Action of truncated quantum groups on quasi-quantum planes and a quasi-associative differential geometry and calculus

Ifq is ap ^th root of unity there exists a quasi-coassociative truncated quantum group algebra whose indecomposable representations are the physical representations ofU _q (sl ₂), whose coproduct yields the truncated tensor product of physical representations ofU _q (sl ₂), and whoseR-matrix satisfies quasi-Yang Baxter equations. These truncated quantum group algebras are examples of weak quasitriangular quasi-Hopf algebras (quasi-quantum group algebras). We describe a space of functions on the quasi quantum plane, i.e. of polynomials in noncommuting complex coordinate functionsz _a , on which multiplication operatorsZ _a and the elements of can act, so thatz _a will transform according to some representation ^f of can be made into a quasi-associative graded algebra on which elements of act as generalized derivations. In the special case of the truncatedU _q (sl ₂) algebra we show that the subspaces of monomials inz _a ofn ^th degree vanish fornp–1, and that carries the 2J+ 1 dimensional irreducible representation of ifn=2J, J=0,1/2, ..., 1/2(p–2). Assuming that the representation ^f of the quasi-quantum group algebra gives rise to anR-matrix with two eigenvalues, we develop a quasi-associative differential calculus on. This implies construction of an exterior differentiation, a graded algebra of forms and partial derivatives. A quasi-associative generalization of noncommutative differential geometry is introduced by defining a covariant exterior differentiation of forms. It is covariant under gauge transformations.     

Lie algebra cohomology and the fusion rules

We prove a vanishing theorem for Lie algebra cohomology which constitutes a loop group analogue of Kostant's Lie algebra version of the Borel-Weil-Bott theorem. Consider a complex semi-simple Lie algebra and an integrable, irreducible, negative energy representation of. Givenn distinct pointsz _k in , with a finite-dimensional irreducible representationV _k of assigned to each, the Lie algebra of-valued polynomials acts on eachV _k , via evaluation atz _k . Then, the relative Lie algebra cohomologyH ^* is concentrated in one degree. As an application, based on an idea of G. Segal's, we prove that a certain homolorphic induction map from representations ofG to representations ofLG at a given level takes the ordinary tensor product into the fusion product. This result had been conjectured by R. Bott.     

Global existence and boundedness of large solutions to nonlinear equations of viscoelasticity with hardening

For the solutions of an initial-boundary value problem for the equations of viscoelasticity with isotropic hardening we derive a uniform bound under a growth condition for the nonlinearities in the case of one-space dimension. Global-in-time existence of solutions to large initial data is a consequence of the existence of this bound. In the most simple form, the equations we consider are with suitable functionsg ₁,g ₂,h satisfyingg ₂0 andXDedicated to Erhard Meister on the occasion of his sixtyfifth birthdayThis research was partially supported by the DFG-Forschergruppe Mathematische und Ingenieurwissenschaftliche Analyse bruchmechanischer und inelastischer Probleme     

On the pointwise behavior of semi-classical measures

In this paper we concern ourselves with the small asymptotics of the inner products of the eigenfunctions of a Schrödinger-type operator with a coherent state. More precisely, let _j and E _j denote the eigenfunctions and eigenvalues of a Schrödinger-type operator H with discrete spectrum. Let _(x,) be a coherent state centered at the point (x, ) in phase space. We estimate as 0 the averages of the squares of the inner products ( _a ^(x,) , _j ) over an energy interval of size around a fixed energy, E. This follows from asymptotic expansions of the form for certain test function and Schwartz amplitudes a of the coherent state. We compute the leading coefficient in the expansion, which depends on whether the classical trajectory through (x, ) is periodic or not. In the periodic case the iterates of the trajectory contribute to the leading coefficient. We also discuss the case of the Laplacian on a compact Riemannian manifold.Research supported in part by NSF grant DMS-9303778     

Stochastic Schrödinger operators and Jacobi matrices on the strip

We discuss stochastic Schrödinger operators and Jacobi matrices with wave functions, taking values in ^l so there are 2l Lyaponov exponents ₁..._l0 _l+1..._2l =–₁. Our results include the fact that if ₁=0 on a set positive measure, thenV is deterministic and one that says that {E|exactly 2j 's are zero} is the essential support of the a.c. spectrum of multiplicity 2j.Research partially supported by USNSF under grant DMS-8416049     

The ground state energy of a Bose gas with Coulomb interaction II

LetH _N be the quantum mechanical Hamiltonian for a neutral system of 2_N charged particles, each of unit charge. The HamiltonianH _N is assumed to act on wave functions inL ²(^6N ) which satisfy Bose statistics. It is shown that if the kinetic energy of is sufficiently small, then |H _N |–CN ^7/5 for some universal constantC.Research supported by U.S. National Science Foundation Grant DMS 8600748     

Entropy,the central limit theorem and the algebra of the canonical commutation relation

The central limit theorem of Cushen and Hudson is reformulated on the algebra of the CCR. Namely, for a gauge invariant state , the weighted convolutions _n of the central limit tend to the quasi-free reduction _Q of pointwise. It is proved that if the initial relative entropy S(, _Q ) is finite, then S( _n , _Q ) goes to 0 and so _n – _Q 0. No restriction on the dimension of the test function space is made.     

Geometric prequantization on the spin bundle based onN-symplectic geometry: The dirac equation

We present preliminary results for a prequantization procedure that leads in a natural way to the Dirac equation. The starting point is the recently introducedn-symplectic geometry on the bundle of linear framesLM of ann-dimensional manifoldM in which the ⁿ-valued soldering 1-form onLM plays the role of then-symplectic potential. On a 4-dimensional spacetime manifold we consider the tensorial ⁴⁴valued function onLM determined by the spacetime metric tensor g as the Hamiltonian for free observers and determine the associated ⁴-valued Hamiltonian vector field , Integration of theX ⁱ yields the dynamics of free observers on spacetime, namely parallel transport of linear frames along spacetime geodesies. In order to obtain a vector field on the spin bundleSM which is a lift of and which is induced by a vector field for an appropriate mapping , it is useful to define a prolongation of some bundleL ^o M of oriented frames ofM. IfGL ⁺(4, ) denotes the identity component ofGL(4, ), thenGL ⁺(4, ) is the structure group ofL ^o M and its double cover is the structure group of. We show that the lift of onL ^o M to induces a natural 4-symplectic potential on. If is the lift of g to, then we find the ⁴-valued Hamiltonian vector field on determined by and show that the vector fieldsX _g ⁱ on are tangent to the subbundleSM. Integration of the restriction of theX ⁱ toSM now yields parallel transport of spin frames and thus tetrads along spacetime geodesies of g. We consider a naive prequantization operator assignment acting on ⁴-spinors in the standard representation ofSL(2, ). The eigenvalue equation for the system of new Hilbert space operators yields the Dirac equation.     

First experiment to test whether space-time is flat. II. Effects of orbit

The gyroscope in an orbiting satellite will be acted on by additional gravitational fields due to the rotation of the earth and due to the orbital velocity of the satellite. According to special relativistic gravitational theory, we deduce _L ^(S) —the gyroscope's precession rate due to the orbital velocity—and _S ^(S) —the gyroscope's precession rate due to the earth's rotation in the polar orbit case. The results are _L ^(S) = (2/3) _L ^(G) , _S ^(S) = (3/2) cos (1 - sin² cos²)^1/2 _S ^(G) , where and are the gyroscope's polar angles, and _L ^(G) and _S ^(G) are the geodetic and frame-dragging precession rates predicted by general relativity, respectively.     

Long range scattering for nonlinear Schrödinger equations in one space dimension

We consider the scattering problem for the nonlinear Schrödinger equation in 1+1 dimensions: where = /x,R{0},R,p>3. We show that modified wave operators for (*) exist on a dense set of a neighborhood of zero in the Lebesgue spaceL ²(R) or in the Sobolev spaceH ¹(R)., The modified wave operators are introduced in order to control the long range nonlinearity |u|² u.Laboratoire associé au Centre National de la Recherche Scientifique     

Spectral properties of a class of operators associated with conformal maps in two dimensions

Iff is a rational map of the Riemann sphere, define the transfer operator by Let also be the Banach space of functions for which the second derivatives are measures. Ifg andg satisfies a simple integrability condition (implying thatg vanishes at critical points and multiple poles off) then is a bounded linear operator on . The essential spectral radius of can be estimated and, under suitable conditions, proved to be strictly less than the spectral radius. Similar estimates for more general operators are also obtained.     

Exponential decay in the stark effect

LetH=–+V+Fx ₁ withV(x ₁,x ) analytic in the first variable andV(x ₁+ia, x ) bounded and decreasing to zero asx for eacha . Let be an eigenvector of –+V with negative eigenvalue. Among our results we show that forF0, (,e ^–H ) decays exponentially at a rate governed by the positions of the resonances ofH. This exponential decay is in marked contrast to conventional atomic resonances for which power law decay is the rule.Research supported by NSF Grant No. MCS 78-00101.     

Duality-invariant orthogonality relation between the magnetic monopole and associated quark charges

Duality invariance of the Dirac-Schwinger charge-symmetric theory for electromagnetism leads one to consider the complex-valued amplitudes ₁ and ₂ for the separation between the magnetic monopole and quarks in the logarithmic charge plane. It is observed that the orthogonality relation on the latter amplitudes, Re( ₁ ^* ₂)=0, is equivalent to the equation (ln 9 ^–1)(ln 2)=(1/2) ², which is indeed satisfied by the experimental value fora to within 0.027%. In addition to fixing the unit of electric charge at a primary physical value, the orientation of ₁, ₂ may also prescribe the Cabibbo angle to have the theoretical value 12.4438.     

Percolation of level sets for two-dimensional random fields with lattice symmetry

Let (x),x², be a random field, which may be viewed as the potential of an incompressible flow for which the trajectories follow the level lines of . Percolation methods are used to analyze the sizes of the connected components of level sets {x:(x)=h} and sets {x:(x)h} in several classes of random fields with lattice symmetry. In typical cases there is a sharp transition at a critical value ofh from exponential boundedness for such components to the existence of an unbounded component. In some examples, however, there is a nondegenerate interval of values ofh where components are bounded but not exponentially so, and in other cases each level set may be a single infinite line which visits every region of the lattice.     

Generalized Bargmann potentials

Conditions are found for the coefficient functions of a linear ordinary differential equation of the kth order ^(k)+u₁^(k-2)+...+u_k-1=^k, when its solution has the following analytical dependence on the parameter: =exp(z _i=1 ⁿ (+a_i)(a_ja_j(Z)). The problem is closely related to the finding of n-soliton solutions of the simplest form for the periodic Toda system, corresponding to A _nand C _nseries.     

Fuzzy quantum logic II. The logics of unsharp quantum mechanics

A survey of the main results of the Italian group about the logics of unsharp quantum mechanics is presented. In particular partial ordered structures playing with respect to effect operators (linear bounded operatorsF on a Hilbert space such that, 0¦F²) the role played by orthomodular posets with respect to orthogonal projections (corresponding to sharp effects) are analyzed. These structures are generally characterized by the splitting of standard orthocomplementation on projectors into two nonusual orthocomplementations (afuzzy-like and anintuitionistic-like) giving rise to different kinds of Brouwer-Zadeh (BZ) posets: de Morgan BZ posets, BZ^* posets, and BZ³ posets. Physically relevant generalizations of ortho-pair semantics (paraconsistent, regular paraconsistent, and minimal quantum logics) are introduced and their relevance with respect to the logic of unsharp quantum mechanics are discussed.     

A matrix integral solution to [P,Q]=P and matrix laplace transforms

In this paper we solve the following problems: (i) find two differential operatorsP andQ satisfying [P, Q]=P, whereP flows according to the KP hierarchy P/t _n =[(P ^n/p )₊,P], withp:=ordP2; (ii) find a matrix a integral representation for the associated -function. First we construct an infinite dimensional spaceW= span{ ₀(z, ₁(z,...)} of functions ofz invariant under the action of two operators, multiplication byz ^p andA _c :=z/z–z+c. This requirement is satisfied, for arbitraryp, if ₀ is a certain function generalizing the classical Hänkel function (forp=2); our representation of the generalized Hänkel function as adouble Laplace transform of a simple function, which was unknown even for thep=2 case, enables us to represent the -function associated with the KP time evolution of the spaceW as a double matrix Laplace transform in two different ways. One representation involves an integration over the space of matrices whose spectrum belongs to a wedge-shaped contour ^-+ ^- defined by ^± = ₊e^±i/p. The new integrals above relate to matrix Laplace transforms, in contrast with matrix Fourier transforms, which generalize the Kontsevich integrals and solve the operator equation [P, Q]=1.The support of a National Science Foundation grant #DMS-95-4-51179 is gratefully acknowledged.The hospitality of the Volterra Center at Brandeis University is gratefully acknowledged.The hospitality of the University of Louvain and Brandeis University is gratefully acknowledged.The support of a National Science Foundation grant #DMS-95-4-51179, a Nato, an FNRS and a Francqui Foundation grant is gratefully acknowledged.     

Common tangent space ℝ 1 4 fromU(2) charges

It is discussed how a common space-time can be constructed from a proposed hiddenU(2) world. Schrödinger's idea to obtain discrete eigenvalues by solving the Maxwell equations for the fieldF on compact spaces without boundaries is modified by orthogonality and identification concepts for the potentialsA. Using residue classes with respect to the metric (Clifford algebra), a common spinor space ⁴=RL and a common Minkowski tangent space ₁ ⁴ are bilinearly constructed from tangent spaces ofU(2) individuals [U(2) manifolds with orthogonal potentials]. The space constructed has the following properties. (1) There are algebraic elements for the identification ofU(2) individuals from ₁ ⁴ as spinors and vectorsA. (2) The transfer of the potentials fromU(2) via ⁴ to ₁ ⁴ is linear. (3) The hiddenU(2) content of the left- and right-handed spaces (L, R) is quite different. The potentials on U(2) individuals are transformed into complex wave functions on the spinor space and into 1-formsA on ₁ ⁴ that can be enlarged to gauge potentials. The construction is discussed from an old point of view of Einstein's, starting with the electric charge as the primary concept for quantum theory. The construction of the tangent space ₁ ⁴ does not depend on a preceding introduction of any points (uncertainty). The identity problem of the interpretation of the quantum theory is discussed in some detail. It is indicated how the algebraic, partiallyad hoc constructions can give a rigid frame for further analytical work.     

Stochastic coupling and thermodynamic inequalities

Let ₁ and ₂ be thermodynamic Gibbs measures on ^m and ⁿ , respectively. Diffusions are constructed having ₁, and ₂ as invariant measures. These diffusions are then coupled; inequalities between expectations of certain random variables on the two spaces result.Partially supported by NSF-MCS 74-07313-A03     

On geometrical interpretation of thep-adic Maslov index

A set of selfdual lattices in a two-dimensionalp-adic symplectic space is provided by an integer valued metricd. A realization of the metric space (,d) as a graph is suggested and this graph has been linked to the Bruhat-Tits tree. An action of symplectic group on a set of cycles of length three of the graph is considered and a geometrical interpretation of thep-adic Maslov index is given in terms of this action.