks. The applications of this study are exceptionally wide such as Astrophysics, cosmology, Quantum gravity, Quantum Mechanics and Multiverse. Mostly this study allows us to know the behaviors of matter in the quantum relativity universe. Universe.展开更多
In the theory of general relativity, the finding of the Einstein Field Equation happens in a complex mathematical operation, a process we don’t need any more. Through a new theory in vector analysis, we’ll see that ...In the theory of general relativity, the finding of the Einstein Field Equation happens in a complex mathematical operation, a process we don’t need any more. Through a new theory in vector analysis, we’ll see that we can calculate the components of the Ricci tensor, Ricci scalar, and Einstein Field Equation directly in an easy way without the need to use general relativity theory hypotheses, principles, and symbols. Formulating the general relativity theory through another theory will make it easier to understand this relativity theory and will help combining it with electromagnetic theory and quantum mechanics easily.展开更多
General relativity of Einstein’s theory and Quantum physics theory are excellent pillars that explain much modern physics. Understanding the relation between these theories is still a theoretical physics central open...General relativity of Einstein’s theory and Quantum physics theory are excellent pillars that explain much modern physics. Understanding the relation between these theories is still a theoretical physics central open question. Over last several decades, works in this direction have led to new physical ideas and mathematical tools broad range. In recent years TOUGMA’s equation is established and solved, one of its solutions such as a real solution is studied in our last article. In this paper, complex TOUGMA’s metric is studied, particularly the physics concepts that this metric implies such as light geodesic and metric’s impacts at r = 0. The first time, we studied the fact of r = 0 and its limits, secondly, we consider a zero length light geodesic is a geodesic and ended by studying it mathematically. These underlying principles study, those various phenomena in universe are interconnected logic leading to develop new technologies for example: news engines, telecommunication networks. This study’s applications are exceptionally wide such as Astrophysics, cosmology, Quantum gravity, Quantum Mechanics, Multiverse. Mostly this study lets us to know the quantum relativity universe behaviors.展开更多
This paper deals with an extension of a previous work [Gravitation & Cosmology, Vol. 4, 1998, pp 107-113] to exact spherical symmetric solutions to the spinor field equations with nonlinear terms which are arbitra...This paper deals with an extension of a previous work [Gravitation & Cosmology, Vol. 4, 1998, pp 107-113] to exact spherical symmetric solutions to the spinor field equations with nonlinear terms which are arbitrary functions of S=ψψ, taking into account their own gravitational field. Equations with power and polynomial nonlinearities are studied in detail. It is shown that the initial set of the Einstein and spinor field equations with a power nonlinearity has regular solutions with spinor field localized energy and charge densities. The total energy and charge are finite. Besides, exact solutions, including soliton-like solutions, to the spinor field equations are also obtained in flat space-time.展开更多
In this paper, we present an efficient approach for unsupervised segmentation of natural and textural images based on the extraction of image features and a fast active contour segmentation model. We address the probl...In this paper, we present an efficient approach for unsupervised segmentation of natural and textural images based on the extraction of image features and a fast active contour segmentation model. We address the problem of textures where neither the gray-level information nor the boundary information is adequate for object extraction. This is often the case of natural images composed of both homogeneous and textured regions. Because these images cannot be in general directly processed by the gray-level information, we propose a new texture descriptor which intrinsically defines the geometry of textures using semi-local image information and tools from differential geometry. Then, we use the popular Kullback-Leibler distance to design an active contour model which distinguishes the background and textures of interest. The existence of a minimizing solution to the proposed segmentation model is proven. Finally, a texture segmentation algorithm based on the Split-Bregrnan method is introduced to extract meaningful objects in a fast way. Promising synthetic and real-world results for gray-scale and color images are presented.展开更多
文摘The quantum metric tensor was introduced for defining the distance in the parameter space of a system. However, it is also useful for other purposes, like predicting quantum phase transitions. Due to the physical information this tensor provides, its gauge independence sounds reasonable. Moreover, its original construction was made by looking for this gauge independence. The aim of this paper, however, is to prove that the quantum metric tensor does depend on the gauge. In addition, a real gauge invariant quantum metric tensor is introduced. A related concept is the quantum fidelity, which is also shown to depend on the gauge in this paper. The gauge dependences are explicitly shown by computing the quantum metric tensor and the quantum fidelity of the Landau problem in different gauges. Then, a real gauge independent metric tensor is proposed and computed for the same Landau problem. Since the gauge dependences have not been observed before, the results of this paper might lead to a new study of topics that are believed to be completely understood.
文摘In the general theory of relativity, the fundamental metric tensor plays a special role, which has its physical basis in the peculiar aspects of gravitation. The fundamental property of gravitational fields provides the possibility of establishing an analogy between the motion in a gravitational field and the motion in any external field considered as a noninertial system of reference. Thus, the properties of the motion in a noninertial frame are the same as those in an inertial system in the presence of a gravitational field. In other words, a noninertial frame of reference is equivalent to a certain gravitational field. This is known as the principle of equivalence. From the mathematical viewpoint, the same special role can be played by the small deformation strain tensor, which describes the geometrical properties of any region deformed because of the effect of some external agent. It can be proved that, from that tensor, all the mathematical structures needed in the general theory of relativity can be constructed.
基金supported by the National Key Research and Development Program of China(Grants No.2017YFA0304202 and No.2017YFA0205700)the NSFC(Grants No.11875231 and No.11935012)the Fundamental Research Funds for the Central Universities through Grant No.2018FZA3005。
文摘Majorana's stellar representation provides an intuitive picture in which quantum states in highdimensional Hilbert space can be observed using the trajectory of Majorana stars.We consider the Majorana's stellar representation of the quantum geometric tensor for a spin state up to spin-3/2.The real and imaginary parts of the quantum geometric tensor,corresponding to the quantum metric tensor and Berry curvature,are therefore obtained in terms of the Majorana stars.Moreover,we work out the expressions of quantum geometric tensor for arbitrary spin in some important cases.Our results will benefit the comprehension of the quantum geometric tensor and provide interesting relations between the quantum geometric tensor and Majorana's stars.
基金supported by a grant from the Research Grants Council of the Hong Kong Special Administration Region,China(Nos.9041637,CiyuU100711)
文摘The authors discuss the existence and uniqueness up to isometries of Enof immersions φ : Ω R^n→ E^n with prescribed metric tensor field(g ij) : Ω→ S^n>, and discuss the continuity of the mapping(gij) →φ defined in this fashion with respect to various topologies. In particular, the case where the function spaces have little regularity is considered. How, in some cases, the continuity of the mapping(gij) →φ can be obtained by means of nonlinear Korn inequalities is shown.
基金Supported by the Department of Education of Hubei Province(B2014281)
文摘Let x : M → R n be an umbilical free hypersurface with non-zero principal curvatures, then x is associated with a Laguerre metric g, a Laguerre tensor L, a Laguerre form C, a Laguerre second fundamental form B, which are invariants of x under Laguerre transformation group. A classical theorem of Laguerre geometry states that M(n > 3) is characterized by g and B up to Laguerre equivalence. A Laguerre isopararmetric hypersurface is defined by satisfying the conditions that C = 0 and all the eigenvalues of B with respect to g are constant. It is easy to see that all Laguerre isopararmetric hypersurfaces are Dupin hypersurfaces. In this paper, we established a complete classification for all Laguerre isopararmetric hypersurfaces with three distinct principal curvatures in R7.
基金the National Natural Science Foundation of China under
文摘This note provides an explicit proof of the equivalence of Belinfante's energy-momentum tensor and metric energy-momentum tensor for general mixed tensor-spinor fields.
文摘We have obtained exact static plane-symmetric solutions to the spinor field equations with nonlinear terms which are arbitrary functions of invariant , taking into account their own gravitational field. It is shown that the initial set of the Einstein and spinor field equations with a power-law nonlinearity have regular solutions with a localized energy density of the spinor field only if m=0 (m is the mass parameter in the spinor field equations). Equations with power and polynomial nonlinearities are studied in detail. In this case, a soliton-like configuration has negative energy. We have also obtained exact static plane-symmetric solutions to the above spinor field equations in flat space-time. It is proved that in this case soliton-like solutions are absent.
基金Research supported by NSFC (10771189 and 10831008)
文摘By studying a fully nonlinear flow deforming conformal metrics on a compact and connected manifold, we prove the long time existence and the exponential convergence of the solutions of the flow for any initial metric g0 with the Schouten tensor Ag0 ∈ Γk.
文摘Einstein’s General relativity theory and Quantum physics are the main pillars for explaining most modern physics. Obtaining these theories relation between them remains a theoretical physics main question. In the last most decades, works are leading to new physical ideas and mathematical tools broad range. In recent years TOUGMA’s equation is established and solved, and one of these solutions, mostly a real solution is studied in our last article. In this work, complex TOUGMA’s metric is studied, such as the physics concepts implied by this metric, mainly material bodies geodesics orbits. We studied the fact material bodies’ orbits and their limits. This study of the underlying principles and various phenomena in universe are interconnected logic leading to new technologies development such as news engines and telecommunication networks. The applications of this study are exceptionally wide such as Astrophysics, cosmology, Quantum gravity, Quantum Mechanics and Multiverse. Mostly this study allows us to know the behaviors of matter in the quantum relativity universe. Universe.
文摘In the theory of general relativity, the finding of the Einstein Field Equation happens in a complex mathematical operation, a process we don’t need any more. Through a new theory in vector analysis, we’ll see that we can calculate the components of the Ricci tensor, Ricci scalar, and Einstein Field Equation directly in an easy way without the need to use general relativity theory hypotheses, principles, and symbols. Formulating the general relativity theory through another theory will make it easier to understand this relativity theory and will help combining it with electromagnetic theory and quantum mechanics easily.
文摘General relativity of Einstein’s theory and Quantum physics theory are excellent pillars that explain much modern physics. Understanding the relation between these theories is still a theoretical physics central open question. Over last several decades, works in this direction have led to new physical ideas and mathematical tools broad range. In recent years TOUGMA’s equation is established and solved, one of its solutions such as a real solution is studied in our last article. In this paper, complex TOUGMA’s metric is studied, particularly the physics concepts that this metric implies such as light geodesic and metric’s impacts at r = 0. The first time, we studied the fact of r = 0 and its limits, secondly, we consider a zero length light geodesic is a geodesic and ended by studying it mathematically. These underlying principles study, those various phenomena in universe are interconnected logic leading to develop new technologies for example: news engines, telecommunication networks. This study’s applications are exceptionally wide such as Astrophysics, cosmology, Quantum gravity, Quantum Mechanics, Multiverse. Mostly this study lets us to know the quantum relativity universe behaviors.
文摘This paper deals with an extension of a previous work [Gravitation & Cosmology, Vol. 4, 1998, pp 107-113] to exact spherical symmetric solutions to the spinor field equations with nonlinear terms which are arbitrary functions of S=ψψ, taking into account their own gravitational field. Equations with power and polynomial nonlinearities are studied in detail. It is shown that the initial set of the Einstein and spinor field equations with a power nonlinearity has regular solutions with spinor field localized energy and charge densities. The total energy and charge are finite. Besides, exact solutions, including soliton-like solutions, to the spinor field equations are also obtained in flat space-time.
基金supported by Swiss National Science Foundation Grant #205320-101621supported by ONR N00014-03-1-0071
文摘In this paper, we present an efficient approach for unsupervised segmentation of natural and textural images based on the extraction of image features and a fast active contour segmentation model. We address the problem of textures where neither the gray-level information nor the boundary information is adequate for object extraction. This is often the case of natural images composed of both homogeneous and textured regions. Because these images cannot be in general directly processed by the gray-level information, we propose a new texture descriptor which intrinsically defines the geometry of textures using semi-local image information and tools from differential geometry. Then, we use the popular Kullback-Leibler distance to design an active contour model which distinguishes the background and textures of interest. The existence of a minimizing solution to the proposed segmentation model is proven. Finally, a texture segmentation algorithm based on the Split-Bregrnan method is introduced to extract meaningful objects in a fast way. Promising synthetic and real-world results for gray-scale and color images are presented.
