Volume 46, Issue 2, February 2005
Index of content:
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Exact solutions of the isoholonomic problem and the optimal control problem in holonomic quantum computation
View Description Hide DescriptionThe isoholonomic problem in a homogeneous bundle is formulated and solved exactly. The problem takes a form of a boundary value problem of a variational equation. The solution is applied to the optimal control problem in holonomic quantum computer. We provide a prescription to construct an optimal controller for an arbitrary unitary gate and apply it to a dimensional unitary gate which operates on an dimensional Hilbert space with . Our construction is applied to several important unitary gates such as the Hadamard gate, the CNOT gate, and the twoqubit discrete Fourier transformation gate. Controllers for these gates are explicitly constructed.

Universal collective rotation channels and quantum error correction
View Description Hide DescriptionWe present and investigate a new class of quantum channels, what we call “universal collective rotation channels,” that includes the class of collective rotation channels as a special case. The fixed point set and noise commutant coincide for a channel in this class. Computing the precise structure of this algebra is a core problem in a particular noiseless subsystem method of quantum error correction. We prove that there is an abundance of noiseless subsystems for every channel in this class and that the Young tableaux combinatorial machine may be used to explicitly compute these subsystems.

Path integral for the radial Dirac equation
View Description Hide DescriptionFor the radial Dirac equation, a countably additive path space measure on the space of continuous paths living on the real halfline is constructed to give a path integral representation of its Green function.

Study of anharmonic singular potentials
View Description Hide DescriptionA simple and efficient variational method is introduced to accelerate the convergence of the eigenenergy computations for a Hamiltonian with singular potentials. Closedform analytic expressions in dimensions are obtained for the matrix elements of with respect to the eigenfunctions of a soluble singular problem with two free parameters and . The matrix eigenvalues are then optimized with respect to and for a given . Applications, convergence rates, and comparisons with earlier work are discussed in detail.

Perturbative and nonperturbative master equations for open quantum systems
View Description Hide DescriptionThis paper develops perturbative and nonperturbative master equations for open quantum systems based on timedependent variational functionals. The perturbative equations are more concise and suitable for dealing with cases of weak systemenvironment coupling for short evolution time scales. The nonperturbative equations are valid for all time and appropriate to treat cases of strong systemenvironment coupling. When a system contains an external control field, both the perturbative and nonperturbative master equations reveal the embedded control field dependence upon the system decoherence, which provides a basis for decoherence management.

There is no generalization of known formulas for mutually unbiased bases
View Description Hide DescriptionIn a quantum system having a finite number of orthogonal states, two orthonormal bases and are called mutually unbiased if all inner products have the same modulus . This concept appears in several quantum information problems. The number of pairwise mutually unbiased bases is at most and various constructions of such bases have been found when is a power of a prime number. We study families of formulas that generalize these constructions to arbitrary dimensions using finite rings. We then prove that there exists a set of mutually unbiased bases described by such formulas, if and only if is a power of a prime number.

 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Second quantization method in the presence of bound states of particles
View Description Hide DescriptionWe develop an approximate second quantization method for describing the manyparticle systems in the presence of bound states of particles at low energies (the kinetic energy of particles is small in comparison to the binding energy of compound particles). In this approximation the compound and elementary particles are considered on an equal basis. This means that creation and annihilation operators of compound particles can be introduced. The Hamiltonians, which specify the interactions between compound and elementary particles and between compound particles themselves, are found in terms of the interaction amplitudes for elementary particles. The nonrelativistic quantum electrodynamics is developed for systems containing both elementary and compound particles. Some applications of this theory are considered.

Algebraic geometry realization of quantum Hall soliton
View Description Hide DescriptionUsing the Iqbal–Netzike–Vafa dictionary giving the correspondence between the homology of del Pezzo surfaces and branes, we develop a way to approach the system of brane bounds in Mtheory on . We first review the structure of 10dimensional quantum Hall soliton (QHS) from the view of Mtheory on . Then, we show how the dissolution in brane is realized in Mtheory language and derive the brane constraint equations used to define appropriately the QHS. Finally, we build an algebraic geometry realization of the QHS in type IIA superstring and show how to get its type IIB dual. Other aspects are also discussed.

Casimir energy for a wedge with three surfaces and for a pyramidal cavity
View Description Hide DescriptionCasimir energy calculations for the conformally coupled massless scalar field for a wedge defined by three intersecting planes and for a pyramid with four triangular surfaces are presented. The group generated by reflections are employed in the formulation of the required Green functions and the wave functions.

Casimir energy in a conical wedge and a conical cavity
View Description Hide DescriptionCasimir energies for a massless scalar field for a conical wedge and a conical cavity are calculated. The group generated by the images is employed in deriving the Green function as well as the wave functions and the energy spectrum.

 GENERAL RELATIVITY AND GRAVITATION


Some remarks on locally conformally flat static space–times
View Description Hide DescriptionNecessary and sufficient conditions for a static space–time to be locally conformally flat are obtained, showing some significant restrictions on the possible warping functions of the space–times. This occurs in opposition to cosmological models, where Robertson–Walker space–times are locally conformally flat for any warping function.

Topological quantization of gravitational fields
View Description Hide DescriptionWe introduce the method of topological quantization for gravitational fields in a systematic manner. First we show that any vacuum solution of Einstein’s equations can be represented in a principal fiber bundle with a connection that takes values in the Lie algebra of the Lorentz group. This result is generalized to include the case of gauge matter fields in multiple principal fiber bundles. We present several examples of gravitational configurations that include a gravitomagnetic monopole in linearized gravity, the Cenergy of cylindrically symmetric fields, the Reissner–Nordström and the Kerr–Newman black holes. As a result of the application of the topological quantization procedure, in all the analyzed examples we obtain conditions implying that the parameters entering the metric in each case satisfy certain discretization relationships.

Asymptotic flatness and Bondi energy in higher dimensional gravity
View Description Hide DescriptionWe give a general geometric definition of asymptotic flatness at null infinity in dimensional general relativity ( even) within the framework of conformal infinity. Our definition is arrived at via an analysis of linear perturbations near null infinity and shown to be stable under such perturbations. The detailed falloff properties of the perturbations, as well as the gauge conditions that need to be imposed to make the perturbations regular at infinity, are qualitatively different in higher dimensions; in particular, the decay rate of a radiating solution at null infinity differs from that of a static solution in higher dimensions. The definition of asymptotic flatness in higher dimensions consequently also differs qualitatively from that in . We then derive an expression for the generator conjugate to an asymptotic time translation symmetry for asymptotically flat space–times in dimensional general relativity ( even) within the Hamiltonian framework, making use especially of a formalism developed by Wald and Zoupas. This generator is given by an integral over a cross section at null infinity of a certain local expression and is taken to be the definition of the Bondi energy in dimensions. Our definition yields a manifestly positive flux of radiated energy. Our definitions and constructions fail in odd space–time dimensions, essentially because the regularity properties of the metric at null infinity seem to be insufficient in that case. We also find that there is no direct analog of the wellknown infinite set of angle dependent translational symmetries in more than four dimensions.

 DYNAMICAL SYSTEMS


Path description of conserved quantities of generalized periodic boxball systems
View Description Hide DescriptionWe investigate conserved quantities of periodic boxball systems (PBBS) with arbitrary kinds of balls and box capacity greater than or equal to 1. We introduce the notion of nonintersecting paths on the two dimensional array of boxes, and give a combinatorial formula for the conserved quantities of the generalized PBBS using these paths.

A revised model for diffusion induced segregation processes
View Description Hide DescriptionA mathematical model for chalcopyrite disease within sphalerite is developed. As one main result, by analyzing the system enthalpy, correct expressions for the reaction terms in a system undergoing phase transitions are worked out. For the resulting equations, the thermodynamical validity is shown and the existence of a unique solution is proved.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Shock waves in the Tamm problem and the possibility of their experimental observation
View Description Hide DescriptionThe position of the singular electromagneticshock waves arising in the smooth Tamm problem is found. They consist of the Cherenkov shock wave and shock waves arising when the charge velocity coincides with the velocity of light in medium. In the limit of small intervals of accelerated and decelerated motion they are not transformed into the singular electromagneticshock waves of the original Tamm problem. The reasons for this and the relation of the arising electromagneticshock waves to the experimental situation are discussed.

 STATISTICAL PHYSICS


An inequality for the Gibbs mean number of clusters
View Description Hide DescriptionThe number of percolation clusters for configurations of the Ising model at zero external field and ferromagnetic first neighbors interaction on a general finite graph is considered. The mean number of clusters with respect to the Gibbs measure at any inverse temperature is proved to be smaller or equal than the one at .

 METHODS OF MATHEMATICAL PHYSICS


On the monotonicity of scalar curvature in classical and quantum information geometry
View Description Hide DescriptionWe study the monotonicity under mixing of the scalar curvature for the geometries on the simplex of probability vectors. From the results obtained and from numerical data, we are led to some conjectures about quantum geometries and Wigner–Yanase–Dyson information. Finally, we show that this last conjecture implies the truth of the Petz conjecture about the monotonicity of the scalar curvature of the Bogoliubov–Kubo–Mori monotone metric.

Largetime asymptotics for nonlinear diffusions: the initialboundary value problem
View Description Hide DescriptionIn this paper we investigate the largetime behavior of solutions to the first initialboundary value problem for the nonlinear diffusion. In particular, we prove exponential decay of towards its own steady state in norm for long times and we give an explicit upper bound for the rate of decay. The result is based on a new application of entropy estimates, and on detailed lower bounds for the entropy production in this situation.

Approximate conditional symmetries and approximate solutions of the perturbed Fitzhugh–Nagumo equation
View Description Hide DescriptionWe consider a perturbation of the Fitzhugh–Nagumo equation. The perturbation is proportional to the electric potential across the cell membrane. The purpose of this investigation is to determine the effects of a change in electric potential across the cell membrane. Exact solutions of the perturbed equation are easily obtained from the wellknown solutions of the unperturbed Fitzhugh–Nagumo equation. The method of approximate conditional symmetries is used to obtain firstorder approximate solutions of the perturbed FitzhughNagumo equation. The approximate solutions are compared with the exact solutions of the perturbed equation. The exact solutions of the perturbed equation do not indicate a change in the wave front connecting one constant state to another. There is only a proportional increase or decrease in the constant nonzero state. The approximate solutions do show a change in the shape of the wave front connecting two constant states as well as a proportional increase or decrease in the constant nonzero state.
