Showing papers in "International Journal of Geometric Methods in Modern Physics in 2006"
TL;DR: In this paper, the Lagrange-Noether formalism is presented in full generality, and the family of quadratic (in the curvature and the torsion) models is analyzed in detail.
Abstract: In the gauge theory of gravity based on the Poincare group (the semidirect product of the Lorentz group and the spacetime translations) the mass (energy–momentum) and the spin are treated on an equal footing as the sources of the gravitational field. The corresponding spacetime manifold carries the Riemann–Cartan geometric structure with the nontrivial curvature and torsion. We describe some aspects of the classical Poincare gauge theory of gravity. Namely, the Lagrange–Noether formalism is presented in full generality, and the family of quadratic (in the curvature and the torsion) models is analyzed in detail. We discuss the special case of the spinless matter and demonstrate that Einstein's theory arises as a degenerate model in the class of the quadratic Poincare theories. Another central point is the overview of the so-called double duality method for constructing of the exact solutions of the classical field equations.
178 citations
TL;DR: In this paper, the unfolded off-shell constraints for symmetric fields of all spins in Minkowski space are shown to have the form of zero curvature and covariant constancy conditions for 1-forms and 0-forms taking values in an appropriate star product algebra.
Abstract: Within unfolded dynamics approach, we represent actions and conserved charges as elements of cohomology of the L∞ algebra underlying the unfolded formulation of a given dynamical system. The unfolded off-shell constraints for symmetric fields of all spins in Minkowski space are shown to have the form of zero curvature and covariant constancy conditions for 1-forms and 0-forms taking values in an appropriate star product algebra. Unfolded formulation of Yang–Mills and Einstein equations is presented in a closed form.
135 citations
TL;DR: In this article, a natural geometric framework is proposed, based on ideas of W. M. Tulczyjew, for constructions of dynamics on general algebroids, which still reproduces the main features of the Analytical Mechanics, like the Euler-Lagrange type equations, the correspondence between the Lagrangian and Hamiltonian functions (Legendre transform) in the hyperregular cases, and a version of the Noether Theorem.
Abstract: A natural geometric framework is proposed, based on ideas of W. M. Tulczyjew, for constructions of dynamics on general algebroids. One obtains formalisms similar to the Lagrangian and the Hamiltonian ones. In contrast with recently studied concepts of Analytical Mechanics on Lie algebroids, this approach requires much less than the presence of a Lie algebroid structure on a vector bundle, but it still reproduces the main features of the Analytical Mechanics, like the Euler–Lagrange-type equations, the correspondence between the Lagrangian and Hamiltonian functions (Legendre transform) in the hyperregular cases, and a version of the Noether Theorem.
126 citations
TL;DR: In this article, the Hamilton-Jacobi problem is revisited bearing in mind the consequences arising from a possible bi-Hamiltonian structure on the tangent bundle for Lagrangian systems.
Abstract: The Hamilton–Jacobi problem is revisited bearing in mind the consequences arising from a possible bi-Hamiltonian structure. The problem is formulated on the tangent bundle for Lagrangian systems in order to avoid the bias of the existence of a natural symplectic structure on the cotangent bundle. First it is developed for systems described by regular Lagrangians and then extended to systems described by singular Lagrangians with no secondary constraints. We also consider the example of the free relativistic particle, the rigid body and the electron-monopole system.
114 citations
TL;DR: In this paper, a geometric description of Lagrangian and Hamiltonian mechanics on Lie algebroids is presented, which allows us to analyze systems subject to nonholonomic constraints, mechanical control systems, Discrete Mechanics and extensions to Classical Field Theory within a single framework.
Abstract: In this survey, we present a geometric description of Lagrangian and Hamiltonian Mechanics on Lie algebroids. The flexibility of the Lie algebroid formalism allows us to analyze systems subject to nonholonomic constraints, mechanical control systems, Discrete Mechanics and extensions to Classical Field Theory within a single framework. Various examples along the discussion illustrate the soundness of the approach.
89 citations
TL;DR: In this paper, a Lie-theoretic expression of the Ricci tensor in a flag manifold is given, which reduces the Einstein equation on flag manifold into an algebraic system of equations, which can be solved in several cases.
Abstract: A flag manifold is a homogeneous space M = G/K, where G is a compact semisimple Lie group, and K the centralizer of a torus in G. Equivalently, M can be identified with the adjoint orbit Ad(G)w of an element w in the Lie algebra of G. We present several aspects of flag manifolds, such as their classification in terms of painted Dynkin diagrams, T-roots and G-invariant metrics, and Kahler metrics. We give a Lie-theoretic expression of the Ricci tensor in M, hence reducing the Einstein equation on flag manifolds into an algebraic system of equations, which can be solved in several cases. A flag manifold is also a complex manifold, and this dual representation as a real and a complex manifold is related to a similar property of an infinite-dimensional manifold, the loop space, which in fact can be viewed as a "universal" flag manifold.
72 citations
TL;DR: In this paper, it was shown that any special para-Kahler manifold is intrinsically an improper affine hypersphere, which is the graph of a real function f of 2n variables.
Abstract: We prove that any special para-Kahler manifold is intrinsically an improper affine hypersphere. As a corollary, any para-holomorphic function F of n para-complex variables satisfying a non-degeneracy condition defines an improper affine hypersphere, which is the graph of a real function f of 2n variables. We give an explicit formula for the function f in terms of the para-holomorphic function F. Necessary and sufficient conditions for an affine hypersphere to admit the structure of a special para-Kahler manifold are given. Finally, it is shown that conical special para-Kahler manifolds are foliated by proper affine hyperspheres of constant mean curvature.
65 citations
TL;DR: In this paper, the weakly-irreducible holonomy algebras of Lorentzian manifolds are described and metrics that realize all these candidates as holonomies are given.
Abstract: All candidates to the weakly-irreducible not irreducible holonomy algebras of Lorentzian manifolds are known. In the present paper metrics that realize all these candidates as holonomy algebras are given. This completes the classification of the Lorentzian holonomy algebras. Also new examples of metrics with the holonomy algebras g2 ⋉ ℝ7 ⊂ 𝔰𝔬(1, 8) and 𝔰𝔭𝔦𝔫(7) ⋉ ℝ8 ⊂ 𝔰𝔬(1, 9) are constructed.
54 citations
TL;DR: In this paper, a new proof of the classical Beltrami Theorem was presented, which states that a metric projectively equivalent to a metric of constant curvature is itself a metric with constant curvatures.
Abstract: We present a new proof of the classical Beltrami Theorem stating that a metric projectively equivalent to a metric of constant curvature is itself a metric of constant curvature.
43 citations
TL;DR: In this paper, the equivalence problem of submanifolds with respect to a transitive pseudogroup action is studied, and the corresponding differential invariants are determined via formal theory and lead to the notions of l-variants and l-covariants.
Abstract: We study the equivalence problem of submanifolds with respect to a transitive pseudogroup action. The corresponding differential invariants are determined via formal theory and lead to the notions of l-variants and l-covariants, even in the case of non-integrable pseudogroup. Their calculation is based on the cohomological machinery: we introduce a complex for covariants, define their cohomology and prove the finiteness theorem. This implies the well-known Lie–Tresse theorem about differential invariants. We also generalize this theorem to the case of pseudogroup action on differential equations.
41 citations
TL;DR: In this paper, the authors summarize the basic facts on the reduction in principal bundles and geometry of Higgs fields and summarize the particular covariant differential in the presence of a Higgs field.
Abstract: In gauge theory, Higgs fields are responsible for spontaneous symmetry breaking. In classical gauge theory on a principal bundle P, a symmetry breaking is defined as the reduction of a structure group of this principal bundle to a subgroup H of exact symmetries. This reduction takes place if and only if there exists a global section of the quotient bundle P/H. It is a classical Higgs field. A metric gravitational field exemplifies such a Higgs field. We summarize the basic facts on the reduction in principal bundles and geometry of Higgs fields. Our goal is the particular covariant differential in the presence of a Higgs field.
TL;DR: In this article, the notion of square integrable group representation modulo a relatively central subgroup is introduced and a generalization of a classical theorem of Duflo and Moore is proved.
Abstract: We introduce the notion of square integrable group representation modulo a relatively central subgroup and, establishing a link with square integrable projective representations, we prove a generalization of a classical theorem of Duflo and Moore. As an example, we apply the results obtained to the Weyl–Heisenberg group.
TL;DR: In this article, the covariant formulation of Chern-Simons theories in a general odd dimension was investigated, which can be obtained by introducing a vacuum connection field as a reference.
Abstract: We investigate the covariant formulation of Chern–Simons theories in a general odd dimension which can be obtained by introducing a vacuum connection field as a reference. Field equations, Nother currents and superpotentials are computed so that results are easily compared with the well-known results in dimension 3. Finally we use this covariant formulation of Chern–Simons theories to investigate their relation with topological BF theories.
TL;DR: In this paper, the basic principles of Affine Quantum Gravity are presented in a pedagogical style with a limited number of equations, and they are used to describe the relationship between affine quantum gravity and affine equilibria.
Abstract: The basic principles of Affine Quantum Gravity are presented in a pedagogical style with a limited number of equations.
TL;DR: In this paper, the physics of long strings that come from infinity are discussed, which are related to non-singlets in the dual matrix model description, and they consider two-dimensional string backgrounds.
Abstract: We consider two-dimensional string backgrounds. We discuss the physics of long strings that come from infinity. These are related to non-singlets in the dual matrix model description.
TL;DR: In this article, the authors estimate the number of isolate invariant holomorphic holomorphic Einstein metrics (up to homothety) on a homogeneous manifold M = G/H, with a simple spectrum of the isotropy representation, a compact convex polytope PM which is the Newton polyto of the rational function s(t) and that to each invariant metric t of M associates its scalar curvature.
Abstract: We associate to a homogeneous manifold M = G/H, with a simple spectrum of the isotropy representation, a compact convex polytope PM which is the Newton polytope of the rational function s(t) and that to each invariant metric t of M associates its scalar curvature. We estimate the number of isolate invariant holomorphic Einstein metrics (up to homothety) on Mℂ = Gℂ/Hℂ. Using the results of A. G. Kouchnirenko and D. N. Bernstein, we prove that , where ν(M) is the integer volume of PM, and give conditions when the defect . In case when G is a compact semisimple Lie group, the positiveness of d(M) is related with the existence of Ricci-flat holomorphic metric on a complexification of a noncompact homogeneous space Mγ = Gγ/HP which is a contraction of M and is associated with a proper face γ of PM.
TL;DR: In this paper, the generalized Pauli matrices were constructed for the case of n = 3, 4, and 5, and the generalized Walsh-Hadamard matrix for n = 5.
Abstract: In the paper (math–ph/0504049) Jarlskog gave an interesting simple parametrization to unitary matrices, which was essentially the canonical coordinate of the second kind in the Lie group theory (math–ph/0505047). In this paper we apply the method to a quantum computation based on multilevel system (qudit theory). Namely, by considering that the parametrization gives a complete set of modules in qudit theory, we construct the generalized Pauli matrices, which play a central role in the theory and also make a comment on the exchange gate of two–qudit systems. Moreover, we give an explicit construction to the generalized Walsh–Hadamard matrix in the case of n = 3, 4, and 5. For the case of n = 5, its calculation is relatively complicated. In general, a calculation to construct it tends to become more and more complicated as n becomes large. To perform a quantum computation the generalized Walsh–Hadamard matrix must be constructed in a quick and clean manner. From our construction it may be possible to say that a qudit theory with n ≥ 5 is not realistic. This paper is an introduction toward Quantum Engineering.
TL;DR: An infinite family of square integrable solutions for the hyperbolic Klein-Gordon equation on Lorentzian manifolds is constructed in this article, and these solutions have a discrete mass spectrum and a finite action.
Abstract: The eigenvalue problem for the square integrable solutions is studied usually for elliptic equations. In this paper we consider such a problem for the hyperbolic Klein–Gordon equation on Lorentzian manifolds. The investigation could help to answer the question why elementary particles have a discrete mass spectrum. An infinite family of square integrable solutions for the Klein–Gordon equation on the Friedman type manifolds is constructed. These solutions have a discrete mass spectrum and a finite action. In particular the solutions on de Sitter space are investigated.
TL;DR: In this paper, the authors extend the classical approach of the R-separation of the Laplace equation Δψ = 0 (as a null eigenvalue problem) to the general steady state Schrodinger equation including cases where a scalar potential V is present and the energy is a fixed constant.
Abstract: We extend the classical approach of the R-separation of the Laplace equation Δψ = 0 (as a null eigenvalue problem) to the general steady state Schrodinger equation including cases where a scalar potential V is present and the energy is a fixed constant.
TL;DR: In this article, the authors define and study non-differentiable deformations of the classical Cartesian space ℝn which can be viewed as the basic bricks to construct irregular objects and derive rigorously the main results of the scalerelativity theory developed by Nottale in the framework of a scale space-time manifold.
Abstract: Many problems of physics or biology involve very irregular objects like the rugged surface of a malignant cell nucleus or the structure of space-time at the atomic scale. We define and study non-differentiable deformations of the classical Cartesian space ℝn which can be viewed as the basic bricks to construct irregular objects. They are obtained by taking the topological product of n-graphs of nowhere differentiable real valued functions. Our point of view is to replace the study of a non-differentiable function by the dynamical study of a one-parameter family of smooth regularization of this function. In particular, this allows us to construct a one-parameter family of smooth coordinates systems on non-differentiable deformations of ℝn, which depend on the smoothing parameter via an explicit differential equation called a scale law. Deformations of ℝn are examples of a new class of geometrical objects called scale manifolds which are defined in this paper. As an application, we derive rigorously the main results of the scale-relativity theory developed by Nottale in the framework of a scale space-time manifold.
TL;DR: The Hamilton-Jacobi equation for a Hamiltonian section on a Lie affgebroid is introduced in this article, and some examples of the Hamiltonian Hamiltonians are discussed.
Abstract: The Hamilton–Jacobi equation for a Hamiltonian section on a Lie affgebroid is introduced and some examples are discussed.
TL;DR: In this article, the authors discuss some ideas and results which might lead to a theory of infinite dimensional symmetric spaces, where the affine Kac-Moody group is the fixed point group of an involution.
Abstract: In this expository article we discuss some ideas and results which might lead to a theory of infinite dimensional symmetric spaces $\hat{G}/\hat{K}$ where $\hat{G}$ is an affine Kac–Moody group and $\hat{K}$ the fixed point group of an involution (of the second kind). We point out several striking similarities of these spaces with their finite dimensional counterparts and discuss their geometry. Furthermore we sketch a classification and show that they are essentially in 1 : 1 correspondence with hyperpolar actions on compact simple Lie groups.
TL;DR: In this paper, the Lie algebra of Hermitian vector fields of a line bundle is analyzed and the base space of the above bundle is specified by considering a Galilei or an Einstein spacetime.
Abstract: We start by analyzing the Lie algebra of Hermitian vector fields of a Hermitian line bundle. Then, we specify the base space of the above bundle by considering a Galilei, or an Einstein spacetime. Namely, in the first case, we consider, a fibred manifold over absolute time equipped with a spacelike Riemannian metric, a spacetime connection (preserving the time fibring and the spacelike metric) and an electromagnetic field. In the second case, we consider a spacetime equipped with a Lorentzian metric and an electromagnetic field. In both cases, we exhibit a natural Lie algebra of special phase functions and show that the Lie algebra of Hermitian vector fields turns out to be naturally isomorphic to the Lie algebra of special phase functions. Eventually, we compare the Galilei and Einstein cases.
TL;DR: In this article, the authors derive standard quantum mechanics, and show how the need for probability amplitudes arises from the use of a standard of measurement in biconformal spaces.
Abstract: Biconformal spaces contain the essential elements of quantum mechanics, making the independent imposition of quantization unnecessary. Based on three postulates characterizing motion and measurement in biconformal geometry, we derive standard quantum mechanics, and show how the need for probability amplitudes arises from the use of a standard of measurement. Additionally, we show that a postulate for unique, classical motion yields Hamiltonian dynamics with no measurable size changes, while a postulate for probabilistic evolution leads to physical dilatations manifested as measurable phase changes. Our results lead to the Feynman path integral formulation, from which follows the Schrodinger equation. We discuss the Heisenberg uncertainty relation and fundamental canonical commutation relations.
TL;DR: In this paper, the authors developed the construction of geometric integrators for higher-order mechanics, which preserve momentum since they have a discrete variational origin and are naturally symplectic and preserve momentum.
Abstract: In this paper we develop the construction of geometric integrators for higher-order mechanics. These integrators are naturally symplectic and preserve momentum since they have a discrete variational origin.
TL;DR: In this article, it was shown that the existence of a PV measure such that the outcomes of the measurement of F can be interpreted as the random diffusion of the outcome of the measurements of E can be inferred.
Abstract: Given a commutative POV measure on the Borel σ-algebra of the reals it is possible to construct a PV measure (the sharp reconstruction of F) such that the outcomes of the measurement of F can be interpreted as the random diffusion of the outcomes of the measurement of E. On the other hand Neumark's theorem ensures the existence of an extended Hilbert space and of a PV measure such that for every . We consider the projection of the self-adjoint operator corresponding to E+ and the self-adjoint operator A corresponding to the sharp reconstruction E and show the existence of a function G: [0,1] → [0,1] such that . An example for which the last relation can be inverted is shown.
TL;DR: In this paper, the authors studied the geometry of hypersurfaces in manifolds with Ricci-flat holonomy group, on which they introduced a G-structure whose intrinsic torsion can be identified with the second fundamental form.
Abstract: We study the geometry of hypersurfaces in manifolds with Ricci-flat holonomy group, on which we introduce a G-structure whose intrinsic torsion can be identified with the second fundamental form. The general problem of extending a manifold with such a G-structure so as to invert this construction is open, but results exist in particular cases, which we review. We list the five-dimensional nilmanifolds carrying invariant SU(2)-structures of this type, and present an example of an associated metric with holonomy SU(3).
TL;DR: An intrinsic description of the Hamilton-Cartan formalism for first-order Berezinian variational problems determined by a submersion of supermanifolds is given in this article.
Abstract: An intrinsic description of the Hamilton–Cartan formalism for first-order Berezinian variational problems determined by a submersion of supermanifolds is given. This is achieved by studying the associated higher-order graded variational problem through the Poincare–Cartan form. Noether theorem and examples from superfield theory and supermechanics are also discussed.
TL;DR: In this article, the formulation of General Relativity and the Hamiltonian formulation of Gauge theories are made to interact and the resulting scheme allows to see general Relativity as a constrained Gauge theory.
Abstract: The formulation of General Relativity presented in [1] and the Hamiltonian formulation of Gauge theories described in [2] are made to interact. The resulting scheme allows to see General Relativity as a constrained Gauge theory.
TL;DR: The idea of applying the gauge principle to formulate the general theory of relativity started with Utiyama in 1956, and various applications of the principle applied to different aspects of the gravitational interactions, are reviewed in this article.
Abstract: The idea of applying the gauge principle to formulate the general theory of relativity started with Utiyama in 1956. In this article, various applications of the gauge principle applied to different aspects of the gravitational interactions, are reviewed.