Volume 44, Issue 2, February 2003
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Scattering on compact manifolds with infinitely thin horns
View Description Hide DescriptionThe quantummechanical scattering on a compact manifold with semiaxes attached to the manifold (“hedgehogshaped manifold”) is considered. The complete description of the spectral structure of Schrödinger operators on such a manifold is done, the proof of existence and uniqueness of scattering states is presented, an explicit form for the scattering matrix is obtained and unitarity of this matrix is proven. It is shown that the positive part of the spectrum of the Schrödinger operator on the initial compact manifold as well as the spectrum of a point perturbation of such an operator may be recovered from the scattering amplitude for one attached halfline. Moreover, the positive part of the spectrum of the initial Schrödinger operator is fully determined by the conductance properties of an “electronic device” consisting of the initial manifold and two “wires” attached to it.

Bound states in one and two spatial dimensions
View Description Hide DescriptionIn this article we study the number of bound states for potentials in one and two spatial dimensions. We first show that in addition to the wellknown fact that an arbitrarily weak attractive potential has a bound state, it is easy to construct examples where weak potentials have an infinite number of bound states. These examples have potentials which decrease at infinity faster than expected. Using somewhat stronger conditions, we derive explicit bounds on the number of bound states in one dimension, using known results for the threedimensional zero angular momentum. A change of variables which allows us to go from the onedimensional case to that of two dimensions results in a bound for the zero angular momentum case. Finally, we obtain a bound on the total number of bound states in two dimensions, first for the radial case and then, under stronger conditions, for the noncentral case.

Fluxacrosssurfaces theorem for a Dirac particle
View Description Hide DescriptionWe consider the asymptotic evolution of a relativistic spin particle, i.e., a particle whose wave function satisfies the Dirac equation with external static potential. We prove that the probability for the particle crossing a (detector)surface converges to the probability, that the direction of the momentum of the particle lies within the solid angle defined by the (detector)surface, as the distance of the surface goes to infinity. This generalizes earlier nonrelativistic results, known as flux across surfaces theorems, to the relativistic regime.

Covariant phase difference observables in quantum mechanics
View Description Hide DescriptionCovariant phase difference observables are determined in two different ways, by a direct computation and by a group theoretical method. A characterization of phase difference observables which can be expressed as the difference of two phase observables is given. The classical limits of such phase difference observables are determined and the PeggBarnett phase difference distribution is obtained from the phase difference representation. The relation of Ban’s theory to the covariant phase theories is exhibited.

Some integrable systems in nonlinear quantum optics
View Description Hide DescriptionIn the paper we investigate the theory of quantum optical systems. As an application we integrate and describe the quantum optical systems which are generically related to the classical orthogonal polynomials. The family of coherent states related to these systems is constructed and described. Some applications are also presented.

Wigner measures and codimension two crossings
View Description Hide DescriptionThis article gives a semiclassical description of nucleonic propagation through codimension two crossings of electronic energy levels. Codimension two crossings are the simplest energy level crossings, which affect the Born–Oppenheimer approximation in the zeroth order term. The model we study is a twolevel Schrödinger equation with a Laplacian as kinetic operator and a matrixvalued linear potential, whose eigenvalues cross, if the two nucleonic coordinates equal zero. We discuss the case of welllocalized initial data and obtain a description of the wavefunction’s twoscaled Wigner measure and of the weak limit of its position density, which is valid globally in time.

Choi’s proof as a recipe for quantum process tomography
View Description Hide DescriptionQuantum process tomography is a procedure by which an unknown quantum operation can be fully experimentally characterized. We reinterpret Choi’s proof [Linear Algebr. Appl. 10, 285 (1975)] of the fact that any completely positive linear map has a Kraus representation as a method for quantum process tomography. The analysis for obtaining the Kraus operators is extremely simple. We discuss the systems in which this tomography method is particularly suitable.

Robust procedures for converting among Lindblad, Kraus and matrix representations of quantum dynamical semigroups
View Description Hide DescriptionGiven a quantum dynamical semigroup expressed as an exponential superoperator acting on a space of dimensional density operators, eigenvalue methods are presented by which canonical Kraus and Lindblad operator sum representations can be computed. These methods provide a mathematical basis on which to develop novel algorithms for quantum process tomography—the statistical estimation of superoperators and their generators—from a wide variety of experimental data. Theoretical arguments and numerical simulations are presented which imply that these algorithms will be quite robust in the presence of random errors in the data.

NonAbelian braid statistics versus projective permutation statistics
View Description Hide DescriptionRecent papers by Finkelstein, Galiautdinov, and coworkers [J. Math. Phys. 42, 1489 (2001); 42, 3299 (2001)] discuss a suggestion by Wilczek that nonAbelian projective representations of the permutation group can be used as a new type of particle statistics, valid in any dimension. Wilczek’s suggestion was based in part on an analysis by Nayak and Wilczek (NW) of the nonAbelian representation of the braid group in a quantum Hall system. We point out that projective permutation statistics is not possible in a local quantum field theory as it violates locality, and show that the NW braid group representation is not equivalent to a projective representation of the permutation group. The structure of the finite image of the braid group in a dimensional representation is obtained.

A connection between distributivity and locality in compound Plattices
View Description Hide DescriptionA Plattice is a σcomplete, orthomodular atomic lattice L which is formed by the set of propositions of a physical system. A composition of physical systems is considered, and some concept of locality in a compound physical system is represented in terms of Plattices. We give a remark toward necessary and sufficient conditions for it to hold, which have been provided implicitly in antecedent studies, and we show that it can be provided under weaker conditions.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


On dimensional regularization of sums
View Description Hide DescriptionWe discuss a systematic way to dimensionally regularize divergent sums arising in field theories with an arbitrary number of physical compact dimensions or finite temperature. The method preserves the same symmetries of the action as the conventional dimensional regularization and allows an easy separation of the regulated divergence from the finite term that depends on the compactification radius (temperature).

Interacting fermions and domain wall defects in dimensions
View Description Hide DescriptionWe consider a Dirac field in dimensions with a domain wall like defect in its mass, minimally coupled to a dynamical Abelian vector field. The mass of the fermionic field is assumed to have just one linear domain wall, which is externally fixed and unaffected by the dynamics. We show that, under some general conditions on the parameters, the localized zero modes predicted by the Callan and Harvey mechanism are stable under the electromagnetic interaction of the fermions.

Casimir energy of a relativistic perfect fluid confined to a dimensional hypercube
View Description Hide DescriptionCompact formulas are obtained for the Casimir energy of a relativistic perfect fluid confined to a dimensional hypercube with von Neumann or Dirichlet boundary conditions. The formulas are conveniently expressed as a finite sum of the wellknown gamma and Riemann zeta functions. Emphasis is placed on the mathematical technique used to extract the Casimir energy from a dimensional infinite sum regularized with an exponential cutoff. Numerical calculations show that initially the Dirichlet energy decreases rapidly in magnitude and oscillates in sign, being positive for even and negative for odd This oscillating pattern stops abruptly at the critical dimension of after which the energy remains negative and the magnitude increases. We show that numerical calculations performed with 16digit precision are inaccurate at higher values of

Proof of a mass singularity free property in high temperature QCD
View Description Hide DescriptionIt is shown that three series of diagrams entering the calculation of some hot QCD process, are mass (or collinear) singularity free, indeed. This generalizes a result which was recently established up to the third nontrivial order of (thermal) perturbation theory.
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 GENERAL RELATIVITY AND GRAVITATION


Geometry of crossing null shells
View Description Hide DescriptionNew geometric objects on null thin layers are introduced and their importance for crossing nulllike shells are discussed. The Barrabès–Israel equations are represented in a new geometric form and they split into a decoupled system of equations for two different geometric objects: tensor density and vector fieldI. Continuity properties of these objects through a crossing sphere are proved. In the case of spherical symmetry Dray–t’Hooft–Redmount formula results from continuity property of the corresponding object.

Scalar and spin particle creation in gravitational and constant electric field backgrounds
View Description Hide DescriptionBy considering the quantum vacuum states in the asymptotic regions of the gravitational background, the production rates of scalar and spin particles created by gravitational fields in specific geometries are computed, with energy distributions shown to be of the Bose–Einstein and Fermi–Dirac types. The analysis is extended in the case of scalar particles to include a constant electric field.
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 DYNAMICAL SYSTEMS


Trigonometric osp(12) Gaudin model
View Description Hide DescriptionThe problems connected with Gaudin models are reviewed by analyzing model related to the trigonometric osp(12) classical matrix. The eigenvectors of the trigonometric osp(12) Gaudin Hamiltonians are found using explicitly constructed creation operators. The commutation relations between the creation operators and the generators of the trigonometric loop superalgebra are calculated. The coordinate representation of the Bethe states is presented. The relation between the Bethe vectors and solutions to the Knizhnik–Zamolodchikov equation yields the norm of the eigenvectors. The generalized Knizhnik–Zamolodchikov system is discussed both in the rational and in the trigonometric case.

Category of nonlinear evolution equations, algebraic structure, and matrix
View Description Hide DescriptionIn this paper we deal with the category of nonlinear evolution equations (NLEEs) associated with the spectral problem and provide an approach for constructing their algebraic structure and matrix. First we introduce the category of NLEEs, which is composed of various positive order and negative order hierarchies of NLEEs both integrable and nonintegrable. The whole category of NLEEs possesses a generalized Lax representation. Next, we present two different Lie algebraic structures of the Lax operator: one of them is universal in the category, i.e., independent of the hierarchy, while the other one is nonuniversal in the hierarchy, i.e., dependent on the underlying hierarchy. Moreover, we find that two kinds of adjoint maps are matrices under the algebraic structures. In particular, the Virasoro algebraic structures without a central extension of isospectral and nonisospectral Lax operators can be viewed as reductions of our algebraic structure. Finally, we give several concrete examples to illustrate our methods. Particularly, the Burgers’ category is linearized when the generator, which generates the category, is chosen to be independent of the potential function. Furthermore, an isospectral negative order hierarchy in the Burgers’ category is solved with its general solution. Additionally, in the KdV category we find an interesting fact: the Harry–Dym hierarchy is contained in this category as well as the wellknown Harry–Dym equation is included in a positive order KdV hierarchy.
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 METHODS OF MATHEMATICAL PHYSICS


Dirichlet forms and symmetric Markovian semigroups on CCR algebras with respect to quasifree states
View Description Hide DescriptionEmploying the construction method of Dirichlet forms on standard forms of von Neumann algebras developed in Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2000, Vol. 3, No. 1, pp. 1–14 (Ref. 1), we construct Dirichlet forms and associated symmetric Markovian semigroups on CCR algebras with respect to quasifree states. More precisely, let be the CCR algebra over a complex separable preHilbert space and let ω be a quasifree state on For any normalized admissible function and complete orthonormal system (CONS) we construct a Dirichlet form and corresponding symmetric Markovian semigroup on the natural standard form associated to the GNS representation of It turns out that the form is independent of admissible function and CONS chosen. By analyzing the spectrum of the generator (Dirichlet operator) of the semigroup, we show that the semigroup is ergodic and tends to the equilibrium exponentially fast.

Finite growth representations of infinite Lie conformal algebras
View Description Hide DescriptionWe classify all finite growth representations of all infinite rank subalgebras of the Lie conformal algebra that contain a Virasoro subalgebra.
