Gravity in non-commutative geometry

We study general relativity in the framework of non-commutative differential geometry. As a prerequisite we develop the basic notions of non-commutative Riemannian geometry, including analogues of Riemannian metric, curvature and scalar curvature. This enables us to introduce a generalized Einstein-Hilbert action for non-commutative Riemannian spaces. As an example we study a space-time which is the product of a four dimensional manifold by a two-point space, using the tools of non-commutative Riemannian geometry, and derive its generalized Einstein-Hilbert action. In the simplest situation, where the Riemannian metric is taken to be the same on the two copies of the manifold, one obtains a model of a scalar field coupled to Einstein gravity. This field is geometrically interpreted as describing the distance between the two points in the internal space.Dedicated to H. ArakiSupported in part by the Swiss National Foundation (SNF)     

Gravity, non-commutative geometry and the Wodzicki residue

We derive an action for gravity in the framework of non-commutative geometry by using the Wodzicki residue. We prove that for a Dirac operator D on an n dimensional compact Riemannian manifold with n ≥ 4, n even, the Wodzicki residue Res(D⁻ⁿ⁺²) is the integral of the second coefficient of the heat kernel expansion of D². We use this result to derive a gravity action for commutative geometry which is the usual Einstein-Hilbert action and we also apply our results to a non-commutative extension which is given by the tensor product of the algebra of smooth functions on a manifold and a finite dimensional matrix algebra. In this case we obtain gravity with a cosmological constant.     

Muong — 2 measurements and non-commutative geometry of quantum beams

We discuss a completely quantum mechanical treatment of the measurement of the anomalous magnetic moment of the muon. A beam of muons move in a strong uniform magnetic field and a weak focusing electrostatic field. Errors in the classical beam analysis are exposed. In the Dirac quantum beam analysis, an important role is played by non-commutative muon beam coordinates leading to a discrepancy between the classical and quantum theories. We obtain a quantum limit to the accuracy achievable in BNL type experiments. Some implications of the quantum corrected data analysis for supersymmetry are briefly mentioned.     

Seiberg-Witten geometry with various matter contents

We obtain the Seiberg-Witten geometry for four-dimensional N = 2 gauge theory with gauge group SO(2N_c) (N_c ⩽ 5) with massive spinor and vector hypermultiplets by considering the gauge symmetry breaking in the N = 2 E₆ theory with massive fundamental hypermultiplets. In a similar way the Seiberg-Witten geometry is determined for N = 2 SU(N_c) (N_c ⩽ 6) gauge theory with massive antisymmetric and fundamental hypermultiplets. Whenever possible we compare our results expressed in the form of ALE fibrations with those obtained by geometric engineering and brane dynamics, and find a remarkable agreement. We also show that these results are reproduced by using N = 1 confining phase superpotentials.     

Weyl-Dirac geometry and dark matter

Weyl proposed a geometry that differed from Riemannian geometry, which underlies general relativity, in that it contained a vector that could be interpreted as describing the electromagnetic field. Dirac modified this geometry to remove certain difficulties and based it on a variational principle which gave satisfactory field equations for gravitation and electromagnetism. However, by changing the value of a parameter appearing in his variational principle one gets, instead of electromagnetism, a field of massive particles of spin 1, which can be assumed to interact with ordinary matter only through gravitation. It is suggested that these bosons, called weylons, provide most of the dark matter in the universe.     

Modified gravity with arbitrary coupling between matter and geometry

The field equations of a generalized f(R)$f(R)$ type gravity model, in which there is an arbitrary coupling between matter and geometry, are obtained. The equations of motion for test particles are derived from a variational principle in the particular case in which the Lagrange density of the matter is an arbitrary function of the energy-density of the matter only. Generally, the motion is non-geodesic, and takes place in the presence of an extra force orthogonal to the four-velocity. The Newtonian limit of the model is also considered. The perihelion precession of an elliptical planetary orbit in the presence of an extra force is obtained in a general form, and the magnitude of the extra gravitational effects is constrained in the case of a constant extra force by using Solar System observations.     

On the local geometry of rotating matter

This paper shows that general relativity and ordinary continuum models of matter imply the presence of Cartan torsion. The key concept is that torsion can be viewed as translational holonomy per unit area, in the limit of very small areas. Translational holonomy is introduced as the nonclosure of the development of a space-time loop into a flat space-time. The translational holonomy around a charged rotating black hole is calculated. If a large collection of small rotating objects is approximated by continuous spinning matter, the resulting torsion and spin have the same relation as in Einstein-Cartan theory, except that the torsion traces remain zero for the simple model of spinning matter used here. Finally, this construction adds torsion to the list of nonpropagating fields which can be viewed as continuum density of holonomy around localized space-time boundaries, or around throats which are connected to further local topological structures.     

Bouncing universe with non-minimally coupled quintom matter

In this Letter, we study the condition for a generalized DBI action providing a quintom scenario of dark energy. We consider a development of string-inspired quintom by introducing non-minimal coupling. Then we show that the bouncing solution can appear in the universe dominated by the non-minimally coupled quintom matter.     

Gravity and duality between coordinates and matter fields

We use the duality between the local Cartezian coordinates and the solutions of the Klein-Gordon equation to parametrize locally the spacetime in terms of wave functions and prepotentials. The components of metric, metric connection, curvature as well as the Einstein equation are given in this parametrization. We also discuss the local duality between coordinates and quantum fields and the metric in this later reparametrization.     

de Sitter invariant gravity coupled with matter and its cosmological consequences

A new mathematical framework for de Sitter invariant gravity is described. Gauge symmetry follows the pattern of symmetry breaking; O(4, 2) → O(4, 1) → O(3, 1). Intimate relationships between spacetime and matter are exhibited via the interaction of gravity with scalar and spinor fields. The relevance of these results to GUT cosmology is discussed. It is shown that the cosmological constant is zero after the phase transition.     

Grading of spinor bundles and gravitating matter in noncommutative geometry

The gravitating matter is studied within the framework of noncommutative geometry. The noncommutative Einstein-Hilbert action on the product of a four-dimensional manifold with discrete space gives models of matter fields coupled to the standard Einstein gravity. The matter multiplet is encoded in the Dirac operator which yields a representation of the algebra of universal forms. The general form of the Dirac operator depends on a choice of the grading of the corresponding spinor bundle. A choice is given, which leads to the nonlinear vectorσ-model coupled to the Einstein gravity.     

Non-local field theories and the non-commutative torus

We argue that by taking a limit of SYM on a non-commutative torus one can obtain a theory on non-compact space with a finite non-locality scale. We also suggest that one can also obtain a similar generalization of the (2,0) field theory in dimensions, and that the DLCQ of this theory is known.     

Strong and electroweak interactions and their unification with non-commutative space-time

Quantum field theories based on non-commutative space-time (NCQFT) have been extensively studied recently. However no NCQFT model which can uniquely describe the strong and electroweak interactions has been constructed. This prevents one to make a consistent and systematic study of non-commutative space-time. In this work we construct a NCQFT model based on the trinification gauge group . A unique feature of this model, that all matter fields (fermions and Higgs bosons) are assigned to (anti-) fundamental representations of the factor SU(3) groups, allows us to construct a NCQFT model for strong and electroweak interactions and their unification without ambiguities. This model provides an example which allows one to make a consistent and systematic study of non-commutative space-time phenomenology. We also comment on some related issues regarding extensions to E₆ and models. Received: 18 September 2002 / Revised version: 21 February 2003 / Published online: 5 May 2003     

Conditions and instability in f(R) gravity with non-minimal coupling between matter and geometry

In this paper, on the basis of the generalized f(R) gravity model with arbitrary coupling between geometry and matter, four classes of f(R) gravity models with non-minimal coupling between geometry and matter will be studied. By means of conditions of power-law expansion and the equation of state of matter less than ?1/3, the relationship among p, w and n, the conditions and the candidate for late-time cosmic accelerated expansion will be discussed in the four classes of f(R) gravity models with non-minimal coupling. Furthermore, in order to keep to considering models that are realistic ones, the Dolgov–Kawasaki instability will be investigated in each of them.     

Semiclassical and quantum motions on the non-commutative plane

We study the canonical and the coherent state quantizations of a particle moving in a magnetic field on the non-commutative plane. Using a θ-modified action, we perform the canonical quantization and analyze the gauge dependence of the theory. We compare coherent states quantizations obtained through Malkin-Man'ko states and circular squeezed states. The relation between these states and the “classical” trajectories is investigated, and we present numerical explorations of some semiclassical quantities.     

Gravitational constant in f(R) theories of gravity with non-minimal coupling between matter and geometry

We study the effect of the non-minimal coupling between matter and geometry on the gravitational constant in the context of f(R) theories of gravity on cosmic scales. For a class of f(R) models,the result shows that the value of the gravitational constant not only changes over time but also has the dampened oscillation behavior.Compared with the result of the standard ACDM model, the consequence suggests that the coupling between matter and geometry should be weak.     

Quasi-particle description of strongly interacting matter: Towards a foundation

We confront our quasi-particle model for the equation of state of strongly interacting matter with recent first-principle QCD calculations. In particular, we test its applicability at finite baryon densities by comparing with Taylor expansion coefficients of the pressure for two quark flavours. We outline a chain of approximations starting from the Φ-functional approach to QCD which motivates the quasi-particle picture.     

Nonlocal vibration of coupled DLGS systems embedded on Visco-Pasternak foundation

In this paper, vibration analysis of the coupled system of double-layered graphene sheets (CS-DLGSs) embedded in a Visco-Pasternak foundation is carried out using the nonlocal elasticity theory of orthotropic plate. The two DLGSs are coupled by an enclosing viscoelastic medium which is simulated as a Visco-Pasternak foundation. Considering the Von Kármán nonlinear strain-displacement-relations, the motion equations are derived using the Hamilton's principle. Differential quadrature method (DQM) is applied to obtain the frequency ratio for various boundary conditions. The detailed parametric study is conducted, focusing on the combined effects of the nonlocal parameter, aspect ratio, graphene sheet's size, boundary conditions and the elastic and viscoelastic medium coefficients on the frequency ratio of CS-DLGSs. In this coupled system, two case of DLGSs vibration are investigated and compared with each other: (1) In-phase vibration (2) Out-of-phase vibration. The results indicate that the frequency ratio of the CS-DLGSs is more than the single-layered graphene sheet (SLGS). The results are in good agreement with the previous researches.     

Representations of non-commutative quantum mechanics and symmetries

We present a unified approach to representations of quantum mechanics on non-commutative spaces with general constant commutators of the phase-space variables. We find two phases and duality relations among them in arbitrary dimensions. Conditions for the physical equivalence of different representations of a given system are analyzed. Symmetries and classification of phase spaces are discussed. Especially, the dynamical symmetry of a physical system is investigated. Finally, we apply our analyses to the two-dimensional harmonic oscillator and the Landau problem. Received: 17 December 2002, Published online: 11 June 2003