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      Quantum-Corrected Einstein Equations for a Noncommutative Spacetime of Lie-Algebraic Type

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          Abstract

          A general formula for the curvature of a central metric, w.r.t a noncommutative spacetime of general Lie-algebraic type is calculated by using the generalized braiding formalism. Furthermore, we calculate geometric quantities such as the Riemann tensor and the Ricci tensor and scalar in order to produce quantum corrections to the Einstein field equations.

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          The quantum structure of spacetime at the Planck scale and quantum fields

          We propose uncertainty relations for the different coordinates of spacetime events, motivated by Heisenberg's principle and by Einstein's theory of classical gravity. A model of Quantum Spacetime is then discussed where the commutation relations exactly implement our uncertainty relations. We outline the definition of free fields and interactions over QST and take the first steps to adapting the usual perturbation theory. The quantum nature of the underlying spacetime replaces a local interaction by a specific nonlocal effective interaction in the ordinary Minkowski space. A detailed study of interacting QFT and of the smoothing of ultraviolet divergences is deferred to a subsequent paper. In the classical limit where the Planck length goes to zero, our Quantum Spacetime reduces to the ordinary Minkowski space times a two component space whose components are homeomorphic to the tangent bundle TS^2 of the 2-sphere. The relations with Connes' theory of the standard model will be studied elsewhere.
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            A Gravity Theory on Noncommutative Spaces

            , , (2005)
            A deformation of the algebra of diffeomorphisms is constructed for canonically deformed spaces with constant deformation parameter theta. The algebraic relations remain the same, whereas the comultiplication rule (Leibniz rule) is different from the undeformed one. Based on this deformed algebra a covariant tensor calculus is constructed and all the concepts like metric, covariant derivatives, curvature and torsion can be defined on the deformed space as well. The construction of these geometric quantities is presented in detail. This leads to an action invariant under the deformed diffeomorphism algebra and can be interpreted as a theta-deformed Einstein-Hilbert action. The metric or the vierbein field will be the dynamical variable as they are in the undeformed theory. The action and all relevant quantities are expanded up to second order in theta.
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              Connections on central bimodules

              We define and study the theory of derivation-based connections on a recently introduced class of bimodules over an algebra which reduces to the category of modules whenever the algebra is commutative. This theory contains, in particular, a noncommutative generalization of linear connections. We also discuss the different noncommutative versions of differential forms based on derivations. Then we investigate reality conditions and a noncommutative generalization of pseudo-riemannian structures.
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                Author and article information

                Journal
                2017-05-09
                Article
                1705.03499
                6329c3d4-902e-4663-b63b-38f1e7556dc7

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                math-ph gr-qc hep-th math.MP

                Mathematical physics,General relativity & Quantum cosmology,High energy & Particle physics,Mathematical & Computational physics

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