Volume 48, Issue 11, November 2007
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Generally covariant quantum mechanics on noncommutative configuration spaces
View Description Hide DescriptionWe generalize the previously given algebraic version of “Feynman’s proof of Maxwell’s equations” to noncommutative configuration spaces. By doing so, we also obtain an axiomatic formulation of nonrelativistic quantum mechanics over such spaces, which, in contrast to most examples discussed in the literature, does not rely on a distinguished set of coordinates. We give a detailed account of several examples, e.g., which leads to nonAbelian YangMillstheories, and of noncommutative tori . Moreover, we examine models over the Moyaldeformed plane . Assuming the conservation of electrical charges, we show that in this case the canonical uncertainty relation with metric is only consistent if is constant.

Generalized EinsteinPodolskyRosen states
View Description Hide DescriptionBased on Bohr’s ideas two families of states and sharing the same perfect correlation of the EinsteinPodolskyRosen state and maximally violating Bell’s inequalities are constructed. Unlike their finitedimensional counterpartner, these states are not unitarily equivalent. Hence they cannot be transformed into each other by physical operations of local algebras. This is a new entanglement phenomenon emerging in infinitedimensional systems. Due to the uncertainty principle the existence of unbounded observables depends on the states and . Therefore the manipulation of unbounded observables in quantum information processes cannot provide information for all states. Especially, the canonical unbounded observables and , of individual particles do not exist in the GNS representations associated with and , and hence properties of individual particles cannot be obtained in such states.

Trace formula for systems with spin from the coherent state propagator
View Description Hide DescriptionWe present a detailed derivation of the trace formula for a general Hamiltonian with two degrees of freedom where one of them is canonical and the other a spin. Our derivation starts from the semiclassical formula for the propagator in a basis formed by the product of a canonical and a spin coherent states and is valid in the limit , with constant. The trace formula, obtained by taking the trace and the Fourier transform of the coherent state propagator, is compared to others found in the literature.

Coulomb potential in one dimension with minimal length: A path integral approach
View Description Hide DescriptionWe solve the path integral in momentum space for a particle in the field of the Coulomb potential in one dimension in the framework of quantum mechanics with the minimal length given by , where is a small positive parameter. From the spectral decomposition of the fixed energy transition amplitude, we obtain the exact energy eigenvalues and momentum space eigenfunctions.

Singlestep controlledNOT logic from any exchange interaction
View Description Hide DescriptionA selfcontained approach to studying the unitary evolution of coupled qubits is introduced, capable of addressing a variety of physical systems described by exchange Hamiltonians containing Rabi terms. The method automatically determines both the Weyl chamber steering trajectory and the accompanying local rotations. Particular attention is paid to the case of anisotropic exchange with tracking controls, which is solved analytically. It is shown that, if computational subspace is well isolated, any exchange interaction can always generate high fidelity, singlestep controlledNOT (CNOT) logic, provided that both qubits can be individually manipulated. The results are then applied to superconductingqubit architectures, for which several CNOT gate implementations are identified. The paper concludes with consideration of two CNOT gate designs having high efficiency and operating with no significant leakage to higherlying noncomputational states.

BarutGirardello and KlauderPerelomov coherent states for the Kravchuk functions
View Description Hide DescriptionThe BarutGirardello and KlauderPerelomov coherent states for the Kravchuk functions are constructed. It is shown that the resolution of unity condition is satisfied for both of the coherent states. Also in the dimensional Hilbert space spanned by the Kravchuk eigenstates, the appropriate measure and analytic function are obtained.

Selfadjointness for Dirac operators via HardyDirac inequalities
View Description Hide DescriptionDistinguished selfadjoint extensions of Dirac operators are constructed for a class of singular potentials including Coulombic ones up to (and including) the critical case, . The method uses HardyDirac inequalities and quadratic form techniques.

Tunneling through inhomogeneous delta barriers as diffraction and scattering
View Description Hide DescriptionWe present a general theory of the quantummechanical tunneling through an inhomogeneous planar delta barrier. The delta barrier means that the potential energy is proportional to the Dirac delta function in the direction perpendicular to the plane of the barrier. The inhomogeneity of the delta barrier means that the delta function is multiplied by a nonnegative function, different from a constant, of lateral coordinates. We assume that this function, being the barrier strength, may be arbitrary. To exemplify our theory, we consider delta barriers that are made inhomogeneous by embedding equal circular windows in the barrier plane, assuming that the barriers are homogeneous both inside and outside the windows. (The value of the barrier strength is taken higher by the side of the windows than inside the windows.) With the inhomogeneous delta barriers of this kind, we show how the tunnelingtheory is related to the theory of diffraction and scattering. Although our general solution of the problem is new in the context of the tunnelingtheory, it is essentially based on a method which was used by Kirchhoff in the 19th century in the theory of waves.

Comment on “A note on the infimum problem of Hilbert space effects” [J. Math. Phys.47, 102103 (2006)]
View Description Hide DescriptionWe show that the two main results of the article [J. Math. Phys.47, 102103 (2006)] have very short proofs as direct consequences of the solution to the infimum problem for bounded nonnegative operators in a Hilbert space given by T. Ando [Analytic and Geometric Inequalities and Applications, Mathematical Applications Vol. 478 (Kluwer Academic, Dordrecht, 1999)] and a formula for the shorted operator obtained by H. Kosaki [“Remarks on Lebesguetype decomposition of positive operators,” J. Oper. Theory11, 137–143 (1984)].

Markovian quasifree states on canonical anticommutation relation algebras
View Description Hide DescriptionThe characterization of quasifree product states on CAR algebras is given. We also prove that the quasifree states on CAR algebra which saturate the strong additivity of von Neumann entropy with equality are product states.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Kähler quantization of and modular forms
View Description Hide DescriptionKähler quantization of is studied. It is shown that this theory corresponds to a fermionic model targeting a noncommutative space. By solving the complexstructure moduli independence conditions, the quantum background independent wave function is obtained. We study the transformation of the wave function under modular transformation. It is shown that the transformation rule is characteristic to the operator ordering. Similar results are obtained for Kähler quantization of .
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 GENERAL RELATIVITY AND GRAVITATION


Some effects on quantum systems due to the gravitational field of a cosmic string
View Description Hide DescriptionWe study the behavior of a nonrelativistic quantum particle interacting with different potentials in the background spacetime generated by a cosmic string. We find the energy spectra for the quantum systems under consideration and discuss how they differ from their flat Minkowski spacetime values.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Extended Hamiltonian systems in multisymplectic field theories
View Description Hide DescriptionWe consider Hamiltonian systems in firstorder multisymplectic field theories. We review the properties of Hamiltonian systems in the socalled restricted multimomentum bundle, including the variational principle which leads to the Hamiltonian field equations. In an analogous way to how these systems are defined in the socalled extended (symplectic) formulation of nonautonomous mechanics, we introduce Hamiltonian systems in the extended multimomentum bundle. The geometric properties of these systems are studied, the Hamiltonian equations are analyzed using integrable multivector fields, the corresponding variational principle is also stated, and the relation between the extended and the restricted Hamiltonian systems is established. All these properties are also adapted to certain kinds of submanifolds of the multimomentum bundles in order to cover the case of almostregular field theories.

Constructing a class of solutions for the HamiltonJacobi equation in field theory
View Description Hide DescriptionA new approach leading to the formulation of the HamiltonJacobi equation for field theories is investigated within the framework of jet bundles and multisymplectic manifolds. An algorithm associating classes of solutions to given sets of boundary conditions of the field equations is provided. The paper also puts into evidence the intrinsic limits of the HamiltonJacobi method as an algorithm to determine families of solutions of the field equations, showing how the choice of the boundary data is often limited by compatibility conditions.

dimensional separation of variables
View Description Hide DescriptionIn this paper we explore general conditions which guarantee that the geodesic flow on a twodimensional manifold with indefinite signature is locally separable. This is equivalent to showing that a twodimensional natural Hamiltonian system on the hyperbolic plane possesses a second integral of motion which is a quadratic polynomial in the momenta associated with a secind rank Killing tensor. We examine the possibility that the integral is preserved by the Hamiltonian flow on a given energy hypersurface only (weak integrability) and derive the additional requirement necessary to have conservation at arbitrary values of the Hamiltonian (strong integrability). Using null coordinates, we show that the leadingorder coefficients of the invariant are arbitrary functions of one variable in the case of weak integrability. These functions are quadratic polynomials in the coordinates in the case of strong integrability. We show that for dimensional systems, there are three possible types of conformal Killing tensors and, therefore, three distinct separability structures in contrast to the single standard HamiltonJacobitype separation in the positive definite case. One of the new separability structures is the complex∕harmonic type which is characterized by complex separation variables. The other new type is the linear∕null separation which occurs when the conformal Killing tensor has a null eigenvector.
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 STATISTICAL PHYSICS


Twoparameter generalization of the logarithm and exponential functions and BoltzmannGibbsShannon entropy
View Description Hide DescriptionThe sum and the product emerge naturally within nonextensive statistical mechanics. We show here how they lead to twoparameter (namely, and ) generalizations of the logarithmic and exponential functions (noted, respectively, and ), as well as of the BoltzmannGibbsShannon entropy (noted ). The remarkable properties of the generalized logarithmic function make the entropic form satisfy, for large regions of , important properties such as expansibility, concavity, and Lesche stability, but not necessarily composability.

Exponential spatial decay of spinspin correlations in translation invariant quasifree states
View Description Hide DescriptionWe study the spatial decay of the transversal spinspin correlations in translation invariant quasifree states on the canonical anticommutation relation algebra on the discrete line. We establish a simple criterion on the spectrum of the density of the quasifree state which ensures the decay to be exponential. Moreover, we illustrate exponential and slower decay with the example of the spin chain out of equilibrium, in thermal equilibrium at positive temperature, in the ground state, and in the chaotic state.

General connectivity distribution functions for growing networks with preferential attachment of fractional power
View Description Hide DescriptionWe study the general connectivity distribution functions for growing networks with preferential attachment (PA) of fractional power, , using Simon’s method. We first show that the heart of the previously known methods of the rate equations for the connectivity distribution functions is nothing but Simon’s method for word problem. Secondly, we show that for the case of fractional , the transformation of the rate equation provides a fractional differential equation of a new type, which coincides with that for PA with linear power, when . We show that to solve such a fractional differential equation, we need to define a transcendental function that we call upsilon function. Most of all the previously known results are obtained consistently in the framework of a unified theory.

solutions to the Cauchy problem of the BGK equation
View Description Hide DescriptionThe BGKmodel of the Boltzmann equation plays an important role in the kinetic theory of rarefied gases. Some existence and uniqueness results of solutions to its Cauchy problem were established for large initial data under various circumstances [see, for example, Perthame, B., “Global existence to the BGKmodel of Boltzmann equation,” J. Differ. Equations82, 191–205 (1989)]. In this paper, by establishing some weighted estimates of the hydrodynamical quantities of a gas, we prove the existence theorem of the solutions to the Cauchy problem and establish the propagation properties of the moments for this kind of solutions.
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 METHODS OF MATHEMATICAL PHYSICS


Symbol calculus and zetafunction regularized determinants
View Description Hide DescriptionIn this work, we use semigroup integral to evaluate zetafunction regularized determinants. This is especially powerful for nonpositive operators such as the Dirac operator. In order to understand fully the quantum effective action, one should know not only the potential term but also the leading kinetic term. In this purpose, we use the Weyl type of symbol calculus to evaluate the determinant as a derivative expansion. The technique is applied both to a spin0 bosonic operator and to the Dirac operator coupled to a scalar field.
