Volume 52, Issue 5, May 2011
 ARTICLES

 Quantum Mechanics (General and Nonrelativistic)

Hypergeometric type operators and their supersymmetric partners
View Description Hide DescriptionThe generalization of the factorization method performed by Mielnik [J. Math. Phys.25, 3387 (1984)] opened new ways to generate exactly solvable potentials in quantum mechanics. We present an application of Mielnik's method to hypergeometric type operators. It is based on some solvable Riccati equations and leads to a unitary description of the quantum systems exactly solvable in terms of orthogonal polynomials or associated special functions.

Levinson's theorem and higher degree traces for AharonovBohm operators
View Description Hide DescriptionWe study Levinsontype theorems for the family of AharonovBohm models from different perspectives. The first one is purely analytical involving the explicit calculation of the waveoperators and allowing to determine precisely the various contributions to the left hand side of Levinson's theorem, namely, those due to the scattering operator, the terms at 0energy and at energy +∞. The second one is based on noncommutative topology revealing the topological nature of Levinson's theorem. We then include the parameters of the family into the topological description obtaining a new type of Levinson's theorem, a higher degree Levinson's theorem. In this context, the Chern number of a bundle defined by a family of projections on bound states is explicitly computed and related to the result of a 3trace applied on the scattering part of the model.

Characterizations of fixed points of quantum operations
View Description Hide DescriptionLet be a general quantum operation. An operator B is said to be a fixed point of if In this note, we shall show conditions under which B, a fixed point implies that B is compatible with the operation element of In particular, we offer an extension of the generalized Lüders theorem.

Semiclassical propagator for SU(n) coherent states
View Description Hide DescriptionWe present a detailed derivation of the semiclassical propagator in the SU(n) coherent state representation. In order to provide support for immediate physical applications, we restrict this work to the fully symmetric irreducible representations, which are suitable for the treatment of bosonic dynamics in n modes, considering systems with conservation of total particle number. The derivation described here can be easily extended to other classes of coherent states, thus representing an alternative approach to previously published methods.

Solutions of D = 2 supersymmetric YangMills quantum mechanics with SU(N) gauge group
View Description Hide DescriptionWe describe the generalization of the recently derived solutions of D = 2 supersymmetric YangMillsquantum mechanics with SU(3) gauge group to the generic case of SU(N) gauge group. We discuss the spectra and eigensolutions in bosonic as well as fermionic sectors.

Exact solution of D _{ N }type quantum Calogero model through a mapping to free harmonic oscillators
View Description Hide DescriptionWe solve the eigenvalue problem of the D _{ N }type of Calogero model by mapping it to a set of decoupled quantum harmonic oscillators through a similarity transformation. In particular, we construct the eigenfunctions of this Calogero model from those of bosonic harmonic oscillators having either all even parity or all odd parity. It turns out that the eigenfunctions of this model are orthogonal with respect to a nontrivial inner product, which can be derived from the quasiHermiticity property of the corresponding conserved quantities.

Duality relations in the auxiliary field method
View Description Hide DescriptionThe eigenenergies ε^{(N)}(m; {n _{ i }, l _{ i }}) of a system of N identical particles with a mass m are functions of the various radial quantum numbers n _{ i } and orbital quantum numbers l _{ i }. Approximations E ^{(N)}(m; Q) of these eigenenergies, depending on a principal quantum number Q({n _{ i }, l _{ i }}), can be obtained in the framework of the auxiliary field method. We demonstrate the existence of numerous exact duality relations linking quantities E ^{(N)}(m; Q) and E ^{(p)}(m′; Q′) for various forms of the potentials (independent of m and N) and for both nonrelativistic and semirelativistic kinematics. As the approximations computed with the auxiliary field method can be very close to the exact results, we show with several examples that these duality relations still hold, with sometimes a good accuracy, for the exact eigenenergies ε^{(N)}(m; {n _{ i }, l _{ i }}).

Rate of convergence in nonlinear Hartree dynamics with factorized initial data
View Description Hide DescriptionThe mean fielddynamics of an Nparticle weekly interactingBoson system can be described by the nonlinear Hartree equation. In this paper, we present estimates on the 1/N rate of convergence of manybody Schrödinger dynamics to the onebody nonlinear Hartree dynamics with factorized initial data with twobody interaction potential V in .

The electric AharonovBohm effect
View Description Hide DescriptionThe seminal paper of Aharonov and Bohm [Phys. Rev.115, 485 (1959)]10.1103/PhysRev.115.485 is at the origin of a very extensive literature in some of the more fundamental issues in physics. They claimed that electromagnetic fields can act at a distance on charged particles even if they are identically zero in the region of space where the particles propagate, that the fundamental electromagnetic quantities in quantum physics are not only the electromagnetic fields but also the circulations of the electromagnetic potentials; what gives them a real physical significance. They proposed two experiments to verify their theoretical conclusions. The magnetic AharonovBohm effect, where an electron is influenced by a magnetic field that is zero in the region of space accessible to the electron, and the electric AharonovBohm effect where an electron is affected by a timedependent electric potential that is constant in the region where the electron is propagating, i.e., such that the electric field vanishes along its trajectory. The AharonovBohm effects imply such a strong departure from the physical intuition coming from classical physics that it is no wonder that they remain a highly controversial issue after more than fifty years, in spite of the fact that they are discussed in most of the text books in quantum mechanics. The magnetic case has been studied extensively. The experimental issues were settled by the remarkable experiments of Tonomura et al. [Phys. Rev. Lett.48, 1443 (1982); Phys. Rev. Lett.56, 792 (1986)] with toroidal magnets, that gave a strong evidence of the existence of the effect, and by the recent experiment of Caprez et al. [Phys. Rev. Lett.99, 210401 (2007)]10.1103/PhysRevLett.99.210401 that shows that the results of the Tonomura et al. experiments cannot be explained by the action of a force. The theoretical issues were settled by Ballesteros and Weder [Commun. Math. Phys.285, 345 (2009)10.1007/s0022000805791; J. Math. Phys.50, 122108 (2009)10.1063/1.3266176; Commun. Math. Phys.303, 175 (2011)]10.1007/s0022001011669 who rigorously proved that quantum mechanics predicts the experimental results of Tonomura et al. and of Caprez et al. The electric AharonovBohm effect has been much less studied. Actually, its existence, that has not been confirmed experimentally, is a very controversial issue. In their 1959 paper Aharonov and Bohm proposed an ansatz for the solution to the Schrödinger equation in regions where there is a timedependent electric potential that is constant in space. It consists in multiplying the free evolution by a phase given by the integral in time of the potential. The validity of this ansatz predicts interference fringes between parts of a coherent electron beam that are subjected to different potentials. In this paper we prove that the exact solution to the Schrödinger equation is given by the AharonovBohm ansatz up to an error bound in norm that is uniform in time and that decays as a constant divided by the velocity. Our results give, for the first time, a rigorous proof that quantum mechanics predicts the existence of the electric AharonovBohm effect, under conditions that we provide. We hope that our results will stimulate the experimental research on the electric AharonovBohm effect.

Exact asymptotic behavior of the Pekar–Tomasevich functional
View Description Hide DescriptionAn explicit asymptotic expression for the groundstate energy of the Pekar–Tomasevich functional for the Npolaron is found, when the repulsion parameter U of the electrons satisfies the inequality 0 ⩽ U ⩽ 2α, where α is the coupling constant of the polaron. If denotes this groundstate energy for the case of Nelectrons and repulsion parameter U, we prove that as N → ∞, where, c _{ p } = 0.10851…. Moreover, we show that , for all N.
 Quantum Information and Computation

An explicit expression for the relative entropy of entanglement in all dimensions
View Description Hide DescriptionThe relative entropy of entanglement is defined in terms of the relative entropy between an entangled state and its closest separable state (CSS). Given a multipartitestate on the boundary of the set of separable states, we find an explicit expression for all the entangled states for which this state is a CSS. Our formula holds for multipartite states in all dimensions. For the bipartite case of two qubits, our formula reduce to the one given by Miranowicz and Ishizaka [Phys. Rev. A78, 032310 (2008)].

Entropy reduction of quantum measurements
View Description Hide DescriptionIt is shown that the entropy reduction (the information gain in the initial terminology) of an efficient (ideal or pure) quantum measurement coincides with the generalized quantum mutual information of a quantumclassical channel mapping an a priori state to the corresponding posterioriprobability distribution of the outcomes of the measurement. As a result the entropy reduction is defined for arbitrary a priori states (not only for states with finite von Neumann entropy) and its analytical properties is studied in detail by using general properties of the quantum mutual information. By using this approach it is shown that the entropy reduction of an efficient quantum measurement is a nonnegative lower semicontinuous concave function on the set of all a priori states having continuous restrictions to subsets on which the von Neumann entropy is continuous. Monotonicity and subadditivity of the entropy reduction are also easily proved by this method. A simple continuity condition for the entropy reduction and for the mean posteriorientropy considered as functions of a pair (a priori state, measurement) is obtained. A characterization of an irreducible measurement (in the Ozawa sense) which is not efficient is considered in the Appendix.
 Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)

Supersymmetry algebra cohomology. III. Primitive elements in four and five dimensions
View Description Hide DescriptionThe primitive elements of the supersymmetryalgebra cohomology as defined in a previous paper are computed for standard supersymmetryalgebras in four and five dimensions, for all signatures of the metric and any number of supersymmetries.

Higher level twisted Zhu algebras
View Description Hide DescriptionThe study of twisted representations of graded vertex algebras is important for understanding orbifold models in conformal field theory. In this paper, we consider the general setup of a vertex algebraV, graded by for some subgroup Γ of containing , and with a Hamiltonian operator H having real (but not necessarily integer) eigenvalues. We construct the directed system of twisted level p Zhu algebras, and we prove the following theorems: For each p, there is a bijection between the irreducible modules and the irreducible Γtwisted positive energy Vmodules, and V is (Γ, H)rational if and only if all its Zhu algebras are finite dimensional and semisimple. The main novelty is the removal of the assumption of integer eigenvalues for H. We provide an explicit description of the level p Zhu algebras of a universal enveloping vertex algebra, in particular of the Virasoro vertex algebra and the universal affine KacMoody vertex algebra at noncritical level. We also compute the inverse limits of these directed systems of algebras.

An approximate κ state solutions of the Dirac equation for the generalized Morse potential under spin and pseudospin symmetry
View Description Hide DescriptionBy using an improved approximation scheme to deal with the centrifugal (pseudocentrifugal) term, we solve the Dirac equation for the generalized Morse potential with arbitrary spinorbit quantum number κ. In the presence of spin and pseudospin symmetry, the analytic bound state energy eigenvalues and the associated upper and lowerspinor components of two Dirac particles are found by using the basic concepts of the NikiforovUvarov method. We study the special cases when κ = ±1 ( swave), the nonrelativistic limit and the limit when α becomes zero (Kratzer potential model). The present solutions are compared with those obtained by other methods.

An infinitedimensional calculus for generalized connections on hypercubic lattices
View Description Hide DescriptionA space for gauge theories is defined, using projective limits as subsets of Cartesian products of homomorphisms from a lattice on the structure group. In this space, noninteracting and interacting measures are defined as well as functions and operators. From projective limits of test functions and distributions on products of compact groups, a projective gauge triplet is obtained, which provides a framework for the infinitedimensional calculus in gauge theories. The gauge measure behavior on nongeneric strata is also obtained.

The uses of the refined matrix model recursion
View Description Hide DescriptionWe study matrix models in the βensemble by building on the refined recursion relation proposed by Chekhov and Eynard. We present explicit results for the first βdeformed corrections in the onecut and the twocut cases, as well as two applications to supersymmetric gauge theories: the calculation of superpotentials in gauge theories, and the calculation of vevs of surface operators in superconformal theories and their Liouville duals. Finally, we study the βdeformation of the Chern–Simons matrix model. Our results indicate that this model does not provide an appropriate description of the Ωdeformed topological string on the resolved conifold, and therefore that the βdeformation might provide a different generalization of topological string theory in toric Calabi–Yau backgrounds.
 General Relativity and Gravitation

Tachyons in general relativity
View Description Hide DescriptionWe consider the motion of tachyons (fasterthanlight particles) in the framework of general relativity. An important feature is the large contribution of low energy tachyons to the energymomentum tensor. We also calculate the gravitational field produced by tachyons in particular geometric arrangements; and it appears that there could be selfcohering bundles of such matter. This leads us to suggest that such theoretical ideas might be relevant to major problems (dark matter and dark energy) in current cosmological models.

U(N) coherent states for loop quantum gravity
View Description Hide DescriptionWe investigate the geometry of the space of Nvalent SU(2) intertwiners. We propose a new set of holomorphic operators acting on this space and a new set of coherent states which are covariant under U(N) transformations. These states are labeled by elements of the Grassmannian Gr _{ N, 2}, they possess a direct geometrical interpretation in terms of framed polyhedra and are shown to be related to the wellknown coherent intertwiners.

LeviCivita cylinders with fractional angular deficit
View Description Hide DescriptionThe angular deficit factor in the LeviCivita vacuum metric has been parametrized using a RiemannLiouville fractional integral. This introduces a new parameter into the general relativistic cylinder description, the fractional index α. When the fractional index is continued into the negative α region, new behavior is found in the GottHiscock cylinder and in an Israel shell.