Covariant Renormalizable Gravity Theories on (Non) Commutative Tangent Bundles

Abstract

The field equations in modified gravity theories possess an important decoupling property with respect to certain classes of nonholonomic frames. This allows us to construct generic off-diagonal solutions depending on all spacetime coordinates via generating and integration functions containing (un-)broken symmetry parameters. Some corresponding analogous models have a nice ultraviolet behavior and seem to be (super) renormalizable in a sense of covariant modifications of Hořava-Lifshits (HL) and ghost free gravity. The apparent noncommutativity and breaking of Lorentz invariance by quantum effects can be encoded into geometric objects and basic equations on noncommutative tangent Lorentz. The constructions can be extended to include conjectured covariant reonormalizable models with effective Einstein fields with (non)commutative variables.We study four equivalent models determined by actions of typewherem 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.Theories with [1] L are useful for constructing generic off-diagonal solutions in general relativity, GR, and modifications. Geometric models with[2] L can be considered as "bridges" between generating functions determining certain classes of generic off-diagonal Einstein manifolds and modified gravity theories. 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 typeL can be quantized using nonperturbative methods for deformation quantization and A-brane quantization, or as perturbative gauge theories. A class of (quantum type) modified noncommutative gravity theories are physically motivated by the Schrödinger type uncertainty relationsû αpβ −p βûα = i θ αβ , where is the Planck constant, i 2 = −1, andû α andp β are, respectively, certain coordinate and momentum type operators. Such geometric models are elaborated on "θ-extensions" of tangent Lorentz bundles T V → θ T V, (there are used also co-tangent bundles T * V → θ T * V etc). The extensions are determined by noncommutative complex distributions stated by "generalized uncertainty" relations u αs u βs − u βs u αs = iθ αsβs , where the antisymmetric matrix θ = (θ αsβs )