Showing papers in "Letters in Mathematical Physics in 1995"
TL;DR: In this paper, it was shown that the generators of two discrete Heisenberg-Weyl groups with irrational rotation numbers θ and −1/θ generate the whole algebraB of operators on L2(R).
Abstract: It is shown that the generators of two discrete Heisenberg-Weyl groups with irrational rotation numbers θ and −1/θ generate the whole algebraB of operators onL2(R). The natural action of the modular group inB is implied. Applications to dynamical algebras appearing in lattice regularization and some duality principles are discussed.
405 citations
TL;DR: In this paper, the geometric data is encoded by a triple (A. H. D) of an algebra, represented in a Hilbert space, with selfadjoint operator D. This point of view is dictated by the general framework of noncommutative geometry and allows us to use geometric ideas in many situations beyond Riemannian geometry.
Abstract: In this Letter, we develop geometry from a spectral point of view, the geometric data being encoded by a triple (A. H. D.) of an algebraA represented in a Hilbert spaceH with selfadjoint operatorD. This point of view is dictated by the general framework of noncommutative geometry and allows us to use geometric ideas in many situations beyond Riemannian geometry.
205 citations
TL;DR: In this article, the fuzzy analog of spinor bundles over the sphere on which the noncommutative analog of the Dirac operator acts was introduced, and the complete set of eigenstates including zero modes were constructed.
Abstract: We introduce the fuzzy analog of spinor bundles over the sphere on which the noncommutative analog of the Dirac operator acts. We construct the complete set of eigenstates including zero modes. In the commutative limit, we recover known results.
159 citations
TL;DR: In this article, the supersymmetric analogies of Darboux-Levi transformations are established for the spectral problems of Manin-Radul-Mathieu sKdV and Manin Radul-Srivastava skdV systems.
Abstract: We consider the Darboux type transformations for the spectral problems of supersymmetric KdV systems. The supersymmetric analogies of Darboux and Darboux-Levi transformations are established for the spectral problems of Manin-Radul-Mathieu sKdV and Manin-Radul sKdV. Several Backlund transformations are derived for the MRM sKdV and MR sKdV systems.
101 citations
TL;DR: In this article, a notion of representation for a star product (equipped with a star compatible trace) is defined, and it is shown that every compact pre-quantizable manifold admits a representable star product.
Abstract: We define in this Letter, a notion of ‘representation’ for a star product (equipped with a star-compatible trace) and show that every compact pre-quantizable symplectic manifold admits a representable star product.
97 citations
TL;DR: In this paper, the authors considered a system of finitely many nonrelativistic electrons bound in an atom or molecule which are coupled to the electromagnetic field via minimal coupling or the dipole approximation.
Abstract: We consider a system of finitely many nonrelativistic electrons bound in an atom or molecule which are coupled to the electromagnetic field via minimal coupling or the dipole approximation Among a variety or results, we give sufficient conditions for the existence of a ground state (an eigenvalue at the bottom of the spectrum) and resonances (eigenvalues of a complex dilated Hamiltonian) of such a system We give a brief outline of the proofs of these statements which will appear at full length in a later work
91 citations
TL;DR: In this paper, the multiplicities of weights in irreducible O(n + 1) -modules were derived in terms of the geometry of the crystal graph attached to corresponding O(U_q (n + n + 1} ) -modules.
Abstract: We give an expression of theq-analogues of the multiplicities of weights in irreducible\(\mathfrak{s}\mathfrak{l}_{n + 1} - modules\) in terms of the geometry of the crystal graph attached to corresponding\(U_q (\mathfrak{s}\mathfrak{l}_{n + 1} ) - modules\). As an application, we describe multivariate polynomial analogues of the multiplicities of the zero weight, refining Kostant's generalized exponents.
89 citations
TL;DR: In this paper, a formal *-product on the algebra of smooth functions on the bounded symmetric domains was defined and proved convergent on a dense subalgebra of the algebra.
Abstract: We use Berezin's dequantization procedure to define a formal *-product on the algebra of smooth functions on the bounded symmetric domains. We prove that this formal *-product is convergent on a dense subalgebra of the algebra of smooth functions.
78 citations
TL;DR: In this article, a cohomology and deformation theory for Poisson algebras is developed, starting with the more general concept of a Leibniz pair, namely of an associative algebra A together with a Lie algebra L mapped into the derivations of A.
Abstract: Cohomology and deformation theories are developed for Poisson algebras starting with the more general concept of a Leibniz pair, namely of an associative algebraA together with a Lie algebraL mapped into the derivations ofA. A bicomplex (with both Hochschild and Chevalley-Eilenberg cohomologies) is essential.
67 citations
TL;DR: In this paper, it was shown that to within normalization there is a unique linear connection on the quantum plane and there is no metric in non-commutative non-convex geometry.
Abstract: A general definition has been proposed recently of a linear connection and a metric in noncommutative geometry. It is shown that to within normalization there is a unique linear connection on the quantum plane and there is no metric.
65 citations
TL;DR: In this article, the minimal affinizations of finite-dimensional complex simple Lie algebras with respect to the set of affinization types defined by Chari are described.
Abstract: If U
q
(g) is a finite-dimensional complex simple Lie algebra, an affinization of a finite-dimensional irreducible representationV of U
q
(g) is a finite-dimensional irreducible representation
$$\hat V$$
of U
q
(ĝ) which containsV with multiplicity one, and is such that all other U
q
(g)-types in
$$\hat V$$
have highest weights strictly smaller than that ofV. There is a natural partial ordering
$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \prec } $$
on the set of affinizations, defined by Chari. In this Letter, we describe the minimal affinizations, with respect to
$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \prec } $$
, when g is not simply-laced.
TL;DR: In this article, a constrained KP hierarchy is discussed that was recently suggested by Aratynet et al. This hierarchy is a restriction of the KP to a submanifold of operators which can be represented as a ratio of two purely differential operators of prescribed orders.
Abstract: A constrained KP hierarchy is discussed that was recently suggested by Aratynet al. and by Bonoraet al. This hierarchy is a restriction of the KP to a submanifold of operators which can be represented as a ratio of two purely differential operators of prescribed orders. Explicit formulas for action of vector fields on these two differential operators are written which gives a new description of the hierarchy and provides a new, more constructive proof of compatibility of the constraint with the hierarchy. Also, the Poisson structure of the constrained hierarchy is discussed.
TL;DR: In this article, the authors reviewed the evidence for S duality and some speculations are presented, and showed that S dualities are rather analogous to T duality symmetries, which hold perturbatively.
Abstract: String theory appears to admit a group of discrete field transformations — calledS dualities — as exact nonperturbative quantum symmetries. Mathematically, they are rather analogous to the better-knownT duality symmetries, which hold perturbatively. In this Letter the evidence forS duality is reviewed and some speculations are presented.
TL;DR: In this article, the Lax pair for the quotient-difference algorithm in the natural Rutishauser variables was constructed using orthogonal polynomial theory, and a Backlund transform that relates these two different discrete-time Toda systems was derived.
Abstract: Using orthogonal polynomial theory, we construct the Lax pair for the quotient-difference algorithm in the natural Rutishauser variables. We start by considering the family of orthogonal polynomials corresponding to a given linear form. Shifts on the linear form give rise to adjacent families. A compatible set of linear problems is made up from two relations connecting adjacent and original polynomials. Lax pairs for several initial boundary-value problems are derived and we recover the discrete-time Toda chain equations of Hirota and of Suris. This approach allows us to derive a Backlund transform that relates these two different discrete-time Toda systems. We also show that they yield the same bilinear equation up to a gauge transformation. The singularity confinement property is discussed as well.
TL;DR: In this paper, the existence of bound states in a plane quantum waveguide is proved under weak conditions: within a bounded set a more general shape than a curved parallel strip is admitted and the curvature of the reference curve need not be differentiable.
Abstract: The existence of bound states in a plane quantum waveguide is proved under weak conditions: Within a bounded set a more general shape than a curved parallel strip is admitted and the curvature of the reference curve need not be differentiable. Furthermore, no upper bound for the width of the strip is required.
TL;DR: In this article, Brzezinski and Majid introduced quantum spheresSq2n-1, projective quantum spaces ℂℙqn- 1, and quantum Grassmann manifoldsGk(ℂqn) to the standard SUq(n) R-matrices.
Abstract: Associated to the standard SUq(n) R-matrices, we introduce quantum spheresSq2n-1, projective quantum spaces ℂℙqn-1, and quantum Grassmann manifoldsGk(ℂqn). These algebras are shown to be homogeneous spaces of standard quantum groups and are also quantum principle bundles in the sense of T. Brzezinski and S. Majid.
TL;DR: In this paper, it was shown that an algebraic structure proposed by Gel'fand and Dorfman in studying Hamiltonian operators is equivalent to an associative algebra with a derivation under a unitary condition.
Abstract: We prove that an algebraic structure proposed by Gel'fand and Dorfman in studying Hamiltonian operators is equivalent to an associative algebra with a derivation under a unitary condition.
TL;DR: In this article, the authors explore the set of noncommutative associative algebra structures on the Schwartz-Weil algebra of any locally compact separable Abelian group, where the products are defined by Fourier integral operators.
Abstract: The theme of doing quantum mechanics on all Abelian groups goes back to Schwinger and Weyl. This theme was studied earlier from the point of view of approximating quantum systems in infinite-dimensional spaces by those associated to finite Abelian groups. This Letter links this theme to deformation quantization, and explores the set of noncommutative associative algebra structures on the Schwartz-Weil algebra of any locally compact separable Abelian group. If the group is a vector space of even dimension over a non-Archimedean local fieldK, there exists a family of noncommutative (Moyal) structures parametrized by the local field and containing membersarbitrarily close to the classical one, although the classical algebra is rigid in the sense of deformation theory. The-products are defined by Fourier integral operators. The problem of constructing sucharithmetic Moyal structures on the algebra of Schwartz-Bruhat functions on manifolds that are locally likeK
2n
is raised.
TL;DR: In this article, some algebraic invariants associated with Bell's inequalities are defined for inclusions of von Neumann algebras and studied within the context of general algebraic quantum theory.
Abstract: Some algebraic invariants associated with Bell's inequalities are defined for inclusions of von Neumann algebras and studied within the context of general algebraic quantum theory. More special results are proven for quantum field theory which establish that these invariants take infinitely many values. Sharp short-distance bounds on the Bell correlations are also demonstrated in the context of relativistic quantum field theory.
TL;DR: In this article, the authors show that the Drinfeld-Sokolov reduction is equivalent to a bi-Hamiltonian reduction, in the sense that these two reductions, although different, lead to the same reduced Poisson structure.
Abstract: We show that the Drinfeld-Sokolov reduction is equivalent to a bi-Hamiltonian reduction, in the sense that these two reductions, although different, lead to the same reduced Poisson (more correctly, bi-Hamiltonian) structure. In order to do this, we heavily use the fact that they are both particular cases of a Marsden-Ratiu reduction.
TL;DR: In this paper, a line-narrowing effect in coincidence interferometry yielding subnatural resolution of atomic transition frequencies is proposed and analyzed using second-order photon correlation properties of the radiation field.
Abstract: A new line-narrowing effect in coincidence interferometry yielding subnatural resolution of atomic transition frequencies is proposed and analyzed. The approach utilizes second-order photon correlation properties of the radiation field. This is in contrast to the first-order measurements associated with time delay spectroscopy, which is known to yield subnatural resolution. Connections between the two techniques are investigated.
TL;DR: In this paper, a complete classification of the class of connected, simply connected Lie groups whose coadjoint orbits are of dimension smaller than or equal to two is given; see Section 2.1.
Abstract: We give a complete classification of the class of connected, simply connected Lie groups whose coadjoint orbits are of dimension smaller or equal to two.
TL;DR: In this paper, an algebraic proof of the spin-statistics connection for the parabosonic and parafermionic quantum topological charges of a theory of local observables with a modular P1 CT-symmetry was given.
Abstract: We give an algebraic proof of the spin-statistics connection for the parabosonic and parafermionic quantum topological charges of a theory of local observables with a modular P1 CT-symmetry. The argument avoids the use of the spinor calculus and also works in 1 + 2 dimensions. It is expected to be a progress towards a general spin-statistics theorem including also (1 + 2)-dimensional theories with braid group statistics.
TL;DR: Fock representations for quantum fields which obey generalized statistics are explicitly constructed and the main features of these representations are investigated in this article, where they are shown to be robust to quantum perturbations.
Abstract: Fock representations for quantum fields which obey generalized statistics are explicitly constructed. The main features of these representations are investigated.
TL;DR: In this paper, a generalized Hitchin system on a cotangent bundle to the moduli of holomorphic principle bundles and an elliptic module is derived, which is a particular example of higher quantum Hitchin integrals which, upon reducing from GHS phase space to the Hitchin phase space, gives a particular extension of the Beilinson-Drinfeld commutative algebra of differential operators on the modulus of holomorph bundles.
Abstract: The Knizhnik–Zamolodchikov–Bernard (KZB) equation on an elliptic curve with a marked point is derived by classical Hamiltonian reduction and further quantization. We consider classical Hamiltonian systems on a cotangent bundle to the loop group L(GL(N, C)) extended by the shift operators, to be related to the elliptic module. After reduction, we obtain a Hamiltonian system on a cotangent bundle to the moduli of holomorphic principle bundles and an elliptic module. It is a particular example of generalized Hitchin systems (GHS) which are defined as Hamiltonian systems on cotangent bundles to the moduli of holomorphic bundles and to the moduli of curves. They are extensions of the Hitchin systems by the inclusion the moduli of curves. In contrast with the Hitchin systems, the algebra of integrals are noncommutative on GHS. We discuss the quantization procedure in our example. The quantization of the quadratic integral leads to the KZB equation. We present an explicit form of higher quantum Hitchin integrals which, upon reducing from GHS phase space to the Hitchin phase space, gives a particular example of the Beilinson–Drinfeld commutative algebra of differential operators on the moduli of holomorphic bundles.
TL;DR: In this article, it was shown that two definitions of dynamical entropy for noncommutative shifts give different results, and two definitions for non-commutativity shifts are defined separately.
Abstract: It is shown that two definitions of dynamical entropy for noncommutative shifts give different results.
TL;DR: In this article, it was shown that the distributions defined on the Gelfand-Shilov spaces with β < 1 and hence more singular than hyperfunctions retain the angular localizability property.
Abstract: We prove that the distributions defined on the Gelfand-Shilov spacesSαβ withβ < 1 and, hence, more singular than hyperfunctions, retain the angular localizability property. Specifically, they have uniquely determined support cones. This result enables one to develop a distribution-theoretic technique suitable for the consistent treatment of quantum fields with arbitrarily singular ultraviolet and infrared behavior. The proof covering the most general and difficult caseβ = 0 is based on the use of the theory of plurisubharmonic functions and Hormander'sL2-estimates.
TL;DR: The representation theory of finite W-algebras has been studied in this paper, where it is shown that the maximal submodule of a Verma module is not generated by singular vectors only.
Abstract: We study the representation theory of finiteW-algebras. After introducing parabolic subalgebras to describe the structure ofW-algebras, we define the Verma modules and give a conjecture for the Kac determinant. This allows us to find the completely degenerate representations of the finiteW-algebras. To extract the irreducible representations we analyse the structure of singular and subsingular vectors and find that, forW-algebras, in general the maximal submodule of a Verma module is not generated by singular vectors only. Surprisingly, the role of the (sub)singular vectors can be encapsulated in terms of a ‘dual’ analogue of the Kazhdan-Lusztig theorem for simple Lie algebras. These involve dual relative Kazhdan-Lusztig polynomials. We support our conjectures with some examples, and briefly discuss applications and the generalisation to infiniteW-algebras.
TL;DR: In this paper, new coherent states of the q-Weyl algebra were constructed, which are related either to the bilateral basic hypergeometric series or to some integrals over them.
Abstract: New coherent states of theq-Weyl algebraAA
† −qA
†
A = 1,0
TL;DR: In this article, it was shown that the Toda chain determines the group of discrete symmetries of the two-dimensional matrix nonlinear Schrodinger equation (the matrix generalization of the Davey-Stewartson system).
Abstract: It is shown that the two-dimensional matrix Toda chain determines the group of discrete symmetries of the two-dimensional matrix nonlinear Schrodinger equation (the matrix generalization of the Davey-Stewartson system). The general solution of this chain with definite boundary conditions is obtained in explicit form.