Volume 44, Issue 6, June 2003
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Controllability properties for finite dimensional quantum Markovian master equations
View Description Hide DescriptionVarious notions from geometric control theory are used to characterize the behavior of the Markovian master equation for level quantum mechanical systems driven by unitary control and to describe the structure of the sets of reachable states. It is shown that the system can be accessible but neither smalltime controllable nor controllable in finite time. In particular, if the generators of quantum dynamical semigroups are unital, then the reachable sets admit easy characterizations as they monotonically grow in time. The two level case is treated in detail.

Dissipative Schrödingertype operators as a model for generation and recombination
View Description Hide DescriptionNonselfadjoint operators play an important role in the modeling of open quantum systems. We consider a onedimensional Schrödingertype operator of the form with dissipative boundary conditions. An explicit description of the characteristic function, the minimal dilation and the generalized eigenfunctions of the dilation is given. The quantities of carrier and current densities are rigorously defined. Furthermore, we will show that the current is not constant and that the variation of the current depend essentially on the chosen density matrix and the imaginary parts of the delta potentials, i.e., This correspondence can be used to model a recombinationgeneration rate in the open quantum system.

Broken symmetries in the entanglement of formation
View Description Hide DescriptionWe compare some recent computations of the entanglement of formation in quantum informationtheory and of the entropy of a subalgebra in quantum ergodic theory. Both notions require optimization over decompositions of quantum states. We show that both functionals are strongly related for some highly symmetric density matrices. Indeed, for certain interesting regions the entanglement of formation can be expressed by the entropy of a commuting subalgebra, and the corresponding optimal decompositions can be obtained one from the other. We discuss the presence of broken symmetries in relation with the structure of the optimal decompositions.

Lower bound for the superheating field in the weakκ limit: The general case
View Description Hide DescriptionWe have constructed asymptotic matched solutions for the one dimensionnal Ginzburg–Landau system when κ is small [Math. Model. Num. Anal. 36, 971–993 (2002)]. We have deduced an expansion in powers of at all orders for the superheating field. In this paper, using these constructions, we propose to show that the superheating field admits for lower bound the expansion of the formal superheating field truncated at order for all We generalize the proof given in Eur. J. Appl. Math. 13, 519–547 (2002), where this result is obtained for Then, we construct solutions of the Ginzburg–Landau system when the exterior magnetic field is near to the superheating field, and we give a localization of these solutions.

Spectral properties of Pauli operators on the Poincaré upperhalf plane
View Description Hide DescriptionWe investigate the essential spectrum of the Pauli operators (and the Dirac and the Schrödinger operators) with magnetic fields on the Poincaré upperhalf plane. The magnetic fields under consideration are asymptotically constant (which may be equal to zero), or diverge at infinity. Moreover, the Aharonov–Casher type result is also considered.

Geometrical phases for the Grassmannian manifold
View Description Hide DescriptionWe generalize the usual Abelian Berry phase generated for example in a system with two nondegenerate states to the case of a system with two doubly degenerate energy eigenspaces. The parametric manifold describing the space of states of the first case is formally given by the Grassmannian manifold, while for the generalized system it is given by the one. For the latter manifold which exhibits a much richer structure than its Abelian counterpart we calculate the connection components, the field strength and the associated geometrical phases that evolve nontrivially both of the degenerate eigenspaces. A simple atomic model is proposed for their physical implementation.

Contextual viewpoint to quantum stochastics
View Description Hide DescriptionWe study the role of context, complex of physical conditions, in quantum as well as classical experiments. It is shown that by taking into account contextual dependence of experimental probabilities we can derive the quantum rule for the addition of probabilities of alternatives. Thus we obtain quantum interference without applying the wave or Hilbert space approach. The Hilbert space representation of contextual probabilities is obtained as a consequence of the elementary geometric fact: costheorem. By using another fact from elementary algebra we obtain complexamplitude representation of probabilities. Finally, we found contextual origin of noncommutativity of incompatible observables.

Generalized coherent states and the diagonal representation for operators
View Description Hide DescriptionWe consider the problem of existence of the diagonal representation for operators in the space of a family of generalized coherent states associated with a unitary irreducible representation of a (compact) Lie group. We show that necessary and sufficient conditions for the possibility of such a representation can be obtained by combining Clebsch–Gordan theory and the reciprocity theorem associated with induced unitary group representations. Applications to several examples involving SU(2), SU(3), and the Heisenberg–Weyl group are presented, showing that there are simple examples of generalized coherent states which do not meet these conditions. Our results are relevant for phase–space description of quantum mechanics and quantum state reconstruction problems.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


On noncommutative orbifolds of K3 surfaces
View Description Hide DescriptionUsing the algebraic geometry method of Berenstein and Leigh for the construction of the toroidal orbifold with discrete torsion and considering local K3 surfaces, we present noncommutative aspects of the orbifolds of product of K3 surfaces. In this way, the ordinary complex deformation of K3 can be identified with the resolution of stringy singularities by noncommutative algebras using crossed products. We give representations and make some comments regarding the fractionation of branes. Illustrating examples are presented.

An algebraic method for solving the SU(3) Gauss law
View Description Hide DescriptionA generalization of existing SU(2) results is obtained. In particular, the sourcefree Gauss law for SU(3)valued gauge fields is solved using a nonAbelian analog of the Poincaré lemma. When sources are present, the colorelectric field is divided into two parts in a way similar to the Hodge decomposition. Singularities due to coinciding eigenvalues of the colormagnetic field are also analyzed.
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 GENERAL RELATIVITY AND GRAVITATION


An invariant action for noncommutative gravity in four dimensions
View Description Hide DescriptionTwo main problems face the construction of noncommutative actions for gravity with star products: the complex metric and finding an invariant measure. The only gauge groups that could be used with star products are the unitary groups. I propose an invariant gravitational action in dimensions based on the constrained gauge group U(2,2) broken to U(1,1)×U(1,1). No metric is used, thus giving a naturally invariant measure. This action is generalized to the noncommutative case by replacing ordinary products with star products. The fourdimensional noncommutative action is studied and the deformed action to first order in deformation parameter is computed.

Cubic algebraic equations in gravity theory, parametrization with the Weierstrass function and nonarithmetic theory of algebraic equations
View Description Hide DescriptionA cubic algebraic equation for the effective parametrizations of the standard gravitational Lagrangian has been obtained without applying any variational principle. It was suggested that such an equation may find application in gravity theory,brane, string and Rundall–Sundrum theories. The obtained algebraic equation was brought by means of a linearfractional transformation to a parametrizable form, expressed through the elliptic Weierstrass function, which was proved to satisfy the standard parametrizable form, but with and functions of a complex variable instead of the definite complex numbers [known from the usual (arithmetic) theory of elliptic functions and curves]. The generally divergent (two) infinite sums of the inverse first and second powers of the poles in the complex plane were shown to be convergent in the investigated particular case. Some relations were found, which ensure the parametrization of the cubic equation in its general form with the Weierstrass function.
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 DYNAMICAL SYSTEMS


Application of Poisson maps on coadjoint orbits of group to many body dynamics
View Description Hide DescriptionThe canonical transformation approach generated by the semisimple subgroup is applied to reduction of the Lie–Poisson bracket on coadjoint orbits of the group and the Poisson coalgebra is determined. Investigating the construction of the particle phase, induced by this reduction, we identify the Poisson coalgebra as the algebra of quadratic invariant forms on symplectic dimensional phase space The general classification scheme of Poissonorbits for is found and applied to the classification of coadjoint orbits of the group occurring in the decomposition of particle phase spaces. We show that the Poisson action on some class of surfaces determined by Casimir invariants is not transitive. The Poisson maps for all classes of orbits and are found. The quantum unitary irreducible representations of are obtained.
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 STATISTICAL PHYSICS


On the convergence to statistical equilibrium for harmonic crystals
View Description Hide DescriptionWe consider the dynamics of a harmonic crystal in dimensions with components, arbitrary, and study the distribution of the solution at time The initial measure has a translationinvariant correlation matrix, zero mean, and finite mean energy density. It also satisfies a Rosenblatt—resp. Ibragimov–Linnik type mixing condition. The main result is the convergence of to a Gaussian measure as The proof is based on the long time asymptotics of the Green’s function and on Bernstein’s “roomcorridors” method.

Hierarchy of equations for reduced density matrices at thermodynamic equilibrium with account taken of the spin of particles
View Description Hide DescriptionThe new approach in quantum statistical mechanics proposed by the author in previous publications is extended to systems of nonzerospin particles. It is shown that properties of particles relevant to their spin can well be incorporated into the approach proposed. In so doing the principal ideas of the approach undergo no changes and lead to a hierarchy of equations for reduced density matrices at thermodynamic equilibrium that generalizes the hierarchy obtained earlier for spinless particles. Thermodynamics based on the hierarchy is worked out as well. The hierarchy and relevant thermodynamics enable one to calculate both thermodynamicalproperties of a system of particles and structural ones. With the help of perturbation expansions it is shown that the hierarchy has a solution and the solution is unique at least in the case of a weakly interacting system. Using the example of a hardsphere system wherein triplet correlations are disregarded it is demonstrated that the hierarchy may serve as basis for deriving various results. Comparison of these results with results of other theoretical treatments evidences complete agreement within the limits allowed by the approximation used.

Superconductivity near critical temperature
View Description Hide DescriptionIn this paper we study the superconductivity for a sample subjected to an applied magnetic field and slightly below the critical temperature We use the Ginzburg–Landau functional to estimate the value of the critical field and examine the superconductivity when temperature is close to and the applied field is below
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 METHODS OF MATHEMATICAL PHYSICS


Tensor operators and constructing indecomposable representations of semidirect product groups
View Description Hide DescriptionConsider a semidirect product group where is reductive and is a vector group. Two irreps and of can be “assembled” into a representation of if it is possible to construct an indecomposable representation Π of whose restriction to is It is shown that this is equivalent to the existence of a tensor operator from to carrying a representation of which is equivalent to a nontrivial quotient of the representation which defines the semidirect product. This provides a systematic method for deciding whether two irreps can be assembled, and, if so, in how many inequivalent ways. The method is applied in many of the standard examples that arise in physical questions.

Extra dimensions and nonlinear equations
View Description Hide DescriptionSolutions of nonlinear multicomponent Euler–Monge partial differential equations are constructed in spatial dimensions by dimensiondoubling, a method that completely linearizes the problem. Nonlocal structures are an essential feature of the method. The Euler–Monge equations may be interpreted as a boundary theory arising from a linearized bulk system such that all boundary solutions follow from simple limits of those for the bulk.

Weak transversality and partially invariant solutions
View Description Hide DescriptionNew exact solutions are obtained for several nonlinear physical equations, namely the Navier–Stokes and Euler systems, an isentropic compressible fluid system and a vector nonlinear Schrödinger equation. The solution method makes use of the symmetry group of the system in situations when the standard Lie method of symmetry reduction is not applicable.
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 COMMENTS AND ERRATA


Publisher’s Note: “On certain geometric aspects of harmonic maps” [J. Math. Phys. 44, 813 (2003)]
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