Volume 37, Issue 9, September 1996
Index of content:

Canonical commutation relations, the Weierstrass Zeta function, and infinite dimensional Hilbert space representations of the quantum group U _{ q }(sl_{2})
View Description Hide DescriptionA two‐dimensional quantum system of a charged particle interacting with a vector potential determined by the Weierstrass Zeta function is considered. The position and the physical momentum operators give a representation of the canonical commutation relations with two degrees of freedom. If the charge of the particle is not an integer (the case corresponding to the Aharonov–Bohm effect), then the representation is inequivalent to the Schrödinger representation. It is shown that the inequivalent representation induces infinite‐dimensional Hilbert space representations of the quantum groupU _{ q }(sl). Some properties of these representations of U _{ q }(sl) are investigated.

The Heisenberg dynamics of spin systems: A quasi*‐algebras approach
View Description Hide DescriptionThe problem of the existence of the thermodynamical limit of the algebraic dynamics for a class of spin systems is considered in the framework of a generalized algebraic approach in terms of a special class of quasi*‐algebras, called CQ*‐algebras. Physical applications to (almost) mean‐field models and to bubble models are discussed.

Gamow states as continuous linear functionals over analytical test functions
View Description Hide DescriptionThe space of analytical test functions ξ, rapidly decreasing on the real axis (i.e., Schwartz test functions of the type S on the real axis), is used to construct the rigged Hilbert space (RHS) (ξ,H,ξ′). Gamow states (GS) can be defined in RHS starting from Dirac’s formula. It is shown that the expectation value of a self‐adjoint operator acting on a GS is real. We have computed exactly the probability of finding a system in a GS and found that it is finite. The validity of recently proposed approximations to calculate the expectation value of self‐adjoint operators in a GS is discussed.

Solvable quantum version of an integrable Hamiltonian system
View Description Hide DescriptionSolvable quantum versions of the classical dynamical system characterized by the Hamiltonian H=∑^{ n } _{ j,k=1} p _{ jp } _{ k }[λ+μ cos(q _{ j }−q _{ k }) ] are presented. The eigenvalues of the quantum Hamiltonians are exhibited, as well as the corresponding eigenfunctions.

Coherent state path‐integral representation of supersymmetric lattice models
View Description Hide DescriptionA kind of high‐temperature superconductivity related lattice model is investigated within the framework of supergroup coherent state path‐integral representation. Symmetry properties are analyzed and the Hamiltonians are written in the symmetric form explicitly in terms of generators of the supergroup U(N/M). By a standard approach, general supergroup coherent states are constructed. Holstein–Primakoff realizations of the supergroup U(N/M) on the coset space U(N/M)/[U(1)⊗U(N−1/M)] are obtained. Vacuum persistence amplitudes are expressed in terms of parameters on the coset space U(2/M)/[U(1)⊗U(1/M)]. Symmetry‐breaking terms in the Hamiltonian are taken into account separately. The Lagrangians of these models are quadratic in Grassmann variables. Thus fermionic fields can be integrated out. The nonlinear σ model is arrived at as effective continuum field theory describing the low‐energy excitations of the supersymmetric lattice models.

Two‐body wave equations in curved space–time
View Description Hide DescriptionTwo scalar particles undergoing mutual interaction are considered in a prescribed curved space–time. Both masses are finite (recoil is not neglected). For a description of this system we propose a pair of coupled Klein–Gordon equations; they involve two‐body‐sector extensions of the Laplace–Beltrami operator, plus a term that takes mutual interaction into account. Besides the problem of compatibility, we discuss several requirements that seem more or less necessary from a physical point of view. Particular attention is devoted to the preservation of space–time symmetries (isometric invariance). Composition of curvature with mutual interaction is a nonlinear problem, but it can be explicitly solved in a toy model of static orthogonal space–time. Moreover, we check that isometric invariance and other physical requirements are satisfied in this example.

Squeezing Bogoliubov transformations on the infinite mode CCR‐algebra
View Description Hide DescriptionA detailed analysis of and a general decomposition theorem for in general unbounded symplectic transformations on an arbitrary complex pre‐Hilbert space (one–boson test function space) are given. The structure of strongly continuous symplectic groups on such spaces is determined. The connection between quadratic Hamiltonians, Bogoliubov transformations, and symplectic transformations is discussed in the Fock representation, and their relevance for squeezing operations in quantum optics is pointed out. The results for this rather general class of transformations are proved in a self‐contained fashion.

Path integral solution of the Schrödinger equation in curvilinear coordinates: A straightforward procedure
View Description Hide DescriptionA new axiomatic formulation of path integrals is used to construct a path integral solution of the Schrödinger equation in curvilinear coordinates. An important feature of the formalism is that a coordinate transformation in the variables of the wavefunction does not imply a change of variable of integration in the path integral. Consequently, a transformation from Euclidean to curvilinear coordinates is simple to handle; there is no need to introduce ‘‘quantum corrections’’ into the action functional. Furthermore, the paths are differentiable: hence, issues related to stochastic paths do not arise. The procedure for constructing the path integral solution of the Schrödinger equation is straightforward. The case of the Schrödinger equation in spherical coordinates for a free particle is presented in detail.

A symmetry of massless fields
View Description Hide DescriptionIt is proved that there exists an additional intrinsic symmetry in the left‐handed and right‐handed fermions (and other fields). The corresponding group of transformations is induced by the Poincaré translations in the space–time manifold. This symmetry predicts an additional intrinsic energy‐momentum for fermions. Considering this symmetry as local leads to introduction of a gauge field and of a nonintegrable phase angle, the corresponding Berry‐type phase depends on the topology of the Riemannian space–time manifold as determined by the vierbein. This additional symmetry provides us with the possibility of considering the fermions as gauge fields on the nonvector bundle.

Quantum Boltzmann equation for photons
View Description Hide DescriptionThe quantum Boltzmann equation is derived for photons. The form of the scattering and absorption terms for the case of photondiffusion is discussed in detail. We show how the structure factor of the scattering centers enters into the scattering rate of the photons.

On the vacuum stability in the Efimov–Fradkin model at finite temperature
View Description Hide DescriptionThe behavior of the nontruncated and truncated Efimov–Fradkin models (L_{int}=−∑^{ N } _{ n=3}λ_{ n }φ^{ n }) at finite temperature in a generic D‐dimensional flat space–time was investigated. The thermal contribution to the renormalized mass and coupling constants are obtained in the one‐loop approximation by the use of a mix between dimensional and the Epstein zeta function analytic regularization and a modified minimal subtraction procedure. We proved that for D _{ c }(N−1)≤D there is not a temperature for which at least one of the renormalized coupling constants becomes zero, where D _{ c }(N−1) is the critical spacetime dimension for the renormalized coupling constant λ_{ N−1}. For D _{ c }(N)≤D<D _{ c }(N−1) only the renormalized coupling constant λ_{ N−1} becomes zero at some temperature β^{−1} _{ N−1}. For D<D _{ c }(N) the renormalized coupling constants λ_{ N−1}(β) and λ_{ N }(β) become zero at temperatures β^{−1} _{ N−1} and β^{−1} _{ N }, respectively. In the latter situation, for temperatures β^{−1} _{ N−1}<β^{−1}<β^{−1} _{ N } the effective potential has a global minimum. For temperatures above β^{−1} _{ N }, the system can develop a first order phase transition, where the origin corresponds to a metastable vacuum. In the nontruncated model, corresponding to a nonpolynomial Lagrange density, for D≥2 all the coupling constants remain positive for any temperature.

The Lax pair by dimensional reduction of Chern–Simons gauge theory
View Description Hide DescriptionWe show that the Nonlinear Schrödinger Equation and the related Lax pair in 1+1 dimensions can be derived from 2+1‐dimensional Chern–Simons Topological Gauge Theory. The spectral parameter, a main object for the Loop algebrastructure and the Inverse Spectral Transform, has to appear as a homogeneous part (condensate) of the statistical gauge field, connected with the compactified extra space coordinate. In terms of solitons, a natural interpretation for the one‐dimensional analog of Chern–Simons Gauss law is given.

Anomalous thresholds and edge singularities in electrical impedance tomography
View Description Hide DescriptionStudies of models of current flow behaviour in electrical impedance tomography(EIT) have shown that the current density distribution varies extremely rapidly near the edge of the electrodes used in the technique. This behaviour imposes severe restrictions on the numerical techniques used in image reconstruction algorithms. In this paper we have considered a simple two dimensional case and we have shown how the theory of end point/pinch singularities which was developed for studying the anomalous thresholds encountered in elementary particle physics can be used to give a complete description of the analytic structure of the current density near to the edge of the electrodes. As a byproduct of this study it was possible to give a complete description of the Riemann sheet manifold of the eigenfunctions of the logarithmic kernel. These methods can be readily extended to other weakly singular kernels.

Localized solutions of the Dirac–Maxwell equations
View Description Hide DescriptionThe full classical Dirac–Maxwell equations are considered in a somewhat novel form and under various simplifying assumptions. A reduction of the equations is performed in the case when the Dirac field is static. A further reduction of the equations is made under the assumption of spherical symmetry. These static spherically symmetric equations are examined in some detail and a numerical solution presented. Some surprising results emerge from this investigation: (i) Spherical symmetry necessitates the existence of a magnetic monopole. (ii) There exists a uniquely defined solution, determined only by the demand that the solution be analytic at infinity. (iii) The equations describe highly compact objects with an inner onion like shell structure.

A manifestly reciprocal theory of scattering in the presence of elastic media
View Description Hide DescriptionThe role of elastic waves in the scattering problem is examined in the context of modern field theory. This effort builds upon a previously published, and since successfully applied formalism for solving the acoustic and electromagneticscattering problems. It specifically addresses the scattering of acoustic waves from a fluid‐solid interface, as well as the scattering of elastodynamic waves from surfaces satisfying the zero‐displacement, stress‐free, and solid–solid boundary conditions. Expressions for the change in the scattering amplitude due to a perturbation in the scattering surface are derived directly from the requirement of time reversal symmetry (also known as reciprocity). These results constitute formal statements of the composite (or two‐scale) model. In a typical application, the perturbation usually corresponds to Bragg scattering and is treated statistically, while the reference surface provides tilt, shadowing, and multiple scattering, and is usually treated deterministically. Used in this way, the new formalism effectively allows existing numerical and operator expansion methods to be used to calculate the scattering from rougher and/or higher dimensional surfaces than would otherwise be possible. An alternate application of the formalism is illustrated using the fluid‐solid boundary as an example. A new manifestly reciprocal expression for the scattering amplitude is presented, as are the small slope and ‘‘local’’ two‐scale approximations for this problem. (By local, it is meant that only local phenomena such as the tilt of the reference surface are automatically included. However, since the result is manifestly reciprocal, it is fairly straightforward to incorporate a non‐local effect such as shadowing.) During the course of the discussion, the classical scattering problem is reexamined from an entirely new perspective.

Invariant of dynamical systems: A generalized entropy
View Description Hide DescriptionIn this work the concept of entropy of a dynamical system, as given by Kolmogorov, is generalized in the sense of Tsallis. It is shown that this entropy is an isomorphism invariant, being complete for Bernoulli schemes.

R‐matrix theory, formal Casimirs and the periodic Toda lattice
View Description Hide DescriptionThe nonunitary r‐matrix theory and the associated bi‐ and triHamiltonian schemes are considered. The language of Poisson pencils and of their formal Casimirs is applied in this framework to characterize the biHamiltonian chains of integrals of motion, pointing out the role of the Schur polynomials in these constructions. This formalism is subsequently applied to the periodic Toda lattice. Some different algebraic settings and Lax formulations proposed in the literature for this system are analyzed in detail, and their full equivalence is exploited. In particular, the equivalence between the loop algebra approach and the method of differential‐difference operators is illustrated; moreover, two alternative Lax formulations are considered, and appropriate reduction algorithms are found in both cases, allowing us to derive the multiHamiltonian formalism from r‐matrix theory. The systems of integrals for the periodic Toda lattice known after Flaschka and Hénon, and their functional relations, are recovered through systematic application of the previously outlined schemes.

Symplectic completion of symplectic jets
View Description Hide DescriptionIn this paper, we outline a method for symplectic integration of three degree‐of‐freedom Hamiltonian systems. We start by representing the Hamiltonian system as a symplectic map. This map (in general) has an infinite Taylor series. In practice, we can compute only a finite number of terms in this series. This gives rise to a truncated map approximation of the original map. This truncated map is however not symplectic and can lead to wrong stability results when iterated. In this paper, following a generalization of the approach pioneered by Irwin (SSC Report No. 228, 1989), we factorize the map as a product of special maps called ‘‘jolt maps’’ in such a manner that symplecticity is maintained.

Solution of Nester’s gauge conditions
View Description Hide DescriptionWe give a rigorous proof of the existence of solutions to the nonlinear gauge conditions on orthonormal frames introduced by Nester to prove the positive energy theorem in general relativity. The proof holds in all dimensions n≥2. If the second de Rham cohomology group vanishes one also proves uniqueness.

The Einstein action for algebras of matrix valued functions—Toy models
View Description Hide DescriptionTwo toy models are considered within the framework of noncommutative differential geometry. In the first one, the Einstein action of the Levi–Civita connection is computed for the algebra of matrix valued functions on a torus. It is shown that, assuming some constraints on the metric, this action splits into a classical‐like, a quantum‐like and a mixed term. In the second model, an analogue of the Palatini method of variation is applied to obtain critical points of the Einstein action functional for M _{4}(R). It is pointed out that a solution to the Palatini variational problem is not necessarily a Levi–Civita connection. In this model, no additional assumptions regarding metrics are made.