Showing papers in "International Journal of Geometric Methods in Modern Physics in 2015"
TL;DR: In this article, the stability of the solutions and the bouncing and ΛCDM models using the Noether symmetries of f(R, T) theories has been investigated.
Abstract: Extended f(R) theories of gravity have been investigated from the symmetry point of view. We briefly has been investigated Noether symmetry of two types of extended f(R) theories: f(R, T) theory, in which curvature is coupled non-minimally to the trace of energy–momentum tensor Tμν and mimetic f(R) gravity, a theory with a scalar field degree of freedom, but ghost-free and with internal conformal symmetry. In both cases we write point-like Lagrangian for flat Friedmann–Lemaitre–Robertson–Walker (FLRW) cosmological background in the presence of ordinary matter. We have been shown that some classes of models existed with Noether symmetry in these viable extensions of f(R) gravity. As a motivated idea, we have been investigating the stability of the solutions and the bouncing and ΛCDM models using the Noether symmetries. We have been shown that in mimetic f(R) gravity bouncing and ΛCDM solutions are possible. Also a class of solutions with future singularities has been investigated.
121 citations
TL;DR: In this paper, a short review of inflation in F(R)-gravity is presented, where the early-time acceleration is analyzed in a higher derivative quantum gravitational model, which mainly reduces to F(r)-gravity.
Abstract: In this short review, we revisit inflation in F(R)-gravity. We find several F(R)-models for viable inflation by applying some reconstruction techniques. A special attention is payed in the reproduction of the last Planck satellite data. The possible generalizations of Starobinsky-like inflation are found and discussed. The early-time acceleration is analyzed in a higher derivative quantum gravitational model which mainly reduces to F(R)-gravity.
108 citations
TL;DR: The notion of a Q-manifold is introduced in this article, which is a graded manifold endowed with a vector field of degree 1 squaring to zero and a fiber bundle in the category of graded manifolds.
Abstract: A Q-manifold is a graded manifold endowed with a vector field of degree 1 squaring to zero. We consider the notion of a Q-bundle, that is, a fiber bundle in the category of Q-manifolds. To each homotopy class of "gauge fields" (sections in the category of graded manifolds) and each cohomology class of a certain subcomplex of forms on the fiber we associate a cohomology class on the base. As any principal bundle yields canonically a Q-bundle, this construction generalizes Chern–Weil classes. Novel examples include cohomology classes that are locally de Rham differential of the integrands of topological sigma models obtained by the AKSZ-formalism in arbitrary dimensions. For Hamiltonian Poisson fibrations one obtains a characteristic 3-class in this manner. We also relate the framework to equivariant cohomology and Lecomte's characteristic classes of exact sequences of Lie algebras.
88 citations
TL;DR: In this paper, the structural identities governing higher gauge theories can be concisely recombined into what is called a Q-structure or, equivalently, an L∞-algebroid.
Abstract: Starting with minimal requirements from the physical experience with higher gauge theories, i.e. gauge theories for a tower of differential forms of different form degrees, we discover that all the structural identities governing such theories can be concisely recombined into what is called a Q-structure or, equivalently, an L∞-algebroid. This has many technical and conceptual advantages: complicated higher bundles become just bundles in the category of Q-manifolds in this approach (the many structural identities being encoded in the one operator Q squaring to zero), gauge transformations are generated by internal vertical automorphisms in these bundles and even for a relatively intricate field content the gauge algebra can be determined in some lines and is given by what is called the derived bracket construction. This paper aims equally at mathematicians and theoretical physicists; each more physical section is followed by a purely mathematical one. While the considerations are valid for arbitrary highest form degree p, we pay particular attention to p = 2, i.e. 1- and 2-form gauge fields coupled nonlinearly to scalar fields (0-form fields). The structural identities of the coupled system correspond to a Lie 2-algebroid in this case and we provide different axiomatic descriptions of those, inspired by the application, including e.g. one as a particular kind of a vector-bundle twisted Courant algebroid.
77 citations
TL;DR: In this paper, the intersection laws for higher WZW-type σ-model branes (open brane ending on background brane) are encoded precisely in (super-)L∞-extension theory and discussed its rationalized/perturbative description in super-Lie n-algebra homotopy theory.
Abstract: We formalize higher-dimensional and higher gauge WZW-type sigma-model local prequantum field theory, and discuss its rationalized/perturbative description in (super-)Lie n-algebra homotopy theory (the true home of the "FDA"-language used in the supergravity literature). We show generally how the intersection laws for such higher WZW-type σ-model branes (open brane ending on background brane) are encoded precisely in (super-)L∞-extension theory and how the resulting "extended (super-)space-times" formalize spacetimes containing σ-model brane condensates. As an application we prove in Lie n-algebra homotopy theory that the complete super-p-brane spectrum of superstring/M-theory is realized this way, including the pure σ-model branes (the "old brane scan") but also the branes with tensor multiplet worldvolume fields, notably the D-branes and the M5-brane. For instance the degree-0 piece of the higher symmetry algebra of 11-dimensional (11D) spacetime with an M2-brane condensate turns out to be the "M-theory super-Lie algebra". We also observe that in this formulation there is a simple formal proof of the fact that type IIA spacetime with a D0-brane condensate is the 11D sugra/M-theory spacetime, and of (prequantum) S-duality for type IIB string theory. Finally we give the non-perturbative description of all this by higher WZW-type σ-models on higher super-orbispaces with higher WZW terms in stacky differential cohomology.
77 citations
TL;DR: In this paper, the authors examined the critical behaviors of a class of 3D black holes with scalar field hair, where the cosmological constant is viewed as a thermodynamic pressure and its conjugate quantity as a volume.
Abstract: The principal focus of the present work concerns the critical behaviors of a class of three-dimensional (3D) black holes with a scalar field hair. Since the cosmological constant is viewed as a thermodynamic pressure and its conjugate quantity as a volume, we examine such properties in terms of two parameters B and a. The latters are related to the scalar field and the angular momentum, respectively. In particular, we give the equation of state predicting a critical universal number depending on the (B, a) moduli space. In the vanishing limit of the B parameter, we recover the usual perfect gas behavior appearing in the case of the non-rotating BTZ black hole. We point out that in a generic region of the (B, a) moduli space, the model behaves like a Van der Waals system.
58 citations
TL;DR: In this article, the stability of the Lorenz system was studied using the Jacobi stability analysis, or the Kosambi-Cartan-Chern (KCC) theory, which is a natural generalization of the (in)stability of the geodesic flow on a differentiable manifold endowed with a Riemannian or Finslerian.
Abstract: We perform the study of the stability of the Lorenz system by using the Jacobi stability analysis, or the Kosambi–Cartan–Chern (KCC) theory. The Lorenz model plays an important role for understanding hydrodynamic instabilities and the nature of the turbulence, also representing a nontrivial testing object for studying nonlinear effects. The KCC theory represents a powerful mathematical method for the analysis of dynamical systems. In this approach, we describe the evolution of the Lorenz system in geometric terms, by considering it as a geodesic in a Finsler space. By associating a nonlinear connection and a Berwald type connection, five geometrical invariants are obtained, with the second invariant giving the Jacobi stability of the system. The Jacobi (in)stability is a natural generalization of the (in)stability of the geodesic flow on a differentiable manifold endowed with a metric (Riemannian or Finslerian) to the non-metric setting. In order to apply the KCC theory, we reformulate the Lorenz system as a set of two second-order nonlinear differential equations. The geometric invariants associated to this system (nonlinear and Berwald connections), and the deviation curvature tensor, as well as its eigenvalues, are explicitly obtained. The Jacobi stability of the equilibrium points of the Lorenz system is studied, and the condition of the stability of the equilibrium points is obtained. Finally, we consider the time evolution of the components of the deviation vector near the equilibrium points.
46 citations
TL;DR: In this paper, the authors consider the most general torsional completion of gravitation together with electrodynamics for the Dirac spinorial material fields and show that consistency arguments constrain torsion to be completely antisymmetric and the dynamics to be parity-invariant and described by actions that are either least-order derivative or renormalizable.
Abstract: We consider the most general torsional completion of gravitation together with electrodynamics for the Dirac spinorial material fields, and we show that consistency arguments constrain torsion to be completely antisymmetric and the dynamics to be parity-invariant and described by actions that are either least-order derivative or renormalizable.
42 citations
TL;DR: In this article, it was shown that the position of the horizon for a particle inside the black hole depends on the energy of that particle and that particle's position outside the horizon.
Abstract: In this paper, we will analyze the gravitational collapse in the framework of gravity's rainbow. We will demonstrate that the position of the horizon for a particle inside the black hole depends on the energy of that particle. It will also be observe that the position of the horizon for a particle falling radially into the black hole also depends on its energy. Thus, it is possible for a particle coming from outside to interact with a particle inside the black, and take some information outside the black hole. This is because for both these particles the position of horizon is different. So, even though the particle from inside the black hole is in its own horizon, it is not in the horizon of the particle coming from outside. Thus, we will demonstrate that in gravity's rainbow information can get out of a black hole.
38 citations
TL;DR: In this article, the cosmological constant was interpreted as a thermodynamic pressure and its conjugate quantity as a temperature volume, and an explicit expression of the universal number in terms of the space dimension n was derived for the (1 + n)-dimensional AdS black hole system.
Abstract: Interpreting the cosmological constant as a thermodynamic pressure and its conjugate quantity as a thermodynamic volume, we reconsider the investigation of P–V critical behaviors of (1 + n)-dimensional AdS black holes in Lovelock–Born–Infeld gravity. In particular, we derive an explicit expression of the universal number $\chi = \frac{P_c v_c}{T_c}$ in terms of the space dimension n. Then, we examine the phase transitions at the critical points of such black holes for 6 ≤ n < 11 as required by the physical condition of the thermodynamical quantities including criticality behaviors. More precisely, the Ehrenfest equations have been checked and they reveal that the black hole system undergoes a second phase transition at the critical points.
38 citations
TL;DR: In this paper, the curvature-teleparallel F(R,T) gravity model was considered and the functional form of the Ricci scalar and the torsion scalar was determined by the presence of symmetries.
Abstract: We consider curvature-teleparallel F(R,T) gravity, where the gravitational Lagrangian density is given by an arbitrary function of the Ricci scalar R and the torsion scalar T. Using the Noether Symmetry Approach, we show that the functional form of the F(R,T) function can be determined by the presence of symmetries. Furthermore, we obtain exact solutions through the presence of conserved quantities and the reduction of cosmological dynamical system. Example of particular cosmological models are considered.
TL;DR: In this paper, it was shown that the existence of a maximum momentum will induce non-local corrections to the first quantized Hamiltonian, which can be effectively treated as local corrections by using the theory of harmonic extensions of functions.
Abstract: In this paper, we will demonstrate that like the existence of a minimum measurable length, the existence of a maximum measurable momentum, also influence all quantum mechanical systems. Beyond the simple one-dimensional case, the existence of a maximum momentum will induce non-local corrections to the first quantized Hamiltonian. However, these non-local corrections can be effectively treated as local corrections by using the theory of harmonic extensions of functions. We will also analyze the second quantization of this deformed first quantized theory. Finally, we will analyze the gauge symmetry corresponding to this deformed theory.
TL;DR: The theory of self-adjoint extensions of first and second-order elliptic differential operators on manifolds with boundary is studied via its most representative instances: Dirac and Laplace operators.
Abstract: The theory of self-adjoint extensions of first- and second-order elliptic differential operators on manifolds with boundary is studied via its most representative instances: Dirac and Laplace operators. The theory is developed by exploiting the geometrical structures attached to them and, by using an adapted Cayley transform on each case, the space of such extensions is shown to have a canonical group composition law structure. The obtained results are compared with von Neumann's theorem characterizing the self-adjoint extensions of densely defined symmetric operators on Hilbert spaces. The 1D case is thoroughly investigated. The geometry of the submanifold of elliptic self-adjoint extensions is studied and it is shown that it is a Lagrangian submanifold of the universal Grassmannian Gr. The topology of is also explored and it is shown that there is a canonical cycle whose dual is the Maslov class of the manifold. Such cycle, called the Cayley surface, plays a relevant role in the study of the phenomena of topology change. Self-adjoint extensions of Laplace operators are discussed in the path integral formalism, identifying a class of them for which both treatments leads to the same results. A theory of dissipative quantum systems is proposed based on this theory and a unitarization theorem for such class of dissipative systems is proved. The theory of self-adjoint extensions with symmetry of Dirac operators is also discussed and a reduction theorem for the self-adjoint elliptic Grassmannian is obtained. Finally, an interpretation of spontaneous symmetry breaking is offered from the point of view of the theory of self-adjoint extensions.
TL;DR: In this paper, the symmetry classification of the Klein-Gordon equation in Bianchi I spacetime has been performed and the results are presented in the form of tables, which can be used for the determination of invariant solutions of the wave equation.
Abstract: In this work we perform the symmetry classification of the Klein–Gordon equation in Bianchi I spacetime. We apply a geometric method which relates the Lie symmetries of the Klein–Gordon equation with the conformal algebra of the underlying geometry. Furthermore, we prove that the Lie symmetries which follow from the conformal algebra are also Noether symmetries for the Klein–Gordon equation. We use these results in order to determine all the potentials in which the Klein–Gordon admits Lie and Noether symmetries. Due to the large number of cases and for easy reference the results are presented in the form of tables. For some of the potentials we use the Lie admitted symmetries to determine the corresponding invariant solution of the Klein–Gordon equation. Finally, we show that the results also solve the problem of classification of Lie/Noether point symmetries of the wave equation in Bianchi I spacetime and can be used for the determination of invariant solutions of the wave equation.
TL;DR: In this paper, the authors consider four-dimensional homogeneous pseudo-Riemannian manifolds with non-trivial isotropy and completely classify the cases giving rise to non-Trivial homogeneous Ricci solitons.
Abstract: We consider four-dimensional homogeneous pseudo-Riemannian manifolds with non-trivial isotropy and completely classify the cases giving rise to non-trivial homogeneous Ricci solitons. In particular, we show the existence of non-compact homogeneous (and also invariant) pseudo-Riemannian Ricci solitons which are not isometric to solvmanifolds, and of conformally flat homogeneous pseudo-Riemannian Ricci solitons which are not symmetric.
TL;DR: In this paper, the authors investigated a new model of nonlinear electromagnetic field coupled with the gravitation field and obtained the black hole solution possessing the asymptotic Reissner-Nordstrom solution.
Abstract: We investigate a new model of nonlinear electromagnetic field coupled with the gravitation field. The black hole solution is obtained possessing the asymptotic Reissner–Nordstrom solution. The electric field has the finite value at the origin and there are not singularities.
TL;DR: In this paper, the authors derived the modified forms of Tolman-Oppenheimer-Volkoff (TOV) equations for a generic function of f(G) gravity.
Abstract: Based on a stringy inspired Gauss–Bonnet (GB) modification of classical gravity, we constructed a model for neutron stars. We derived the modified forms of Tolman–Oppenheimer–Volkoff (TOV) equations for a generic function of f(G) gravity. The hydrostatic equations remained unchanged but the dynamical equations for metric functions are modified due to the effects of GB term.
TL;DR: In this article, Grothendieck's dessins d'enfants are found to be related to the geometry of two-and three-qubit Pauli groups.
Abstract: We point out an explicit connection between graphs drawn on compact Riemann surfaces defined over the field $\bar{{\mathbb Q}}$ of algebraic numbers — the so-called Grothendieck's dessins d'enfants — and a wealth of distinguished point-line configurations. These include simplices, cross-polytopes, several notable projective configurations, a number of multipartite graphs and some "exotic" geometries. Among them, remarkably, we find not only those underlying Mermin's magic square and magic pentagram, but also those related to the geometry of two- and three-qubit Pauli groups. Of particular interest is the occurrence of all the three types of slim generalized quadrangles, namely GQ(2, 1), GQ(2, 2) and GQ(2, 4), and a couple of closely related graphs, namely the Schlafli and Clebsch ones. These findings seem to indicate that dessins d'enfants may provide us with a new powerful tool for gaining deeper insight into the nature of finite-dimensional Hilbert spaces and their associated groups, with a special emph...
TL;DR: In this paper, a relation between octonion basis elements with Dirac (gamma), Weyl and Majorana representations has been obtained, which leads to nice representations of the corresponding Clifford algebras.
Abstract: Starting with octonion algebra and its 4 × 4 matrix representation, we have made an attempt to write the extension of Pauli's matrices in terms of division algebra (octonion). The octonion generalization of Pauli's matrices shows the counterpart of Pauli's spin and isospin matrices. In this paper, we also have obtained the relationship between Clifford algebras and the division algebras, i.e. a relation between octonion basis elements with Dirac (gamma), Weyl and Majorana representations. The division algebra structure leads to nice representations of the corresponding Clifford algebras. We have made an attempt to investigate the octonion formulation of Dirac wave equations, conserved current and weak isospin in simple, compact, consistent and manifestly covariant manner.
TL;DR: In this paper, the Noether Symmetry Approach for a cosmological model derived from a tachyon scalar field T with a Dirac-Born-Infeld Lagrangian and a potential V(T) was considered.
Abstract: We consider the Noether Symmetry Approach for a cosmological model derived from a tachyon scalar field T with a Dirac–Born–Infeld Lagrangian and a potential V(T). Furthermore, we assume a coupled canonical scalar field ϕ with an arbitrary interaction potential B(T, ϕ). Exact solutions are derived consistent with the accelerated behavior of cosmic fluid.
TL;DR: In this article, the authors presented the analytic Lifshitzitz solutions for a scalar field model minimally coupled with the abelian gauge field in N-dimensions.
Abstract: We present the analytic Lifshitz solutions for a scalar field model minimally coupled with the abelian gauge field in N-dimensions. We also consider the presence of cosmological constant Λ. The Lifshitz parameter z appearing in the solution plays the role of the Lorentz breaking parameter of the model. We investigate the thermodynamical properties of the solutions and discuss the energy issue. Furthermore, we study the hairy black hole solutions in which the abelian gauge field breaks the symmetry near to the horizon. In the holographic picture, it is equivalent to a second-order phase transition. Explicitly we show that there exists a critical temperature which is a function of the Lifshitz parameter z. The system below the critical temperature becomes superconductor, but the critical exponent of the model remains the same of the usual holographic superconductors without the higher-order gravitational corrections, in agreement with Ginzburg–Landau theories.
TL;DR: In this paper, the authors used a class of analytical f(R)-theories to construct the first time derivative of the orbital period of binary systems starting from a quadrupolar gravitational emission.
Abstract: There are several approaches to extend General Relativity in order to explain the phenomena related to the Dark Matter and Dark Energy. These theories, generally called Extended Theories of Gravity, can be tested using observations coming from relativistic binary systems as PSR J0348 + 0432. Using a class of analytical f(R)-theories, one can construct the first time derivative of orbital period of the binary systems starting from a quadrupolar gravitational emission. Our aim is to set boundaries on the parameters of the theory in order to understand if they are ruled out, or not, by the observations on PSR J0348 + 0432. Finally, we have computed an upper limit on the graviton mass showing that agree with constraint coming from other observations.
TL;DR: In this paper, a series of five lectures around the common subject of self-adjoint extensions of symmetric operators and its applications to Quantum Physics is presented, with a brief account of some recent ideas in the theory of selfadjoint extension on Hilbert spaces and their applications to a few specific problems in Quantum Mechanics.
Abstract: This is a series of five lectures around the common subject of the construction of self-adjoint extensions of symmetric operators and its applications to Quantum Physics. We will try to offer a brief account of some recent ideas in the theory of self-adjoint extensions of symmetric operators on Hilbert spaces and their applications to a few specific problems in Quantum Mechanics.
TL;DR: In this paper, a set of lecture notes on quantum systems with time-dependent boundaries is presented, where the dynamics of a non-relativistic particle in a bounded domain of physical space are analyzed when the boundaries are moving or changing.
Abstract: We present here a set of lecture notes on quantum systems with time-dependent boundaries. In particular, we analyze the dynamics of a non-relativistic particle in a bounded domain of physical space, when the boundaries are moving or changing. In all cases, unitarity is preserved and the change of boundaries does not introduce any decoherence in the system.
TL;DR: In this article, a review of canonical ADM tetrad gravity in a family of globally hyperbolic asymptotically Minkowskian space-times without super-translations is presented.
Abstract: In this updated review of canonical ADM tetrad gravity in a family of globally hyperbolic asymptotically Minkowskian space-times without super-translations I show which is the status-of-the-art in the search of a canonical basis adapted to the first-class Dirac constraints and of the Dirac observables of general relativity (GR) describing the tidal degrees of freedom of the gravitational field. In these space-times the asymptotic ADM Poincare group replaces the Poincare group of particle physics, there is a York canonical basis diagonalizing the York–Lichnerowicz approach and a post-Minkowskian linearization is possible with the associated description of gravitational waves in the family of non-harmonic 3-orthogonal Schwinger time gauges. Moreover I show that every fixation of the inertial gauge variables (i.e. the choice of a non-inertial frame) of every generally covariant formulation of GR is equivalent to a set of conventions for the metrology of the space-time (like the GPS ones near the Earth): for instance the freedom in clock synchronization is described by the inertial gauge variable York time (the trace of the extrinsic curvature of the instantaneous 3-spaces). This inertial gauge freedom and the non-Euclidean nature of the instantaneous 3-spaces required by the equivalence principle are connected with the dark side of the universe and could explain the presence of dark matter or at least part of it by means of the adoption of suitable metrical conventions for the ICRS celestial reference system. Also some comments on a canonical quantization of GR coherent with this viewpoint are done.
TL;DR: In this paper, the authors introduce three new transformations and establish connections between moving non-null curves and soliton equations according to Bishop frame in Minkowski 3-space, and find formulas for differentials associated with the nonlinear heat system and repulsive type nonlinear Schrodinger equation.
Abstract: In this paper, we introduce three new transformations and establish connections between moving non-null curves and soliton equations according to Bishop frame in Minkowski 3-space. Later we find formulas for differentials of these three new transformations associated with the nonlinear heat system and repulsive type nonlinear Schrodinger equation.
TL;DR: In this paper, the critical phase of a mixed system of U(2) gauge fields and global symmetry on the boundary using gauge/gravity was analyzed and a variational minimization problem was formulated.
Abstract: We analytically study the critical phase of a mixed system of the U(2) gauge fields and global symmetry on the boundary using gauge/gravity. A variational minimization problem has been formulated. The numerical results pertinently show that there exists a minimum chemical potential in which both scalar (s-wave) and vector (p-wave) condensates exist in a mixture, as well as in the distinct s- and p-phases. This result is obtained by breaking the symmetry into U(1) symmetry and rotational symmetry. While the analytical solutions of condensates and charge densities are achieved in both cases: the balanced and unbalanced holographic superconductors. This is the first analytical study of the coexistence of two modes of the superconductivity with different order parameters. The realistic model consists of two different phases of the superfluidity in helium.
TL;DR: In this article, the authors revisited their joint work with Antonio Siconolfi on time functions and showed how to construct a Lipschitz time function in a simplified setting.
Abstract: In this paper we revisit our joint work with Antonio Siconolfi on time functions. We will give a brief introduction to the subject. We will then show how to construct a Lipschitz time function in a simplified setting. We will end with a new result showing that the Aubry set is not an artifact of our proof of existence of time functions for stably causal manifolds.
TL;DR: This work shows how to use boundary conditions to drive the evolution on a Quantum Mechanical system and will see how this problem can be expressed in terms of a time-dependent Schr\"{o}dinger equation.
Abstract: We show how to use boundary conditions to drive the evolution on a quantum mechanical system. We will see how this problem can be expressed in terms of a time-dependent Schrodinger equation. In particular, we will need the theory of self-adjoint extensions of differential operators in manifolds with boundary. An introduction of the latter as well as meaningful examples will be given. It is known that different boundary conditions can be used to describe different topologies of the associated quantum systems. We will use the previous results to study the topology change and to obtain necessary conditions to accomplish it in a dynamical way.
TL;DR: The oscillator algebra acting on pairs of long rows or long columns in the Young diagrams of the operators can be reached by a Inonu-Wigner contraction of the u(2) algebra inside of the U(p) algebra of p giant gravitons as mentioned in this paper.
Abstract: The operators with large scaling dimensions can be labeled by Young diagrams. Among other bases, the operators using restricted Schur polynomials have been known to have a large N but nonplanar limit under which they map to states of a system of harmonic oscillators. We analyze the oscillator algebra acting on pairs of long rows or long columns in the Young diagrams of the operators. The oscillator algebra can be reached by a Inonu–Wigner contraction of the u(2) algebra inside of the u(p) algebra of p giant gravitons. We present evidences that integrability in this case can persist at higher loops due to the presence of the oscillator algebra which is expected to be robust under loop corrections in the nonplanar large N limit.