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     

Symmetry theory in a two-level quantum system

We develop the theory of symmetry for a two-level quantum system in oder to illustrate the main ideas of the general theory of symmetry in quantum theory. It is based on the diffeomorphism of the two-dimensional sphere S ² onto the space of states P ¹ and the isomorphism between the groups P(2) and SO ₃ (). In particular, rotational invariance leads to the appearance of the spin1/2 in a natural way.     

Conformal blocks and generalized theta functions

LetSU _X r be the moduli space of rankr vector bundles with trivial determinant on a Riemann surfaceX. This space carries a natural line bundle, the determinant line bundleL. We describe a canonical isomorphism of the space of global sections ofL ^k with the space of conformal blocks defined in terms of representations of the Lie algebrasl _r (C((z))). It follows in particular that the dimension ofH ⁰(SU _X r,L ^k ) is given by the Verlinde formula.Both authors were partially supported by the European Science Project Geometry of Algebraic Varieties, Contract no. SCI-0398-C(A)     

The extremal microscopic coherent boson states

In a recent paper, the Fock-normal (=microscopic) fully-coherent statesL _L ⁽⁾ (L denotes the factorizing linear form) have been completely characterized, and the existence of nonpure elements in the extreme boundary of the weak-*-compact, convex setL _L has been shown. This letter is devoted to an analysis of the extremal fully-coherent states, especially those which are not pure states. An affine isomorphism is constructed betweenL _L ⁽⁾ and a certain convex subset of the normal completely positive maps on the bounded operators on Fock space. Then the extreme boundary ofL _L ⁽⁾ is determined by results and techniques from the theory of completely positive operators.     

Star-polarization: A natural link between phase space representation and operator representation of quantum mechanics

The method of *-polarization connects phase space mechanics to the usual operator formulation of quantum theory. A *-polarization is a linear submanifold of the space of C functions on phase space. Elements of a *-polarization are in direct correspondence with the Schroedinger wave functions and this correspondence induces the Weyl correspondence between classical observables and operators. All generalized Moyal algebras admit *-polarizations and a general method is thus available for translating *-quantization into operator language.     

Compositions of Sasaki projections

In an orthomodular lattice (abbreviated OML) L, a Sasaki projection is a mappinga _x(a)=x(x va) fromL toL, wherexL. We study compositions of finite numbers of Sasaki projections and of the same Sasaki projections composed in inverse order. By using the Baer-semigroup of all finite compositions of Sasaki projections, we establish a new characterization of kernels of congruences in OMLs and a generalization of the parallelogram law for dimension OMLs. Our results are also related to quantum measurements via Pool's definition of the change of the support of a state after a measurement.     

Band-gap structure of the spectrum of periodic Maxwell operators

We investigate the band-gap structure of some second-order differential operators associated with the propagation of waves in periodic two-component media. Particularly, the operator associated with the Maxwell equations with position-dependent dielectric constant (x),xR ³, is considered. The medium is assumed to consist of two components: the background, where (x) = _b , and the embedded component composed of periodically positioned disjoint cubes, where (x) = _a . We show that the spectrum of the relevant operator has gaps provided some reasonable conditions are imposed on the parameters of the medium. Particularly, we show that one can open up at least one gap in the spectrum at any preassigned point provided that the size of cubesL, the distancel=L betwen them, and the contrast = _b / _a are chosen in such a way thatL ^–2, and quantities ^-1-3/2 and ² are small enough. If these conditions are satisfied, the spectrum is located in a vicinity of widthw(^3/2)^-1 of the set {² L ^-2 k ²:kZ³}. This means, in particular, that any finite number of gaps between the elements of this discrete set can be opened simultaneously, and the corresponding bands of the spectrum can be made arbitrarily narrow. The method developed shows that if the embedded component consists of periodically positioned balls or other domains which cannot pack the space without overlapping, one should expect pseudogaps rather than real gaps in the spectrum.     

A theory of generally invariant lagrangians for the metric fields. I

The second-order generally invariant Lagrangians for the metric fields are studied within the framework of the Ehresmann theory of jets. Such a Lagrangian is a function on an appropriate fiber bundle whose structure group is the groupL _n ³ of invertible 3-jets with source and target at the origin 0 of the real,n-dimensional Euclidean spaceR ⁿ, and whose type fiber is the manifold T_n ²(R^n* R ^n*) of 2-jets with source at 0 R ⁿ and target in the symmetric tensor productR ^n* R^n*. Explicit formulas for the action ofL _n ³ onT _n ²(R^n* R ^n*) are considered, and a complete system of differential identities for the generally invariant Lagrangians is obtained.     

Quantum Kinematic Theory of the Poincare Group in Two-Dimensional Spacetime

Non-Abelian quantum kinematics is applied to thePoincare group P ₊ (1, 1),as an example of the quantization-through-the-symmetryapproach to quantum mechanics. Upon quantizing thegroup, generalized Heisenberg commutation relations are obtained, and aclosed Heisenberg–Weyl algebra follows. Then,according to the general theory, the three basicquantum-kinematic invariant operators are calculated;these afford the superselection rules for diagonalizing theincoherent rigged Hilbert space H(P ₊ ) of the regularrepresentation. This paper examines only one of thesediagonalization schemes, while introducing a irreducible spacetime representation carried by isotopicplane-wave eigenvectors of two compatible superselectionoperators (which define a Poincare-invariant linear2-momentum). Thereafter, the principle of microcausality produces massive 2-spinor isotopic states in 1+ 1 Minkowski space. The Dirac equation is thus deducedwithin the quantum kinematic formalism, and the familiarJordan–Pauli propagation kernel in 2-dimensional spacetime is also obtained as a Hurwitzinvariant integral over the group manifold. The maininterest of this approach lies in the adoptedgroup-quantization technique, which is a strictlydeductive method and uses exclusively the assumed Poincaresymmetry.     

Stochastic phase spaces and master Liouville spaces in statistical mechanics

The concept of probability space is generalized to that of stochastic probability space. This enables the introduction of representations of quantum mechanics on stochastic phase spaces. The resulting formulation of quantum statistical mechanics in terms of -distribution functions bears a remarkable resemblance to its classical counterpart. Furthermore, both classical and quantum statistical mechanics can be formulated in one and the same master Liouville space overL ²(). A joint derivation of a classical and quantum Boltzman equation provides an illustration of the practical uses of these formalisms.Supported in part by an NRC grant.     

The Spectral Gap for the Kawasaki Dynamics at Low Temperature

In this paper we analyze the convergence to equilibrium of Kawasaki dynamics for the Ising model in the phase coexistence region. First we show, in strict analogy with the nonconservative case, that in any lattice dimension, for any boundary condition and any positive temperature and particle density, the spectral gap in a box of side L does not shrink faster than a negative exponential of the surface L ^d–1. Then we prove that, in two dimensions and for free boundary condition, the spectral gap in a box of side L is smaller than a negative exponential of L provided that the temperature is below the critical one and the particle density satisfies (*_–, *₊), where *_± represents the particle density of the plus and minus phase, respectively.     

Statistical mechanics with homogeneous first-degree Lagrangians

The equilibrium statistical mechanics is investigated of any system whose LagrangianL ₀(v, q) is a convex homogeneous function of generalized velocitiesv, with coordinatesq in a bounded setD. A member of a canonical ensemble, the system has a conjugate HamiltonianH ₀(p, q) that vanishes identically in some subsetC×D of its phase space. The subsetC may also be specified, in some systems with a finite functionf(p, q), convex inpL ₀/, and thenL ₀ is also convex and homogeneous inv. In either case, ifC is bounded and convex, thenC or the convex functionf constitutes the fundamental constraint on the system. Under this fundamental constraint, it is shown that the so-called partition function becomes a phase-space volumeG (classical) or a numberW of microstates (quantum) from which follows the thermodynamic fundamental relation, entropySk InG (ork lnW).     

Further extension of the Gauss-Hertz principle

The continuum form of the Gauss-Hertz principle is extended to include the time domain as well as space. The Schrödinger equation and general relativity are derived by this method. The equivalence of the principle is shown to that of the Hamiltonian method where the energy is the expression –[ ² +A·² A], with being the difference between the acceleration potential and potential energy density, andA being the difference between the vector potentials of the acceleration field and the force field. The goal of Hertz to demonstrate a third arrangement of the principles of mechanics...which starts with... time, space and mass has apparently been achieved for relativity and for quantum mechanics, in addition to those classical equations previously found.     

Quantum relativistic action at a distance

A well-known relativistic action at a distance interaction of two unequal masses is altered so as to yield purely Newtonian radial forces with fixed particle rest masses in the system center-of-momentum inertial frame. Although particle masses experience no kinematic mass increase in this frame, speeds are naturally restricted to less than the speed of light. We derive a relation between the center-of-momentum frame total Newtonian energy and the composite rest mass. In a new proper time quantum formalism, we obtain an L²(R⁴ R⁴, C) Hilbert space by varying individual particle rest masses. We propose the use of density operators, recognizing that the auxiliary proper time parameter is not an observable. The quantum formalism is applied to our altered version of the relativistic harmonic oscillator. Our generalized coherent states yield four-dimensional wave packets which follow the correct classical world lines. Appendices contain reviews of classical Hamiltonian reparametrization (incorporating our notion of manifest covariance), and a comparison of this work with the literature.     

A comment on an exact solution with a stiff equation of state

Recently, an analytical stellar model with a stiff equation of state and density distribution= _c (1-r ²/r _o ² ) was presented. We show that such a solution cannot exist.     

Braided matrix structure of the Sklyanin algebra and of the quantum Lorentz group

Braided groups and braided matrices are novel algebraic structures living in braided or quasitensor categories. As such they are a generalization of super-groups and super-matrices to the case of braid statistics. Here we construct braided group versions of the standard quantum groupsU _q (g). They have the same FRT generatorsl ^± but a matrix braided-coproductL=LL, whereL=l ⁺ Sl ^–, and are self-dual. As an application, the degenerate Sklyanin algebra is shown to be isomorphic to the braided matricesBM _q(2); it is a braided-commutative bialgebra in a braided category. As a second application, we show that the quantum doubleD(U _q (sl ₂)) (also known as the quantum Lorentz group) is the semidirect product as an algebra of two copies ofU _q (sl ₂), and also a semidirect product as a coalgebra if we use braid statistics. We find various results of this type for the doubles of general quantum groups and their semi-classical limits as doubles of the Lie algebras of Poisson Lie groups.     

Radiactiv corrections to deeply virtual compton scattering

The non-commuting matrix elements of matrices from the quantum group GL _q(2;C) with q = being the n-th root of unity are given a representation as operators in Hilbert space with help of C ₄ ⁽ⁿ⁾ generalized Clifford algebra generators.The case of q C, |q| = 1 is treated parallelly.     

Hamiltonians defined as quadratic forms

We present a complete mathematical theory of two-body quantum mechanics for a class of potentials which is larger than the usualL ²-classes and which includes potentials with singularities as bad asr ^–2+. The basic idea is to defineH _o +V as a sum of quadratic forms rather than as an operator sum.Based on a thesis submitted to Princeton University in partial fulfillment of the degree of Doctor of Philosophy.     

The classical limit of quantum spin systems

We derive a classical integral representation for the partition function,Z ^Q , of a quantum spin system. With it we can obtain upper and lower bounds to the quantum free energy (or ground state energy) in terms of two classical free energies (or ground state energies). These bounds permit us to prove that when the spin angular momentumJ (but after the thermodynamic limit) the quantum free energy (or ground state energy) is equal to the classical value. In normal cases, our inequality isZ ^C (J)Z ^Q (J)Z ^C (J+1).On leave from the Department of Mathematics, M.I.T., Cambridge, Mass. 02139, USA. Work partially supported by National Science Foundation Grant GP-31674X and by a Guggenheim Memorial Foundation Fellowship.     

A Particle in a Magnetic Field of an Infinite Rectilinear Current

We consider the Schrödinger operator H=(i+A)² in the space L ₂(R ³) with a magnetic potential A created by an infinite rectilinear current. We show that the operator H is absolutely continuous, its spectrum has infinite multiplicity and coincides with the positive half-axis. Then we find the large-time behavior of solutions exp(–i H t)f of the time dependent Schrödinger equation. Our main observation is that a quantum particle has always a preferable (depending on its charge) direction of propagation along the current. Similar result is true in classical mechanics.