Volume 51, Issue 5, May 2010
 ARTICLES

 Quantum Mechanics (General and Nonrelativistic)

Phases of modified Stockwell transforms and instantaneous frequencies
View Description Hide DescriptionThe phase of a signal is analyzed using the Stockwell transform. In particular, the relationships between the instantaneous frequencies of a signal in polar form and the phase of the corresponding Stockwell transform are given. The corresponding results using a reciprocal Morlet wavelet transform are given for comparisons.

Hamiltonians of quantum systems with positions and momenta in
View Description Hide DescriptionA quantum system with positions and momenta in is considered. Such a system can be constructed from smaller systems, in which the positions and momenta take values in , if the Hamiltonian of this partite system is compatible with . The concept of compatibility of a Hamiltonian with allows the quantum formalism in the partite system to be expressed in terms of Galois arithmetic. Transformations of the basis in produce unitary transformations of the quantum states, which form a representation of . They are used to define which subset of the general set of Hamiltonians in the partite system is compatible with .

On the existence of quantum representations for two dichotomic measurements
View Description Hide DescriptionUnder which conditions do outcome probabilities of measurements possess a quantummechanical model? This kind of problem is solved here for the case of two dichotomic von Neumann measurements which can be applied repeatedly to a quantum system with trivial dynamics. The solution uses methods from the theory of operator algebras and the theory of moment problems. The ensuing conditions reveal surprisingly simple relations between certain quantummechanical probabilities. It also shown that generally, none of these relations holds in general probabilistic models. This result might facilitate further experimental discrimination between quantum mechanics and other general probabilistic theories.

Spectral singularities for nonHermitian onedimensional Hamiltonians: Puzzles with resolution of identity
View Description Hide DescriptionWe examine the completeness of biorthogonal sets of eigenfunctions for nonHermitian Hamiltonians possessing a spectral singularity. The correct resolutions of identity are constructed for deltalike and smooth potentials. Their form and the contribution of a spectral singularity depend on the class of functions employed for physical states. With this specification there is no obstruction to completeness originating from a spectral singularity.

Spectroscopy of drums and quantum billiards: Perturbative and nonperturbative results
View Description Hide DescriptionWe develop powerful numerical and analytical techniques for the solution of the Helmholtz equation on general domains. We prove two theorems: the first theorem provides an exact formula for the ground state of an arbitrary membrane, while the second theorem generalizes this result to any excited state of the membrane. We also develop a systematic perturbative scheme which can be used to study the small deformations of a membrane of circular or square shapes. We discuss several applications, obtaining numerical and analytical results.

On the construction of coherent states of position dependent mass Schrödinger equation endowed with effective potential
View Description Hide DescriptionIn this paper, we propose an algorithm to construct coherent states for an exactly solvable position dependent mass Schrödinger equation. We use point canonical transformation method and obtain ground state eigenfunction of the position dependent mass Schrödinger equation. We fix the ladder operators in the deformed form and obtain explicit expression of the deformed superpotential in terms of mass distribution and its derivative. We also prove that these deformed operators lead to minimum uncertainty relations. Further, we illustrate our algorithm with two examples, in which the coherent states given for the second example are new.

Callan–Symanzik equation and asymptotic freedom in the Marr–Shimamoto model
View Description Hide DescriptionThe exactly soluble nonrelativistic Marr–Shimamoto model was introduced in 1964 as an example of the Lee model with a propagator and a nontrivial vertex function. An exactly soluble relativistic version of this model, known as the Zachariasen model, has been found to be asymptotically free in terms of coupling constant renormalization at an arbitrary spacelike momentum and on the basis of exact solutions of the Gell–Mann–Low equations. This is accomplished with conventional cutoff regularization by setting up the Yukawa and Fermi coupling constants at Euclidean momenta in terms of on massshell couplings and then taking the asymptotic limit. In view of this background, it may be expected that an investigation of the nonrelativistic Marr–Shimamoto theory may also exhibit asymptotic freedom in view of its manifest mathematical similarity to that of the Zachariasen model. To prove this point, the present paper prefers to examine asymptotic freedom in the nonrelativistic Marr–Shimamoto theory using the powerful concepts of the renormalization group and the Callan–Symanzik equation, in conjunction with the specificity of dimensional regularization and onshell renormalization. This approach is based on calculations of the Callan–Symanzik coefficients and determinations of the effective coupling constants. It is shown that the Marr–Shimamoto theory is asymptotically free for dimensions and for values of occurring in periodic intervals over the range of of particular interest. This differs from the original Lee model which has been shown by several authors, using these same concepts, to be asymptotically free only for .

A quantum logical and geometrical approach to the study of improper mixtures
View Description Hide DescriptionWe study improper mixtures from a quantum logical and geometrical point of view. Taking into account the fact that improper mixtures do not admit an ignorance interpretation and must be considered as states in their own right, we do not follow the standard approach which considers improper mixtures as measures over the algebra of projections. Instead of it, we use the convex set of states in order to construct a new lattice whose atoms are all physical states: pure states and improper mixtures. This is done in order to overcome one of the problems which appear in the standard quantum logical formalism, namely, that for a subsystem of a larger system in an entangled state, the conjunction of all actual properties of the subsystem does not yield its actual state. In fact, its state is an improper mixture and cannot be represented in the von Neumann lattice as a minimal property which determines all other properties as is the case for pure states or classical systems. The new lattice also contains all propositions of the von Neumann lattice. We argue that this extension expresses in an algebraic form the fact that—alike the classical case—quantum interactions produce nontrivial correlations between the systems. Finally, we study the maps which can be defined between the extended lattice of a compound system and the lattices of its subsystems.
 Quantum Information and Computation

Unique decompositions, faces, and automorphisms of separable states
View Description Hide DescriptionLet be the set of separable states on admitting a representation as a convex combination of pure product states, or fewer. If , , and , we show that admits a subset such that is dense and open in , and such that each state in has a unique decomposition as a convex combination of pure product states, and we describe all possible convex decompositions for a set of separable states that properly contains . In both cases we describe the associated faces of the space of separable states, which in the first case are simplexes, and in the second case are direct convex sums of faces that are isomorphic to state spaces of full matrix algebras. As an application of these results, we characterize all affine automorphisms of the convex set of separable states and all automorphisms of the state space of that preserve entanglement and separability.

Noncommutative Poisson boundaries of unital quantum operations
View Description Hide DescriptionIn this paper, Poisson boundaries of unital quantum operations (also called Markov operators) are investigated. In the case of unital quantum channels, compact operators belonging to Poisson boundaries are characterized. Using the characterization of amenable groups by the injectivity of their von Neumann algebras, we will answer negatively some conjectures appearing in the work of Arias et al. [“Fixed points of quantum operations,” J. Math. Phys.43, 5872 (2002)] about injectivity of the commuting algebra of the Kraus operators of unital quantum operations and their injective envelopes.
 Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)

A superspace formulation of Yang–Mills theory on sphere
View Description Hide DescriptionA superspace approach to the Becchi–Rouet–Stora–Tyutin (BRST) formalism for the Yang–Mills theory on an dimensional unit sphere is developed in a manifestly covariant manner based on the rotational supersymmetry characterized by the supergroup . This is done by employing an dimensional unit supersphere parametrized by commutative and two anticommutative coordinate variables so that it includes as a subspace and realizes the supersymmetry. In this superspace formulation, referred to as the supersphere formulation, the socalled horizontality condition is concisely expressed in terms of the rank3 field strength tensor of a Yang–Mills superfield on . The supersphere formulation completely covers the BRST gaugefixing procedure for the Yang–Mills theory on provided by us [R. Banerjee and S. Deguchi, Phys. Lett. B632, 579 (2006); arXiv:hepth/0509161]. Furthermore, this formulation admits the (massive) Curci–Ferrari model defined on , describing the gaugefixing and mass terms on together as a mass term on .

Realizations of conformal currenttype Lie algebras
View Description Hide DescriptionIn this paper we obtain the realizations of some infinitedimensional Lie algebras, named “conformal currenttype Lie algebras,” in terms of a twodimensional Novikov algebra and its deformations. Furthermore, Ovsienko and Roger’s loop cotangent Virasoro algebra, which can be regarded as a nice generalization of the Virasoro algebra with two space variables, is naturally realized as an affinization of the tensor product of a deformation of the twodimensional Novikov algebra and the Laurent polynomialalgebra. These realizations shed new light on various aspects of the structure and representation theory of the corresponding infinitedimensional Lie algebras.

Spinor algebra and null solutions of the wave equation
View Description Hide DescriptionIn this paper we exploit the ideas and formalisms of twistor theory, to show how, on Minkowski space, given a null solution of the wave equation, there are precisely two null directions in , at least one of which is a shearfree ray congruence.

New fourdimensional integrals by Mellin–Barnes transform
View Description Hide DescriptionThis paper is devoted to the calculation of a special class of integrals by Mellin–Barnes transform. It contains double integrals in the position space in dimensions, where is parameter of dimensional regularization. These integrals contribute to the effective action of the supersymmetric Yang–Mills theory. The integrand is a fraction in which the numerator is the logarithm of the ratio of spacetime intervals, and the denominator is the product of powers of spacetime intervals. According to the method developed in the previous papers, in order to make use of the uniqueness technique for one of two integrations, we shift exponents in powers in the denominator of integrands by some multiples of . As the next step, the second integration in the position space is done by Mellin–Barnes transform. For normalizing procedure, we reproduce first the known result obtained earlier by Gegenbauer polynomial technique. Then, we make another shift of exponents in powers in the denominator to create the logarithm in the numerator as the derivative with respect to the shift parameter . We show that the technique of work with the contour of the integral modified in this way by using Mellin–Barnes transform repeats the technique of work with the contour of the integral without such a modification. In particular, all the operations with a shift of contour of integration over complex variables of twofold Mellin–Barnes transform are the same as before the modification of indices, and even the poles of residues coincide. This confirms the observation made in the previous papers that in the position space all the Green’s function of supersymmetric Yang–Mills theory can be expressed in terms of Usyukina–Davydychev functions.

BogomolnyPrasadSommerfeld state counting in local obstructed curves from quiver theory and Seiberg duality
View Description Hide DescriptionIn this paper, we study the BogomolnyPrasadSommerfeld (BPS) state counting in the geometry of local obstructed curve with normal bundle . We find that the BPS states have a framed quiver description. Using this quiver description along with the Seiberg duality and the localization techniques, we can compute the BPS state indices in different chambers dictated by stability parameter assignments. This provides a welldefined method to compute the generalized Donaldson–Thomas invariants. This method can be generalized to other affine ADE quiver theories.

Extremal asymmetric universal cloning machines
View Description Hide DescriptionThe tradeoffs among various output fidelities of asymmetric universal cloning machines are investigated. First we find out all the attainable optimal output fidelities for the 1 to 3 asymmetric universal cloning machine and it turns out that there are two kinds of extremal machines which have to cooperate in order to achieve some of the optimal output fidelities. Second we construct a family of extremal cloning machines that includes the universal symmetric cloning machine as well as an asymmetric 1 to cloning machine for qudits with two different output fidelities such that the optimal tradeoff between the measurement disturbance and state estimation is attained in the limit of infinite .

Affine extension of Galilean conformal algebra in dimensions
View Description Hide DescriptionWe show that a class of nonrelativistic algebras including noncentrally extended Schrödinger algebra and Galilean conformal algebra (GCA) has an affine extension in hitherto unknown. This extension arises out of the conformal symmetries of the two dimensional complex plane. We suggest that this affine form may be the symmetry that explains the relaxation of some classical phenomena toward their critical point. This affine algebra admits a central extension and maybe realized in the bulk. The bulk realization suggests that this algebra may be derived by looking at the asymptotic symmetry of an Antide Sitter (AdS) theory. This suggests that AdS/CFT (conformal field theory) duality may take on a special form in four dimensions.
 General Relativity and Gravitation

A note on wave equation in Einstein and de Sitter spacetime
View Description Hide DescriptionWe consider the wave propagating in the Einstein and de Sitter spacetime. The covariant d’Alembert’s operator in the Einstein and de Sitter spacetime belongs to the family of the nonFuchsian partial differential operators. We introduce the initial value problem for this equation and give the explicit representation formulas for the solutions. We also show the estimates for solutions.
 Dynamical Systems

Asymptotic stability of secondorder neutral stochastic differential equations
View Description Hide DescriptionIn this paper, we study the existence and asymptotic stability in moment of mild solutions to secondorder nonlinear neutral stochastic differential equations. Further, this result is extended to establish stability criterion for stochastic equations with impulsive effects. With the help of fixed point strategy, stochastic analysis technique, and semigroup theory, a set of novel sufficient conditions are derived for achieving the required result. Finally, an example is provided to illustrate the obtained result.

Asymptotic regularity for Laplacian equation
View Description Hide DescriptionThis paper is devoted to proving some asymptotic regularity of the solutions of the Laplacian equation considered on a bounded domain . The nonlinear term satisfies the polynomial growth condition of arbitrary order , where is arbitrary. As an application of the asymptotic regularity results, we not only can obtain the existence of a global attractor immediately but also can show further that attracts every bounded subsets of under the norm for any . Furthermore, the fractal dimension of is finite in for any .