Zero Modes’ Fusion Ring and Braid Group Representations for the Extended Chiral <Emphasis Type="Italic">su</Emphasis>(2) WZNW Model

A zero modes’ Fock space is constructed for the extended chiral WZNW model. It gives room to a realization of the fusion ring of representations of the restricted quantum universal enveloping algebra at an even root of unity, and of its infinite dimensional extension by the Lusztig operators We provide a streamlined derivation of the characteristic equation for the Casimir invariant from the defining relations of A central result is the characterization of the Grothendieck ring of both and in Theorem 3.1. The properties of the fusion ring in are related to the braiding properties of correlation functions of primary fields of the conformal current algebra model.      

Cohomology of $${\mathfrak{osp}}(1|2)$$ Acting on Linear Differential Operators on the Supercircle S1|1

We compute the first cohomology spaces of the Lie superalgebra with coefficients in the superspace of linear differential operators acting on weighted densities on the supercircle S ^1|1. The structure of these spaces was conjectured in (Gargoubi et al. in Lett Math Phys 79:5165, 2007). In fact, we prove here that the situation is a little bit more complicated.      

Spectrum of Time Operators

Let H be a self-adjoint operator on a complex Hilbert space . A symmetric operator T on is called a time operator of H if, for all , (D(T) denotes the domain of T) and . In this paper, spectral properties of T are investigated. The following results are obtained: (i) If H is bounded below, then σ(T), the spectrum of T, is either (the set of complex numbers) or . (ii) If H is bounded above, then is either or . (iii) If H is bounded, then . The spectrum of time operators of free Hamiltonians for both nonrelativistic and relativistic particles is exactly identified. Moreover spectral analysis is made on a generalized time operator. This work is supported by the Grant-in-Aid No.17340032 for Scientific Research from the JSPS.     

Quantization of Semi-Classical Twists and Noncommutative Geometry

A problem of defining the quantum analogues for semi-classical twists in U()[[t]] is considered. First, we study specialization at q = 1 of singular coboundary twists defined in Uq ())[[t]] for g being a nonexceptional Lie algebra, then we consider specialization of noncoboundary twists when = and obtain q-deformation of the semiclassical twist introduced by Connes and Moscovici in noncommutative geometry. Mathematics Subject Classification: 16W30, 17B37, 81R50     

Ion Mobility Measurements and Ion Chemical Reaction Studies at Heavy Elements in a Buffer Gas Cell

Drift time measurements of ions in a buffer gas cell filled with argon have been performed from which changes of the ion mobility and ionic radii for various heavy elements and their compounds were determined. The ionic radius of americium shrinks by (3.1 1.3)% with respect to that of plutonium, and an increase of the radius by (28 2)% of plutonium oxide with respect to plutonium was found. Ion chemical reactions of erbium ions were studied online in an argon buffer gas cell to which the reaction gases oxygen (O) and methane (CH) were added. The erbium ions were implanted into the buffer gas cell with an energy of 50 MeV. The online measured reaction constant = (3.2 0.4) 10 cm/(molecule s) for the reaction Er + O ErO + O agrees with a reference measurement = (3.6 0.4) 10 cm/(molecule s), performed with a Fourier-Transform-Mass-Spectrometer.     

Non-relativistic Limit of a Dirac Polaron in Relativistic Quantum Electrodynamics

A quantum system of a Dirac particle interacting with the quantum radiation field is considered in the case where no external potentials exist. Then the total momentum of the system is conserved and the total Hamiltonian is unitarily equivalent to the direct integral of a family of self-adjoint operators acting in the Hilbert space , where is the Hilbert space of the quantum radiation field. The fiber operator is called the Hamiltonian of the Dirac polaron with total momentum . The main result of this paper is concerned with the non-relativistic (scaling) limit of . It is proven that the non-relativistic limit of yields a self-adjoint extension of a Hamiltonian of a polaron with spin 1/2 in non-relativistic quantum electrodynamics.     

Dynamical Noncommutativity and Noether Theorem in Twisted $${\phi^{\star 4}}$$ Theory

A -product is defined via a set of commuting vector fields , and used in a theory coupled to the fields. The -product is dynamical, and the vacuum solution , reproduces the usual Moyal product. The action is invariant under rigid translations and Lorentz rotations, and the conserved energy–momentum and angular momentum tensors are explicitly derived.      

On the Uniqueness of Weak Weyl Representations of the Canonical Commutation Relation

Let (T, H) be a weak Weyl representation of the canonical commutation relation (CCR) with one degree of freedom. Namely T is a symmetric operator and H is a self-adjoint operator on a complex Hilbert space satisfying the weak Weyl relation: for all (the set of real numbers), e^−itH D(T) ⊂ D(T) (i is the imaginary unit and D(T) denotes the domain of T) and . In the context of quantum theory where H is a Hamiltonian, T is called a strong time operator of H. In this paper we prove the following theorem on uniqueness of weak Weyl representations: Let be separable. Assume that H is bounded below with and , where is the set of complex numbers and, for a linear operator A on a Hilbert space, σ(A) denotes the spectrum of A. Then ( is the closure of T) is unitarily equivalent to a direct sum of the weak Weyl representation on the Hilbert space , where is the multiplication operator by the variable and with . Using this theorem, we construct a Weyl representation of the CCR from the weak Weyl representation . This work is supported by the Grant-in-Aid No.17340032 for Scientific Research from Japan Society for the Promotion of Science (JSPS).     

Elliptic Equation with Singular Terms on Riemannian Manifold

In this paper we deal with the following equation: on a three-dimensional Riemannian manifold . We assume that the volume of Σ, the norm , and are small enough. Using a rather simple argument we show the existence of solution to the problem. Dedicated to Gosia and Basia.     

Renormalized Traces and Cocycles on the Algebra of S 1-Pseudo-differential Operators

Using renormalized (or weighted) traces of classical pseudo-differential operators and calculus on formal symbols. We exhibit three cocycles on the Lie algebra of classical pseudo-differential operators $Cl(S^1,\mathbb{C}^n)Using renormalized (or weighted) traces of classical pseudo-differential operators and calculus on formal symbols. We exhibit three cocycles on the Lie algebra of classical pseudo-differential operators acting on . We first show that the Schwinger functional associated to the Dirac operator is a cocycle on , and not only on a restricted algebra Then, we investigate two bilinear functionals and , which satisfies We show that and are two cocycles in , and and have the same nonvanishing cohomology class. We finaly calculate on classical pseudo-differential operators of order 1 and on differential operators of order 1, in terms of partial symbols. By this last computation, we recover the Virasoro cocyle and the K?hler form of the loop group. Mathematics Subject Classification (1991). 47G30, 47N50     

Integrable Discrete Nets in Grassmannians

We consider discrete nets in Grassmannians , which generalize Q-nets (maps with planar elementary quadrilaterals) and Darboux nets (-valued maps defined on the edges of such that quadruples of points corresponding to elementary squares are all collinear). We give a geometric proof of integrability (multidimensional consistency) of these novel nets, and show that they are analytically described by the noncommutative discrete Darboux system.      

Logarithmic Sobolev inequality for lattice gases with mixing conditions

Let denote the grand canonical Gibbs measure of a lattice gas in a cube of sizeL with the chemical potential and a fixed boundary condition. Let be the corresponding canonical measure defined by conditioning on . Consider the lattice gas dynamics for which each particle performs random walk with rates depending on near-by particles. The rates are chosen such that, for everyn andL fixed, is a reversible measure. Suppose that the Dobrushin-Shlosman mixing conditions holds for forall chemical potentials . We prove that for any probability densityf with respect to ; here the constant is independent ofn orL andD denotes the Dirichlet form of the dynamics. The dependence onL is optimal.Research partially supported by U.S. National Science Foundations grant 9403462, Sloan Foundation Fellowship and David and Lucile Packard Foundation Fellowship.     

Hidden Grassmann Structure in the XXZ Model

For the critical XXZ model, we consider the space of operators which are products of local operators with a disorder operator. We introduce two anti-commutative families of operators which act on . These operators are constructed as traces over representations of the q-oscillator algebra, in close analogy with Baxter’s Q-operators. We show that the vacuum expectation values of operators in can be expressed in terms of an exponential of a quadratic form of . On leave of absence from Skobeltsyn Institute of Nuclear Physics, MSU, 119992, Moscow, Russia Membre du CNRS     

A Space–Time Integral Estimate For A Large Data Semi-linear Wave Equation on the Schwarzschild Manifold

We consider the wave equation with in . The wave equation on a spherically symmetric manifold with a single closed geodesic surface or on the exterior of the Schwarzschild manifold can be reduced to this form. Using a smoothed Morawetz estimate which does not require a spherical harmonic decomposition, we show that there is decay in for initial data in the energy class, even if the initial data is large. This requires certain conditions on the potentials V, V _L and f. We show that a key condition on the weight in the smoothed Morawetz estimate can be reduced to an ODE condition, which is verified numerically.      

Trigonometric Dynamical <Emphasis Type="Italic">r</Emphasis>-Matrices Over Poisson Lie Base

Let be a finite dimensional complex Lie algebra and a Lie subalgebra equipped with the structure of a factorizable quasitriangular Lie bialgebra. Consider the Lie group Exp with the Semenov-Tjan-Shansky Poisson bracket as a Poisson Lie manifold for the double Lie bialgebra . Let be an open domain parameterizing a neighborhood of the identity in Exp by the exponential map. We present dynamical r-matrices with values in over the Poisson Lie base manifold .*This research is partially supported by the Emmy Noether Research Institute for Mathematics, the Minerva Foundation of Germany, the Excellency Center Group Theoretic Methods in the study of Algebraic Varieties of the Israel Science foundation, and by the RFBR grant no. 03-01-00593.     

Singular Spectrum Near a Singular Point of Friedrichs Model Operators of Absolute Type

In we consider a family of selfadjoint operators of the Friedrichs model: . Here is the operator of multiplication by the corresponding function of the independent variable , and (perturbation) is a trace-class integral operator with a continuous Hermitian kernel satisfying some smoothness condition. These absolute type operators have one singular point of order . Conditions on the kernel are found guaranteeing the absence of the point spectrum and the singular continuous one of such operators near the origin. These conditions are actually necessary and sufficient. They depend on the finiteness of the rank of a perturbation operator and on the order of singularity . The sharpness of these conditions is confirmed by counterexamples.     

Universal R-Matrix for Esoteric Quantum Groups

The universal R-matrix for a class of esoteric (nonstandard) quantum groups q(gl(2N+1)) is constructed as a twisting of the universal R-matrix S of the Drinfeld–Nimbo quantum algebras. The main part of the twisting cocycle is chosen to be the canonical element of an appropriate pair of separated Hopf subalgebras (quantized Borel's (N) q (gl(2N+1))), providing the factorization property of . As a result, the esoteric quantum group generators can be expressed in terms of Drinfeld and Jimbo.     

A Uniform Quantum Version of the Cherry Theorem

Consider in the operator family . P ₀ is the quantum harmonic oscillator with diophantine frequency vector ω, F ₀ a bounded pseudodifferential operator with symbol decreasing to zero at infinity in phase space, and . Then there exist independent of and an open set such that if and , the quantum normal form near P ₀ converges uniformly with respect to . This yields an exact quantization formula for the eigenvalues, and for the classical Cherry theorem on convergence of Birkhoff’s normal form for complex frequencies is recovered. Partially supported by PAPIIT-UNAM IN106106-2.     

Why is \mathcal{CPT} Fundamental?

Lüders and Pauli proved the theorem based on Lagrangian quantum field theory almost half a century ago. Jost gave a more general proof based on “axiomatic” field theory nearly as long ago. The axiomatic point of view has two advantages over the Lagrangian one. First, the axiomatic point of view makes clear why is fundamental—because it is intimately related to Lorentz invariance. Secondly, the axiomatic proof gives a simple way to calculate the transform of any relativistic field without calculating , and separately and then multiplying them. The purpose of this pedagogical paper is to “deaxiomatize” the theorem by explaining it in a few simple steps. We use theorems of distribution theory and of several complex variables without proof to make the exposition elementary.