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We develop the Seiberg-Witten map using the gauge-covariant star product with the noncommutativity tensor $\theta^{\mu\nu}(x)$. The latter guarantees the Lorentz invariance of the theory. The usual form of this map and its other recent generalizations do not consider such a covariant star product. We construct the Seiberg-Witten map for the gauge parameter, the gauge field and the strength tensor to the first order in the noncommutativity parameter $\theta^{\mu\nu}(x)$. Prescription for the generalization of the map to higher orders is also given. Interestingly, the associativity of the covariant star product both in the first and second orders requires the same constraints, namely, on the $\theta^{\mu\nu}(x)$ and on the space-time connection. This fact suggests that the same constraints could be enough to ensure the associativity in all orders. The resulting Seiberg-Witten map applies both to the internal and space-time gauge theories. Comparisons with the Seiberg-Witten map based on other (non-covariant) star products are given and some characteristic properties are also presented. As an application, we consider the $GL(2, C)$ noncommutative gauge theory of gravitation, in which it is shown that the connection determines a space-time with symplectic structure (as proposed by Zumino et al [AIP Conf. Proc. 1200 (2010), 204, arXiv:0910.0459]). This example shows that the constraints required for the associativity of the gauge-covariant star product can be satisfied. The presented $GL(2, C)$ noncommutative gauge theory of gravitation is also compared to the one (given by Chamseddine [Phys. Rev. D 69 (2004), 024015, hep-th/0309166]) with non-covariant star product., Comment: 32 pages

We establish the Hamiltonian analysis and the canonical path integral for a local formulation of vacuum energy sequestering. In particular, by considering the state of the universe as a superposition of vacuum states corresponding to different values of the cosmological and gravitational constants, the path integral is extended to include integrations over the cosmological and gravitational constants. The result is an extension of the Ng-van Dam form of the path integral of unimodular gravity. It is argued to imply a relation between the fraction of the most likely values of the gravitational and cosmological constants and the average values of the energy density and pressure of matter over spacetime. Finally, we construct and analyze a BRST-exact formulation of the theory, which can be considered as a topological field theory., Comment: 30 pages, V2: Version published in PRD

We study the quantization of two versions of unimodular gravity, namely, fully diffeomorphism-invariant unimodular gravity and unimodular gravity with fixed metric determinant utilizing standard path integral approach. We derive the BRST symmetry of effective actions corresponding to several relevant gauge conditions. We observe that for some gauge conditions, the restricted gauge structure may complicate the formulation and effective actions, in particular, if the chosen gauge conditions involve the canonical momentum conjugate to the induced metric on the spatial hypersurface. The BRST symmetry is extended further to the finite field-dependent BRST transformation, in order to establish the mapping between different gauge conditions in each of the two versions of unimodular gravity., Comment: 26 pages, minor modification in texts

We investigate path integral quantization of two versions of unimodular gravity. First a fully diffeomorphism-invariant theory is analyzed, which does not include a unimodular condition on the metric, while still being equivalent to other unimodular gravity theories at the classical level. The path integral has the same form as in general relativity (GR), except that the cosmological constant is an unspecified value of a variable, and it thus is unrelated to any coupling constant. When the state of the universe is a superposition of vacuum states, the path integral is extended to include an integral over the cosmological constant. Second, we analyze the standard unimodular theory of gravity, where the metric determinant is fixed by a constraint. Its path integral differs from the one of GR in two ways: the metric of spacetime satisfies the unimodular condition only in average over space, and both the Hamiltonian constraint and the associated gauge condition have zero average over space. Finally, the canonical relation between the given unimodular theories of gravity is established., Comment: 35 pages; v4: included a full treatment of nonlocally linearly dependent generators as Appendix A, Refs. 28 and 29 added, extended the discussion on physical degrees of freedom, boundary terms, and conclusions

Cavity optomechanics has served as a platform for studying the interaction between light and micromechanical motion via radiation pressure. Here we observe such phenomena with a graphene mechanical resonator coupled to an electromagnetic mode. We measure thermal motion and back-action cooling in a bilayer graphene resonator coupled to a microwave on-chip cavity. We detect the lowest flexural mode at 24 MHz down to 50 mK, corresponding to roughly mechanical 40 quanta, representing nearly three orders of magnitude lower phonon occupation than recorded to date with graphene resonators., Comment: 5 pages

Most conventional magnetic materials used in the electronic devices are ferrites, which are composed of micrometer-size grains. But ferrites have small saturation magnetization, therefore the performance at GHz frequencies is rather poor. That is why functionalized nanocomposites comprising magnetic nanoparticles (e.g. Fe, Co) with dimensions ranging from a few nm to 100 nm, and embedded in dielectric matrices (e.g. silicon oxide, aluminium oxide) have a significant potential for the electronics industry. When the size of the nanoparticles is smaller than the critical size for multidomain formation, these nanocomposites can be regarded as an ensemble of particles in single-domain states and the losses (due for example to eddy currents) are expected to be relatively small. Here we review the theory of magnetism in such materials, and we present a novel measurement method used for the characterization of the electromagnetic properties of composites with nanomagnetic insertions. We also present a few experimental results obtained on composites consisting of iron nanoparticles in a dielectric matrix., Comment: 20 pages, 10 figures, 5 tables

A covariant Poisson bracket and an associated covariant star product in the sense of deformation quantization are defined on the algebra of tensor-valued differential forms on a symplectic manifold, as a generalization of similar structures that were recently defined on the algebra of (scalar-valued) differential forms. A covariant star product of arbitrary smooth tensor fields is obtained as a special case. Finally, we study covariant star products on a more general Poisson manifold with a linear connection, first for smooth functions and then for smooth tensor fields of any type. Some observations on possible applications of the covariant star products to gravity and gauge theory are made., Comment: AMS-LaTeX, 27 pages. v2: minor corrections in presentation and language, added one reference

We develop an internal gauge theory using a covariant star product. The space-time is a symplectic manifold endowed only with torsion but no curvature. It is shown that, in order to assure the restrictions imposed by the associativity property of the star product, the torsion of the space-time has to be covariant constant. An illustrative example is given and it is concluded that in this case the conditions necessary to define a covariant star product on a symplectic manifold completely determine its connection., Comment: AMS-LaTeX 19 pages. v2: corrections in language and equations (typos), expanded sections 3-5, added references. v3: minor presentational and grammatical corrections, completed, corrected and reordered some references.

Einstein's Theory of General Relativity was formulated as a gauge theory of Lorentz symmetry by Utiyama in 1956, while the Einstein-Cartan gravitational theory was formulated by Kibble in 1961 as the gauge theory of Poincare transformations. In a noncommutative space-time with canonical commutation relations between the coordintes, Lorentz symmetry is violated and field theories constructed on such space-times have instead the so-called twisted Poincare invariance. In this paper a gauge theory formulation of noncommutative gravity is proposed based on the twisted Poincare symmetry together with the requirement of covariance under the general coordinate transformations, an essential ingredient of the theory of general relativity. The advantages of such a formulation as well as the related problems are discussed and possible ways out are outlined., Comment: 21 pages