Bosonization and Integral Representation of Solutions¶of the Knizhnik–Zamolodchikov–Bernard Equations

We construct an integral representation of solutions of the Knizhnik–Zamolodchikov–Bernard equations, using the Wakimoto modules. Received: 5 October 1998 / Accepted: 8 February 1999     

The Complex Geometry of Weak Piecewise Smooth Solutions of Integrable Nonlinear PDE's¶of Shallow Water and Dym Type

An extension of the algebraic-geometric method for nonlinear integrable PDE's is shown to lead to new piecewise smooth weak solutions of a class of N-component systems of nonlinear evolution equations. This class includes, among others, equations from the Dym and shallow water equation hierarchies. The main goal of the paper is to give explicit theta-functional expressions for piecewise smooth weak solutions of these nonlinear PDE's, which are associated to nonlinear subvarieties of hyperelliptic Jacobians. The main results of the present paper are twofold. First, we exhibit some of the special features of integrable PDE's that admit piecewise smooth weak solutions, which make them different from equations whose solutions are globally meromorphic, such as the KdV equation. Second, we blend the techniques of algebraic geometry and weak solutions of PDE's to gain further insight into, and explicit formulas for, piecewise-smooth finite-gap solutions. The basic technique used to achieve these aims is rather different from earlier papers dealing with peaked solutions. First, profiles of the finite-gap piecewise smooth solutions are linked to certain finite dimensional billiard dynamical systems and ellipsoidal billiards. Second, after reducing the solution of certain finite dimensional Hamiltonian systems on Riemann surfaces to the solution of a nonstandard Jacobi inversion problem, this is resolved by introducing new parametrizations. Amongst other natural consequences of the algebraic-geometric approach, we find finite dimensional integrable Hamiltonian dynamical systems describing the motion of peaks in the finite-gap as well as the limiting (soliton) cases, and solve them exactly. The dynamics of the peaks is also obtained by using Jacobi inversion problems. Finally, we relate our method to the shock wave approach for weak solutions of wave equations by determining jump conditions at the peak location. Received: 16 February 1999 / Accepted: 10 April 2001     

Application of Coordinate-Momentum Intermediate Representation in Solving Eigenstate Problem of a Coupled Oscillator System

We study the eigenstate problem of a kind of coupled oscillators in the new quantum mechanical representation ｜q,μ,υ〉, which is defined as the eigenvector of the operator （μQ ＋ υP）, whereμ and υ are two real parameters. We also use the U operator transformation method to deal with the same problem. We obtain the normally ordered product expressions of U operator and eigenvector. It is shown that the ground state of system Hamiltonian is a squeezed state.     

A Fleming–Viot Particle Representation¶of the Dirichlet Laplacian

We consider a model with a large number N of particles which move according to independent Brownian motions. A particle which leaves a domain D is killed; at the same time, a different particle splits into two particles. For large N, the particle distribution density converges to the normalized heat equation solution in D with Dirichlet boundary conditions. The stationary distributions converge as N→∞ to the first eigenfunction of the Laplacian in D with the same boundary conditions. Received: 11 November 1999 / Accepted: 19 May 2000     

Quantum Vertex Representations via Finite Groups¶and the McKay Correspondence

We establish a q-analog of our recent work on vertex representations and the McKay correspondence. For each finite group Γ we construct a Fock space and associated vertex operators in terms of wreath products of $Γ×ℂ^× and the symmetric groups. An important special case is obtained when Γ is a finite subgroup of SU ₂, where our construction yields a group theoretic realization of the representations of the quantum affine and quantum toroidal algebras of ADE type. Received: 17 August 1999 / Accepted: 2 December 1999     

In Appreciation¶The Depth and Breadth of John Bell's Physics

This essay surveys the work of John Stewart Bell, one of the great physicists of the twentieth century. Section 1 is a brief biography, tracing his career from working-class origins and undergraduate training in Belfast, Northern Ireland, to research in accelerator and nuclear physics in the British national laboratories at Harwell and Malvern, to his profound research on elementary particle physics as a member of the Theory Group at CERN and his equally profound "hobby" of investigating the foundations of quantum mechanics. Section 2 concerns this hobby, which began in his discontent with Bohr's and Heisenberg's analyses of the measurement process. He was attracted to the program of hidden variables interpretations, but he revolutionized the foundations of quantum mechanics by a powerful negative result: that no hidden variables theory that is "local" (in a clear and well-motivated sense) can agree with all the correlations predicted by quantum mechanics regarding well-separated systems. He further deepened the foundations of quantum mechanics by penetrating conceptual analyses of results concerning measurement theory of von Neumann, de Broglie and Bohm, Gleason, Jauch and Piron, Everett, and Ghirardi-Rimini-Weber. Bell's work in particle theory (Section 3) began with a proof of the CPT theorem in his doctoral dissertation, followed by investigations of the phenomenology of CP-violating experiments. At CERN Bell investigated the commutation relations in current algebras from various standpoints. The failure of current algebra combined with partially conserved current algebra to permit the experimentally observed decay of the neutral pi-meson into two photons stimulated the discovery by Bell and Jackiw of anomalous or quantal symmetry breaking, which has numerous implications for elementary particle phenomena. Other late investigations of Bell on elementary particle physics were bound states in quantum chromodynamics (in collaboration with Bertlmann) and estimates for the anomalous magnetic moment of the muon (in collaboration with de Rafael). Section 4 concerns accelerations, starting at Harwell with the algebra of strong focusing and the stability of orbits in linear accelerators and synchrotrons. At CERN he continued to contribute to accelerator physics, and with his wife Mary Bell he wrote on electron cooling and Beamstrahlung. A spectacular late achievement in accelerator physics was the demonstration (in collaboration with Leinaas) that the effective black-body radiation seen by an accelerated observer in an electromagnetic vacuum - the "Unruh effect" - had already been observed experimentally in the partial depolarization of electrons traversing circular orbits.     

Certain Extensions¶of Vertex Operator Algebras of Affine Type

We generalize Feigin and Miwa's construction of extended vertex operator (super)algebras A _k (sl(2)) for other types of simple Lie algebras. For all the constructed extended vertex operator (super)algebras, irreducible modules are classified, complete reducibility of every module is proved and fusion rules are determined modulo the fusion rules for vertex operator algebras of affine type. Received: 7 March 2000 / Accepted: 10 November 2000     

The Singular Kane Oscillator

The energy spectrum of carriers in narrow band gap semiconductor nanocrystals are studied theoretically taking into account the nonparabolicity of charge carriers dispersion laws. The confinement potential of nanocrystals is approximated to be λγ^2 ＋ λ1γ^-2, and the dispersion laws are considered within the framework of the three-band Kane model.     

Absolute Continuity of the Floquet Spectrum¶for a Nonlinearly Forced Harmonic Oscillator

We prove that the Floquet spectrum of the time periodic Schr?dinger equation corresponding to a mildly nonlinear resonant forcing, is purely absolutely continuous for μ suitably small. Received: 23 March 2000 / Accepted: 24 May 2000     

Cyclic Representation of the gl√q (n)-Covariant Oscillator Algebra

The cyclic representation ofgl_q (n)-covariant oscillator algebrais discussed by use of some noncommutating cyclicvariables.     

A Generalized Hypergeometric Function¶Satisfying Four Analytic Difference Equations¶of Askey--Wilson Type

The hypergeometric function ₂ F ₁ can be written in terms of a contour integral involving gamma functions. We generalize this (Barnes) representation by using a certain generalized gamma function as a building block. In this way we obtain a new ₂ F ₁-generalization with various symmetry features. We determine the analyticity properties of the R-function in all of its eight arguments, and show that it is a joint eigenfunction of four distinct Askey–Wilson type difference operators, two acting on v and two on . The Askey–Wilson polynomials can be obtained by a suitable discretization of v or . Received: 21 December 1998 / Accepted: 14 April 1999     

Energy Correlations in O(N) Models¶and the Wolff Representation

We prove that energy functions are positively correlated in isotropic, ferromagnetic O(N) models on an arbitrary graph. In our inductive proof, this is used to prove the strong FKG property of the Wolff representation for isotropic, ferromagnetic O(N+ 1) models. This strong FKG property is then used to prove energy correlations for the O(N+ 1) model. Furthermore, percolation in the Wolff representation is proved to be a necessary and sufficient condition for positivity of the spontaneous magnetization (previously known only for N= 3). Received: 7 March 2000 / Accepted: 31 October 2000     

The Ponzano–Regge Model and Parametric Representation

We give a parametric representation of the effective noncommutative field theory derived from a ${\kappa}$ -deformation of the Ponzano–Regge model and define a generalized Kirchhoff polynomial with ${\kappa}$ -correction terms, obtained in a ${\kappa}$ -linear approximation. We then consider the corresponding graph hypersurfaces and the question of how the presence of the correction term affects their motivic nature. We look in particular at the tetrahedron graph, which is the basic case of relevance to quantum gravity. With the help of computer calculations, we verify that the number of points over finite fields of the corresponding hypersurface does not fit polynomials with integer coefficients, hence the hypersurface of the tetrahedron is not polynomially countable. This shows that the correction term can change significantly the motivic properties of the hypersurfaces, with respect to the classical case.     

The Split Property and the Symmetry Breaking¶of the Quantum Spin Chain

We consider the relationship between the symmetry breaking and the split property of pure states of quantum spin chains. We obtain a representation theoretic condition implying that the half-sided uniform mixing condition leads to symmetry breaking of translationally invariant pure states. This is a mathematical generalization of Dichotomy previously found by I. Affleck and E. Lieb and M. Aizenman and B. Nachtergaele for ground states of a special class of Hamiltonians. Received: 1 February 1999 / Accepted: 5 December 2000     

Representation and Dimensional Reduction of the Universe

The external space we live in or the apparent dimension in the Kaluza-Klein model can be identified by using the right representation in quantum cosmology. The external dimension of the Freund-Rubin model is rain(s, n-s), where s is the rank of the antisymmetric field strength in the model.     

The ‘Miracle’ of Applicability? The Curious Case of the Simple Harmonic Oscillator

The paper discusses to what extent the conceptual issues involved in solving the simple harmonic oscillator model fit Wigner’s famous point that the applicability of mathematics borders on the miraculous. We argue that although there is ultimately nothing mysterious here, as is to be expected, a careful demonstration that this is so involves unexpected difficulties. Consequently, through the lens of this simple case we derive some insight into what is responsible for the appearance of mystery in more sophisticated examples of the Wigner problem.     

The Extended Moduli Space of¶Special Lagrangian Submanifolds

It is well known that the moduli space of all deformations of a compact special Lagrangian submanifold X in a Calabi–Yau manifold Y within the class of special Lagrangian submanifolds is isomorphic to the first de Rham cohomology group of X. Reinterpreting the embedding data X⊂Y within the mathematical framework of the Batalin–Vilkovisky quantization, we find a natural deformation problem which extends the above moduli space to the full de Rham cohomology group of X. Received: 29 June 1998 / Accepted: 7 June 1999     

The Construction of Frobenius Manifolds¶from KP tau-Functions

Frobenius manifolds (solutions of WDVV equations) in canonical coordinates are determined by the system of Darboux–Egoroff equations. This system of partial differential equations appears as a specific subset of the n-component KP hierarchy. KP representation theory and the related Sato infinite Grassmannian are used to construct solutions of this Darboux–Egoroff system and the related Frobenius manifolds. Finally we show that for these solutions Dubrovin's isomonodromy tau-function can be expressed in the KP tau-function. Received: 1 September 1998 / Accepted: 7 March 1999     

The Action of Outer Automorphisms on Bundles¶of Chiral Blocks

On the bundles of WZW chiral blocks over the moduli space of a punctured rational curve we construct isomorphisms that implement the action of outer automorphisms of the underlying affine Lie algebra. These bundle-isomorphisms respect the Knizhnik–Zamolodchikov connection and have finite order. When all primary fields are fixed points, the isomorphisms are endomorphisms; in this case, the bundle of chiral blocks is typically a reducible vector bundle. A conjecture for the trace of such endomorphisms is presented; the proposed relation generalizes the Verlinde formula. Our results have applications to conformal field theories based on non-simply connected groups and to the classification of boundary conditions in such theories. Received: 11 May 1998 / Accepted: 17 April 1999     

Discrete Dynamical Systems¶Associated with Root Systems of Indefinite Type

A geometric charactrization of the equation found by Hietarinta and Viallet, which satisfies the singularity confinement criterion but which exhibits chaotic behavior, is presented. It is shown that this equation can be lifted to an automorphism of a certain rational surface and can therefore be considered to be a realization of a Cremona isometry on the Picard group of the surface. It is also shown that the group of Cremona isometries is isomorphic to an extended Weyl group of indefinite type. A method to construct the mappings associated with some root systems of indefinite type is also presented. Received: 19 March 2001 / Accepted: 11 July 2001