About: Tensor density is a research topic. Over the lifetime, 2988 publications have been published within this topic receiving 73870 citations.
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TL;DR: In this paper, the boundary stress tensor associated with a gravitating system in asymptotically anti-de Sitter space is computed, and the conformal anomalies in two and four dimensions are recovered.
Abstract: We propose a procedure for computing the boundary stress tensor associated with a gravitating system in asymptotically anti-de Sitter space. Our definition is free of ambiguities encountered by previous attempts, and correctly reproduces the masses and angular momenta of various spacetimes. Via the AdS/CFT correspondence, our classical result is interpretable as the expectation value of the stress tensor in a quantum conformal field theory. We demonstrate that the conformal anomalies in two and four dimensions are recovered. The two dimensional stress tensor transforms with a Schwarzian derivative and the expected central charge. We also find a nonzero ground state energy for global AdS5, and show that it exactly matches the Casimir energy of the dual super Yang–Mills theory on S 3×R.
TL;DR: It is shown that eigenvalues are roots of a one-dimensional polynomial, and when the order of the tensor is even, E-eigenvaluesare roots of another one- dimensional polynomials associated with the symmetric hyperdeterminant.
Abstract: In this paper, we define the symmetric hyperdeterminant, eigenvalues and E-eigenvalues of a real supersymmetric tensor. We show that eigenvalues are roots of a one-dimensional polynomial, and when the order of the tensor is even, E-eigenvalues are roots of another one-dimensional polynomial. These two one-dimensional polynomials are associated with the symmetric hyperdeterminant. We call them the characteristic polynomial and the E-characteristic polynomial of that supersymmetric tensor. Real eigenvalues (E-eigenvalues) with real eigenvectors (E-eigenvectors) are called H-eigenvalues (Z-eigenvalues). When the order of the supersymmetric tensor is even, H-eigenvalues (Z-eigenvalues) exist and the supersymmetric tensor is positive definite if and only if all of its H-eigenvalues (Z-eigenvalues) are positive. An mth-order n-dimensional supersymmetric tensor where m is even has exactly n(m-1)^n^-^1 eigenvalues, and the number of its E-eigenvalues is strictly less than n(m-1)^n^-^1 when m>=4. We show that the product of all the eigenvalues is equal to the value of the symmetric hyperdeterminant, while the sum of all the eigenvalues is equal to the sum of the diagonal elements of that supersymmetric tensor, multiplied by (m-1)^n^-^1. The n(m-1)^n^-^1 eigenvalues are distributed in n disks in C. The centers and radii of these n disks are the diagonal elements, and the sums of the absolute values of the corresponding off-diagonal elements, of that supersymmetric tensor. On the other hand, E-eigenvalues are invariant under orthogonal transformations.
TL;DR: In this paper, it was shown that the matrix elements of the symmetric energy-momentum tensor are cut-off dependent in renormalized perturbation theory for most renormalizable field theories.
Abstract: We show that the matrix elements of the conventional symmetric energy-momentum tensor are cut-off dependent in renormalized perturbation theory for most renormalizable field theories. However, we argue that, for any renormalizable field theory, it is possible to construct a new energy-momentum tensor, such that the new tensor defines the same four-momentum and Lorentz generators as the conventional tensor, and, further, has finite matrix elements in every order of renormalized perturbation theory. (“Finite” means independent of the cut-off in the limit of large cut-off.) We explicitly construct this tensor in the most general case. The new tensor is an improvement over the old for another reason: the currents associated with scale transformations and conformal transformations have very simple expressions in terms of the new tensor, rather than the very complicated ones they have in terms of the old. We also show how to alter general relativity in such a way that the new tensor becomes the source of the gravitational field, and demonstrate that the new gravitation theory obtained in this way meets all the epxerimental tests that have been applied to general relativity.
TL;DR: In this article, the phase relationship contained in the magnetotelluric (MT) impedance tensor is shown to be a second-rank tensor and the phase tensor can be depicted graphically as an ellipse, the major and minor axes representing the principal axes of the tensor.
Abstract: SUMMARY The phase relationships contained in the magnetotelluric (MT) impedance tensor are shown to be a second-rank tensor. This tensor expresses how the phase relationships change with polarization in the general case where the conductivity structure is 3-D. Where galvanic effects produced by heterogeneities in near-surface conductivity distort the regional MT response the phase tensor preserves the regional phase information. Calculation of the phase tensor requires no assumption about the dimensionality of the underlying conductivity distribution and is applicable where both the heterogeneity and regional structure are 3-D. For 1-D regional conductivity structures, the phase tensor is characterized by a single coordinate invariant phase equal to the 1-D impedance tensor phase. If the regional conductivity structure is 2-D, the phase tensor is symmetric with one of its principal axes aligned parallel to the strike axis of the regional structure. In the 2-D case, the principal values (coordinate invariants) of the phase tensor are the transverse electric and magnetic polarization phases. The orientation of the phase tensor’s principal axes can be determined directly from the impedance tensor components in both 2-D and 3-D situations. In the 3-D case, the phase tensor is nonsymmetric and has a third coordinate invariant that is a distortion-free measure of the asymmetry of the regional MT response. The phase tensor can be depicted graphically as an ellipse, the major and minor axes representing the principal axes of the tensor. 3-D model studies show that the orientations of the phase tensor principal axes reflect lateral variations (gradients) in the underlying regional conductivity structure. Maps of the phase tensor ellipses provide a method of visualizing this variation.
TL;DR: In this article, the correctness of appropriate string diagrams for various kinds of monoidal categories with duals has been proved for various classes of classes of subject classes, including algebra, geometry, physics, and astronomy.
Abstract: This paper defines and proves the correctness of the appropriate string diagrams for various kinds of monoidal categories with duals. Mathematics Subject Classifications (1991). 18D10, 52B11, 53A45 , 57M25, 68Q10, 82B23.
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