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Volume 37, Issue 8, August 1996

Surface‐embeddability approach to the dynamics of the inhomogeneous Heisenberg spin chain
View Description Hide DescriptionThe surface‐embeddability approach of Lund and Regge is applied to the classical, inhomogeneous Heisenberg spin chain to study the class of inhomogeneity functions f for which the spin evolution equation and its gauge‐equivalent generalized nonlinear Schrödinger equation (GNLSE) are exactly solvable. Writing the spin vector S(x,t) as ∂_{ x } r and identifying r(x,t) with a position vector generating a surface, we show that the kinematic equation satisfied by r implies certain constraints on the admissible geometries of this surface. These constraints, together with the Gauss–Mainardi–Codazzi equations, enable us to express the coefficient of the second fundamental form as well as f in terms of the metric coefficients G and its derivatives, for arbitrary time‐independent G. Explicit solutions for the GNLSE can also be found in terms of the same quantities. Of the admissible surfaces generated by r, a special class that emerges naturally is that of surfaces of revolution: Explicit solutions for r and S are found and discussed for this class of surfaces.

The Casimir energy of the twisted string loop: Uniform and two segment loops
View Description Hide DescriptionWe calculate the Casimir energy of a two segment loop of string with one normal boundary point and one twisted boundary point. The energy is renormalized relative to the twisted uniform loop. The use of the twisted loop in simplifying untwisted loop calculations is discussed.

The q‐Coulomb problem in configuration space
View Description Hide DescriptionWe formulate the q‐Coulomb problem in configuration space with the aid of ladder operators for the radial wave function. The highest angular momentum state corresponding to the principal quantum number n is found to be the monomial r ^{ n−1} multiplied by a q‐exponential. The states of lower angular momentum are q‐associated Laguerre polynomials multiplied by the same q‐exponential. The state functions all lie in the complex plane and may be interpreted in the standard way. The energy levels are again given by a Balmer formula with n replaced by the basic n.

Cabled Wilson loops in BF theories
View Description Hide DescriptionA generatingfunction for cabled Wilson loops in three‐dimensional BF theories is defined, and a careful study of its behavior for vanishing cosmological constant is performed. This allows an exhaustive description of the unframed knot invariants coming from the pure BF theory based on SU(2), and, in particular, it proves a conjecture relating them to the Alexander–Conway polynomial.

The KZ equation and the quantum‐group difference equation in quantum self‐dual Yang–Mills theory
View Description Hide DescriptionFrom the time‐independent current T̃(ȳ,k̄) in the quantum self‐dual Yang–Mills (SDYM) theory, we construct new group‐valued quantum fields Ũ(ȳ,k̄) and Ũ^{−1}(ȳ,k̄) which satisfy a set of exchange algebras such that fields of T̃(ȳ,k̄)∼Ũ(ȳ,k̄)∂ȳŨ^{−1}(ȳ,k̄) satisfy the original time‐independent current algebra. For the correlation functions of the products of the Ũ(ȳ,k̄) and Ũ^{−1}(ȳ,k̄) fields defined in the invariant state constructed through the current T̃(ȳ,k̄) we can derive the Knizhnik–Zamolodchikov (KZ) equations with an additional spatial dependence on k̄. From the Ũ(ȳ,k̄) and Ũ^{−1}(ȳ,k̄) fields we construct the quantum‐group generators, local, global and semi‐local, and their algebraic relations. For the correlation functions of the products of the Ũ and Ũ^{−1} fields defined in the invariant state constructed through the semi‐local quantum‐group generators we obtain the quantum‐group difference equations. We give the explicit solution to the two point function.

Batalin–Vilkovisky formalism and integration theory on manifolds
View Description Hide DescriptionThe correspondence between the BV formalism and integration theory on supermanifolds is established. An explicit formula for the density on a Lagrangiansurface in a superspace provided with an odd symplectic structure and a volume form is proposed.

BRST quantization of gauge theory in noncommutative geometry: Matrix derivative approach
View Description Hide DescriptionThe Becchi–Rouet–Stora–Tyutin (BRST) quantization of a gauge theory in noncommutative geometry is carried out in the ‘‘matrix derivative’’ approach. BRST/anti‐BRST transformation rules are obtained by applying the horizontality condition, in the superconnection formalism. A BRST/anti‐BRST invariant quantum action is then constructed, using an adaptation of the method devised by Baulieu and Thierry‐Mieg for the Yang–Mills case. The resulting quantum action turns out to be the same as that of a gauge theory in the ’t Hooft gauge with spontaneously broken symmetry. Our result shows that only the even part of the supergroup acts as a gauge symmetry, while the odd part effectively provides a global symmetry. We treat the general formalism first, then work out the SU(2/1) and SU(2/2) cases explicitly.

A natural extension of coherent state path integrals
View Description Hide DescriptionWe develop the formulation of the path integrals via the coherent states based on general starting vectors. In this paper we treat the case of the ordinary canonical (Heisenberg‐Weyl) coherent state. We find that an additional term appears in the action in the path integral expression. Geometric phases associated with the path integrals, including the possibilities of the experimental detection, are also discussed.

Liouville vortex and φ^{4} kink solutions of the Seiberg–Witten equations
View Description Hide DescriptionThe Seiberg–Witten equations, when dimensionally reduced to R ^{2}, naturally yield the Liouville equation, whose solutions are parametrized by an arbitrary analytic function g(z). The magnetic flux Φ is the integral of a singular Kaehler form involving g(z); for an appropriate choice of g(z), N coaxial or separated vortex configurations with Φ=2πN/e are obtained when the integral is regularized. The regularized connection in the R ^{1} case coincides with the kink solution of φ^{4} theory.

Transformation bracket for 2D harmonic oscillator functions and its application to few‐electron quantum dots
View Description Hide DescriptionTransformation bracket relating 2D harmonic oscillator product states with different sets of Jacobian coordinates is derived for systems composed of an arbitrary number of particles with arbitrary masses. The numerical diagonalization of the Hamiltonian of a three‐electron quantum dot is given as an example to illustrate its applications.

Multiple condensate solutions for the Chern–Simons–Higgs theory
View Description Hide DescriptionWe study the existence of condensate solutions for the Chern‐Simons‐Higgs model with the choice of a potential field where both the symmetric and asymmetric vacua occur as ground states [see Hong, Kim, and Pac, Phys. Rev. Lett. 64, 2230 (1990) and Jackiw and Weinberg, ibid. 64, 2234 (1990)]. We show that if the Chern‐Simons coupling parameter k is above a critical value, no such solutions can exist, while for k≳0 below this critical value there exist at least two condensate solutions carrying the same quantized energy, as well as electric and magnetic charge. This multiplicity result accounts for the two vacua states present in the model. In fact, as k→0^{+} it is shown that the two solutions found ‘‘bifurcate’’ from the asymmetric and symmetric vacuum states respectively.

Deriving the standard model from the simplest two‐point K cycle
View Description Hide DescriptionBasing on a differential algebra over the simplest two‐point K cycle and graded Lie algebras of homomorphisms of finite projective modules, we derive the classical action of the standard model. This construction uses both the general framework of noncommutative geometry developed by Connes and ideas of the Mainz–Marseille approach to model building. We get a prediction of the Weinberg angle and constraints between the fermion masses and the masses of the W and Higgs bosons on tree level, which differ from the relations obtained by Kastler and Schücker for the quaternionic K cycle.

Discrete Weyl–Heisenberg transforms
View Description Hide DescriptionThe concepts of von Neumann lattices and tight frames are used for defining discrete quantum mechanical transforms in the phase plane. These transforms are obtained by finite shifts la and mb in the coordinate x and momentum p, respectively, of the Weyl–Heisenberg group, and they are called the discrete Weyl–Heisenberg transforms ψ(la,mb). Here ab=h/N with h the Planck constant,l and m integers, and N a positive integer. A construction is carried out of ψ(la,mb) for a general Weyl–Heisenberg set by using the kq‐representation, in which a useful formula is established for the frame operator. The construction is illustrated on an example of the ground state of a harmonic oscillator. It is shown that any physical quantity can be described by the discrete Weyl–Heisenberg transform. Connections are established between ψ(la,mb), the Bargmann representation, and the Husimi distribution function.

On the Green‐functions technique and phase velocity approximation of axially symmetric fields in stratified media
View Description Hide DescriptionIn this paper a numerical method is developed to solve a one‐dimensional inverse scattering problem associated with a pair of two coupled partial differential equations. These equations are for the Green functions that can be used for the optimization of electromagnetic energy within selected regions of a medium in which the propagation is taking place. The method is based upon constructing the Nth degree interpolationpolynomial to approximate the Green functions and the phase velocity function using Legendre–Gauss–Lobatto collocation points. An example is given to demonstrate the accuracy of the developed method.

Self‐dual Maxwell fields on curved space–times
View Description Hide DescriptionWe present a manifestly conformally invariant formulation of Maxwell equations on asymptotically flat space–times. It is shown how to construct regular self‐dual and antiself‐dual fields from suitable radiation data, and the general solution as a sum of fields with both types of duality. The basic variable in this formalism is a scalar fieldF defined as the phase of the parallel propagator (associated with the Maxwell potential) from interior points to future null infinity along null geodesics. Field equations equivalent to the source free Maxwell’sequations are derived for F. A perturbative solution based on Huygens’ principle is proposed. Exact solutions are found for H‐spaces. The use of these results on gravitational lensing is discussed.

An invariant imbedding analysis of general wave scattering problems
View Description Hide DescriptionThe invariant imbedding technique, via the solution of a Riccati‐type equation, is modified to calculate the wave fields inside and scattered from a strongly (laterally and vertically) heterogenous, anisotropic inclusion, which may be large but remains compact. The factorization underlying this approach is carried out with respect to direction of average power flow rather than the more conventional factorization with respect to local direction of propagation. The solution of the operator Riccati equation is related to the Dirichlet‐to‐Neumann map. The formulation is robust in the sense that it can handle a rather extreme range of modal wave speeds, and allows continuous as well as discontinuous medium variations on different (wave) length scales. It also, inherently, takes care of critical‐angle phenomena. The algorithm, based on the invariant imbedding approach, yields the internal fields for a full survey of sources and receivers simultaneously. The wave field solution in the inclusion is coupled to the external field via a boundary element approach.

The construction of spinor fields on manifolds with smooth degenerate metrics
View Description Hide DescriptionWe examine some of the subtleties inherent in formulating a theory of spinors on a manifold with a smooth degenerate metric. We concentrate on the case where the metric is singular on a hypersurface that partitions the manifold into Lorentzian and Euclidean domains. We introduce the notion of a complex spinor fibration to make precise the meaning of continuity of a spinor field and give an express‐ ion for the components of a local spinor connection that is valid in the absence of a frame of local orthonormal vectors. These considerations enable one to construct a Dirac equation for the discussion of the behavior of spinors in the vicinity of the metric degeneracy. We conclude that the theory contains more freedom than the spacetime Dirac theory and we discuss some of the implications of this for the continuity of conserved currents.

Master equation based formulation of nonequilibrium statistical mechanics
View Description Hide DescriptionFor a nonequilibrium system characterized by its state space, by a dynamics defined by a transfer matrix and by a reference equilibrium dynamics given by a detailed‐balance transfer matrix, we define various nonequilibrium concepts: relative entropy, dissipation during the relaxation to the stationary state, path entropy, cost for maintaining the system in a nonequilibrium state, fluctuation‐dissipation theory, and finally a tree integral formula for the stationary state.

Scattering matrix in external field problems
View Description Hide DescriptionWe discuss several aspects of second quantized scattering operators Ŝ for fermions in external time‐dependent fields. We derive our results on a general, abstract level having in mind as a main application potentials of the Yang‐Mills type and in various dimensions. We present a new and powerful method for proving the existence of Ŝ which is also applicable to other situations like external gravitational fields. We also give two complementary derivations of the change of phase of the scattering matrix under generalized gauge transformations which can be used whenever our method of proving the existence of Ŝ applies. The first is based on a causality argument, i.e., Ŝ (including phase) is determined from a time evolution, and the second exploits the geometry of certain infinite‐dimensional group extensions associated with the second quantization of one‐particle operators. As a special case we obtain a Hamiltonian derivation of the axial fermion‐Yang‐Mills anomaly and the Schwinger terms related to it via the descent equations, which is on the same footing and traces them back to a common root.

Quasi‐exactly solvable potentials on the line and orthogonal polynomials
View Description Hide DescriptionIn this paper we show that a quasi‐exactly solvable (normalizable or periodic) one‐dimensional Hamiltonian satisfying very mild conditions defines a family of weakly orthogonal polynomials which obey a three‐term recursion relation. In particular, we prove that (normalizable) exactly solvable one‐dimensional systems are characterized by the fact that their associated polynomials satisfy a two‐term recursion relation. We study the properties of the family of weakly orthogonal polynomials defined by an arbitrary one‐dimensional quasi‐exactly solvable Hamiltonian, showing in particular that its associated Stieltjes measure is supported on a finite set. From this we deduce that the corresponding moment problem is determined, and that the kth moment grows like the kth power of a constant as k tends to infinity. We also show that the moments satisfy a constant coefficient linear difference equation, and that this property actually characterizes weakly orthogonal polynomial systems.