Volume 40, Issue 10, October 1999
Index of content:
 QUANTUM PHYSICS; PARTICLES AND FIELDS


Extended gauge theories starting from the matter field Lagrangian
View Description Hide DescriptionWe obtain an extended gauge theory by imposing that the matter field Lagrangian is invariant under a local gauge transformation that also contains a vector parameter besides the usual scalar one.

Largeorder perturbation theory for a nonHermitian PTsymmetric Hamiltonian
View Description Hide DescriptionA precise calculation of the groundstate energy of the complex PTsymmetric Hamiltonian is performed using highorder Rayleigh–Schrödinger perturbation theory. The energy spectrum of this Hamiltonian has recently been shown to be real using numerical methods. Here we present convincing numerical evidence that the Rayleigh–Schrödinger perturbation series is Borel summable, and show that Padé summation provides excellent agreement with the real energy spectrum. Padé analysis provides strong numerical evidence that the oncesubtracted groundstate energy considered as a function of is a Stieltjes function. The analyticity properties of this Stieltjes function lead to a dispersion relation that can be used to compute the imaginary part of the energy for the related real but unstable Hamiltonian

A conserved current for the field perturbations in the Einstein–Yang–Millsdilaton–axion theory
View Description Hide DescriptionUsing the selfadjoint character of the operators governing the field perturbations in the Einstein–Yang–Millsdilaton–axion theory, we demonstrate that a covariantly (both gauge covariant and space–time covariant) conserved current associated with the coupled field perturbations arises, by contrast to other approaches, in a natural way. Our results cover, in particular, the bosonic lowenergy limit of the string theory. We discuss how the present results can serve as a starting point for future investigations.

On the number of particles that a curved quantum waveguide can bind
View Description Hide DescriptionWe discuss the discrete spectrum of particles in a curved planar waveguide. If they are neutral fermions, the maximum number of particles that the waveguide can bind is given by a oneparticle Birman–Schwinger bound in combination with the Pauli principle. On the other hand, if they are charged, e.g., electrons in a bent quantum wire, the Coulomb repulsion plays a crucial role. We prove a sufficient condition under which the discrete spectrum of such a system is empty.

A general BRST approach to string theories with zeta function regularizations
View Description Hide DescriptionWe propose a new general BRST approach to string and stringlike theories that have a wider range of applicability than, e.g., the conventional conformal field theory method. The method involves a simple general regularization of all basic commutators, which makes all divergent sums expressible in terms of zeta functions from which finite values then may be extracted in a rigorous manner. The method is particularly useful in order to investigate possible state space representations to a given model. The method is applied to three string models: The ordinary bosonic string, the tensionless string, and the conformal tensionless string. We also investigate different state spaces for these models. The tensionless string models are treated in detail. Although we mostly rederive known results, they appear in a new fashion that deepens our understanding of these models. Furthermore, we believe that our treatment is more rigorous than most of the previous ones. In the case of the conformal tensionless string we find a new solution for

Time dependence of operators in anharmonic quantum oscillators: Explicit perturbative analysis
View Description Hide DescriptionAn explicit, orderbyorder perturbative solution, valid over extended time scales, for the time dependence of operators of anharmonic oscillators, is developed within the framework of the method of normal forms. The freedom of choice of the zerothorder term and, concurrently in the higherorder corrections, is exploited to develop a minimal normal form (MNF). The expansion for the eigenvalues of the perturbed Hamiltonian in a standard form is independent of the choice. However, the simple form obtained for the time dependence of the perturbative solution is more suitable than any other choice for application to highlying excited states, as it offers a renormalized form for the propagator.

Nonassociative structure of quantum mechanics in curved space–time
View Description Hide Descriptionde Sitter space gives a local, osculating (and therefore prototypical) representation of a curved space–time. The de Sitter geodesics allow a description as uniform straightline motion with respect to a family of preferred frames that are inertial frames interconnected by a projective (fractionallinear) transformation. Use of these frames overcomes the ambiguities of general covariance in stating commutation rules and Hamiltonian structure. But the Hamiltonian for a particle running on a geodesic then necessitates nonassociative elements. The latter are worked out as elements of a Cayley–Dickson algebra of 16 dimensions (doubled octonions, or decahexions), wherein a standard Hilbert space type of format for quantum mechanics is ruled out. A formal Schrödinger wave equation is devised, and exhibits antilinear as well as linear terms in the decahexion components of the wave function. All the same, the Heisenberg laws of motion for a dynamical variable are neatly and unambiguously formulated, giving a full account of the quantum time evolution of the dynamical variable (although the Heisenberg program of diagonalizing the Hamiltonian cannot be executed).

Absence of confinement in the absence of vortices
View Description Hide DescriptionWe consider the Wilson loop expectation in lattice gauge theory in the presence of constraints. The constraints eliminate gauge field configurations, which, in physical terms, allow the presence of thick center vortices linking with the loop. We prove that, for dimension the soconstrained Wilson loop follows perimeter law, i.e., nonconfining behavior, at weak coupling (low temperature).

Covariant phase observables in quantum mechanics
View Description Hide DescriptionIn this paper we characterize all the phase shift covariant normalized positive operator measures, i.e., phase observables, and we investigate some of their examples. We also characterize those phase observables which arise from the phase space observables as their polar coordinate angle margins.

The semispin groups in string theory
View Description Hide DescriptionIn string theory, an important role is played by certain Lie groups which are locally isomorphic to It has long been known that these groups are actually isomorphic not to but rather to the groups for which the halfspin representations are faithful, which we propose to call (They are known in the physics literature by the ambiguous name of “”) Recent work on string duality has shown that the distinction between and can have a definite physical significance. This work is a survey of the relevant properties of and its subgroups.

Quantum superstring field theory in the gauge and the physical scattering amplitudes
View Description Hide DescriptionWe propose the (BRSTinvariant) quantum open superstring field theory in the “gauge,” based on Neveu–Schwarz (NS) strings in 1 picture and Ramond (R) strings in picture. We give the propagators of these open NS and R superstrings. In order to obtain the BRSTinvariant interaction terms among these superstrings, we modify the interaction terms among three superstrings (i.e., among NS–NS–NS and R–R–NS) by subtracting the infinite number of counter terms, each of which involves interaction terms among “more than four superstrings.” The modified action can be obtained successively, so that resulting amplitudes in gloops should become BRST invariant. Thus obtained amplitudes are referred to as the “amputated scatts,” with the help of which the physicalscattering amplitudes can be expressed. These physical scattering amplitudes among bosonic ( fermionic) particles are calculated by using the analytic inlint gluing operator (which has already been proposed and used in the quantum bosonic string field theory “in the gauge”).

Asymptotics of Clebsch–Gordan coefficients
View Description Hide DescriptionAsymptotic expressions for Clebsch–Gordan coefficients are derived from an exact integral representation. Both the classically allowed and forbidden regions are analyzed. Higherorder approximations are calculated. These give, for example, six digit accuracy when the quantum numbers are in the hundreds.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Exact solution of the Herrera equation of motion in classical electrodynamics
View Description Hide DescriptionThe Landau–Lifshitz equation is derived as a firstorder iteration of the Lorentz–Dirac equation for the charged particle. In those cases with null electromagnetic field’s gradient, the LandauLifshitz gives the so named Herrera equation. A general method for the solution of the latter equation is presented and applied to the motion of a particle in a uniform electromagnetic field.

Diffusive energy scattering from weakly random surfaces
View Description Hide DescriptionWe derive transport theoretic boundary conditions for acoustic wave reflection at a weakly rough boundary in an inhomogeneous half space. We use the Wigner distribution to go from waves to energytransport in the high frequency limit. We generalize known results on the reflection of acoustic plane waves in a homogeneous medium. We analyze higher order corrections, which include a enhanced backscatteringeffect in the back direction.

Transport equations for a general class of evolution equations with random perturbations
View Description Hide DescriptionWe derive transport equations from a general class of equations of form where and are pseudodifferential operators (Weyl operator) with symbols and where being polynomial in k and smooth in is a mean zero random function and is stationary in space variable. We also consider system of equations in the above form. Such equations cover many of the equations that arise in wave propagations, such as those considered in a paper by Ryzhik, Papanicolaou, and Keller [Wave Motion 24, 327–370 (1996)]. Our results generalize those by Ryzhik, Papanicolau, and Keller.

Time harmonic electromagnetic scattering from a bounded obstacle: An existence theorem and a computational method
View Description Hide DescriptionLet be a bounded simply connected obstacle with boundary ∂Ω locally Lipschitz, we consider the scattering of a time harmonic electromagnetic wave that hits Ω when ∂Ω is assumed to be perfectly conducting. The scattered electromagnetic field is the solution of an exterior boundary value problem for the vector Helmholtz equation. Under suitable hypotheses we prove the existence and uniqueness of the solution of this boundary value problem and we give a new numerical method to compute this solution. The numerical method proposed is based on a perturbative series and is highly parallelizable. Some numerical results obtained with the numerical method proposed on test problems are presented and discussed from the numerical and the physical point of view.

On the electromagnetic scattering problem for an infinite dielectric cylinder of an arbitrary cross section located in the wedge
View Description Hide DescriptionDiffraction of timeharmonic Epolarized electromagnetic waves by an infinite dielectric cylinder of arbitrary cross section located inside a wedge parallel to its axis is considered. By the methods of potential theory, the transmission problem is reduced to a system of two onedimensional Fredholm integral equations with the kernels having logarithmic singularities; integration is performed over the boundary of the cylinder cross section. Existence and uniqueness of solutions are proved both for the system of integral equations and the transmission problem. The kernels of integral equations are represented as rapidly convergent series.

Jacobi’s principle for magnetic interactions
View Description Hide DescriptionIt is shown that the trajectories of a charged particle in a static magnetic field and a velocityindependent potential in a threedimensional space are (the projection of) the geodesics of a suitably defined metric in a fourdimensional space. It is shown that each oneparameter group of isometries of the original configuration space that leaves the magnetic field invariant gives rise to a oneparameter group of isometries of the metric defined on the fourdimensional space and, hence, to a constant of the motion. It is also shown, similarly, that the Schrödinger equation for a charged particle in a static magnetic field is equivalent to the Schrödinger equation for a free particle in the fourdimensional space mentioned above.

Uniqueness theorems for classical fourvector fields in Euclidean and Minkowski spaces
View Description Hide DescriptionEuclidean and Minkowski fourspace uniqueness theorems are derived which yield a new perspective of classical fourvector fields. The Euclidean fourspace uniqueness theorem is based on a Euclidean fourvector identity which is analogous to an identity used in Helmholtz’s theorem on the uniqueness of threevector fields. A Minkowski space identity and uniqueness theorem can be formulated from first principles and the space components of this identity turn out to reduce to the threevector Helmholtz’s identity in a static Newtonian limit. A further result is a uniqueness theorem for scalar fields based on an identity which is proved to be a static Newtonian limit of the zeroth or scalar component of the Minkowski space extension of the Helmholtz identity. Last, the threevector Helmholtz identity and uniqueness theorem and their fourspace extensions to Minkowski space are generalized to mass damped fields.

 STATISTICAL PHYSICS AND STOCHASTIC PROCESSES


Grassmann algebra and fermions at finite temperature
View Description Hide DescriptionFor any ddimensional selfinteracting fermionic model, all coefficients in the hightemperature expansion of its grand canonical partition function can be put in terms of multivariable Grassmann integrals. A new approach to calculate such coefficients, based on direct exploitation of the Grassmannian nature of fermionic operators, is presented. We apply the method to the soluble Hatsugai–Kohmoto model, reobtaining wellknown results.
