Diffeomorphism covariant star products and noncommutative gravity

Abstract

The use of a diffeomorphism covariant star product enables us to construct diffeomorphism invariant gravities on noncommutative symplectic manifolds without twisting the symmetries. As an example, we construct noncommutative deformations of all two-dimensional dilaton gravity models thus overcoming some difficulties of earlier approaches. One of such models appears to be integrable. We find all classical solutions of this model and discuss their properties.

Field Theory with Coordinate Dependent Noncommutativity

“…With the description of gravity in mind, the formulation of noncommutative field theories (and in particular of gauge theories) on generic symplectic manifolds with curvature and/or torsion has been addressed by various authors using diverse approaches, e.g., see [3,4,5,11,12,24,28,29,31,32,33,43,44,46,51,59,63,64,81,94,104,105,106] as well as [86,95] for some nice introductions and overviews of the literature up to the year 2010. In relationship with the main subject of the present work (in particular the conservation laws for field theories on flat noncommutative space-time) we note that it should also be possible to obtain the energymomentum tensor (EMT) of matter fields in flat space-time by coupling these fields to a metric tensor field: the EMT is then given by the flat space limit of the curved space EMT defined as the variational derivative of the matter field action with respect to the metric tensor (see [20] and references therein for a justification of this procedure).…”

Canonical quantum gravity on noncommutative space–time

“…Since the canonical quantization of general relativity is a very natural way to try to obtain a quantum description of general relativity, under this assumption it would indeed seem to be necessary to formulate canonical quantum gravity on noncommutative space-time. Various theories of gravity in the context of noncommutative geometry have amongst others been studied in [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26] and cosmology on noncommutative space-time has been explored in [27]. Even certain ideas referring to quantum gravity have been explored with respect to noncommutative geometry, see [28], [29], [30], [31], [32], [33], [34], [35] for example.…”

Covariant Renormalizable Gravity Theories on (Non) Commutative Tangent Bundles

“…where [1] [2] L = R + L, [3] [4] L = f (Ȓ)+ m L are considered for the same metric g = {g µν } on a four dimensional (4d) Lorentz manifold V but for different connections and respective scalar curvatures and/ or f -modified Lagrange densities 1 ; m L and L are respectively the Lagrange densities of matter fields and effective matter. The spacetime axiomatic can be extended 2 for Einstein -Finsler like models on tangent Lorentz bundle T V.…”

“…The effective Lagrange density [3] L is similar to that introduced for covariant renormalization and HL gravity. 1 The models with [4] L transfers us into the field of "standard" modified theories with scalar curvature determined by the Levi-Civita (LC) connection. In almost Kähler variables, theories of type [1] L- [3] L can be quantized using nonperturbative methods for deformation quantization and A-brane quantization, or as perturbative gauge theories.…”

“…We use Υ = − ⋄ n L which results in a different class of generating functionsn φ from ∂4(n φ)∂4h3 2h3h4 = − ⋄ n L. A new class of off-diagonal solutions (4) can be generated for data φ →n φ, φ →nφ and ⋄ Υ → − ⋄ n L. Twisted (non) commutative extensions to higher "velocity/ momentum" type fibers can be performed in a standard manner as we considered in the previous section. We can apply the analysis provided in 1,3 which states that the analogous [3] L and [4] L models derived for (5) are renormalizable ifn = 1 and super-renormalizable forn = 2. Such properties can be preserved for theories generalized on tangent Lorentz bundles.…”