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Ricci decomposition
About: Ricci decomposition is a research topic. Over the lifetime, 1972 publications have been published within this topic receiving 45295 citations.
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TL;DR: In this paper, alternative gravitational theories based on Lagrangian densities that depend in a nonlinear way on the Ricci tensor of a metric are considered, and it is shown that, provided certain weak regularity conditions are met, any such theory is equivalent, from the Hamiltonian point of view, to the standard Einstein theory for a new metric.
Abstract: “Alternative gravitational theories” based on Lagrangian densities that depend in a nonlinear way on the Ricci tensor of a metric are considered. It is shown that, provided certain weak regularity conditions are met, any such theory is equivalent, from the Hamiltonian point of view, to the standard Einstein theory for a new metric (which, roughly speaking, coincides with the momentum canonically conjugated to the original metric), interacting with external matterfields whose nature depends on the original Lagrangian density.
245 citations
TL;DR: This work discusses how to incorporate a global symmetry, given by a compact, completely reducible group G, in tensor network decompositions and algorithms, by considering tensors that are invariant under the action of the group G.
Abstract: Tensor network decompositions offer an efficient description of certain many-body states of a lattice system and are the basis of a wealth of numerical simulation algorithms. We discuss how to incorporate a global symmetry, given by a compact, completely reducible group G, in tensor network decompositions and algorithms. This is achieved by considering tensors that are invariant under the action of the group G. Each symmetric tensor decomposes into two types of tensors: degeneracy tensors, containing all the degrees of freedom, and structural tensors, which only depend on the symmetry group. In numerical calculations, the use of symmetric tensors ensures the preservation of the symmetry, allows selection of a specific symmetry sector, and significantly reduces computational costs. On the other hand, the resulting tensor network can be interpreted as a superposition of exponentially many spin networks. Spin networks are used extensively in loop quantum gravity, where they represent states of quantum geometry. Our work highlights their importance in the context of tensor network algorithms as well, thus setting the stage for cross-fertilization between these two areas of research.
235 citations
TL;DR: In this article, the existence of a metric whose Schouten tensor satisfies a quadratic inequality was shown to imply that the eigenvalues of the Ricci tensor are positively pinched.
Abstract: We formulate natural conformally invariant conditions on a 4-manifold for the existence of a metric whose Schouten tensor satisfies a quadratic inequality. This inequality implies that the eigenvalues of the Ricci tensor are positively pinched.
231 citations
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225 citations
214 citations