Volume 46, Issue 4, April 2005
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Inversion time of large spins
View Description Hide DescriptionIn order to find an accurate expression for the inversion probability of a large spin, an asymptotic expansion in series of Bessel functions is found, and a formula for the inversion time is obtained.

Quantum kinematics of bosonic vortex loops
View Description Hide DescriptionIn the framework of geometric quantization, filaments of vorticity in a twodimensional, ideal incompressible superfluid belong to certain coadjoint orbits of the group of areapreserving diffeomorphisms. The Poisson structure for such vortex strings is analyzed in detail. While the Lie algebra associated with areapreserving diffeomorphisms is noncanonical, we can nevertheless find canonical coordinates and their conjugate momenta that describe these systems. We then introduce a Focklike space of quantum states for the simplest case of bosonic vortex loops, with natural, nonlocal creation and annihilation operators for the quantized vortex filaments.

Bounds on integrals of the Wigner function: The hyperbolic case
View Description Hide DescriptionWigner functions play a central role in the phase space formulation of quantum mechanics. Although closely related to classical Liouville densities, Wigner functions are not positive definite and may take negative values on subregions of phase space. We investigate the accumulation of these negative values by studying bounds on the integral of an arbitrary Wigner function over noncompact subregions of the phase plane with hyperbolic boundaries. We show using symmetry techniques that this problem reduces to computing the bounds on the spectrum associated with an exactly solvable eigenvalue problem and that the bounds differ from those on classical Liouville distributions. In particular, we show that the total “quasiprobability” on such a region can be greater than 1 or less than zero.

Alternative descriptions in quaternionic quantum mechanics
View Description Hide DescriptionWe characterize the quasiantiHermitian quaternionic operators in quaternionic quantum mechanics by means of their spectra; moreover, we state a necessary and sufficient condition for a set of quasiantiHermitian quaternionic operators to be antiHermitian with respect to a uniquely defined positive scalar product in a infinite dimensional (right) quaternionic Hilbert space. According to such results we obtain two alternative descriptions of a quantum optical physical system, in the realm of quaternionic quantum mechanics, while no alternative can exist in complex quantum mechanics, and we discuss some differences between them.

Conditions for multiplicativity of maximal norms of channels for fixed integer
View Description Hide DescriptionWe introduce a condition for memoryless quantum channels which, when satisfied guarantees the multiplicativity of the maximal norm with a fixed integer. By applying the condition to qubit channels, it can be shown that it is not a necessary condition, although some known results for qubits can be recovered. When applied to the WernerHolevo channel, which is known to violate multiplicativity when is large relative to the dimension , the condition suggests that multiplicativity holds when . This conjecture is proved explicitly for . Finally, a new class of channels is considered which generalizes the depolarizing channel to maps which are combinations of the identity channel and a noisy one whose image is an arbitrary density matrix. It is shown that these channels are multiplicative for .

A rigorous proof of the Landauer–Büttiker formula
View Description Hide DescriptionRecently, Avron et al. in a series of papers shed new light on the question of quantum transport in mesoscopic samples coupled to particle reservoirs by semiinfinite leads. They rigorously treat the case, when the sample undergoes an adiabatic evolution thus generating a current through the leads, and prove the socalled BPT formula. Using a discrete model, we complement their work by giving a rigorous proof of the Landauer–Büttiker formula, which deals with the current generated by an adiabatic evolution on the leads. As is well known from physics, both of these formulas link the conductance coefficients for such systems to the matrix of the associated scattering problem. As an application, we discuss resonant transport through a quantum dot. The single charge tunneling processes are mediated by extended edge states, simultaneously localized near several leads.

Moment operators of the Cartesian margins of the phase space observables
View Description Hide DescriptionThe theory of operator integrals is used to determine the moment operators of the Cartesian margins of the phase space observables generated by the mixtures of the number states. The moments of the margin are polynomials of the position operator and those of the margin are polynomials of the momentum operator.

Initial states and decoherence of histories
View Description Hide DescriptionWe study decoherence properties of arbitrarily long histories constructed from a fixed projective partition of a finite dimensional Hilbert space. We show that decoherence of such histories for all initial states that are naturally induced by the projective partition implies decoherence for arbitrary initial states. In addition we generalize the simple necessary decoherence condition [Scherer et al., Phys. Lett. A326, 307 (2004)] for such histories to the case of arbitrary coarse graining.

Improvement of uncertainty relations for mixed states
View Description Hide DescriptionWe study a possible improvement of uncertainty relations. The Heisenberg uncertainty relation employs commutator of a pair of conjugate observables to set the limit of quantum measurement of the observables. The Schrödinger uncertainty relation improves the Heisenberg uncertainty relation by adding the correlation in terms of anticommutator. However both relations are insensitive whether the state used is pure or mixed. We improve the uncertainty relations by introducing additional terms which measure the mixtureness of the state. For the momentum and position operators as conjugate observables and for the thermal state of quantum harmonic oscillator, it turns out that the equalities in the improved uncertainty relations hold.

Construction of the dual family of Gazeau–Klauder coherent states via temporally stable nonlinear coherent states
View Description Hide DescriptionUsing the analytic representation of the socalled Gazeau–Klauder coherent states (CSs), we shall demonstrate that how a new class of generalized CSs, namely the family of dual states associated with theses states, can be constructed through viewing these states as temporally stable nonlinear CSs. Also we find that the ladder operators, as well as the displacement type operator corresponding to these two pairs of generalized CSs, may be easily obtained using our formalism, without employing the supersymmetric quantum mechanics (SUSYQM) techniques. Then, we have applied this method to some physical systems with known spectrum, such as Pöschl–Teller, infinite well, Morse potential and hydrogenlike spectrum as some quantum mechanical systems. Finally, we propose the generalized form of the Gazeau–Klauder CS and the corresponding dual family.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Estimates on Green functions of second order differential operators with singular coefficients
View Description Hide DescriptionWe investigate the Green functions of some second order differential operators on with singular coefficients depending only on one coordinate . We express the Green functions by means of the Brownian motion. Applying probabilistic methods we prove that when and (here ) lie on the singular hyperplanes, then is more regular than the Green function of operators with regular coefficients.

Mass renormalization in nonrelativistic quantum electrodynamics
View Description Hide DescriptionIn nonrelativistic quantum electrodynamics the charge of an electron equals its bare value, whereas the selfenergy and the mass must be renormalized. In our contribution we study perturbative mass renormalization, including second order in the fine structure constant , in the case of a single, spinless electron. As is well known, if denotes the bare mass and the mass computed from the theory, to order one has which suggests that for small . If correct, in order the leading term should be . To check this point we expand to order . The result is as leading term, suggesting a more complicated dependence of on .

Existence of the semilocal Chern–Simons vortices
View Description Hide DescriptionWe consider the Bogomol’nyi equations of the Abelian Chern–Simons–Higgs model with symmetry. This is a generalization of the wellknown Abelian Chern–Simons–Higgs model with symmetry. We prove existence of both topological and nontopological multivortex solutions of the system on the plane.

The topology of the electroweak interaction
View Description Hide DescriptionIn this paper we show that the Higgs boson of the (minimal) standard model has at most three gauge inequivalent ground states. One of these states is related to ordinary electromagnetism and the other two to electromagnetism within magnetically charged vacua. If space–time is assumed to be rotationally symmetric then the chargedelectroweak vacua may be identified with Dirac monopoles of magnetic charge. This offers a physical interpretation of magnetic monopoles and Dirac’s quantization condition of electric charge in terms of the electroweak interaction. Moreover, in the case of the (minimal) standard model the three possible gauge inequivalent ground states of the Higgs boson are shown to fully determine the topological structure of the gauge bundle which underlies the electroweak interaction.

Integral equations for heat kernel in compound media
View Description Hide DescriptionBy making use of the potentials of the heat conduction equation the integral equations are derived which determine the heat kernel for the Laplace operator in the case of compound media. In each of the media the parameter acquires a certain constant value. At the interface of the media the conditions are imposed which demand the continuity of the temperature and the heat flows. The integration in the equations is spread out only over the interface of the media. As a result the dimension of the initial problem is reduced by 1. The perturbation series for the integral equations derived are nothing else as the multiple scatteringexpansions for the relevant heat kernels. Thus a rigorous derivation of these expansions is given. In the one dimensional case the integral equations at hand are solved explicitly (Abel equations) and the exact expressions for the regarding heat kernels are obtained for diverse matching conditions. Derivation of the asymptotic expansion of the integrated heat kernel for a compound media is considered by making use of the perturbation series for the integral equations obtained. The method proposed is also applicable to the configurations when the same medium is divided, by a smooth compact surface, into internal and external regions, or when only the region inside (or outside) this surface is considered with appropriate boundary conditions.

Universality in a class of Qball solutions: An analytic approach
View Description Hide DescriptionThe properties of Qballs in the general case of a sixth order potential have been studied using analytic methods. In particular, for a given potential, the initial field value that leads to the soliton solution has been derived and the corresponding energy and charge have been explicitly evaluated. The proposed scheme is found to work reasonably well for all allowed values of the model parameters.

Relation between generalized Bogoliubov and Bogoliubov–de Gennes approaches including Nambu–Goldstone mode
View Description Hide DescriptionThe two approaches of consistent quantum field theory for systems of the trapped Bose–Einstein condensates are known, one is the Bogoliubov–de Gennes approach and the other is the generalized Bogoliubov approach. In this paper, we investigate the relation between the two approaches and show that they are formally equivalent to each other. To do this one must carefully treat the Nambu–Goldstone mode which plays a crucial role in the condensation. It is emphasized that the choice of vacuum is physically relevant.

On extensions of superconformal algebras
View Description Hide DescriptionStarting from vector fields that preserve a differential form on a Riemann sphere with Grassmann variables, one can construct a superconformal algebra by considering central extensions of the algebra of vector fields. In this paper, the case is analyzed closely, where the presence of weight zero operators in the field theory forces the introduction of noncentral extensions. How this modifies the existing field theory, representation theory, and Gelfand–Fuchs constructions is discussed. It is also discussed how graded Riemann sphere geometry can be used to give a geometrical description of the central charge in the theory.

Functional determinants for the Dirac equation with mixed pseudodifferential boundary conditions over finite cylinders
View Description Hide DescriptionIn this note, we explicitly compute the functional determinant of a Dirac Laplacian with nonlocal pseudodifferentialboundary conditions over a finite cylinder in terms of the function of the Dirac operator on the cross section and the pseudodifferential operators defining the boundary conditions. In particular, this result reduces to our previous formula [J. Phys. AJPHAC537, 7381 (2004)] for the special case of generalized Atiyah–Patodi–Singer conditions. To prove our main result, we use the gluing and comparison formulas established by the present authors in Refs 14 and 15.

Nontopological bare solutions in the relativistic selfdual Maxwell–Chern–Simons–Higgs model
View Description Hide DescriptionIn this paper we prove the existence of the radially symmetric nontopological bare solutions in the relativistic selfdual Maxwell–Chern–Simons–Higgsmodel. We also verify the Chern–Simons limit for those solutions.
