Volume 56, Issue 4, April 2015
Index of content:
In usual realistic classical dynamical systems, the Hamiltonian depends explicitly on time. In this work, a class of classical systems with time dependent nonlinear Hamiltonians is analyzed. This type of problems allows to find invariants by a family of Veronese maps. The motivation to develop this method results from the observation that the Poisson-Lie algebra of monomials in the coordinates and momenta is clearly defined in terms of its brackets and leads naturally to an infinite linear set of differential equations, under certain circumstances. To perform explicit analytic and numerical calculations, two examples are presented to estimate the trajectories, the first given by a nonlinear problem and the second by a quadratic Hamiltonian with three time dependent parameters. In the nonlinear problem, the Veronese approach using jets is shown to be equivalent to a direct procedure using elliptic functions identities, and linear invariants are constructed. For the second example, linear and quadratic invariants as well as stability conditions are given. Explicit solutions are also obtained for stepwise constant forces. For the quadratic Hamiltonian, an appropriated set of coordinates relates the geometric setting to that of the three dimensional manifold of central conic sections. It is shown further that the quantum mechanical problem of scattering in a superlattice leads to mathematically equivalent equations for the wave function, if the classical time is replaced by the space coordinate along a superlattice. The mathematical method used to compute the trajectories for stepwise constant parameters can be applied to both problems. It is the standard method in quantum scattering calculations, as known for locally periodic systems including a space dependent effective mass.
- Partial Differential Equations
56(2015); http://dx.doi.org/10.1063/1.4916290View Description Hide Description
In this paper, we consider a skew-symmetric Chern-Simons system problem with a coupling parameter. Our main goal is that, when the coupling parameter is small, the topological type solutions to this system problem are uniquely determined by the location of their vortex points. This result follows by the bubbling analysis and the non-degeneracy of linearized equations.
- Representation Theory and Algebraic Methods
56(2015); http://dx.doi.org/10.1063/1.4916889View Description Hide Description
The Wigner–Eckart theorem is a well known result for tensor operators of 𝔰𝔲(2) and, more generally, any compact Lie algebra. In this paper, the theorem will be generalized to the particular non-compact case of 𝔰𝔩(2, ℝ). In order to do so, recoupling theory between representations that are not necessarily unitary will be studied, namely, between finite-dimensional and infinite-dimensional representations. As an application, the Wigner–Eckart theorem will be used to construct an analogue of the Jordan–Schwinger representation, previously known only for representations in the discrete class, which also covers the continuous class.
56(2015); http://dx.doi.org/10.1063/1.4917054View Description Hide Description
We use the generating function of the characters of C 2 to obtain a generating function for the multiplicities of the weights entering in the irreducible representations of that simple Lie algebra. From this generating function, we derive some recurrence relations among the multiplicities and a simple graphical recipe to compute them.
56(2015); http://dx.doi.org/10.1063/1.4917197View Description Hide Description
We construct a new cohomology complex of Leibniz conformal algebras with coefficients in a representation instead of a module. The low-dimensional cohomology groups of this complex are computed. Meanwhile, we construct a Leibniz algebra from a Leibniz conformal algebra and prove that the category of Leibniz conformal algebras is equivalent to the category of equivalence classes of formal distribution Leibniz algebras.
56(2015); http://dx.doi.org/10.1063/1.4917545View Description Hide Description
We introduce excited coherent states, , where n is an integer and states denote the coherent states of a charged particle in a uniform magnetic field. States minimize the Schrödinger-Robertson uncertainty relation while having the nonclassical properties. It has been shown that the resolution of identity condition is realized with respect to an appropriate measure on the complex plane. Some of the nonclassical features such as sub-Poissonian statistics and quadrature squeezing of these states are investigated. Our results are compared with similar Agarwal’s type photon added coherent states (PACSs) and it is shown that, while photon-counting statistics of are the same as PACSs, their squeezing properties are different. It is also shown that for large values of , while they are squeezed, they minimize the uncertainty condition. Additionally, it has been demonstrated that by changing the magnitude of the external magnetic field, Bext , the squeezing effect is transferred from one component to another. Finally, a new scheme is proposed to generate states in cavities.
56(2015); http://dx.doi.org/10.1063/1.4917549View Description Hide Description
In this paper, we first construct a class of weight modules, called φ-imaginary Verma modules, for the toroidal Lie algebras. Then a criterion for the irreducibility of the φ-imaginary Verma modules is obtained and the irreducible quotients for the reducible ones are studied. Furthermore, we construct a more general class of ℤ n -graded modules for the toroidal Lie algebras and we discuss their irreducibility. This class of modules includes the φ-imaginary Verma modules as special examples.
56(2015); http://dx.doi.org/10.1063/1.4918773View Description Hide Description
Let H 1 be a quantum group and f : H 1⟶H 2 a Hopf algebra homomorphism. Assume that B is some braided group obtained by Majid’s transmutation process. We first show that there is a tensor equivalence between the category of comodules over the braided group B and that of relative Yetter-Drinfeld modules. Next, we prove that the Drinfeld centers of the two categories mentioned above are equivalent to the category of modules over some quantum double, namely, the category of ordinary Yetter-Drinfeld modules over some Radford’s biproduct Hopf algebra. Importantly, the above results not only hold for a finite dimensional quantum group but also for an infinite dimensional one.
56(2015); http://dx.doi.org/10.1063/1.4918648View Description Hide Description
An algorithm to compute Connes spectral distance, adaptable to the Hilbert-Schmidt operatorial formulation of non-commutative quantum mechanics, was developed earlier by introducing the appropriate spectral triple and used to compute infinitesimal distances in the Moyal plane, revealing a deep connection between geometry and statistics. In this paper, using the same algorithm, the Connes spectral distance has been calculated in the Hilbert-Schmidt operatorial formulation for the fuzzy sphere whose spatial coordinates satisfy the su(2) algebra. This has been computed for both the discrete and the Perelemov’s SU(2) coherent state. Here also, we get a connection between geometry and statistics which is shown by computing the infinitesimal distance between mixed states on the quantum Hilbert space of a particular fuzzy sphere, indexed by n ∈ ℤ/2.
- Quantum Mechanics
56(2015); http://dx.doi.org/10.1063/1.4908107View Description Hide Description
First order integrals of motion for Schrödinger equations with position dependent masses are classified. Eighteen classes of such equations with non-equivalent symmetries are specified. They include integrable, superintegrable, and maximally superintegrable systems. Among them is a system invariant with respect to the Lie algebra of Lorentz group and a system whose integrals of motion form algebra so(4). Three of the obtained systems are solved exactly.
56(2015); http://dx.doi.org/10.1063/1.4918551View Description Hide Description
In variational calculations with harmonic oscillator wavefunctions as trial bases, the transformation coefficients that relate harmonic oscillator wavefunctions in two different sets of internal coordinates are convenient to the evaluation of some matrix elements. Here, we present the explicit expression of these transformation coefficients for five-body systems. These transformation coefficients can be collected in a matrix according to the quantum number N of harmonic oscillator shell and can be programmed for arbitrary N.
56(2015); http://dx.doi.org/10.1063/1.4918367View Description Hide Description
To understand the propagation behavior of an oblique incident matter wave in a three-dimensional non-spherical quantum cloak, we perform the transformation design for the prolate spheroidal coordinate system and obtain a quantum cloak with an ellipsoidal shape. The mass parameters and effective potential for the creation of a perfect prolate spheroidal invisibility region are given. The analytic representations of the cloaked matter wave and probability current in the cloaking shell are presented. Special attention is paid to the discussions of the probability current in the cloaking shell for only that current can manifestly exhibit how the wave vector of the matter wave is curved, rotated, and guided in the cloaking shell to flow around the non-spherically invisible region. With the current analysis, one shows that the presented cloak can perfectly guide the matter wave in the situation of any oblique incidence. The proposed prolate spheroidal cloak for matter waves provides the first non-spherically three-dimensional setup for quantum cloaking.
56(2015); http://dx.doi.org/10.1063/1.4918547View Description Hide Description
The present article is concerned with the use of approximations in the calculation of the many-body density of levels ρ mb(E, N) of a system with total energy E, composed by N bosons. In the mean-field framework, an integral expression for ρ mb, which is proper to be performed by asymptotic expansions, can be derived. However, the standard second order steepest descent method cannot be applied to this integral when the ground-state is sufficiently populated. Alternatively, we derive a uniform formula for ρ mb, which is potentially able to deal with this regime. In the case of the one-dimensional harmonic oscillator, using results found in the number theory literature, we show that the uniform formula improves the standard expression achieved by means of the second order method.
Formulation of a unified method for low- and high-energy expansions in the analysis of reflection coefficients for one-dimensional Schrödinger equation56(2015); http://dx.doi.org/10.1063/1.4918552View Description Hide Description
We study low-energy expansion and high-energy expansion of reflection coefficients for one-dimensional Schrödinger equation, from which expansions of the Green function can be obtained. Making use of the equivalent Fokker-Planck equation, we develop a generalized formulation of a method for deriving these expansions in a unified manner. In this formalism, the underlying algebraic structure of the problem can be clearly understood, and the basic formulas necessary for the expansions can be derived in a natural way. We also examine the validity of the expansions for various asymptotic behaviors of the potential at spatial infinity.
56(2015); http://dx.doi.org/10.1063/1.4918707View Description Hide Description
Starting from degree solutions of a time dependent Schrödinger-like equation for classical orthogonal polynomials, a linear matrix equation describing perturbations around the zeros of the polynomial is derived. The matrix has remarkable Diophantine properties. Its eigenvalues are independent of the zeros. The corresponding eigenvectors provide the representations of the lower degree ( ) polynomials in terms of the zeros of the degree polynomial. The results are valid universally for all the classical orthogonal polynomials, including the Askey scheme of hypergeometric orthogonal polynomials and its q-analogues.
- Quantum Information and Computation
56(2015); http://dx.doi.org/10.1063/1.4917056View Description Hide Description
We consider the transformations of quantum states obtainable by a process of the following sort. Combine the given input state with a specially prepared initial state of an auxiliary system. Apply a unitary transformation to the combined system. Measure the state of the auxiliary subsystem. If (and only if) it is in a specified final state, consider the process successful, and take the resulting state of the original (principal) system as the result of the process. We review known information about exact realization of transformations by such a process. Then we present results about approximate realization of finite partial transformations. We not only consider primarily the issue of approximation to within a specified positive ε, but also address the question of arbitrarily close approximation.
- Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory
A relativistically interacting exactly solvable multi-time model for two massless Dirac particles in 1 + 1 dimensions56(2015); http://dx.doi.org/10.1063/1.4915952View Description Hide Description
The question how to Lorentz transform an N-particle wave function naturally leads to the concept of a so-called multi-time wave function, i.e., a map from (space-time) N to a spin space. This concept was originally proposed by Dirac as the basis of relativistic quantum mechanics. In such a view, interaction potentials are mathematically inconsistent. This fact motivates the search for new mechanisms for relativistic interactions. In this paper, we explore the idea that relativistic interaction can be described by boundary conditions on the set of coincidence points of two particles in space-time. This extends ideas from zero-range physics to a relativistic setting. We illustrate the idea at the simplest model which still possesses essential physical properties like Lorentz invariance and a positive definite density: two-time equations for massless Dirac particles in 1 + 1 dimensions. In order to deal with a spatio-temporally non-trivial domain, a necessity in the multi-time picture, we develop a new method to prove existence and uniqueness of classical solutions: a generalized version of the method of characteristics. Both mathematical and physical considerations are combined to precisely formulate and answer the questions of probability conservation, Lorentz invariance, interaction, and antisymmetry.
56(2015); http://dx.doi.org/10.1063/1.4916291View Description Hide Description
This paper defines a generalization of the Connes-Moscovici Hopf algebra, , that contains the entire Hopf algebra of rooted trees. A relationship between the former, a much studied object in non-commutative geometry, and the latter, a much studied object in perturbative quantum field theory, has been established by Connes and Kreimer. The results of this paper open the door to study the cohomology of the Hopf algebra of rooted trees.
- General Relativity and Gravitation
56(2015); http://dx.doi.org/10.1063/1.4915290View Description Hide Description
We discuss the limit of vanishing G (Newton’s constant of universal gravitation) of the maximal analytically extended Kerr–Newman electrovacuum spacetimes represented in Boyer–Lindquist coordinates. We investigate the topologically nontrivial spacetime emerging in this limit and show that it consists of two copies of flat Minkowski spacetime cross-linked at a timelike solid cylinder (spacelike 2-disk × timelike ℝ). As G → 0, the electromagnetic fields of the Kerr–Newman spacetimes converge to nontrivial solutions of Maxwell’s equations on this background spacetime . We show how to obtain these fields by solving Maxwell’s equations with singular sources supported only on a circle in a spacelike slice of . These sources do not suffer from any of the pathologies that plague the alternate sources found in previous attempts to interpret the Kerr–Newman fields on the topologically simple Minkowski spacetime. We characterize the singular behavior of these sources and prove that the Kerr–Newman electrostatic potential and magnetic scalar potential are the unique solutions of the Maxwell equations among all functions that have the same blow-up behavior at the ring singularity.
56(2015); http://dx.doi.org/10.1063/1.4918368View Description Hide Description
We prove the existence and the uniqueness of the static dyonic black holes in four dimensional N = 1 supergravity theory coupled vector and scalar multiplets. We set the near-horizon geometry to be a product of two Einstein surfaces, whereas the asymptotic geometry has to be a space of constant scalar curvature. Using these data, we show that there exists a unique solution for scalar fields which interpolates these regions.
- Dynamical Systems
56(2015); http://dx.doi.org/10.1063/1.4916289View Description Hide Description
The subject of this investigation is the blow-up phenomena of the positive solutions of the mixed problem for the one-dimensional polytropic filtration equation with inhomogeneous density and source. It is shown that under certain conditions on the nonlinearities and data, blow-up will occur at some finite time. Note that the technique applied for the proof does not use the Zel’dovich-Kompaneets-Barenblatt solutions, since the construction of such type of function is more complicated in our case. Therefore, we obtain a result by multiplying on a special factor which has convenient properties. In particular, by choosing the parameters of the factor and using the properties of the solution, we obtain the inequality which allows us to show the blow up phenomena.