Volume 46, Issue 6, June 2005
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Scattering kernel for polyatomic molecules
View Description Hide DescriptionA polyatomic scattering kernel phenomenologically presented in a previous paper is derived from an integral operator formulation. The five parameters involved in the scattering kernel expression are shown to be equal to the accommodation coefficients of various fluxes at the wall, namely, the fluxes of the three components of the momentum and the fluxes of the rotational and vibrational energies of molecules. Under its present form the model is especially convenient for the diatomic molecules.

On the infimum of quantum effects
View Description Hide DescriptionThe quantum effects for a physical system can be described by the set of positive operators on a complex Hilbert space that are bounded above by the identity operator. While a general effect may be unsharp, the collection of sharp effects is described by the set of orthogonal projections . Under the natural order, becomes a partially ordered set that is not a lattice if . A physically significant and useful characterization of the pairs such that the infimum exists is called the infimum problem. We show that exists for all , and give an explicit expression for . We also give a characterization of when exists in terms of the location of the spectrum of . We present a counterexample which shows that a recent conjecture concerning the infimum problem is false. Finally, we compare our results with the work of Ando on the infimum problem.

Distinguishing bipartitite orthogonal states using LOCC: Best and worst cases
View Description Hide DescriptionTwo types of results are presented for distinguishing pure bipartite quantum states using local operations and classical communications. We examine sets of states that can be perfectly distinguished, in particular showing that any three orthogonal maximally entangled states in form such a set. In cases where orthogonal states cannot be distinguished, we obtain upper bounds for the probability of error using LOCC taken over all sets of orthogonal states in . In the process of proving these bounds, we identify some sets of orthogonal states for which perfect distinguishability is not possible.

Time reversal and qubit canonical decompositions
View Description Hide DescriptionOn pure states of quantum bits, the concurrence entanglement monotone returns the norm of the inner product of a pure state with its spinflip. The monotone vanishes for odd, but for even there is an explicit formula for its value on mixed states, i.e., a closedform expression computes the minimum over all ensemble decompositions of a given density. For even a matrix decomposition of the unitary group is explicitly computable and allows for study of the monotone’s dynamics. The side factors and of this concurrence canonical decomposition (CCD) are concurrence symmetries, so the dynamics reduce to consideration of the factor. This unitary phases a basis of entangled states, and the concurrence dynamics of are determined by these relative phases. In this work, we provide an explicit numerical algorithm computing for odd. Further, in the odd case we lift the monotone to a twoargument function. The concurrence capacity of according to the double argument lift may be nontrivial for odd and reduces to the usual concurrence capacity in the literature for even. The generalization may also be studied using the CCD, leading again to maximal capacity for most unitaries. The capacity of is at least that of , so oddqubit capacities have implications for evenqubit entanglement. The generalizations require considering the spinflip as a time reversal symmetry operator in Wigner’s axiomatization, and the original Lie algebra homomorphism defining the CCD may be restated entirely in terms of this time reversal. The polar decomposition related to the CCD then writes any unitary evolution as the product of a timesymmetric and timeantisymmetric evolution with respect to the spinflip. En route we observe a Kramers’ nondegeneracy: the existence of a nondegenerate eigenstate of any time reversal symmetric qubit Hamiltonian demands (i) even and (ii) maximal concurrence of said eigenstate. We provide examples of how to apply this work to study the kinematics and dynamics of entanglement in spin chain Hamiltonians.

An isoperimetric problem for leaky loops and related meanchord inequalities
View Description Hide DescriptionWe consider a class of Hamiltonians in with attractive interaction supported by piecewise smooth loops of a fixed length , formally given by with . It is shown that the ground state of this operator is locally maximized by a circular . We also conjecture that this property holds globally and show that the problem is related to an interesting family of geometric inequalities concerning mean values of chords of .

Operational distance and fidelity for quantum channels
View Description Hide DescriptionWe define and study a fidelity criterion for quantum channels, which we term the minimax fidelity, through a noncommutative generalization of maximal Hellinger distance between two positive kernels in classical probability theory. Like other known fidelities for quantum channels, the minimax fidelity is well defined for channels between finitedimensional algebras, but it also applies to a certain class of channels between infinitedimensional algebras (explicitly, those channels that possess an operatorvalued RadonNikodym density with respect to the trace in the sense of BelavkinStaszewski) and induces a metric on the set of quantum channels that is topologically equivalent to the CBnorm distance between channels, precisely in the same way as the Bures metric on the density operators associated with statistical states of quantummechanical systems, derived from the wellknown fidelity (“generalized transition probability”) of Uhlmann, is topologically equivalent to the tracenorm distance.

Generalized density functional theories using the electron densities: Development of kinetic energy functionals
View Description Hide DescriptionSeveral explicit formulas for the kinetic energy of a manyelectron system as a functional of the electron density are derived, with emphasis on the electron pair density. The emphasis is on general techniques for deriving approximate kinetic energy functionals and features generalized Weisacker bounds and methods using densitymatrix reconstruction. Adapting results from statistical mechanics, a hierarchy of equations is derived that links electron pairs, triplets, quadruplets, etc.; this may be used to derive more accurate approximations. Several methods for defining the exact kinetic energy functional are presented, including the generalizations of the Levy and Lieb formulations of densityfunctional theory. Together with representability constraints on the density, this paper provides the basis for “generalized density functional theories” based on the electron pair density. There are also implications for conventional densityfunctional theory, notably regarding the development of more accurate density functionals for the kinetic energy.

Analytic solution of the Schrödinger equation for the Coulombpluslinear potential. I. The wave functions
View Description Hide DescriptionWe solve the Schrödinger equation for a quark–antiquark system interacting via a Coulombpluslinear potential, and obtain the wave functions as power series, with their coefficients given in terms of the combinatorics functions.

Solvable symmetric model with a tunable interspersion of nonmerging levels
View Description Hide DescriptionWe study the spectrum in such a symmetric square well (of a diameter ) where the “strength of the nonHermiticity” is controlled by the two parameters, viz., by an imaginary coupling and by the distance of its onset from the origin. We solve this problem and confirm that the spectrum is discrete and real in a nonempty interval of . Surprisingly, a specific distinction between the bound states is found in their asymptotic stability∕instability with respect to an unlimited growth of beyond . In our model,all of the lowlying levels remain asymptotically unstable at the small and finite while only the stable levels survive near or in the purely imaginary force limit with . In between these two extremes, an unusual and tunable, variable pattern of the interspersed “robust” and “fragile” subspectra of the real levels is obtained.

Superconductivity by means of the subquantum medium coherence
View Description Hide DescriptionIn the hydrodynamic formulation of the scale relativity theory one shows that a stable vortices distribution of bipolaron type induces superconducting pairs by means of the quantum potential. Then, usual mechanisms (as, for example, the exchange interaction used in the bipolarontheory) are reduced to the coherence on the subquantum medium, the superconducting pairs resulting as a onedimensional projection of a fractal. The temperature dependences of the superconducting parameters (coherence length, critical speed, pair breaking time, carriers concentration, penetration depth, critical field, critical current) and the concordance with the experimental data and other theories are analyzed.

Representation of the contextual statistical model by hyperbolic amplitudes
View Description Hide DescriptionWe continue the development of a socalled contextual statistical model (here context has the meaning of a complex of physical conditions). It is shown that, besides contexts producing the conventional trigonometric cosinterference, there exist contexts producing the hyperbolic cosinterference. Starting with the corresponding interference formula of total probability we represent such contexts by hyperbolic probabilistic amplitudes or in the abstract formalism by normalized vectors of a hyperbolic analogue of the Hilbert space. There is obtained a hyperbolic Born’s rule. Incompatible observables are represented by noncommutative operators. This paper can be considered as the first step towards hyperbolic quantum probability. We also discuss possibilities of experimental verification of hyperbolic quantum mechanics: in physics of elementary particles, string theory as well as in experiments with nonphysical systems, e.g., in psychology, cognitive sciences, and economy.

Lifting Bell inequalities
View Description Hide DescriptionA Bell inequality defined for a specific experimental configuration can always be extended to a situation involving more observers, measurement settings, or measurement outcomes. In this article, such “liftings” of Bell inequalities are studied. It is shown that if the original inequality defines a facet of the polytope of local joint outcome probabilities then the lifted one also defines a facet of the more complex polytope.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Neveu–Schwarz fivebrane and tachyon condensation
View Description Hide DescriptionWe argue that a semiinfinite D6brane ending on an NS5brane can be obtained from the condensation of the tachyon on the unstable D9brane of type IIA theory. The construction uses a combination of the descriptions of these branes as solitons of the worldvolume theory of the D9brane. The NS5brane, in particular, involves a gauge bundle which is operator valued, and hence is better thought of as a gerbe.

Normal ordering and boundary conditions in open bosonic strings
View Description Hide DescriptionBoundary conditions play a nontrivial role in string theory. For instance, the rich structure of Dbranes is generated by choosing appropriate combinations of Dirichlet and Neumann boundary conditions. Furthermore, when an antisymmetric background is present at the string end points (corresponding to mixed boundary conditions) space time becomes noncommutative there. We show here how to build up normal ordered products for bosonic string position operators that satisfy both equations of motion and open string boundary conditions at the quantum level. We also calculate the equal time commutator of these normal ordered products in the presence of an antisymmetric tensor background.

Bose–Einstein condensate and spontaneous breaking of conformal symmetry on Killing horizons
View Description Hide DescriptionLocal scalar quantum field theory (in Weyl algebraic approach) is constructed on degenerate semiRiemannian manifolds corresponding to Killing horizons in spacetime. Covariance properties of the algebra of observables with respect to the conformal group are studied. It is shown that, in addition to the state studied by Guido, Longo, Roberts, and Verch for bifurcated Killing horizons, which is conformally invariant and KMS at Hawking temperature with respect to the Killing flow and defines a conformal net of von Neumann algebras, there is a further wide class of algebraic (coherent) states representing spontaneous breaking of symmetry. This class is labeled by functions in a suitable Hilbert space and their GNS representations enjoy remarkable properties. The states are nonequivalent extremal KMS states at Hawking temperature with respect to the residual oneparameter subgroup of associated with the Killing flow. The KMS property is valid for the two local subalgebras of observables uniquely determined by covariance and invariance under the residual symmetry unitarily represented. These algebras rely on the physical region of the manifold corresponding to a Killing horizon cleaned up by removing the unphysical points at infinity [necessary to describe the whole action]. Each of the found states can be interpreted as a different thermodynamic phase, containing Bose–Einstein condensate, for the considered quantum field. It is finally suggested that the found states could describe different black holes.

The Epstein–Glaser approach to perturbative quantum field theory: graphs and Hopf algebras
View Description Hide DescriptionThe paper aims at investigating perturbative quantum field theory in the approach of Epstein and Glaser (EG) and, in particular, its formulation in the language of graphs and Hopf algebras (HAs). Various HAs are encountered, each one associated with a special combination of physical concepts such as normalization, localization, pseudounitarity, causal regularization, and renormalization. The algebraic structures, representing the perturbative expansion of the matrix, are imposed on operatorvalued distributions equipped with appropriate graph indices. Translation invariance ensures the algebras to be analytically well defined and graded total symmetry allows to formulate bialgebras. The algebraic results are given embedded in the corresponding physical framework, covering the two EG versions by Fredenhagen and Scharf that differ with respect to the concrete recursive implementation of causality. Besides, the ultraviolet divergences occurring in Feynman’s representation are mathematically reasoned. As a final result, the change of the renormalization scheme in the context of EG is modeled via a HA and interpreted as the EG analog of Kreimer’s HA.

Derivation of particle, string, and membrane motions from the Born–Infeld electromagnetism
View Description Hide DescriptionWe derive classical particle, string, and membrane motion equations from a rigorous asymptotic analysis of the Born–Infeld nonlinear electromagnetictheory. We first add to the Born–Infeld equations the corresponding energymomentum conservation laws and write the resulting system as a nonconservative symmetric system of firstorder PDEs. Then we show that four rescaled versions of the system have smooth solutions existing in the (finite) time interval where the corresponding limit problems have smooth solutions. Our analysis is based on a continuation principle previously formulated by Yong for (singular) limit problems.

Asymptotics of regulated field commutators for EinsteinRosen waves
View Description Hide DescriptionWe discuss the asymptotic behavior of regulated field commutators for linearly polarized, cylindrically symmetric gravitational waves and the mathematical techniques needed for this analysis. We concentrate our attention on the effects brought about by the introduction of a physical cutoff in the study of the microcausality of the model and describe how the different physically relevant regimes are affected by its presence. Specifically we discuss how genuine quantum gravityeffects can be disentangled from those originating in the introduction of a regulator.

Covariant differential operators and unitary highest weight representations for
View Description Hide DescriptionWe investigate a oneparameter family of quantum Harish–Chandra modules of . This family is an analog of the holomorphic discrete series of representations of the group for the quantum group. We introduce a analog of “the wave” operator (a determinanttype differential operator) and prove certain covariance property of its powers. This result is applied to the study of some quotients of the abovementioned quantum Harish–Chandra modules. We also prove an analog of a known result by J. Faraut and A. Koranyi on the expansion of reproducing kernels which determines the analytic continuation of the holomorphic discrete series.
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 GENERAL RELATIVITY AND GRAVITATION


Energy transport in the Vaidya system
View Description Hide DescriptionEnergy transport mechanisms can be generated by imposing relations between null tetrad Ricci components. Several kinds of mass and density transport generated by these relations are studied for the generalized Vaidya system.
