Volume 41, Issue 1, January 2000
 QUANTUM PHYSICS; PARTICLES AND FIELDS


The oneparticle energy spectrum of weakly coupled quantum rotators
View Description Hide DescriptionThe ground state of a lattice model of weakly interacting quantum rigid rotators is analyzed by the cluster expansion method applied to its Feynman–Kac representation. The Hamiltonian of the infinite crystal in the ground state is shown to have a branch of absolutely continuous spectrum separated by gaps from the rest of the spectrum, describing the oneparticle excitations.

Relativistic scattering theory for a δ sphere plus a Coulomb interaction with boundary conditions of second type
View Description Hide DescriptionWe study stationary relativistic scattering theory for a δsphere interaction formally given by the Hamiltonian with the boundary conditions of second type. First we give the mathematical definition of the model, selfadjointness of the Hamiltonian, indicial equation, stationary scattering theory and the spectral properties. Next we extend the model by adding a Coulomb potential and provide useful mathematical definitions and corresponding stationary scattering elements.

Locality in free string field theory
View Description Hide DescriptionFree string field operators are constructed for the open bosonic string in the light cone gauge in any dimension. These are naturally localized by the center of mass coordinate. Relative to this localization they are shown to have a causal commutator provided there are no tachyons. For the critical string in the result still holds if the tachyon is suppressed. We also show a causal commutator relative to the “string light cone.”

Semiclassical study of the origin of quantized Hall conductance in periodic potentials
View Description Hide DescriptionThe semiclassical study of the integer quantum Hall conductivity is investigated for electrons in a biperiodic potential The Hall conductivity is due to the tunnelling effect and we concentrate our study on potentials having three wells in a periodic cell. We show that a nonzero topological conductivity requires special conditions for the positions and shapes of the wells. The results are derived by changing the potential, using the topological nature of Chern indices. Our numerical calculations show that these semiclassical results are still valid for small value of B.

A new type of loop independence and quantum Yang–Mills theory in two dimensions
View Description Hide DescriptionThe expectation values of Wilson loop products for the pure Euclidean Yang–Mills theory on given by Ashtekar et al. in the article “ Quantum Yang–Mills Theory in Two Dimensions: A Complete Solution” [J. Math. Phys. 38, 5453 (1997)] are determined directly for all piecewise analytic loops. For that purpose we enlarge their calculations from quadratic lattices to general floating lattices introducing a new kind of loop independence and slightly modifying the regularization scheme.

Resonances in a box
View Description Hide DescriptionWe investigate a numerical method for studying resonances in quantum mechanics. We prove rigorously that this method yields accurate approximations to resonance energies and widths for shape resonances in the semiclassical limit.

Friedrichs model with virtual transitions. Exact solution and indirect spectroscopy
View Description Hide DescriptionThe Friedrichstype model of interaction between matter (multilevel system) and radiation including virtual transitions is considered. The canonical Bogolubov transformation diagonalizing the total Hamiltonian is constructed. It is pointed out that the transformation is improper when the discrete part of the spectrum of system is “dissolved” in the continuous one. The new vacuum state for the total Hamiltonian is obtained. The time evolution of the bare vacuum and the bare operators is calculated. Using the exact solution, the result of Passante, Petrosky, and Prigogine [Physica A 218, 437 (1995)] that the transition from the bare vacuum state to the true vacuum leads to the emission of real photons is confirmed. The dressing of the bare vacuum at the presence of resonances is an irreversible process. The relation of the result with the idea of “indirect spectroscopy” is discussed.

Refined algebraic quantization in the oscillator representation of SL(2, R)
View Description Hide DescriptionWe investigate refined algebraic quantization (RAQ) with group averaging in a constrained Hamiltonian system with unreduced phase space and gauge group SL(2, R). The reduced phase space M is connected and contains four mutually disconnected “regular” sectors with topology but these sectors are connected to each other through an exceptional set, where M is not a manifold and where M has nonHausdorff topology. The RAQ physical Hilbert space decomposes as where the four subspaces naturally correspond to the four regular sectors of M. The RAQ observable algebra represented on contains natural subalgebras represented on each The group averaging takes place in the oscillator representation of SL(2, R) on and ensuring convergence requires a subtle choice for the test state space: the classical analog of this choice is to excise from M the exceptional set while nevertheless retaining information about the connections between the regular sectors. A quantum theory with the Hilbert space and a finitely generated observable subalgebra of is recovered through both Ashtekar’s algebraic quantization and Isham’s group theoretic quantization.

Geometry, stochastic calculus, and quantum fields in a noncommutative space–time
View Description Hide DescriptionThe algebras of nonrelativistic and of classical mechanics are unstable algebraic structures. Their deformation towards stable structures leads, respectively, to relativity and to quantum mechanics. Likewise, the combined relativistic quantum mechanicsalgebra is also unstable. Its stabilization requires the noncommutativity of the space–time coordinates and the existence of a fundamental length constant. The new relativistic quantum mechanicsalgebra has important consequences on the geometry of space–time, on quantum stochastic calculus, and on the construction of quantum fields. Some of these effects are studied in this paper.

Green’s function for the fivedimensional SU(2) MIC–Kepler problem
View Description Hide DescriptionThe Green’s function for the fivedimensional counterpart of the MIC–Kepler problem [Kepler potential plus SU(2) Yang–Mills instanton plus Zwanzigerlike centrifugal term] is constructed on the basis of the Green’s function for the eightdimensional harmonic oscillator.

Single point interactions, quantization by parts, and boundary conditions
View Description Hide DescriptionTraditionally point interaction problems in quantum mechanics are treated in the physics literature by solving the Schrödinger equation containing an appropriate potential term. In this paper we discuss a systematic quantization scheme, known as the method of quantization by parts, which provides a general method for dealing with single point interactions in one dimension, namely by reducing the problem to boundary conditions. Our method can also be applied to superconducting systems like those concerning Josephson and the recently discovered πjunctions, and single electron circuit systems.

The averaged null energy condition for general quantum field theories in two dimensions
View Description Hide DescriptionIt is shown that in any local quantum field theory in twodimensional Minkowski space–time possessing a mass gap and an energymomentum tensor, the averaged null energy condition is fulfilled for the set of those vector states which correspond to energetically strongly damped, local excitations of the vacuum. This set of physical vector states is translation invariant and dense. The energymomentum tensor of the theory is assumed to be a Wightman field which is local relative to the observables, generates locally the translations, is divergencefree, and energetically bounded. Thus the averaged null energy condition can be deduced from completely generic, standard assumptions for general quantum field theory in twodimensional flat space–time.

Branchpoint structure and the energy level characterization of avoided crossings
View Description Hide DescriptionThe appearance of avoided crossings among energy levels as a system parameter is varied is signaled by the presence of squareroot branch points in the complex parameterplane. Even hidden crossings, which are so gradual as to be difficult to resolve experimentally, can be uncovered by the knowledge of the locations of these branch points. As shown in this paper, there are two different analytic structures that feature squareroot branch points and give rise to avoided crossings in energy. Either may be present in an actual quantummechanical problem. This poses special problems in perturbation theory since the analytic structure of the energy is not readily apparent from the perturbation series, and yet the analytic structure must be known beforehand if the perturbation series is to be summed to high accuracy. Determining which analytic structure is present from the perturbation series is illustrated here with the example of a dimensional perturbation treatment of the diamagnetic hydrogen problem. The branch point trajectories for this system in the complex plane of the perturbation parameter δ (related to the magnetic quantum number and the dimensionality) as the magnetic field strength is varied are also examined. It is shown how the trajectories of the two branchpoint pairs as the magnetic field strength varies are a natural consequence of the particular analytic structure the energy manifests in the complex δplane. There is no need to invoke any additional analytic structures as a function of the field strength parameter.

On the universality of almost every quantum logic gate
View Description Hide DescriptionLloyd [Phys. Rev. Lett. 75, 346 (1995)] showed that almost every quantum logic gate is universal in the sense that it can be used to approximate any unitary transformation. The argument relied on a more general fact whose proof was not given in detail. We give a complete proof of this more general fact.

A transformation of a Feynman–Kac formula for holomorphic families of type B
View Description Hide DescriptionA transformation formula for resolvents of families of Schrödinger operators which are assumed to be holomorphic of type B, is proved. It can be used to derive the wellknown correspondence between threedimensional Coulomb problem and fourdimensional harmonic oscillator.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Free energy of a screened ion pair
View Description Hide DescriptionWe calculate the effect of weak electrostatic screening of ions in a plasma. The original calculation by Salpeter is based on a linearization of the Poisson–Boltzmann equation for the screened electrostatic potential. This approximation is valid where the potential is small, but is formally invalid in the vicinity of the ions where the solutions for the potential and the associated charge distribution diverge. In reality, quantum exclusion must prevent the divergence of the charge density of the screening cloud. Nevertheless, in the limit in which screening is weak, Salpeter’s value for the total screening energy is essentially correct. Here we extend Salpeter’s calculation to account approximately for both quantummechanical exclusion in the vicinity of the ions, using what we call the Poisson–Boltzmann–Fermi–Dirac approximation, and the polarization of the screening cloud. By matching the solution onto an expression for the twocenter Poisson–Boltzmann charge distribution far from the ions we are able to construct a consistent solution over all space. We obtain the firstorder term in the expansion of that solution, from which we calculate the freeenergy associated with the screened ion pair.
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 DYNAMICAL SYSTEMS AND FLUID DYNAMICS


On solitontype solutions of equations associated with component systems
View Description Hide DescriptionThe algebraic geometric approach to Ncomponent systems of nonlinear integrable PDE’s is used to obtain and analyze explicit solutions of the coupled KdV and Dym equations. Detailed analysis of soliton fission, kink to antikink transitions and multipeaked soliton solutions is carried out. Transformations are used to connect these solutions to several other equations that model physical phenomena in fluid dynamics and nonlinear optics.

Contraction of superintegrable Hamiltonian systems
View Description Hide DescriptionWe investigate the contraction of a class of superintegrable Hamiltonians by implementing the contraction of the underlying Lie groups. We also discuss the behavior of the coordinate systems that separate their equations of motion, the motion constants, as well as the corresponding solutions along such a process.

Algebrogeometric solutions of dimensional coupled modified Kadomtsev–Petviashvili equations
View Description Hide DescriptionNew dimensional integrable coupled modified Kadomtsev–Petviashvili (mKP) equations are proposed with the help of known dimensional soliton equations. The dimensional coupled mKP equations are decomposed into systems of solvable ordinary differential equations. The Abel–Jacobi coordinates are introduced to straighten the flows, from which new algebrogeometric solutions of the dimensional mKP equation and algebrogeometric solutions of the dimensional coupled mKP equations are obtained in terms of the Riemann theta functions.

Algebraic decay for the solution to a class system of nonNewtonian fluid in
View Description Hide DescriptionIn this paper, we investigate the decay rates for the solution to a class system of nonNewtonian fluid in The decay rates are optimal in the sense that they coincide with the decay rates of the solution to the heat system.
