Space Of Constant Negative Curvature Research Articles

On the symmetry algebra of a one-dimensional quantum-mechanical oscillator on a hyperbola

The quantum-mechanical problem of a harmonic oscillator on a hyperbola as a one-dimensional space of constant negative curvature is considered in this article. A generalization to the singular oscillator model in the context of one-dimensional Cayley – Klein geometries is given by the factorization method. The energy spectrum and wave functions of stationary states are found having the curvature of space as a parameter. For the energy levels of the singular oscillator, the effect of non-zero curvature is clearly manifested through a positive or negative term, depending on the sign of the curvature, which is quadratic in the level number. The results obtained are consistent with those previously published. The dynamical symmetry of the problem is shown explicitly as a quadratic Hahn algebra QH(3) or its isomorphic Higgs algebra.

Superintegrability, symmetry and point particle T-duality

We show that the ideas related to integrability and symmetry play an important role not only in the string T-duality story but also in its point particle counterpart. Applying those ideas, we find that the T-duality seems to be a more widespread phenomenon in the context of the point particle dynamics than it is in the string one; moreover, it concerns physically very relevant point particle dynamical systems and not just somewhat exotic ones fabricated for the purpose. As a source of T-duality examples, we consider maximally superintegrable spherically symmetric electro-gravitational backgrounds in [Formula: see text] dimensions. We then describe in detail four such spherically symmetric dynamical systems which are all mutually interconnected by a web of point particle T-dualities. In particular, the dynamics of a charged particle scattered by a repulsive Coulomb potential in a flat space is T-dual to the dynamics of the Coulomb scattering in the space of constant negative curvature, but it is also T-dual to the (conformal) Calogero–Moser inverse square dynamics both in flat and hyperbolic spaces. Thus knowing just the Hamiltonian dynamics of the scattered particle cannot give us information about the curvature of the space in which the particle moves.

К динамике твердого стержня в пространстве Лобачевского

An absolutely solid infinitely thin rectilinear immaterial finite length rod helical motion problem is considered. Motion is heical relative to the inertia center. Located at rod ends point masses are identiall. The rod movement occurs in the constant negative curvature space (Lobachevsky space) under the gyroscopic forces system influence and a constant tracking force. The gyroscopic forces structure is set with using special conditions, which contain six characteristic constant parameters (gyroscopic coefficients). The characteristic motion properties are given. Components analytical dependense on the kinematic rod screw, rod orientation parameters is founded. Expressions of these parameters as time function, are obtained.

New improved massive gravity and three-dimensional spacetimes of constant curvature and constant torsion

We derive the field equations for topologically massive gravity coupled with the most general quadratic curvature terms using the language of exterior differential forms and a first order constrained variational principle. We find variational field equations both in the presence and absence of torsion. We then show that spaces of constant negative curvature (i.e. the anti de-Sitter space $AdS_3$) and constant torsion provide exact solutions.

Computing hyperbolic choreographies

An algorithm is presented for numerical computation of choreographies in spaces of constant negative curvature in a hyperbolic cotangent potential, extending the ideas given in a companion paper for computing choreographies in the plane in a Newtonian potential and on a sphere in a cotangent potential. Following an idea of Diacu, P\'{e}rez-Chavela and Reyes Victoria, we use stereographic projection and study the problem in the Poincar\'{e} disk. Using approximation by trigonometric polynomials and optimization methods with exact gradient and exact Hessian matrix, we find new choreographies, hyperbolic analogues of the ones presented in the companion paper. The algorithm proceeds in two phases: first BFGS quasi-Newton iteration to get close to a solution, then Newton iteration for high accuracy.

New improved massive gravity

We derive the field equations for topologically massive gravity coupled with the most general quadratic curvature terms using the language of exterior differential forms and a first-order constrained variational principle. We find variational field equations both in the presence and absence of torsion. We then show that spaces of constant negative curvature (i.e. the anti de-Sitter space AdS3) and constant torsion provide exact solutions.

Thinking Outside the Euclidean Box: Riemannian Geometry and Inter-Temporal Decision-Making.

Inter-temporal decisions involves assigning values to various payoffs occurring at different temporal distances. Past research has used different approaches to study these decisions made by humans and animals. For instance, considering that people discount future payoffs at a constant rate (e.g., exponential discounting) or at variable rate (e.g., hyperbolic discounting). In this research, we question the widely assumed, but seldom questioned, notion across many of the existing approaches that the decision space, where the decision-maker perceives time and monetary payoffs, is a Euclidean space. By relaxing the rigid assumption of Euclidean space, we propose that the decision space is a more flexible Riemannian space of Constant Negative Curvature. We test our proposal by deriving a discount function, which uses the distance in the Negative Curvature space instead of Euclidean temporal distance. The distance function includes both perceived values of time as well as money, unlike past work which has considered just time. By doing so we are able to explain many of the empirical findings in inter-temporal decision-making literature. We provide converging evidence for our proposal by estimating the curvature of the decision space utilizing manifold learning algorithm and showing that the characteristics (i.e., metric properties) of the decision space resembles those of the Negative Curvature space rather than the Euclidean space. We conclude by presenting new theoretical predictions derived from our proposal and implications for how non-normative behavior is defined.

Sigma models with negative curvature

We construct Higgs Effective Field Theory (HEFT) based on the scalar manifold H^n, which is a hyperbolic space of constant negative curvature. The Lagrangian has a non-compact O(n,1) global symmetry group, but it gives a unitary theory as long as only a compact subgroup of the global symmetry is gauged. Whether the HEFT manifold has positive or negative curvature can be tested by measuring the S-parameter, and the cross sections for longitudinal gauge boson and Higgs boson scattering, since the curvature (including its sign) determines deviations from Standard Model values.

Lower bounds for the first Dirichlet eigenvalue of the Laplacian for domains in hyperbolic space

AbstractWe consider domains in a simply connected space of constant negative curvature and develop a new technique that improves existing classical lower bound for Dirichlet eigenvalues obtained by H. P. McKean as well as the lower bounds recently obtained by A. Savo.

The Higgs oscillator on the hyperbolic plane and light-front holography

The Light Front Holographic (LFH) wave equation, which is the conformal scalar equation on the plane, is revisited from the perspective of the supersymmetric quantum mechanics, and attention is drawn to the fact that it naturally emerges in the small hyperbolic angle approximation to the "curved" Higgs oscillator on the hyperbolic plane, i.e. on the upper part of the two-dimensional hyperboloid of two sheets, a space of constant negative curvature. Such occurs because the particle dynamics under consideration reduces to the one dimensional Schr\"odinger equation with the second hyperbolic P\"oschl-Teller potential, whose flat-space (small-angle) limit reduces to the conformally invariant inverse square distance plus harmonic oscillator interaction, on which LFH is based. In consequence, energies and wave functions of the LFH spectrum can be approached by the solutions of the Higgs oscillator on the hyperbolic plane in employing its curvature and the potential strength as fitting parameters. Also the proton electric charge form factor is well reproduced within this scheme by means of a Fourier-Helgason hyperbolic wave transform of the charge density. In conclusion, in the small angle approximation, the Higgs oscillator on the hyperbolic plane is demonstrated to satisfactory parallel essential outcomes of the Light Front Holographic QCD. The findings are suggestive of associating the hyperboloid curvature of the with a second scale in LFH, which then could be employed in the definition of a chemical potential.

Wightman function and the Casimir effect for a Robin sphere in a constant curvature space

We evaluate the Wightman function, the mean field squared and the vacuum expectation value of the energy–momentum tensor for a scalar field with the Robin boundary condition on a spherical shell in the background of a constant negative curvature space. For the coefficient in the boundary condition there is a critical value above which the scalar vacuum becomes unstable. In both the interior and the exterior regions, the vacuum expectation values are decomposed into the boundary-free and sphere-induced contributions. For the latter, rapidly convergent integral representations are provided. In the region inside the sphere, the eigenvalues are expressed in terms of the zeros of the combination of the associated Legendre function and its derivative and the decomposition is achieved by making use of the Abel–Plana type summation formula for the series over these zeros. The sphere-induced contribution to the vacuum expectation value of the field squared is negative for the Dirichlet boundary condition and positive for the Neumann one. At distances from the sphere larger than the curvature scale of the background space the suppression of the vacuum fluctuations in the gravitational field corresponding to the negative curvature space is stronger compared with the case of the Minkowskian bulk. In particular, the decay of the vacuum expectation values with the distance is exponential for both massive and massless fields. The corresponding results are generalized for spaces with spherical bubbles and for cosmological models with negative curvature spaces.

A summation formula over the zeros of a combination of the associated Legendre functions with a physical application

By using the generalized Abel–Plana formula, we derive a summation formula for the series over the zeros of a combination of the associated Legendre functions with respect to the degree. The summation formula for the series over the zeros of the combination of the Bessel functions, previously discussed in the literature, is obtained as a limiting case. As an application we evaluate the Wightman function for a scalar field with a general curvature coupling parameter in the region between concentric spherical shells on a background of constant negative curvature space. For the Dirichlet boundary conditions the corresponding mode-sum contains the series over the zeros of the combination of the associated Legendre functions. The application of the summation formula allows us to present the Wightman function in the form of the sum of two integrals. The first one corresponds to the Wightman function for the geometry of a single spherical shell and the second one is induced by the presence of the second shell. The boundary-induced part in the vacuum expectation value of the field squared is investigated. For points away from the boundaries the corresponding renormalization procedure is reduced to that for the boundary-free part.

A summation formula over the zeros of the associated Legendre function with a physical application

Associated Legendre functions arise in many problems of mathematical physics. By using the generalized Abel–Plana formula, in this paper we derive a summation formula for the series over the zeros of the associated Legendre function of the first kind with respect to the degree. The summation formula for the series over the zeros of the Bessel function, previously discussed in the literature, is obtained as a limiting case. The Wightman function for a scalar field with a general curvature coupling parameter is considered inside a spherical boundary on the background of a constant negative curvature space. The corresponding mode sum contains the series over the zeros of the associated Legendre function. The application of the summation formula allows us to present the Wightman function in the form of the sum of two integrals. The first one corresponds to the Wightman function for the bulk geometry without boundaries and the second one is induced by the presence of the spherical shell. For points away from the boundary the latter is finite in the coincidence limit. In this way the renormalization of the vacuum expectation value of the field squared is reduced to that for the boundary-free part.

Gap generation for Dirac fermions on Lobachevsky plane in a magnetic field

We study symmetry breaking and gap generation for fermions in the 2D space of constant negative curvature (the Lobachevsky plane) in an external covariantly constant magnetic field in a four-fermion model. It is shown that due to the magnetic and negative curvature catalyses phenomena the critical coupling constant is zero and there is a symmetry breaking condensate in the chiral limit even in free theory. We analyze solutions of the gap equation in the cases of zero, weak, and strong magnetic fields. As a byproduct, we calculate the density of states and the Hall conductivity for noninteracting fermions that may be relevant for studies of graphene.

Periodic boundary conditions on the pseudosphere

We provide a framework for building periodic boundary conditions on the pseudosphere (or hyperbolic plane), the infinite two-dimensional Riemannian space of constant negative curvature. Starting from the common case of periodic boundary conditions in the Euclidean plane, we introduce all the required mathematical notions and sketch a classification of periodic boundary conditions on the hyperbolic plane. We stress the possible applications in statistical mechanics for studying the bulk behavior of physical systems, and illustrate how to implement such periodic boundary conditions in two examples, the dynamics of particles on the pseudosphere and the study of classical spins on hyperbolic lattices.

Kubo–Mori–Bogoliubov Fisher information on the quantum Gaussian model and violation of the scale invariance

The classical Gaussian model, a parametric family of the Gaussian distribution, is known to be a space of constant negative curvature if one regards the Fisher information on the model as a Riemannian metric. Constant curvature reflects the scale invariance of the classical Gaussian model, which is well known in information geometry. However, it is shown that if the Kubo–Mori–Bogoliubov Fisher information on the quantum Gaussian model is adopted as a Riemannian metric, then this scale invariance on the Gaussian model is broken due to the quantum effect. In the present study, the connection between the geometry of the classical Gaussian model and its quantum counterpart is clarified using the Taylor expansion with respect to the Planck constant. It is further shown that such a method is not applicable to a finite-dimensional system such as the spin system.

Kaluza-Klein black hole with negatively curved extra dimensions in string-generated gravity models

We obtain a new exact black-hole solution in Einstein-Gauss-Bonnet gravity with a cosmological constant which bears a specific relation to the Gauss-Bonnet coupling constant. The spacetime is a product of the usual 4-dimensional manifold with a $(n-4)$-dimensional space of constant negative curvature, i.e., its topology is locally ${\ma M}^n \approx {\ma M}^4 \times {\ma H}^{n-4}$. The solution has two parameters and asymptotically approximates to the field of a charged black hole in anti-de Sitter spacetime. The most interesting and remarkable feature is that the Gauss-Bonnet term acts like a Maxwell source for large $r$ while at the other end it regularizes the metric and weakens the central singularity.

Regge trajectories of the Coulomb potential in the space of constant negative curvature S31

Analytic properties of the scattering amplitude for Coulomb potential on the background of the space of constant negative curvature are studied. Special attention is given to the comparison of the Regge trajectories for curved and flat spaces. We show that there exist considerably differences in the behavior of the Regge trajectories in these spaces.

Topological Analysis of Integrable Problems of Celestial Mechanics on a Sphere and Pseudosphere

In this paper, we consider the problem of motion of a point on a two-dimensional sphere and pseudosphere (with metric of constant curvature) in the field generated by two attracting centers. This problem is a natural analog of the classical plane two-center problem integrated even by Euler. However, the integrability of similar problems on the sphere and the Lobachevskii plane was established only recently. Therefore, the topological properties of these problems are poorly studied as yet. The present paper fills this gap. We also study the Lagrange problem on the pseudosphere under the action of the Newtonian attraction of the stationary center and one more force constant in magnitude and direction. This problem is a limit case of the two-center problem. In the process of passing to the limit, the second center moves at infinity in the direction of the propulsion force (moreover, its mass grows so that the constancy of the propulsion is ensured, i.e., proportionally to the squared distance). This problem was considered by Lagrange for the first time and reduces to quadratures. The question on gravitational interactions in spaces of constant curvature was posed for the first time by N. N. Lobachevskii, who studied generalizations of the attraction law to the space of constant negative curvature. This theme was further developed in the work of N. E. Zhukovskii [18] on the motion of a pseudospheric (“plane”) plate on the Lobachevskii plane. On a three-dimensional sphere (of constant curvature 1), one also considers an analog of the Newtonian potential generated by a gravitational center, which in this case is cotan θ, where the angle θ is equal to the distance to the center. The integrability of the Kepler problem on the three-dimensional sphere S3 was proved by E. Schrodinger [12]. Schrodinger considered the motion of an electron in the potential field of the nucleus of a hydrogen-like atom from the standpoint of quantum mechanics. Three classical Kepler laws have natural generalizations for spaces of constant curvature. For the Lobachevskii space and for the sphere, these generalizations were formulated by A. N. Chernikov [5] and V. V. Kozlov [8]. In particular, any finite orbit here is an ellipse in one of whose foci the gravitational center (the Sun) is located. Along with this, V. V. Kozlov and O. A. Kharin proved the integrability of the problem on the motion of a material particle on the sphere S2 in the field of two stationary Newtonian centers [9]. In [14–17], the problem of two centers on the three-dimensional sphere S3 and in the three-dimensional Lobachevskii space H3 and also the Lagrange problem (the limit case of the two-center problem when one of the centers is moved off at infinity) in the Lobachevskii space H3 were studied. In these works, the integrabiity of the problems is demonstrated, bifurcation diagrams are constructed, and the classification of regions of feasible motion is carried out. The first step in studying these problems consists of reducing the problem from the three-dimensional to the two-dimensional case, i.e., reducing the number of degrees of freedom. The next step consists of passing to convenient coordinates (in the problems under consideration, for instance, to spherical-conical and pseudo-spherical-conical coordinates which are defined as roots of certain rational functions) in order to reduce the system to the Liouville form.

SHARP BOUNDS ON THE NUMBER OF RESONANCES FOR CONFORMALLY COMPACT MANIFOLDS WITH CONSTANT NEGATIVE CURVATURE NEAR INFINITY

Abstract The purpose of this article is to prove a sharp bound on the number of resonances for the Laplacian on conformally compact manifolds with constant negative curvature near infinity, thus improving the polynomial bound of Guillope and Zworki (Guillope, L., Zworski, M. ([1995b]). Polynomial bound on the number of resonances for some complete spaces of constant negative curvature near infinity. Asympt. Anal. 11:1–22).