Volume 49, Issue 4, April 2008
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Comment on “Barut–Girardello and Klauder–Perelomov coherent states for the Kravchuk functions” [J. Math. Phys.48, 112106 (2007)]
View Description Hide DescriptionWe call attention to the misconstructions in a paper recently published in this journal [A. Chenaghlou and O. Faizy, J. Math. Phys.48, 112106 (2007)]. It is shown that the constructed Barut–Girardello coherent states are problematic from the view points of the definition and the measure. The claimed coherencies for the Kravchuk functions cannot actually exist.

Correlated entanglement distillation and the structure of the set of undistillable states
View Description Hide DescriptionWe consider entanglement distillation under the assumption that the input states are allowed to be correlated among each other. We hence replace the usually considered independent and identically distributed hypothesis by the weaker assumption of merely having identical reductions. We find that whether a state is then distillable or not is only a property of these reductions, and not of the correlations that are present in the input state. This is shown by establishing an appealing relation between the set of copycorrelated undistillable states and the standard set of undistillable states: The former turns out to be the convex hull of the latter. As an example of the usefulness of our approach to the study of entanglement distillation, we prove a new activation result, which generalizes earlier findings: It is shown that for every entangled state and every , there exists a copycorrelated undistillable state such that is singlecopy distillable. Finally, the relation of our results to the conjecture about the existence of bound entangled states with a nonpositive partial transpose is discussed.

Environment induced bipartite entanglement
View Description Hide DescriptionRecently, a sufficient condition on the structure of the Kossakowski–Lindblad master equation has been given such that the generated reduced dynamics of two qubits results entangling for at least one among their initial separable pure states. In this paper we study to which extent this condition is also necessary. Further, we find sufficient conditions for bathmediated entanglement generation in higher dimensional bipartite open quantum systems.

Supersymmetry versus ghosts
View Description Hide DescriptionWe consider the simplest nontrivial supersymmetric quantum mechanical system involving higher derivatives. We unravel the existence of additional bosonic and fermionic integrals of motion forming a nontrivial algebra. This allows one to obtain the exact solution both in the classical and quantum cases. The supercharges and are not Hermitially conjugate to each other anymore, which allows for the presence of negative energies in the spectrum. We show that the spectrum of the Hamiltonian is unbounded from below. It is discrete and infinitely degenerate in the free oscillatorlike case and becomes continuous running from to when interactions are added. Notwithstanding the absence of the ground state, there is no collapse, which suggests that a unitary evolution operator may be defined.

Joint excitation probability for two harmonic oscillators in one dimension and the Mott problem
View Description Hide DescriptionWe analyze a one dimensional quantum system consisting of a test particle interacting with two harmonic oscillators placed at the positions and , with and , in the two possible situations: and . At time zero, the harmonic oscillators are in their ground state and the test particle is in a superposition state of two wave packets centered in the origin with opposite mean momentum. Under suitable assumptions on the physical parameters of the model, we consider the time evolution of the wave function and we compute the probability that both oscillators are in the excited states labeled by and at time when . We prove that is negligible with respect to up to second order in time dependent perturbation theory. The system we consider is a simplified, one dimensional version of the original model of a cloud chamber introduced by Mott [“The wave mechanics of ray tracks,” Proc. R. Soc. London, Ser. A126, 79 (1929)], where the result was argued using euristic arguments in the framework of the time independent perturbation theory for the stationary Schrödinger equation. The method of the proof is entirely elementary and it is essentially based on a stationary phase argument. We also remark that all the computations refer to the Schrödinger equation for the threeparticle system, with no reference to the wave packet collapse postulate.

Dynamical decoupling schemes derived from Hamilton cycles
View Description Hide DescriptionWe address the problem of decoupling the interactions in a spin network governed by a pairinteraction Hamiltonian. Combinatorial schemes for decoupling and for manipulating the couplings of Hamiltonians have been developed, which use selective pulses. In this paper, we consider an additional requirement on these pulse sequences: as few different control operations as possible should be used. This requirement is motivated by the fact that to find an optimal implementation of each individual selective pulse will be expensive since it requires to solve a pulse shaping problem. Hence, it is desirable to use as few different selective pulses as possible. As a first result, we show that for dimensional systems, where , the ability to implement only two control operations is sufficient to turn off the time evolution. Next, we address the case of a bipartite system with local control and show that four different control operations are sufficient. Finally, turning to networks consisting of several dimensional nodes, we show that decoupling can be achieved if one is able to control a number of different control operations, which is logarithmic in the number of nodes. We give an explicit family of efficient decoupling schemes with logarithmic number of different pulses based on the classic Hamming codes. We also provide a table of the best known decoupling schemes for small networks of qubits.

Timedependent Pauli equation in the presence of the Aharonov–Bohm effect
View Description Hide DescriptionWe use the Lewis–Riesenfeld theory to determine the exact form of the wavefunctions of a twodimensional Pauli equation of a charged spin particle with timedependent mass and frequency in the presence of the Aharonov–Bohm effect and a twodimensional timedependent harmonic oscillator. We find that the irregular solution at the origin as well as the regular one contributes to the phase of the wavefunction.

Nonassociativity, supersymmetry, and hidden variables
View Description Hide DescriptionIt is shown that the supersymmetric quantum mechanics has an octonionic generalization. The generalization is based on the inclusion of quaternions into octonions. The elements from the coset octonions∕quaternions are unobservables because they cannot be considered as quantum operators as a consequence of their nonassociative properties. The idea that the octonionic generalization of the supersymmetric quantum mechanics describes an observable particle formed with unobservable “particles” is presented.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Physical subspace in a model of the quantized electromagnetic field coupled to an external field with an indefinite metric
View Description Hide DescriptionWe study a model of the quantized electromagnetic field interacting with an external static source in the Feynman (Lorentz) gauge and construct the quantized radiation field as an operatorvalued distribution acting on the Fock space with an indefinite metric. By using the Gupta subsidiary condition , one can select the physical subspace. According to the Gupta–Bleuler formalism, is a nonnegative subspace so that elements of , called physical states, can be probabilistically interpretable. Indeed, assuming that the external source is infrared regular, i.e., , we can characterize the physical subspace and show that is nonnegative. In addition, we find that the Hamiltonian of the model is reduced to the Hamiltonian of the transverse photons with the Coulomb interaction. We, however, prove that the physical subspace is trivial, i.e., , if and only if the external source is infrared singular, i.e., . We also discuss a representation different from the above representation such that the physical subspace is not trivial under the infrared singular condition.

Noncommutative fields and actions of twisted Poincaré algebra
View Description Hide DescriptionWithin the context of the twisted Poincaré algebra, there exists no noncommutative analog of the Minkowski space interpreted as the homogeneous space of the Poincaré group quotiented by the Lorentz group. The usual definition of commutative classical fields as sections of associated vector bundles on the homogeneous space does not generalize to the noncommutative setting, and the twisted Poincaré algebra does not act on noncommutative fields in a canonical way. We make a tentative proposal for the definition of noncommutative classical fields of any spin over the Moyal space, which has the desired representation theoretical properties. We also suggest a way to search for noncommutative Minkowski spaces suitable for studying noncommutative field theory with deformed Poincaré symmetries.

Eightfold way from dynamical first principles in strongly coupled lattice quantum chromodynamics
View Description Hide DescriptionWe obtain from first principles, i.e., from the quarkgluon dynamics, the Gell’MannNe’eman baryonic eightfold way energy momentum spectrum exactly in an imaginarytime functional integral formulation of strongly coupled lattice quantum chromodynamics in dimensions, with local gauge and global flavor symmetries. We take the hopping parameter κ and the pure gauge coupling β satisfying the strong coupling regime condition . The form of the 56 baryon fields emerges naturally from the dynamics and is unveiled using the hyperplane decoupling method. There is no a priori guesswork. In the associated physical quantum mechanical Hilbert space, spectral representations are derived for the twobaryon functions, which are used to rigorously detect the particles in the energymomentum spectrum. Using the symmetry, the 56 baryon states admit a decomposition into states associated with a spin octet and states associated with a spin decuplet. The states are labeled by the quantum numbers of total hypercharge , total isospin , its third component , and the value of the quadratic Casimir of ; they also carry a label of total spin and its component . The total spin operators are defined using rotations about the spatial coordinate axes and for improper zero momentum baryon states agree with the infinitesimal generators of the continuum. We show there is a partial restoration of continuous rotational symmetry which implies that all the octet (decuplet) masses are the same. For , the masses of the 56 baryon states have the form , with analytic. There is no mass splitting within the octet (decuplet). However, we find an octetdecuplet mass splitting given by . For , the nonsingular part of the masses, is analytic in κ and β and the mass splitting persists for . For spatial momentum , , the 56 baryondispersion curves have the form , where is of . For the octet, is jointly analytic in and in each for small . For each baryon, there is an antibaryon related to it by charge conjugation and with identical spectral properties. It is shown that the spectrum associated with baryons and antibaryons is the only spectrum in the subspace of with an odd number of quarks, up to near the mesonbaryon energy threshold of . A new time reflection is found which is used to define a local spin flip symmetry. The spin flip symmetry, together with the usual parity, time reversal, and spatial rotation symmetries and analytic implicit function arguments, are used to obtain these results. Our method extends to the case to uncover baryon states and also to treat mesons. Coupling our baryon results with our similar results for the eightfold mesons (of asymptotic mass ) shows that the model exhibits confinement up to near the twomeson threshold.
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 GENERAL RELATIVITY AND GRAVITATION


The Euclidean gravitational action as black hole entropy, singularities, and spacetime voids
View Description Hide DescriptionWe argue why the static spherically symmetric vacuum solutions of Einstein’s equations described by the textbook Hilbert metric is not diffeomorphic to the metric corresponding to the gravitational field of a point mass delta function source at . By choosing a judicious radial function involving the Heaviside step function, one has the correct boundary condition , while displacing the horizon from to a location arbitrarily close to as one desires, , where stringy geometry and quantum gravitational effects begin to take place. We solve the field equations due to a delta function point mass source at , and show that the Euclidean gravitational action (in units) is precisely equal to the black holeentropy (in Planck area units). This result holds in any dimensions . In the Reissner–Nordstrom (massive charged) and Kerr–Newman black hole case (massive rotating charged) we show that the Euclidean action in a bulk domain bounded by the inner and outer horizons is the same as the black holeentropy. When one smears out the pointmass and pointcharge delta function distributions by a Gaussian distribution, the areaentropy relation is modified. We postulate why these modifications should furnish the logarithmic corrections (and higher inverse powers of the area) to the entropy of these smeared black holes. To finalize, we analyze the Bars–Witten stringy black hole in dimension and its relation to the maximal acceleration principle in phase spaces and Finsler geometries.

Gravitational solitons and vacuum metrics in fivedimensional Lovelock gravity
View Description Hide DescriptionJunction conditions for vacuum solutions in fivedimensional Einstein–Gauss–Bonnet gravity are studied. We focus on those cases where two spherically symmetric regions of spacetime are joined in such a way that the induced stress tensor on the junction surface vanishes. So a spherical vacuum shell, containing no matter, arises as a boundary between two regions of the spacetime. A general analysis is given of solutions that can be constructed by this method of geometric surgery. Such solutions are a generalized kind of spherically symmetric empty space solutions, described by metric functions of the class . New global structures arise with surprising features. In particular, we show that vacuum spherically symmetric wormholes do exist in this theory. These can be regarded as gravitational solitons, which connect two asymptotically (anti)de Sitter spaces with different masses and/or different effective cosmological constants. We prove the existence of both static and dynamical solutions and discuss their (in)stability under perturbations that preserve the symmetry. This leads us to discuss a new type of instability that arises in fivedimensional Lovelock theory of gravity for certain values of the coupling of the Gauss–Bonnet term. The issues of existence and uniqueness of solutions and determinism in the dynamical evolution are also discussed.

The symmetries of fivedimensional minimal supergravity reduced to three dimensions
View Description Hide DescriptionThe 14 Killing vectors of the target space for fivedimensional minimal supergravity reduced to three dimensions are explicitly constructed in terms of the original field variables. These vectors generate the Lie algebra of . We also construct a symmetrical matrix representative of the coset as a function of the same fields.

Genericity of black hole formation in the gravitational collapse of homogeneous selfinteracting scalar fields
View Description Hide DescriptionThe gravitational collapse of a wide class of selfinteracting homogeneous scalar fields models is analyzed. The class is characterized by certain general conditions on the scalar field potential, which, in particular, include both asymptotically polynomial and exponential behaviors. Within this class, we show that the generic evolution is always divergent in a finite time, and then make use of this result to construct radiating star models of the Vaidya type. It turns out that blackholes are generically formed in such models.
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 DYNAMICAL SYSTEMS


Fractional Hamiltonian monodromy from a Gauss–Manin monodromy
View Description Hide DescriptionFractional Hamiltonian monodromy is a generalization of the notion of Hamiltonian monodromy, recently introduced by [Nekhoroshev, Sadovskií, and Zhilinskií, C. R. Acad. Sci. Paris, Ser. 1335, 985 (2002); Nekhoroshev, Sadovskií, and Zhilinskií, Ann. Henri Poincare7, 1099 (2006)] for energymomentum maps whose image has a particular type of nonisolated singularities. In this paper, we analyze the notion of fractional Hamiltonian monodromy in terms of the Gauss–Manin monodromy of a Riemann surface constructed from the energymomentum map and associated with a loop in complex space which bypasses the line of singularities. We also prove some propositions on fractional Hamiltonian monodromy for and resonant systems.

Quantuminspired maximizer
View Description Hide DescriptionThe objective of this paper is to create a new kind of dynamical systems—a quantumclassical hybrid—that would preserve superposition and entanglement of random solutions while allowing one to measure their state variables by using classical methods. Such an optimal combination of characteristics is a perfect match for quantuminspired computing. The model is represented by a modified Madelung equation in which the quantum potential is replaced by a different, specially chosen “computational” potential. As a result, the dynamics attains both quantum and classical properties. Similarities and differences of the proposed model with quantum systems are outlined. As an application, an algorithm for the global maximum of an arbitrary integrable function is proposed. The idea of the proposed algorithms is very simple: based on the quantuminspired maximizer, introduce a positive function to be maximized as the probability density to which the solution is attracted. Then, the larger value of this function will have the higher probability to appear. Special attention is paid to the simulation of integer programming, NPcomplete problems and information retrieval.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Surface charge algebra in gauge theories and thermodynamic integrability
View Description Hide DescriptionSurface charges and their algebra in interacting Lagrangiangauge field theories are constructed out of the underlying linearized theory using techniques from the variational calculus. In the case of exact solutions and symmetries, the surface charges are interpreted as a Pfaff system. Integrability is governed by Frobenius’ theorem and the charges associated with the derived symmetry algebra are shown to vanish. In the asymptotic context, we provide a generalized covariant derivation of the result that the representation of the asymptotic symmetry algebra through charges may be centrally extended. Comparison with Hamiltonian and covariant phase space methods is made. All approaches are shown to agree for exact solutions and symmetries while there are differences in the asymptotic context.

Nonideal unilateral constraints in impulsive mechanics: A geometric approach
View Description Hide DescriptionWe use the differential geometric framework of the first jet bundle of the classical spacetime bundle to study the impulsive behavior of a mechanical system with a finite number of degrees of freedom subject to nonideal unilateral constraints. We show that this framework allows deeper insights on the concepts of nonideal constitutive characterization and of coefficient of restitution of the constraints. We study the relations among Newton, Poisson, and Stronge definitions of coefficient of restitution: we reveal the inconsistency of the criticisms based on the energy balance of the impact for the Newton definition; we show the equivalence of the three definitions in the nonideal smooth case; we prove the equivalence of Newton and Poisson ones and the insufficiency of the Stronge one in the nonideal rough case. We analyze the relation between coefficient of restitution and Coulomb’s friction coefficient in the rough case. We present also several physically meaningful examples.

Gravitational and harmonic oscillator potentials on surfaces of revolution
View Description Hide DescriptionIn this paper, we consider the motion of a particle on a surface of revolution under the influence of a central force field. We prove that there are at most two analytic central potentials for which all the bounded, nonsingular orbits are closed and that there are exactly two on some surfaces with constant Gaussian curvature. The two potentials leading to closed orbits are suitable generalizations of the gravitational and harmonic oscillator potential. We also show that there could be surfaces admitting only one potential that leads to closed orbits. In this case, the potential is a generalized harmonic oscillator. In the special case of surfaces of revolution with constant Gaussian curvature, we prove a generalization of the wellknown Bertrand theorem.
