Volume 51, Issue 12, December 2010
Index of content:
 ARTICLES

 Quantum Mechanics (General and Nonrelativistic)

Collocation method for fractional quantum mechanics
View Description Hide DescriptionWe show that it is possible to obtain numerical solutions to quantum mechanical problems involving a fractional Laplacian, using a collocation approach based on little sinc functions, which discretizes the Schrödinger equation on a uniform grid. The different boundary conditions are naturally implemented using sets of functions with the appropriate behavior. Good convergence properties are observed. A comparison with results based on a Wentzel–Kramers–Brillouin analysis is performed.

Quasiclassical asymptotics and coherent states for bounded discrete spectra
View Description Hide DescriptionWe consider discrete spectra of bound states for nonrelativistic motion in attractive potentials . For these potentials the quasiclassical approximation for n → ∞ predicts quantized energy levels of a bounded spectrum varying as . We construct collective quantum states using the set of wavefunctions of the discrete spectrum assuming this asymptotic behavior. We give examples of states that are normalizable and satisfy the resolution of unity, using explicit positive functions. These are coherent states in the sense of Klauder and their completeness is achieved via exact solutions of Hausdorff moment problems, obtained by combining Laplace and Mellin transform methods. For σ in the range 0 < σ ⩽ 2/3 we present exact implementations of such states for the parametrization σ = 2(k − l)/(3k − l) with k and l positive integers satisfying .

The formal path integral and quantum mechanics
View Description Hide DescriptionGiven an arbitrary Lagrangian function on and a choice of classical path, one can try to define Feynman's path integral supported near the classical path as a formal power series parameterized by “Feynman diagrams,” although these diagrams may diverge. We compute this expansion and show that it is (formally, if there are ultraviolet divergences) invariant under volumepreserving changes of coordinates. We prove that if the ultraviolet divergences cancel at each order, then our formal path integral satisfies a “Fubini theorem” expressing the standard composition law for the time evolution operator in quantum mechanics. Moreover, we show that when the Lagrangian is inhomogeneous quadratic in velocity such that its homogeneousquadratic part is given by a matrix with constant determinant, then the divergences cancel at each order. Thus, by “cutting and pasting” and choosing volumecompatible local coordinates, our construction defines a Feynmandiagrammatic “formal path integral” for the nonrelativistic quantum mechanics of a charged particle moving in a Riemannian manifold with an external electromagnetic field.

Wave functions and energy spectra for the hydrogenic atom in
View Description Hide DescriptionWe consider the hydrogenic atom in a space of the form , where may be a generalized manifold obeying certain properties. We separate the solution to the governing timeindependent Schrödinger equation into a component over and a component over . Upon obtaining a solution to the relevant eigenvalue problems, we recover both the wave functions and energyspectrum for the hydrogenic atom over . We consider some specific examples of , including the fairly simple Ddimensional torus and the more complicated Kähler conifold in order to illustrate the method. In the examples considered, we see that the corrections to the standard energyspectrum for the hydrogen atom due to the addition of higher dimensions scale as a constant times , where L denotes the size of the additional dimensions. Thus, under the assumption of small compact extra dimensions, even the first energy corrections to the standard spectrum will be quite large.

The U(1)Kepler Problems
View Description Hide DescriptionLet n ⩾ 2 be a positive integer. To each irreducible representation σ of U(1), a U(1)Kepler problem in dimension (2n − 1) is constructed and analyzed. This system is superintegrable and when n = 2 it is equivalent to a MICZKepler problem. The dynamical symmetry group of this system is , and the Hilbert space of bound states is the unitary highest weight representation of with the minimal positive GelfandKirillov dimension. Furthermore, it is shown that the correspondence between (the dual of σ) and is the thetacorrespondence for dual pair .

Riemann surface approach to bound and resonant states for a nonlocal potential
View Description Hide DescriptionThe Riemann surface approach to bound and resonant states is extended to the case of a separable nonlocal potential that is constant on a certain domain of the inner region and vanishes in the rest of the domain. The approach consists in the construction of the Riemann surface of the Smatrix pole function k = k(g) over the gplane, where g is the strength of the complex nonlocal potential. On the Riemann surface the pole function k = k(g) is singlevalued and analytic. The branch points of the pole function k = k(g) and their kplane images are determined and analyzed as a function of the position of the region of nonlocality. The Riemann surface of the Smatrix pole function is constructed. According to the Riemann surface approach to each bound or resonant state a sheet of the Riemann surface is associated. All the natural modes (bound and resonant states) of the system are identified and treated in a unified way. The nonlocal potential generates narrow resonant states that cannot be produced by a local potential.

On the absence of absolutely continuous spectra for Schrödinger operators on radial tree graphs
View Description Hide DescriptionThe subject of the paper is Schrödinger operators on tree graphs which are radial, having the branching number at all the vertices at the distance from the root. We consider a family of coupling conditions at the vertices characterized by real parameters. We prove that if the graph is sparse so that there is a subsequence of growing to infinity, in the absence of the potential the absolutely continuous spectrum is empty for a large subset of these vertex couplings, but on the the other hand, there are cases when the spectrum of such a Schrödinger operator can be purely absolutely continuous.

Madelung representation of damped parametric quantum oscillator and exactly solvable Schrödinger–Burgers equations
View Description Hide DescriptionWe construct a Madelung fluid model with time variable parameters as a dissipative quantum fluid and linearize it in terms of Schrödinger equation with timedependent parameters. It allows us to find exact solutions of the nonlinear Madelung system in terms of solutions of the Schrödinger equation and the corresponding classical linear ordinary differential equation with variable frequency and damping. For the complex velocity field, the Madelung system takes the form of a nonlinear complex Schrödinger–Burgers equation, for which we obtain exact solutions using complex Cole–Hopf transformation. In particular, we give exact results for nonlinear Madelung systems related with Caldirola–Kanaitype dissipative harmonic oscillator. Collapse of the wave function in dissipative models and possible implications for the quantum cosmology are discussed.
 Quantum Information and Computation

The divergence and mixing times of quantum Markov processes
View Description Hide DescriptionWe introduce quantum versions of the divergence, provide a detailed analysis of their properties, and apply them in the investigation of mixing times of quantum Markov processes. An approach similar to the one presented in the literature for classical Markov chains is taken to bound the tracedistance from the steady state of a quantum processes. A strict spectral bound to the convergence rate can be given for timediscrete as well as for timecontinuous quantum Markov processes. Furthermore, the contractive behavior of the divergence under the action of a completely positive map is investigated and contrasted to the contraction of the trace norm. In this context we analyze different versions of quantum detailed balance and, finally, give a geometric conductance bound to the convergence rate for unital quantum Markov processes.

Universal coding for transmission of private information
View Description Hide DescriptionWe consider the scenario in which Alice transmits private classical messages to Bob via a classicalquantum channel, part of whose output is intercepted by an eavesdropper Eve. We prove the existence of a universal coding scheme under which Alice's messages can be inferred correctly by Bob, and yet Eve learns nothing about them. The code is universal in the sense that it does not depend on specific knowledge of the channel. Prior knowledge of the probability distribution on the input alphabet of the channel, and bounds on the corresponding Holevo quantities of the output ensembles at Bob's and Eve's end suffice.

From qubits to E7
View Description Hide DescriptionThere is an intriguing relation between quantum informationtheory and super gravity, discovered by M. J. Duff and S. Ferrara. It relates entanglement measures for qubits to black hole entropy, which in a certain case involves the quartic invariant on the 56dimensional representation of the Lie group E7. In this paper we recall the relatively straightforward manner in which threequbits lead to E7, or at least to the Weyl group of E7. We also show how the Fano plane emerges in this context.

Chain of Hardytype local reality constraints for n qubits
View Description Hide DescriptionNonlocality without inequality is an elegant argument introduced by Hardy for two qubit systems, and later generalised to nqubits, to establish contradiction of quantum theory with local realism. Interestingly, for n = 2 this argument is actually a corollary of Belltype inequalities, viz., the CH–Hardy inequality involving Bell correlations, but for n greater than two it involves nparticle probabilities more general than Bellcorrelations. In this paper, we first derive a chain of completely new local realistic inequalities involving joint probabilities for nqubits and then associated with each such inequality, we provide a new Hardytype local reality constraint without inequalities. Quantum mechanical maximal violations of the chain of inequalities and of the associated constraints are also studied by deriving appropriate Cirel'sontype theorems. These results involving joint probabilities more general than Bell correlations are expected to provide a new systematic tool to investigate entanglement.

The extension problem for partial Boolean structures in quantum mechanics
View Description Hide DescriptionAlternative partial Boolean structures, implicit in the discussion of classical representability of sets of quantum mechanical predictions, are characterized, with definite general conclusions on the equivalence of the approaches going back to Bell and Kochen–Specker. An algebraic approach is presented, allowing for a discussion of partial classical extension, amounting to reduction of the “number of contexts,” classical representability arising as a special case. As a result, known techniques are generalized and some of the associated computational difficulties overcome. The implications on the discussion of Boole–Bell inequalities are indicated.
 Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)

Timeevolution of the external field problem in Quantum Electrodynamics
View Description Hide DescriptionWe construct the timeevolution for the secondquantized Dirac equation subject to a smooth, compactly supported, time dependent electromagnetic potential and identify the degrees of freedom involved. Earlier works on this (e.g., Ruijsenaars) observed the Shale–Stinespring condition and showed that the oneparticle timeevolution can be lifted to Fock space if and only if the external field had zero magnetic components. We scrutinize the idea, observed earlier by Fierz and Scharf, that the timeevolution can be implemented between time varying Fock spaces. In order to define these Fock spaces we are led to consider classes of reference vacua and polarizations. We show that this implementation is up to a phase independent of the chosen reference vacuum or polarization and that all induced transition probabilities are welldefined and unique.

Supersymmetry algebra cohomology. I. Definition and general structure
View Description Hide DescriptionThis paper concerns standard supersymmetryalgebras in diverse dimensions, involving bosonic translational generators and fermionic supersymmetry generators. A cohomology related to these supersymmetryalgebras, termed supersymmetryalgebra cohomology, and corresponding “primitive elements” are defined by means of a BRST (BecchiRouetStoraTyutin)type coboundary operator. A method to systematically compute this cohomology is outlined and illustrated by simple examples.

Quantized Nambu–Poisson manifolds and nLie algebras
View Description Hide DescriptionWe investigate the geometric interpretation of quantized Nambu–Poisson structures in terms of noncommutative geometries. We describe an extension of the usual axioms of quantization in which classical Nambu–Poisson structures are translated to nLie algebras at quantum level. We demonstrate that this generalized procedure matches an extension of Berezin–Toeplitz quantization yielding quantized spheres, hyperboloids, and superspheres. The extended Berezin quantization of spheres is closely related to a deformation quantization of nLie algebras as well as the approach based on harmonic analysis. We find an interpretation of Nambu–Heisenberg nLie algebras in terms of foliations of by fuzzy spheres, fuzzy hyperboloids, and noncommutative hyperplanes. Some applications to the quantum geometry of branes in Mtheory are also briefly discussed.

Klein–Gordon equation in hydrodynamical form
View Description Hide DescriptionWe follow and modify the Feshbach–Villars formalism by separating the Klein–Gordon equation into two coupled timedependent Schrödinger equations for particle and antiparticlewave function components with positive probability densities. We find that the equation of motion for the probability densities is in the form of relativistic hydrodynamics where various forces have their classical counterparts, with the additional element of the quantum stress tensor that depends on the derivatives of the amplitude of the wave function. We derive the equation of motion for the Wigner function and we find that its approximate classical weakfield limit coincides with the equation of motion for the distribution function in the collisionless kinetic theory.

Hopf solitons in the Nicole model
View Description Hide DescriptionThe Nicole model is a conformal field theory in a threedimensional space. It has topological soliton solutions classified by the integervalued Hopf charge, and all currently known solitons are axially symmetric. A volumepreserving flow is used to construct soliton solutions numerically for all Hopf charges from 1 to 8. It is found that the known axially symmetric solutions are unstable for Hopf charges greater than 2 and new lower energy solutions are obtained that include knots and links. A comparison with the Skyrme–Faddeev model suggests many universal features, though there are some differences in the link types obtained in the two theories.

Can we make a Bohmian electron reach the speed of light, at least for one instant?
View Description Hide DescriptionIn Bohmian mechanics, a version of quantum mechanics that ascribes world lines to electrons, we can meaningfully ask about an electron's instantaneous speed relative to a given inertial frame. Interestingly, according to the relativistic version of Bohmian mechanics using the Dirac equation, a massive particle's speed is less than or equal to the speed of light, but not necessarily less. That is, there are situations in which the particle actually reaches the speed of light—a very nonclassical behavior. That leads us to the question of whether such situations can be arranged experimentally. We prove a theorem,Theorem 5, implying that for generic initial wave functions the probability that the particle ever reaches the speed of light, even if at only one point in time, is zero. We conclude that the answer to the question is no. Since a trajectory reaches the speed of light whenever the quantum probability current is a lightlike 4vector, our analysis concerns the current vector field of a generic wave function and may thus be of interest also independently of Bohmian mechanics. The fact that the current is never spacelike has been used to argue against the possibility of fasterthanlight tunneling through a barrier, a somewhat similar question. Theorem 5, as well as a more general version provided by Theorem 6, are also interesting in their own right. They concern a certain property of a function that is crucial to the question of reaching the speed of light, namely being transverse to a certain submanifold of along a given compact subset of spacetime. While it follows from the known transversality theorem of differential topology that this property is generic among smooth functions , Theorem 5 asserts that it is also generic among smooth solutions of the Dirac equation.

Curvature function and coarse graining
View Description Hide DescriptionA classic theorem in the theory of connections on principal fiber bundles states that the evaluation of all holonomy functions gives enough information to characterize the bundle structure (among those sharing the same structure group and base manifold) and the connection up to a bundle equivalence map. This result and other important properties of holonomy functions have encouraged their use as the primary ingredient for the construction of families of quantum gauge theories. However, in these applications often the set of holonomy functions used is a discrete proper subset of the set of holonomy functions needed for the characterization theorem to hold. We show that the evaluation of a discrete set of holonomy functions does not characterize the bundle and does not constrain the connection modulo gauge appropriately. We exhibit a discrete set of functions of the connection and prove that in the abelian case their evaluation characterizes the bundle structure (up to equivalence), and constrains the connection modulo gauge up to “local details” ignored when working at a given scale. The main ingredient is the Lie algebra valued curvature function defined below. It covers the holonomy function in the sense that .