gl q (n) and quantum monodromy

A generalized Toda lattice based on gl(n) is considered. The Poisson brackets are expressed in terms of a Lax connection, L=–() and a classical r-matrix, {₁,₂}=[r,₁+₂}. The essential point is that the local lattice transfer matrix is taken to be the ordinary exponential, T=e; this assures the intepretation of the local and the global transfer matrices in terms of monodromy, which is not true of the T-matrix used for the sl(n) Toda lattice. To relate this exponential transfer matrix to the more manageable and traditional factorized form, it is necessary to make specific assumptions about the equal time operator product expansions. The simplest possible assumptions lead to an equivalent, factorized expression for T, in terms of operators in (an extension of) the enveloping algebra of gl(n). Restricted to sl(n), and to multiplicity-free representations, these operators satisfy the commutation relations of sl _q (n), which provides a very simple injection of sl _q (n) into the enveloping algebra of sl(n). A deformed coproduct, similar in form to the familiar coproduct on sl _q (n), turns gl(n) into a deformed Hopf algebra gl _q (n). It contains sl _q (n) as a subalgebra, but not as a sub-Hopf algebra.     

Z 3-Graded exterior differential calculus and gauge theories of higher order

We present a possible generalization of the exterior differential calculus, based on the operator d such that d³=0, but d²0. The entities dx ⁱ and d² x ^k generate an associative algebra; we shall suppose that the products dx ⁱ dx ^k are independent of dx ^k dx ⁱ , while theternary products will satisfy the relation: dx ⁱ dx ^k dx ^m =jdx ^k dx ^m dx ⁱ =j ²dx ^m dx ^m dx ⁱ dx ^k , complemented by the relation dx ⁱ d² x ^k =jd² x ^k dx ⁱ , withj:=e^2i/3.We shall attribute grade 1 to the differentials dx ⁱ and grade 2 to the second differentials d² x ^k ; under the associative multiplication law the grades add up modulo 3.We show how the notion ofcovariant derivation can be generalized with a 1-formA so thatD:=d+A, and we give the expression in local coordinates of thecurvature 3-form defined as :=d² A+d(A ²)+AdA+A ³.Finally, the introduction of notions of a scalar product and integration of theZ ₃-graded exterior forms enables us to define the variational principle and to derive the differential equations satisfied by the 3-form . The Lagrangian obtained in this way contains the invariants of the ordinary gauge field tensorF _ik and its covariant derivativesD _i F _km .     

On the linearization of second-order ordinary differential equations

In this Letter, the problem of characterizing all second-order ordinary differential equations y=f(x, y, y) which are locally linearizable by a change of dependent and independent variables (x, y)(X, Y) is considered. Since all second-order linear equations are locally equivalent to y=0, the problem amounts to finding necessary and sufficient conditions for y=f(x, y, y) to be locally equivalent to y=0. It turns out that two apparently different criteria for linearizability have been formulated in the literature: the one found by M. Tresse and later rederived by É. Cartan, and the criterion recently given by Arnol'd [Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, New York, 1983]. It is shown here that these two sets of linearizability conditions are actually equivalent. As a matter of fact, since Arnol'd's criterion is stated without proof in the latter reference, this work can be alternatively considered as a proof of Arnol'd's linearizability conditions based on Cartan-Tresse's. Some further points in connection with the relationship between Arnol'd's and Cartan-Tresse's treatment of the linearization problem are also discussed and illustrated with several examples.     

A generalized consistency condition for massless fields

Using a generalized consistency condition (gcc), we construct couplings between massless scalar fields and the spin 2 gravitational field. Specifically, we consider all possible third-order interaction terms for scalar fields and a and use the gcc to single out one. We also find three generalized current identities associated with the massless gauge fields.     

Berenstein-Zelevinsky triangles,elementary couplings,and fusion rules

We present a general scheme for describing (N) _k fusion rules in terms of elementary couplings, using Berenstein-Zelevinsky triangles. A fusion coupling is characterized by its corresponding tensor product coupling (i.e. its Berenstein-Zelevinsky triangle) and the threshold level at which it first appears. We show that a closed expression for this threshold level is encoded in the Berenstein-Zelevinsky triangle and an explicit method to calculate it is presented. In this way, a complete solution of (4) _k fusion rules is obtained.Work supported by NSERC (Canada).Work supported by NSERC (Canada) and FCAR (Québec).     

Asymptotics for Bosonic atoms

It was proved by Benguria and Lieb that for an atom where the electrons do not satisfy the exclusion principle, the critical electron number N _c, i.e., the maximal number of electrons the atom can bind, satisfies lim inf_zN_c/Z 1 + , where Z is the nuclear charge. Here is a positive constant derived from the Hartree model. We complete this result by proving that the correct asymptotics for N _c(Z) is indeed _zN_c/Z = 1 + .This work was done while the author was a graduate student at Princeton University supported by a Danish Research Academy fellowship and U.S. National Science Foundation grant PHY-85-15288-A03.     

Equilibrium polymerization in a solvent: Solution on the Bethe lattice

The lattice model for equilibrium polymerization in a solvent proposed by Wheeler and Pfeuty is solved exactly on a Bethe lattice (core of a Caylay tree) with general coordination numberq. Earlier mean-field results are reobtained in the limitq, but the phase diagrams show deviations from them for finiteq. Whenq=2, our results turn into the solution of the one-dimensional problem. Although the model is solved directly, without the use of the correspondence between the equilibrium polymerization model and the diluten0 model, we verified that the latter model may also be solved on the Bethe lattice, its solution being identical to the direct solution in all parameter space. As observed in earlier studies of the puren0 vector model, the free energy is not always convex. We obtain the region of negative susceptibility for our solution and compare this result with mean field and renormalization group (-expansion) calculations.     

Symmetries of theq-difference heat equation

The symmetry operators of aq-difference analog of the heat equation in one space dimension are determined. They are seen to generate aq-deformation of the semidirect product of sl(2, ) with the three-dimensional Weyl algebra. It is shown that this algebraic structure is preserved if differentq-analogs of the heat equation are considered. The separation of variables associated to the dilatation symmetry is performed and solutions involving discreteq-Hermite polynomials are obtained.     

Lie-Poisson groups: Remarks and examples

The aim of this Letter is twofold. On the one hand, we discuss two possible definitions of complex structures on Poisson-Lie groups and we give a complete classification of the isomorphism classes of complex Lie-Poisson structures on the group SL(2, ). On the other hand, we give an algebraic characterization of a class of solutions of the Yang-Baxter equations which contains the well-known Drinfeld solutions [1]; in particular, we prove the existence of a nontrivial Lie-Poisson structure on any simply connected real semi-simple Lie Group G. Other low dimensional examples will appear elsewhere.Chercheur qualifié au FNRS.     

Darboux coordinates and isospectral Hamiltonian flows for the massive thirring model

The two-dimensional massive Thirring model is described as the integrability condition of a pair of commuting completely integrable isospectral Hamiltonian flows in the dual (2)^+* of the positive part (2)⁺ of the twisted loop algebra (2). Action-angle coordinates corresponding to the spectral invariants are derived on rational coadjoint orbits and a linearization of the flows obtained in the Jacobi variety of the underlying invariant spectral curve through a Liouville generating function for canonical coordinates.Research supported in part by the Natural Sciences Engineering Research Council of Canada and the Fonds FCAR du Québec.     

Algebraic quantization

The different correspondences (or orderings) used in quantum mechanics and the associated deformations, are both seen from an algebraic viewpoint. The deformations which are compatible with the diagonal map (the ₀-deformations) are introduced and connected to the formal groups. A very straighforward example of a ₀-deformation (the multiplicative deformation) appears in the normal quantization of the harmonic oscillator.     

Normal ordering and quantum groups

The irreducible R-matrices associated with the quantum Liouville and sine-Gordon equations were classified by the su(2) index l, 2l integer. We find that the associated quantum field theories must have the following equal time operator product expansions in the lattice approximation

Finite-element lattice Hamiltonian matrix elements: Anharmonic oscillators

The finite-element approach to lattice field theory is both highly accurate (relative errors 1/N ², whereN is the number of lattice points) and exactly unitary (in the sense that canonical commutation relations are exactly preserved at the lattice sites). In this Letter, we construct matrix elements for the time evolution operator for the anharmonic oscillator, for which the continuum Hamiltonian isH=p ²/2+q ^2k /2k. Construction of such matrix elements does not require solving the implicit equations of motion. Low-order approximations turn out to be quite accurate. For example, the matrix element of the time evolution operator in the harmonic oscillator groundstate gives a result for thek=2 anharmonic oscillator groundstate energy accurate to better than 1% while a two-state approximation reduces the error to less than 0.1%. Accurate wavefunctions are also extracted. Analogous results may be obtained in the continuum, but there the computation is more difficult, and not generalizable to field theories in more dimensions.     

Mutual Annihilation of Two Diffusing Particles in One- and Two-Dimensional Lattices

The probabilistic dynamics of a pair of particles which can mutually annihilate in the course of their random walk on a lattice is considered and analytically found for d=1 and d=2. In view of available recent experiments achieved on the femtosecond scale, emphasis is put on the necessity of a full continuous-time, discrete-space solution at all times. Quantities of physical interest are calculated at any time, including the total pair survival probability N(t) and the two-particle correlation function. As a by-product, the lattice version allows for a precise regularization of the continuous-space framework, which is ill-conditionned for d2; this being done, formal generalization to any real dimensionality can be straightforwardly performed.     

On a Mathematical Definition of a Renormalization Transformation for Lattice Gauge Theories

In lattice gauge theories, the renormalization transformation and its properties are formally defined and formally proved by making use of Dirac's function and its properties. In this Letter, we shall give a mathematically rigorous definition of a renormalization transformation for lattice pure gauge field theories and show the required properties, which are use to show ultraviolet stability of lattice gauge theories.     

Classical Dynamical r-Matrices,Poisson Homogeneous Spaces,and Lagrangian Subalgebras

Lu has shown that any dynamical r-matrix for the pair ( , ) naturally induces a Poisson homogeneous structure on G/U. She also proved that if is complex simple, is its Cartan subalgebra and r is quasitriangular, then this correspondence is in fact one-to-one. In this Letter we find some general conditions under which the Lu correspondence is one-to-one. Then we apply this result to describe all triangular Poisson homogeneous structures on G/U for a simple complex group G and its reductive subgroup U containing a Cartan subgroup.     

The Level Two and Three Modular Invariants of SU (n)

In this Letter, we explicitly classify all modular invariant partition functions for at levels 2 and 3. Previously, these were known only for level 1. Level 2 exceptions exist at r=9, 15, and 27;level 3 exceptions exist at r=4, 8, and 20. One of these is new, but the others were all anticipated by the rank-level duality relating level k and level r+1. The main recent result which this Letter rests on is the classification of -type invariants.     

A tri-Hamiltonian formulation of the full Kostant-Toda lattice

We define Poisson structures which lead to a tri-Hamiltonian formulation for the full Kostant-Toda lattice. In addition, a hierarchy of vector fields called master symmetries are constructed and they are used to generate the nonlinear Poisson brackets and other invariants. Various deformation relations are investigated. The results are analogous to results for the finite nonperiodic Toda lattice.     

Z 4-symmetric factorizedS-matrix in two space-time dimensions

The factorizedS-matrix with internal symmetryZ ₄ is constructed in two space-time dimensions. The two-particle amplitudes are obtained by means of solving the factorization, unitarity and analyticity equations. The solution of factorization equations can be expressed in terms of elliptic functions. TheS-matrix contains the resonance poles naturally. The simple formal relation between the general factorizedS-matrices and the Baxter-type lattice transfer matrices is found. In the sense of this relation theZ ₄-symmetricS-matrix corresponds to the Baxter transfer matrix itself.     

Lattice and continuum wavelets and the block renormalization group

We obtain a resolution of the identity operator, for functions on a latticeZ ^d, which is derived from the block renormalization group. We use eigenfunctions of the terms of the decomposition to form a basis forl ₂(Z^d) and show how the basis is generated from lattice wavelets. The lattice spacing is taken to zero and continuum wavelets are obtained.