scholarly journals CLASSICAL TENSORS AND QUANTUM ENTANGLEMENT I: PURE STATES

2010 ◽  
Vol 07 (03) ◽  
pp. 485-503 ◽  
Author(s):  
P. ANIELLO ◽  
J. CLEMENTE-GALLARDO ◽  
G. MARMO ◽  
G. F. VOLKERT
Keyword(s):  
Hilbert Space ◽  
Symplectic Geometry ◽  
Vector Fields ◽  
Riemannian Metric ◽  
Inner Product ◽  
Tensor Fields ◽  
Metric Tensor ◽  
Separable States ◽  
Pure States ◽  
Complex Projective

The geometrical description of a Hilbert space associated with a quantum system considers a Hermitian tensor to describe the scalar inner product of vectors which are now described by vector fields. The real part of this tensor represents a flat Riemannian metric tensor while the imaginary part represents a symplectic two-form. The immersion of classical manifolds in the complex projective space associated with the Hilbert space allows to pull-back tensor fields related to previous ones, via the immersion map. This makes available, on these selected manifolds of states, methods of usual Riemannian and symplectic geometry. Here, we consider these pulled-back tensor fields when the immersed submanifold contains separable states or entangled states. Geometrical tensors are shown to encode some properties of these states. These results are not unrelated with criteria already available in the literature. We explicitly deal with some of these relations.

Differential geometry

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Differential Geometry ◽  
Vector Fields ◽  
Differential Forms ◽  
Curvilinear Coordinates ◽  
Moving Frames ◽  
Tensor Fields ◽  
Metric Tensor ◽  
Vector Calculus ◽  
Vector Version ◽  
Definition Of

This chapter presents some elements of differential geometry, the ‘vector’ version of Euclidean geometry in curvilinear coordinates. In doing so, it provides an intrinsic definition of the covariant derivative and establishes a relation between the moving frames attached to a trajectory introduced in Chapter 2 and the moving frames of Cartan associated with curvilinear coordinates. It illustrates a differential framework based on formulas drawn from Chapter 2, before discussing cotangent spaces and differential forms. The chapter then turns to the metric tensor, triads, and frame fields as well as vector fields, form fields, and tensor fields. Finally, it performs some vector calculus.

scholarly journals On Differential Equations Derived from the Pseudospherical Surfaces

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Hongwei Yang ◽  
Xiangrong Wang ◽  
Baoshu Yin
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Gordon Equation ◽  
Curvature Scalar ◽  
Riemann Curvature ◽  
Tensor Fields ◽  
Nonlinear Superposition ◽  
Metric Tensor ◽  
New Exact Solutions ◽  
Pseudospherical Surfaces

We construct two metric tensor fields; by means of these metric tensor fields, sinh-Gordon equation and elliptic sinh-Gordon equation are obtained, which describe pseudospherical surfaces of constant negative Riemann curvature scalarσ= −2,σ= −1, respectively. By employing the Bäcklund transformation, nonlinear superposition formulas of sinh-Gordon equation and elliptic sinh-Gordon equation are derived; various new exact solutions of the equations are obtained.

scholarly journals New Constraints on Preferred Frame Effects from Binary Pulsars

2012 ◽  
Vol 8 (S291) ◽  
pp. 496-498 ◽  
Author(s):  
Lijing Shao ◽  
Norbert Wex ◽  
Michael Kramer
Keyword(s):  
Strong Field ◽  
Weak Field ◽  
Optical Observations ◽  
Orbital Eccentricity ◽  
Tensor Fields ◽  
Metric Tensor ◽  
Preferred Frame ◽  
Binary Pulsars ◽  
Canonical Metric ◽  
Ppn Parameters

AbstractPreferred frame effects (PFEs) are predicted by a number of alternative gravity theories which include vector or additional tensor fields, besides the canonical metric tensor. In the framework of parametrized post-Newtonian (PPN) formalism, we investigate PFEs in the orbital dynamics of binary pulsars, characterized by the two strong-field PPN parameters, and . In the limit of a small orbital eccentricity, and contributions decouple. By utilizing recent radio timing results and optical observations of PSRs J1012+5307 and J1738+0333, we obtained the best limits of and in the strong-field regime. The constraint on also surpasses its counterpart in the weak-field regime.

Estimates for the constant in two nonlinear Korn inequalities

2020 ◽  
pp. 108128652095770
Author(s):  
Maria Malin ◽  
Cristinel Mardare
Keyword(s):  
Euclidean Space ◽  
Open Subset ◽  
Rigid Motion ◽  
Tensor Fields ◽  
Metric Tensor ◽  
Square Roots ◽  
Korn Inequality ◽  
New Inequalities

A nonlinear Korn inequality estimates the distance between two immersions from an open subset of [Formula: see text] into the Euclidean space [Formula: see text], [Formula: see text], in terms of the distance between specific tensor fields that determine the two immersions up to a rigid motion in [Formula: see text]. We establish new inequalities of this type in two cases: when k = n, in which case the tensor fields are the square roots of the metric tensor fields induced by the two immersions, and when k = 3 and n = 2, in which case the tensor fields are defined in terms of the fundamental forms induced by the immersions. These inequalities have the property that their constants depend only on the open subset over which the immersions are defined and on three scalar parameters defining the regularity of the immersions, instead of constants depending on one of the immersions, considered as fixed, as up to now.

A NEW ALGORITHM FOR COMPUTING RIEMANNIAN GEODESIC DISTANCE IN RECTANGULAR 2-D AND 3-D GRIDS

2013 ◽  
Vol 22 (06) ◽  
pp. 1360020
Author(s):  
OLA NILSSON ◽  
MARTIN REIMERS ◽  
KEN MUSETH ◽  
ANDERS BRUN
Keyword(s):  
Diffusion Tensor ◽  
Geodesic Distance ◽  
Three Dimensional ◽  
Conference Paper ◽  
Tensor Fields ◽  
Metric Tensor ◽  
Benchmark Study ◽  
Rectangular Grids ◽  
And Diffusion ◽  
Dimensional Domain

We present a novel way to efficiently compute Riemannian geodesic distance over a two- or three-dimensional domain. It is based on a previously presented method for computation of geodesic distances on surface meshes. Our method is adapted for rectangular grids, equipped with a variable anisotropic metric tensor. Processing and visualization of such tensor fields is common in certain applications, for instance structure tensor fields in image analysis and diffusion tensor fields in medical imaging. The included benchmark study shows that our method provides significantly better results in anisotropic regions in 2-D and 3-D and is faster than current stat-of-the-art solvers in 2-D grids. Additionally, our method is straightforward to code; the test implementation is less than 150 lines of C++ code. The paper is an extension of a previously presented conference paper and includes new sections on 3-D grids in particular.

scholarly journals A note on recurrent bivectors in 4-dimensional Lorentz Manifolds

2018 ◽  
Vol 103 (117) ◽  
pp. 103-112
Author(s):  
Bahar Kırık
Keyword(s):  
Classification Scheme ◽  
Holonomy Group ◽  
Second Order ◽  
Dimensional Manifold ◽  
Tensor Fields ◽  
Metric Tensor ◽  
Lorentz Manifolds

We study recurrence properties of the second order skew-sym- metric tensor fields, which are referred to as bivectors, on a 4-dimensional manifold admitting a Lorentz metric. Considering the known classification scheme for these tensor fields, recurrent bivectors which can be scaled to be parallel are first determined and these results are associated with the holonomy theory. This examination then identifies proper recurrence of such bivectors on the manifold. The link between these bivectors and the holonomy group is investigated and some theorems are proved.

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