Volume 52, Issue 10, October 2011
Index of content:
- Quantum Mechanics (General and Nonrelativistic)
52(2011); http://dx.doi.org/10.1063/1.3640029View Description Hide Description
The matrix Schrödinger equation with a self-adjoint matrix potential is considered on the half line with the most general self-adjoint boundary condition at the origin. When the matrix potential is integrable and has a first moment, it is shown that the corresponding scattering matrix is continuous at zero energy. An explicit formula is provided for the scattering matrix at zero energy. The small-energy asymptotics are established also for the related Jost matrix, its inverse, and various other quantities relevant to the corresponding direct and inverse scattering problems.
52(2011); http://dx.doi.org/10.1063/1.3646371View Description Hide Description
The concept of the elegant work introduced by Lévai [J. Phys. A22, 689 (1989)] is extended for the solutions of the Schrödinger equation with more realistic other potentials used in different disciplines of physics. The connection between the present model and the other alternative algebraic technique in the literature is discussed.
- Quantum Information and Computation
52(2011); http://dx.doi.org/10.1063/1.3643768View Description Hide Description
We study a spin--particle moving on a one-dimensional lattice subject to disorder induced by a random, space-dependent quantum coin. The discrete time evolution is given by a family of random unitary quantum walk operators, where the shift operation is assumed to be deterministic. Each coin is an independent identically distributed random variable with values in the group of two-dimensional unitary matrices. We derive sufficient conditions on the probability distribution of the coins such that the system exhibits dynamical localization. Put differently, the tunneling probability between two lattice sites decays rapidly for almost all choices of random coins and after arbitrary many time steps with increasing distance. Our findings imply that this effect takes place if the coin is chosen at random from the Haar measure, or some measure continuous with respect to it, but also for a class of discrete probability measures which support consists of two coins, one of them being the Hadamard coin.
52(2011); http://dx.doi.org/10.1063/1.3652924View Description Hide Description
We give an explicit family of XOR games with O(n)-bit questions requiring 2 n ebits to play near-optimally. More generally, we introduce a new technique for proving lower bounds on the amount of entanglement required by an XOR game: we show that near-optimal strategies for an XOR game G correspond to approximate representations of a certain C*-algebra associated to G. Our results extend an earlier theorem of Tsirelson characterising the set of quantum strategies which implement extremal quantum correlations.
- Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)
52(2011); http://dx.doi.org/10.1063/1.3629472View Description Hide Description
The computation of Feynman integrals in massive higher order perturbative calculations in renormalizablequantum field theories requires extensions of multiply nested harmonic sums, which can be generated as real representations by Mellin transforms of Poincaré–iterated integrals, including denominators of higher cyclotomic polynomials. We derive the cyclotomic harmonic polylogarithms and harmonic sums and study their algebraic and structural relations. The analytic continuation of cyclotomic harmonic sums to complex values of N is performed using analytic representations. We also consider special values of the cyclotomic harmonic polylogarithms at argument x = 1, respectively, for the cyclotomic harmonic sums at N → ∞, which are related to colored multiple zeta values, deriving various of their relations, based on the stuffle and shuffle algebras and three multiple argument relations. We also consider infinite generalized nested harmonic sums at roots of unity which are related to the infinite cyclotomic harmonic sums. Basis representations are derived for weight sums up to cyclotomy . This paper is dedicated to Martinus Veltman on the occasion of his 80th birthday.
52(2011); http://dx.doi.org/10.1063/1.3652696View Description Hide Description
The time-dependent field equations of the nonlinear field systems, whose static soliton solutions are (global) vortex strings, are studied by a numerical approach. They concern (i) the theory of a single complex scalar field with a spontaneously broken U(1) symmetry, and (ii) the system of a complex scalar field doublet with an approximate U(2) symmetry. The obtained numerical solutions allow to clarify the dynamical behaviors of the systems under fluctuations. The systems are shown to have order-chaos phase transitions, but, despite phase transitions and deformations in field profiles by fluctuations, the shapes of the total field energy density distributions are rather stable.
52(2011); http://dx.doi.org/10.1063/1.3653483View Description Hide Description
We recall the Einstein velocity addition on the open unit ball of and its algebraic structure, called the Einstein gyrogroup. We establish an isomorphism between the Einstein gyrogroup on and the set of all qubit density matrices representing mixed states endowed with an appropriate addition. Our main result establishes a relation between the trace metric for the qubit density matrices and the rapidity metric on the Einstein gyrogroup .
52(2011); http://dx.doi.org/10.1063/1.3653545View Description Hide Description
Quantum field theories on de Sitter spacetime with global U(1) gauge symmetry are deformed using the joint action of the internal symmetry group and a one-parameter group of boosts. The resulting theory turns out to be wedge-local and non-isomorphic to the initial one for a class of theories, including the free charged Dirac field. The properties of deformed models coming from inclusions of CAR-algebras are studied in detail.
52(2011); http://dx.doi.org/10.1063/1.3653550View Description Hide Description
We find the explicit form of the volume form on the moduli space of non-hyperelliptic Riemann surfaces induced by the Siegel metric, a long-standing question in string theory. This question is related to the explicit form of the (g−2)(g−3)/2 linearly independent relations among the 2-fold products of holomorphic abelian differentials, that are provided in the case of canonical curves of genus g ⩾ 4. Such relations can be completely expressed in terms of determinants of the standard normalized holomorphic abelian differentials. Remarkably, it turns out that the induced volume form is the Kodaira-Spencer map of the square of the Bergman reproducing kernel.
- General Relativity and Gravitation
52(2011); http://dx.doi.org/10.1063/1.3642616View Description Hide Description
In this second paper, we develop an analytical theory of quasi-equatorial lensing by Kerrblack holes. In this setting we solve perturbatively our general lens equation with displacement given in Paper I, going beyond weak-deflection Kerr lensing to third order in our expansion parameter ε, which is the ratio of the angular gravitational radius to the angular Einstein radius. We obtain new formulas and results for the bending angle, image positions, image magnifications, total unsigned magnification, and centroid, all to third order in ε and including the displacement. New results on the time delay between images are also given to second order in ε, again including displacement. For all lensing observables we show that the displacement begins to appear only at second order in ε. When there is no spin, we obtain new results on the lensing observables for Schwarzschild lensing with displacement.
52(2011); http://dx.doi.org/10.1063/1.3653840View Description Hide Description
The current early stage in the investigation of the stability of the Kerr metric is characterized by the study of appropriate model problems. Particularly interesting is the problem of the stability of the solutions of the Klein-Gordon equation, describing the propagation of a scalar field in the background of a rotating (Kerr-) black hole. Results suggest that the stability of the field depends crucially on its mass μ. Among others, the paper provides an improved bound for μ above which the solutions of the reduced, by separation in the azimuth angle in Boyer-Lindquist coordinates, Klein-Gordon equation are stable. Finally, it gives new formulations of the reduced equation, in particular, in form of a time-dependent wave equation that is governed by a family of unitarily equivalent positive self-adjoint operators. The latter formulation might turn out useful for further investigation. On the other hand, it is proved that from the abstract properties of this family alone it cannot be concluded that the corresponding solutions are stable.
52(2011); http://dx.doi.org/10.1063/1.3654158View Description Hide Description
We describe the stress energy of a fluid with two unequal stresses and heat flow in terms of two perfect fluid components. The description is in terms of the fluid velocity overlap of the components, and makes no assumptions about the equations of state of the perfect fluids. The description is applied to the metrics of a conformally flat system and a black string.
- Dynamical Systems
52(2011); http://dx.doi.org/10.1063/1.3644343View Description Hide Description
We consider the non-autonomous linear Dirac equation on a time scale containing important discrete, continuous, and quantum time scales. A representation of the solutions is established via an approximate solutions in terms of unknown phase functions with the error estimates. JWKB and other asymptotic representations are discussed. The adiabatic invariants of the Dirac equation are described by using a small parameter method. We also calculate the transition probabilities for the Dirac equation. Using the asymptotic solutions we show that the electron-positron transition probability during a long period of time is about 1/3. Since this probability is high, there is a simple explanation of the stability of the revolution of an electron about the proton only by the electromagnetic field. Indeed when the electron is far from the proton, it is attracted by the electromagnetic field of the proton. When the electron approaches closer to the proton, it turns to the positron which is repelling from the proton by the same electromagnetic field.
52(2011); http://dx.doi.org/10.1063/1.3638052View Description Hide Description
Some years ago, Mikhailov and Sokolov identified as integrable the neat system of two evolution equations , where U ≡ U(t) and V ≡ V(t) are two N × N matrices, N is an arbitrarypositive integer, t (“time”) is the independent variable, and superimposed dots indicate the time derivatives. This entails, rather trivially, that the genericsolution of the modified version of this model reading , with ω an arbitrary positive constant, is completely periodicwith period T = 2π/ω (or possibly a period which is an integer multiple of T): “isochrony.” Another, less trivial, consequence of their finding is the observation that the solution of the many-body problem characterized by the Hamiltonian system of N Newtonian evolution equations,, where x n ≡ x n (t) are N scalar dependent variables and a, g are two arbitrary constants, is simply related to the evolution of the (appropriately rescaled) eigenvalues of a matrix simply related to an appropriate solution of the original Mikhailov-Sokolov integrablematrix evolution system, hence is itself integrable.
- Classical Mechanics and Classical Fields
52(2011); http://dx.doi.org/10.1063/1.3653196View Description Hide Description
In this paper the conditions, under which non-topological solitons are absent in the Yang-Millstheory coupled to a nonlinear scalar field in Minkowski space, are obtained in a very simple way. It is also shown that non-topological solitons are absent in the theory describing the massive complex vector field coupled to the electromagnetic field in Minkowski space.
52(2011); http://dx.doi.org/10.1063/1.3653510View Description Hide Description
In this work, we analyze the canonical structure of Podolsky's generalized electrodynamics on the null-plane. We show that the constraint structure presents a set of second-class constraints, which are exclusive of the analysis on the null-plane, and an expected set of first-class ones. An inspection on the field equations leads to the generalized radiation gauge on the null-plane. Dirac Brackets are then introduced by considering the problem of uniqueness under the chosen initial-boundary conditions of the theory.
52(2011); http://dx.doi.org/10.1063/1.3646115View Description Hide Description
Helmholtz's equations provide the motion of a system of Nvortices which describes a planar incompressible fluid with zero viscosity. A relative equilibrium is a particular solution of these equations for which the distances between the vortices are invariant during the motion. In this article, we first show that a relative equilibrium formed of a regular polygon and a possible vortex at the center, with more than three vertices on the polygon (two if there is a vortex at the center), requires equal vorticities on the polygon. We also provide an 8-vortex configuration, formed of two concentric squares making an angle of 45°, with uniform vorticity on each square, which is in relative equilibrium for any value of the vorticities.
- Methods of Mathematical Physics
52(2011); http://dx.doi.org/10.1063/1.3644345View Description Hide Description
The Gaudin model based on the sl 2-invariant r-matrix with an extra Jordanian term depending on the spectral parameters is considered. The appropriate creation operators defining the Bethe states of the system are constructed through a recurrence relation. The commutation relations between the generating function t(λ) of the integrals of motion and the creation operators are calculated and therefore the algebraic Bethe ansatz is fully implemented. The energy spectrum as well as the corresponding Bethe equations of the system coincide with the ones of the sl 2-invariant Gaudin model. As opposed to the sl 2-invariant case, the operator t(λ) and the Gaudin Hamiltonians are not Hermitian. Finally, the inner products and norms of the Bethe states are studied.
52(2011); http://dx.doi.org/10.1063/1.3651477View Description Hide Description
We consider a generalized Raychaudhuri equation which has appeared in modern string cosmology. This is a system of polynomialdifferential equations in depending on four parameters. We study the existence of analytic first integrals of this model for all values of the parameters.