Showing papers in "International Journal of Geometric Methods in Modern Physics in 2011"
TL;DR: In this paper, the product of real spectral triples of arbitrary finite dimension and arbitrary parity was constructed, taking into account the fact that in the odd case there are two inequivalent representations of the gamma matrices (Clifford algebra), and in the even even case there were two natural candidates for the Dirac operator of the product triple.
Abstract: We construct the product of real spectral triples of arbitrary finite dimension (and arbitrary parity) taking into account the fact that in the even case there are two possible real structures, in the odd case there are two inequivalent representations of the gamma matrices (Clifford algebra), and in the even-even case there are two natural candidates for the Dirac operator of the product triple.
45 citations
TL;DR: In this article, the authors show how the Einstein equations with cosmological constant (and/or various types of matter field sources) can be integrated in a very general form following the anholonomic deformation method for constructing exact solutions in four and five-dimensional gravity.
Abstract: We show how the Einstein equations with cosmological constant (and/or various types of matter field sources) can be integrated in a very general form following the anholonomic deformation method for constructing exact solutions in four- and five-dimensional gravity (S. Vacaru, IJGMMP 4 (2007) 1285). In this paper, we prove that such a geometric method can be used for constructing general non-Killing solutions. The key idea is to introduce an auxiliary linear connection which is also metric compatible and completely defined by the metric structure but contains some torsion terms induced nonholonomically by generic off-diagonal coefficients of metric. There are some classes of nonholonomic frames with respect to which the Einstein equations (for such an auxiliary connection) split into an integrable system of partial differential equations. We have to impose additional constraints on generating and integration functions in order to transform the auxiliary connection into the Levi-Civita one. This way, we ex...
45 citations
TL;DR: In this article, the initial value formulation of metric-affine f(R)-gravity in the presence of a Klein-Gordon scalar field acting as the source of the field equations is studied.
Abstract: We study the initial value formulation of metric-affine f(R)-gravity in presence of a Klein–Gordon scalar field acting as source of the field equations. Sufficient conditions for the well-posedness of the Cauchy problem are formulated. This result completes the analysis of the same problem already considered for other sources.
38 citations
TL;DR: In this paper, it was shown that if a 3D Sasakian metric is a non-trivial Ricci soliton, then it is expanding and homothetic to the standard Sasakians on the Heisenberg group nil3.
Abstract: We show that, if a 3-dimensional Sasakian metric is a non-trivial Ricci soliton, then it is expanding and homothetic to the standard Sasakian metric on the Heisenberg group nil3. We have also discussed properties of the Ricci soliton potential vector field that relate to the underlying contact structure.
33 citations
TL;DR: In this paper, the authors give a general procedure to construct "algebro-geometric Feynman rules", that is, characters of the Connes-Kreimer Hopf algebra of Feynmann graphs that factor through a Grothendieck ring of immersed conical varieties, via the class of the complement of the affine graph hypersurface.
Abstract: We give a general procedure to construct "algebro-geometric Feynman rules", that is, characters of the Connes–Kreimer Hopf algebra of Feynman graphs that factor through a Grothendieck ring of immersed conical varieties, via the class of the complement of the affine graph hypersurface. In particular, this maps to the usual Grothendieck ring of varieties, defining "motivic Feynman rules". We also construct an algebro-geometric Feynman rule with values in a polynomial ring, which does not factor through the usual Grothendieck ring, and which is defined in terms of characteristic classes of singular varieties. This invariant recovers, as a special value, the Euler characteristic of the projective graph hypersurface complement. The main result underlying the construction of this invariant is a formula for the characteristic classes of the join of two projective varieties. We discuss the BPHZ renormalization procedure in this algebro-geometric context and some motivic zeta functions arising from the partition functions associated to motivic Feynman rules.
31 citations
TL;DR: In this paper, the authors developed a geometric setting that also allows us to assign a canonical nonlinear connection to a system of higher-order ordinary differential equations (HODE), and explicitly computed all curvature components of the corresponding Jacobi endomorphism.
Abstract: To a system of second-order ordinary differential equations one can assign a canonical nonlinear connection that describes the geometry of the system. In this paper, we develop a geometric setting that also allows us to assign a canonical nonlinear connection to a system of higher-order ordinary differential equations (HODE). For this nonlinear connection we develop its geometry, and explicitly compute all curvature components of the corresponding Jacobi endomorphism. Using these curvature components we derive a Jacobi equation that describes the behavior of nearby geodesics to a HODE. We motivate the applicability of this nonlinear connection using examples from the equivalence problem, the inverse problem of the calculus of variations, and biharmonicity. For example, using components of the Jacobi endomorphism we express two Wuenschmann-type invariants that appear in the study of scalar third- or fourth-order ordinary differential equations.
29 citations
TL;DR: In this article, a formal definition of a general reference frame in general spacetime, as an equivalence class of charts, was proposed, which corresponds with the notion of a reference frame as being a (fictitious) deformable body.
Abstract: We propose a formal definition of a general reference frame in a general spacetime, as an equivalence class of charts. This formal definition corresponds with the notion of a reference frame as being a (fictitious) deformable body, but we assume, moreover, that the time coordinate is fixed. This is necessary for quantum mechanics, because the Hamiltonian operator depends on the choice of the time coordinate. Our definition allows us to associate rigorously with each reference frame F, a unique "space" (a three-dimensional differentiable manifold), which is the set of the world lines bound to F. This also is very useful for quantum mechanics. We briefly discuss the application of these concepts to Godel's universe.
28 citations
TL;DR: In this paper, the authors studied the holomorphic unitary representations of the Jacobi group based on Siegel-Jacobi domains and obtained explicit polynomial orthonormal bases of the Fock spaces.
Abstract: We study the holomorphic unitary representations of the Jacobi group based on Siegel–Jacobi domains. Explicit polynomial orthonormal bases of the Fock spaces based on the Siegel–Jacobi disk are obtained. The scalar holomorphic discrete series of the Jacobi group for the Siegel–Jacobi disk is constructed and polynomial orthonormal bases of the representation spaces are given.
25 citations
TL;DR: In this paper, it was shown that quantum mechanics can be interpreted as a modification of the Euclidean nature of 3D space into a particular affine space, which is called Q-wis.
Abstract: We show that quantum mechanics can be interpreted as a modification of the Euclidean nature of 3-d space into a particular affine space, which we call Q-wis. This is proved using the Bohm–de Broglie causal formulation of quantum mechanics. In the Q-wis geometry, the length of extended objects changes from point to point. In this formulation, deformation of physical distances are in the core of quantum effects allowing a geometrical formulation of the uncertainty principle.
24 citations
TL;DR: In this article, the generalized Hyres-Ulam-Rassias stability and Bourgin-type superstability of higher derivations in non-Archimedean Banach algebras were investigated using a version of fixed-point theorem.
Abstract: In this paper, we investigate the generalized Hyres–Ulam–Rassias stability and Bourgin-type superstability of higher derivations in non-Archimedean Banach algebras by using a version of fixed-point theorem.
23 citations
TL;DR: In this article, the authors studied the energy-momentum tensor associated with a spinor field and derived a spinorial Gauss type formula for oriented hypersurfaces of a spin^c manifold.
Abstract: On Spin^c manifolds, we study the Energy-Momentum tensor associated with a spinor field. First, we give a spinorial Gauss type formula for oriented hypersurfaces of a Spin^c manifold. Using the notion of generalized cylinders, we derive the variationnal formula for the Dirac operator under metric deformation and point out that the Energy-Momentum tensor appears naturally as the second fundamental form of an isometric immersion. Finally, we show that generalized Spin^c Killing spinors for Codazzi Energy-Momentum tensor are restrictions of parallel spinors.
TL;DR: In this article, the existence of covariant conserved quantities in terms of principal Cartan connections on gauge-natural prolongations was studied and the implications of a reductive split structure associated with canonical Lagrangian conserved quantity on gauge natural bundles.
Abstract: We study geometric aspects concerned with symmetries and conserved quantities in gauge-natural invariant variational problems and investigate implications of the existence of a reductive split structure associated with canonical Lagrangian conserved quantities on gauge-natural bundles. In particular, we characterize the existence of covariant conserved quantities in terms of principal Cartan connections on gauge-natural prolongations.
TL;DR: In this article, the authors propose new structures called Membrane-, Membranec, StringK(ℤ,3) and FivebraneK(β,4) structures, which can also have twisted counterparts.
Abstract: Studying the M-branes leads us naturally to new structures that we call Membrane-, Membranec, StringK(ℤ,3) and FivebraneK(ℤ,4) structures, which we show can also have twisted counterparts. We study some of their basic properties, highlight analogies with structures associated with lower levels of the Whitehead tower of the orthogonal group, and demonstrate the relations to M-branes.
TL;DR: In this article, the existence problem of periodic first integrals for periodic Hamiltonian systems of Lie type was studied and the existence of Poisson algebras of periodic integrals was proved under different criteria based on properties for the Killing form of the adjoint group.
Abstract: In this paper, we study the existence problem of periodic first integrals for periodic Hamiltonian systems of Lie type. From a natural ansatz for time-dependent first integrals, we refer their existence to the existence of periodic solutions for a periodic Euler equation on the Lie algebra associated to the original system. Under different criteria based on properties for the Killing form or on exponential properties for the adjoint group, we prove the existence of Poisson algebras of periodic first integrals for the class of Hamiltonian systems considered. We include an application for a nonlinear oscillator having relevance in some modern physics applications.
TL;DR: In this article, the authors introduce a new notion of umbilical Riemannian maps and give a method on how to construct examples of such maps. And they also give necessary and sufficient conditions for such maps to be totally geodesic.
Abstract: This paper has two aims. First, we show that the usual notion of umbilical maps between Riemannian manifolds does not work for Riemannian maps. Then we introduce a new notion of umbilical Riemannian maps between Riemannian manifolds and give a method on how to construct examples of umbilical Riemannian maps. In the second part, as a generalization of CR-submanifolds, holomorphic submersions, anti-invariant submersions, invariant Riemannian maps and anti-invariant Riemannian maps, we introduce semi-invariant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds, give examples and investigate the geometry of distributions which are arisen from definition. We also obtain a decomposition theorem and give necessary and sufficient conditions for a semi-invariant Riemannian map to be totally geodesic. Then we study the geometry of umbilical semi-invariant Riemannian maps and obtain a classification theorem for such Riemannian maps.
TL;DR: Classical gravitation theory is formulated as gauge theory on natural bundles where gauge symmetries are general covariant transformations and a gravitational field is a Higgs field responsible for their spontaneous symmetry breaking as mentioned in this paper.
Abstract: Classical gravitation theory is formulated as gauge theory on natural bundles where gauge symmetries are general covariant transformations and a gravitational field is a Higgs field responsible for their spontaneous symmetry breaking.
TL;DR: In this article, a linear connection D is introduced such that the structure of these manifolds is parallel with respect to D and its torsion is totally skew-symmetric.
Abstract: The subject of investigations are the almost hypercomplex manifolds with Hermitian and anti-Hermitian (Norden) metrics. A linear connection D is introduced such that the structure of these manifolds is parallel with respect to D and its torsion is totally skew-symmetric. The class of the nearly Kahler manifolds with respect to the first almost complex structure is of special interest. It is proved that D has a D-parallel torsion and is weak if it is not flat. Some curvature properties of these manifolds are studied.
TL;DR: In this article, the effect of exotic smooth structures on typical observables in Euclidean quantum gravity was investigated in the semiclassical regime for several gravitational free-field actions and the results were similar, independent of the particular action that is chosen.
Abstract: In this paper we calculate the effect of the inclusion of exotic smooth structures on typical observables in Euclidean quantum gravity. We do this in the semiclassical regime for several gravitational free-field actions and find that the results are similar, independent of the particular action that is chosen. These are the first results of their kind in dimension four, which we extend to include one-loop contributions as well. We find these topological features can have physically significant results without the need for additional exotic physics.
TL;DR: In this paper, complete cotangent lifts of vector fields, their decomposition into vertical representative and holonomic part provide a geometrical framework underlying Eulerian equations of continuum mechanics.
Abstract: We show that complete cotangent lifts of vector fields, their decomposition into vertical representative and holonomic part provide a geometrical framework underlying Eulerian equations of continuum mechanics. We discuss Euler equations for ideal incompressible fluid and momentum-Vlasov equations of plasma dynamics in connection with the lifts of divergence-free and Hamiltonian vector fields, respectively. As a further application, we obtain kinetic equations of particles moving with the flow of contact vector fields both from Lie–Poisson reductions and with the techniques of present framework.
TL;DR: In this article, a partly original description of gauge fields and electroweak geometry is proposed, and a discussion of the breaking of conformal symmetry and the nature of the dilaton in the proposed setting is discussed.
Abstract: A partly original description of gauge fields and electroweak geometry is proposed. A discussion of the breaking of conformal symmetry and the nature of the dilaton in the proposed setting indicates that such questions cannot be definitely answered in the context of electroweak geometry.
TL;DR: In this paper, a necessary and sufficient condition for an n-dimensional Riemannian manifold to be isometrically immersed into one of the Lorentzian products was given.
Abstract: We give a necessary and sufficient condition for an n-dimensional Riemannian manifold to be isometrically immersed into one of the Lorentzian products 𝕊n × ℝ1 or ℍn × ℝ1. This condition is expressed in terms of its first and second fundamental forms, the tangent and normal projections of the vertical vector field. As applications, we give an equivalent condition in a spinorial way and we deduce the existence of a one-parameter family of isometric maximal deformation of a given maximal surface obtained by rotating the shape operator.
TL;DR: In this paper, a survey of non-commutative cosmological models of the very early universe is presented, based on the asymptotic expansion of the spectral action functional and on renormalization group analysis of the associated particle physics model.
Abstract: This is an overview of new and ongoing research developments aimed at constructing cosmological
models based on noncommutative geometry, via the spectral action functional,
thought of as a modified gravity action, which includes the coupling with matter when
computed on an almost commutative geometry. This survey is mostly based on recent
results obtained in collaboration with Elena Pierpaoli and Kevin Teh. We describe various
aspects of cosmological models of the very early universe, developed by the author
and Pierpaoli, based on the asymptotic expansion of the spectral action functional and
on renormalization group analysis of the associated particle physics model (an extension
of the standard model with right-handed neutrinos and Majorana mass terms previously
developed in collaboration with Chamseddine and Connes). We also describe nonperturbative
results, more recently obtained by Pierpaoli, Teh, and the author, which extend
to the more modern universe, which show that, for different candidate cosmic topologies,
the form of the slow-roll inflation potentials obtained from the nonperturbative calculation
of the spectral action are strongly coupled to the underlying geometry and topology.
We discuss some ongoing directions of research and open questions in this new field of
“noncommutative cosmology”. The paper is based on the talk given by the author at
the conference “Geometry and Quantum Field Theory” at the MPI, in honor of Alan
Carey.
TL;DR: In this paper, the authors treated the Jaynes-Cummings model with dissipation and gave an approximate solution to the master equation for the density operator under the general setting by making use of the Zassenhaus expansion.
Abstract: In this paper we treat the Jaynes–Cummings model with dissipation and give an approximate solution to the master equation for the density operator under the general setting by making use of the Zassenhaus expansion.
TL;DR: In this paper, the conservation laws for mechanical systems with nonholonomic constraints were studied and applied to Riemannian submanifolds, Lagrangians of mechanical type, and the determination of quadratic first integrals.
Abstract: This paper deals with conservation laws for mechanical systems with nonholonomic constraints. It uses a Lagrangian formulation of nonholonomic systems and a Cartan form approach. We present what we believe to be the most general relations between symmetries and first integrals. We discuss the so-called nonholonomic Noether theorem in terms of our formalism, and we give applications to Riemannian submanifolds, to Lagrangians of mechanical type, and to the determination of quadratic first integrals.
TL;DR: In this article, the first-order Seiberg-Witten map was applied to evaluate the first order non-commutative Kerr tetrad for rotational energy extraction from a black hole.
Abstract: This paper applies the first-order Seiberg–Witten map to evaluate the first-order non-commutative Kerr tetrad. The classical tetrad is taken to follow the locally non-rotating frame prescription. We also evaluate the tiny effect of non-commutativity on the efficiency of the Penrose process of rotational energy extraction from a black hole.
TL;DR: In this article, the Levi-Civita connection, curvature tensor, sectional and scalar curvatures of these spaces were studied and it was shown that the only space which admits left-invariant Randers metric of Berwald type has three-dimensional center.
Abstract: In this paper we study the geometry of simply connected two-step nilpotent Lie groups of dimension five. We give the Levi–Civita connection, curvature tensor, sectional and scalar curvatures of these spaces and show that they have constant negative scalar curvature. Also we show that the only space which admits left-invariant Randers metric of Berwald type has three-dimensional center. In this case the explicit formula for computing flag curvature is obtained and it is shown that flag curvature and sectional curvature have the same sign.
TL;DR: In this article, the authors analyze Lemaitre-Tolman-Bondi models in the presence of the cosmological constant Λ through the classical Weierstrass criterion.
Abstract: We analyze Lemaitre–Tolman–Bondi models in presence of the cosmological constant Λ through the classical Weierstrass criterion. Precisely, we show that the Weierstrass approach allows us to classify the dynamics of these inhomogeneous spherically symmetric Universes taking into account their relationship with the sign of Λ.
TL;DR: In this paper, U(1)n Chern-Simons type theories describing fractional quantum Hall solitons (QHS) in 1 + 2 dimensions from M-theory compactified on eight-dimensional hyper-Kahler manifolds as target space of N = 4 sigma model.
Abstract: We engineer U(1)n Chern–Simons type theories describing fractional quantum Hall solitons (QHS) in 1 + 2 dimensions from M-theory compactified on eight-dimensional hyper-Kahler manifolds as target space of N = 4 sigma model. Based on M-theory/type IIA duality, the systems can be modeled by considering D6-branes wrapping intersecting Hirzebruch surfaces F0's arranged as ADE Dynkin Diagrams and interacting with higher-dimensional R-R gauge fields. In the case of finite Dynkin quivers, we recover well known values of the filling factor observed experimentally including Laughlin, Haldane and Jain series.
TL;DR: In this paper, the generalized-quaternionic Kahler analogue of Lagrangian and Hamiltonian mechanical systems is introduced and the geometrical-physical results related to generalized quaternionic kahler mechanical systems are also given.
Abstract: In this paper, we introduce generalized-quaternionic Kahler analogue of Lagrangian and Hamiltonian mechanical systems. Finally, the geometrical-physical results related to generalized-quaternionic Kahler mechanical systems are also given.
TL;DR: In this paper, the quadratic and homogeneous Hamiltonian systems were analyzed from the Poisson dynamics and geometry point of view from the perspective of homogeneous geometry and poisson dynamics.
Abstract: In this paper we analyze the quadratic and homogeneous Hamiltonian systems on (𝔰𝔬(3))* from the Poisson dynamics and geometry point of view.