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Showing papers on "Ladder operator published in 1998"


Book
01 Jan 1998
TL;DR: In the last 20 years, the study of operator algebras has developed from a branch of functional analysis to a central field of mathematics with applications and connections with different areas in both pure mathematics (foliations, index theory, K-theory, cyclic homology, affine Kac--Moody algesbras, quantum groups, low dimensional topology) and mathematical physics (integrable theories, statistical mechanics, conformal field theories and the string theories of elementary particles).
Abstract: In the last 20 years, the study of operator algebras has developed from a branch of functional analysis to a central field of mathematics with applications and connections with different areas in both pure mathematics (foliations, index theory, K-theory, cyclic homology, affine Kac--Moody algebras, quantum groups, low dimensional topology) and mathematical physics (integrable theories, statistical mechanics, conformal field theories and the string theories of elementary particles). The theory of operator algebras was initiated by von Neumann and Murray as a tool for studying group representations and as a framework for quantum mechanics, and has since kept in touch with its roots in physics as a framework for quantum statistical mechanics and the formalism of algebraic quantum field theory. However, in 1981, the study of operator algebras took a new turn with the introduction by Vaughan Jones of subfactor theory and remarkable connections were found with knot theory, 3-manifolds, quantum groups and integrable systems in statistical mechanics and conformal field theory. The purpose of this book, one of the first in the area, is to look at these combinatorial-algebraic developments from the perspective of operator algebras; to bring the reader to the frontline of research with the minimum of prerequisites from classical theory.

420 citations


Journal ArticleDOI
TL;DR: In this paper, the authors constructed an operator that measures the length of a curve in four-dimensional Lorentzian vacuum quantum gravity, in which an SU(2) connection is diagonal and it is therefore surprising that the operator obtained after regularization is densely defined, does not suffer from factor ordering singularities, and does not require any renormalization.
Abstract: We construct an operator that measures the length of a curve in four-dimensional Lorentzian vacuum quantum gravity. We work in a representation in which an SU(2) connection is diagonal and it is therefore surprising that the operator obtained after regularization is densely defined, does not suffer from factor ordering singularities, and does not require any renormalization. We show that the length operator admits self-adjoint extensions and compute part of its spectrum which, like its companions, the volume and area operators already constructed in the literature, is purely discrete and roughly quantized in units of the Planck length. The length operator contains full and direct information about all the components of the metric tensor which facilitates the construction of so-called weave states which approximate a given classical three-geometry.

236 citations


Journal ArticleDOI
George A. Hagedorn1
TL;DR: In this paper, the authors constructed raising and lowering operators for certain orthonormal bases of L2(R n) consisting of quantum mechanical wave packets that can be used to develop asymptotic expansions for solutions to the time-dependent Schrodinger equation in the semiclassical limit.

174 citations


Journal ArticleDOI
TL;DR: In this paper, a closed formula for the matrix elements of the volume operator for canonical Lorentzian quantum gravity in four space-time dimensions in the continuum in a spin-network basis was derived.
Abstract: We derive a closed formula for the matrix elements of the volume operator for canonical Lorentzian quantum gravity in four space–time dimensions in the continuum in a spin-network basis. We also display a new technique of regularization which is state dependent but we are forced to it in order to maintain diffeomorphism covariance and in that sense it is natural. We arrive naturally at the expression for the volume operator as defined by Ashtekar and Lewandowski up to a state-independent factor.

161 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that for spatial dimensions less than or equal to 3, the normal operator remains a Fourier integral operator, be it not a pseudo-differential operator anymore.

94 citations



Journal ArticleDOI
TL;DR: In this article, a perturbation theory expansion for the projection operator about the parabolic barrier limit and the classical limit was derived, and the expansion was applied to a symmetric and an asymmetric Eckart barrier.
Abstract: The exact quantum expression for the thermal rate of reaction is the trace of a product of two operators It may therefore be written exactly as a phase space integral over the Wigner phase space representations of the two operators The two are a projection operator onto the product’s space, which is difficult to compute, and the symmetrized thermal flux operator, which can be computed using Monte Carlo methods A quantum transition state theory was presented recently, in which the exact projection operator was replaced by its parabolic barrier limit Alternatively, the exact projection operator may be replaced by its classical limit Both approximations give thermodynamic estimates for the quantum rates In this paper, we derive a perturbation theory expansion for the projection operator about the parabolic barrier limit and the classical limit The correction terms are then used to evaluate the leading order corrections to the rate estimates based on the parabolic barrier or classical limits of the projection operator The expansion is applied to a symmetric and an asymmetric Eckart barrier The first two terms in the expansion give excellent results for temperatures above the crossover between quantum tunneling and thermal activation For deep tunneling and asymmetric systems, the use of variational transition state theory, the classical limit, and perturbation theory leads to significant improvement in the estimate of the tunneling rate Multidimensional extensions are presented and discussed

72 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the notion of intertwining operator algebras of central charge is isomorphic to rational genus-zero modular functors (certain analytic partial operads) satisfying a certain generalized meromorphicity property.
Abstract: In [7] and [9], the author introduced the notion of intertwining operator algebra, a nonmeromorphic generalization of the notion of vertex operator algebra involving monodromies. The problem of constructing intertwining operator algebras from representations of suitable vertex operator algebras was solved implicitly earlier in [5]. In the present paper, we generalize the geometric and operadic formulation of the notion of vertex operator algebra given in [3, 4, 11, 12, 8] to the notion of intertwining operator algebra. We show that the category of intertwining operator algebras of central charge is isomorphic to the category of algebras over rational genus-zero modular functors (certain analytic partial operads) of central charge c satisfying a certain generalized meromorphicity property. This result is one main step in the construction of genus-zero conformal field theories from representations of vertex operator algebras announced in [7]. One byproduct of the proof of the present isomorphism theorem is a geometric construction of (framed) braid group representations from intertwining operator algebras and, in particular, from representations of suitable vertex operator algebras.

43 citations


Journal ArticleDOI
TL;DR: In this paper, the Euler angle parametrization of SU(3) is used to find the highest weight state of an arbitrary irreducible representation and a description of the calculation of Clebsch-Gordon coefficients is given.
Abstract: The `D' matrices for all states of the two fundamental representations and octet are given in the Euler angle parametrization of SU(3). The raising and lowering operators are given in terms of linear combinations of the left-invariant vector fields of the group manifold in this parametrization. Using these differential operators the highest weight state of an arbitrary irreducible representation is found and a description of the calculation of Clebsch-Gordon coefficients is given.

38 citations


Journal ArticleDOI
TL;DR: In this paper, the norm of an integral operator occurring in the partial wave decomposition of an operator B introduced by Brown and Ravenhall in a model for relativistic one-electron atoms is determined.
Abstract: The norm of an integral operator occurring in the partial wave decomposition of an operator B introduced by Brown and Ravenhall in a model for relativistic one-electron atoms is determined. The result implies that B is non-negative and has no eigenvalue at 0 when the nuclear charge does not exceed a specified critical value.

38 citations


Journal ArticleDOI
TL;DR: In this paper, nonlinear deformation algebra is realized in a physical system with P?schl-Teller potential, from which the eigenproblem of the system can be exactly solved by the operator method.
Abstract: Nonlinear deformation algebra are realized in a physical system with P?schl-Teller potential. The raising and lowering operators satisfying this algebra are constructed, from which the eigenproblem of the system can be exactly solved by the operator method. The physical meaning of two deforming functions involved in this algebra is also found. In addition, algebra is obtained naturally, and discussions on the coherent state are also made.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the difference between the two operators is due to the non-commutativity that is known to arise in the quantum theory, and the choice of which one to use depends on the context of the physical problem of interest.
Abstract: We show that, apart from the usual area operator of non-perturbative quantum gravity, there exists another, closely related, operator that measures areas of surfaces. Both corresponding classical expressions yield the area. Quantum mechanically, however, the spectra of the two operators are different, coinciding only in the limit when the spins labelling the state are large. We argue that both operators are legitimate quantum operators, and the choice of which one to use depends on the context of the physical problem of interest. Thus, for example, we argue that it is the operator proposed here that is relevant for use in the context of black holes as measuring the area of the black-hole horizon. We show that the difference between the two operators is due to the non-commutativity that is known to arise in the quantum theory. We give a heuristic picture explaining the difference between the two area spectra in terms of quantum fluctuations of the surface whose area is being measured.


Journal ArticleDOI
TL;DR: In this article, a representation-theoretic aspect of a two-dimensional quantum system of a charged particle in a vector potential A, which may be singular on an infinite discrete subset D of R2 is investigated, and an explicit formula is derived for the strongly continuous one-parameter unitary group generated by the self-adjoint operator Pv (the closure of Pv), i.e., the magnetic translation to the direction of the vector v.
Abstract: Some representation-theoretic aspects of a two-dimensional quantum system of a charged particle in a vector potential A, which may be singular on an infinite discrete subset D of R2 are investigated. For each vector v in a set V(D)⊂R2\{0}, the projection Pv of the physical momentum operator P≔p−αA to the direction of v is defined by Pv≔v⋅P as an operator acting in L2(R2), where p=(−iDx,−iDy)[(x,y)∈R2] with Dx (resp., Dy) being the generalized partial differential operator in the variable x (resp., y) and α∈R is a parameter denoting the charge of the particle. It is proven that Pv is essentially self-adjoint and an explicit formula is derived for the strongly continuous one-parameter unitary group {eitPv}t∈R generated by the self-adjoint operator Pv (the closure of Pv), i.e., the magnetic translation to the direction of the vector v. The magnetic translations along curves in R2\D are also considered. Conjugately to Pv and Pw [w∈V(D)], a self-adjoint multiplication operator Qv,w is introduced, which is a ...

Journal ArticleDOI
TL;DR: In this paper, it was shown that the moment operators of any of the phase space observables associated with the number states are the powers of the lowering operator, and that all moment operators are integer valued polynomials of the number operator.

Journal ArticleDOI
TL;DR: A new 5-point finite difference operator is developed for integrated optics simulations and implementation in a beam propagator leads to a highly efficient computation scheme.
Abstract: A new 5-point finite difference operator is developed for integrated optics simulations. Interfaces are taken into account accurately. The eigenvalues of the discretization matrix converge rapidly, according to O(Δx4). Implementation of the operator in a beam propagator leads to a highly efficient computation scheme.

Journal ArticleDOI
TL;DR: Using algebraic tools of supersymmetric quantum mechanics, this article constructed classes of conditionally exactly solvable potentials, being the supersymmetric partners of the linear or radial harmonic oscillator.
Abstract: Using algebraic tools of supersymmetric quantum mechanics, we construct classes of conditionally exactly solvable potentials being the supersymmetric partners of the linear or radial harmonic oscillator. With the aid of the raising and lowering operators of these harmonic oscillators and the SUSY operators, we construct ladder operators for these new conditionally solvable systems. It is found that these ladder operators, together with the Hamilton operator, form a nonlinear algebra, which is of the quadratic and cubic types for the SUSY partners of the linear and radial harmonic oscillators, respectively.

Journal ArticleDOI
TL;DR: In this paper, the failure of conventional perturbation theory due to secularity is considered with renormalization-group tecniques in two operator problems, namely the quantum anharmonic oscillator and quantum parametric resonance.
Abstract: The failure of conventional perturbation theory due to secularity is considered with renormalization-group tecniques in two operator problems. Specifically, some results concerning the quantum anharmonic oscillator and quantum parametric resonance are obtained with a rather modest effort in comparison to other methods.

Journal ArticleDOI
TL;DR: In this paper, a non-unitary representation of the SU(2) algebra for the Dirac equation with a Coulomb potential is introduced. But the authors do not define a new set of operators for the relativistic hydrogen atom.
Abstract: New non-unitary representations of the SU(2) algebra are introduced for the case of the Dirac equation with a Coulomb potential; an extra phase, needed to close the algebra, is also introduced. The new representations does not require integer or half-integer labels. The set of operators defined are used to span the complete space of bound-state eigenstates of the problem thus solving it in an essentially algebraic way. Hydrogen-like atoms are some of the most important quantum systems solved. Even for describing stabilization properties and for testing QED and weak interaction theories a great deal can be done at the relativistic atomic physics level (Greiner 1991, Kylstra et al 1997, Quiney et al 1997). It is, therefore, very important to extend our insight into the properties of hydrogen-like systems. An important tool has been the algebraic properties of the set of operators defining the system; these are not only connected with the corresponding group and its symmetry algebra but often offer simplified methods for carrying out some calculations. It is the purpose of this letter to define a new set of operators for the Dirac relativistic hydrogen atom. This comprises a non-unitary representation of the SU(2) algebra and defines ladder operators for the problem. An extra phase is needed to close the algebra but this allows us to solve the Dirac hydrogen atom in a neat algebraic way. The Dirac Hamiltonian for a hydrogen-like atom is

Journal ArticleDOI
TL;DR: In this paper, the degeneracy structure of the eigenspace of the N-particle Calogero-Sutherland model is studied from an algebraic point of view, and suitable operators satisfying SU(2) algebras and acting on the degenerate eIGenspace are explicitly constructed for the twoparticle case and then appropriately generalized to the Nparticle model.
Abstract: The degeneracy structure of the eigenspace of the N-particle Calogero–Sutherland model is studied from an algebraic point of view. Suitable operators satisfying SU(2) algebras and acting on the degenerate eigenspace are explicitly constructed for the two-particle case and then appropriately generalized to the N-particle model. The raising and lowering operators of these algebras connect the states, in a subset of the degenerate eigenspace, with each other.

Posted Content
TL;DR: In this article, it was shown that if there exists a operator in the form of a sum of two vertex operators which has the simplest correlation functions with the quantized screen operator, namely a function with one pole and one zero, then, the screen operator and this operator are uniquely determined, and that this operator is quantized virasoro algebra.
Abstract: Starting from bosonization, we study the operator that commute or commute up-to a total difference with of any quantized screen operator of a free field. We show that if there exists a operator in the form of a sum of two vertex operators which has the simplest correlation functions with the quantized screen operator, namely a function with one pole and one zero, then, the screen operator and this operator are uniquely determined, and this operator is the quantized virasoro algebra. For the case when the screen is a fermion, there are a family of this kind of operator, which give new algebraic structures. Similarly we study the case of two quantized screen operator, which uniquely gives us the quantized W-algebra corresponding to $sl(3)$ for the generic case, and a new algebra, which is a quantized W-algebra corresponding to ${\frak sl}(2,1)$, for the case that one of the two screening operators is or both are fermions.

Book
01 Sep 1998
TL;DR: Bosonic construction of symplectic affine Kac-Moody algebras and modules as discussed by the authors is a well-known technique for algebraic construction of vertex operator para-algeses.
Abstract: Introduction Bosonic construction of symplectic affine Kac-Moody algebras Bosonic construction of symplectic vertex operator algebras and modules Bosonic construction of vertex operator para-algebras Appendix Bibliography.

Journal ArticleDOI
TL;DR: In this article, the authors studied the algebra of difference operators that commute with the two-body Ruijsenaars operator, a $q$-deformation of the Lam\'e differential operator for generic values of the deformation parameter.
Abstract: We study the algebra of difference operators that commute with the two-body Ruijsenaars operator, a $q$-deformation of the Lam\'e differential operator, for generic values of the deformation parameter. The algebra is commutative. It is the algebra of polynomial functions on an affine hyperelliptic curve $Y^2=P(X^2)$. We also compute the difference Galois group of the eigenvalue problem.


Journal ArticleDOI
TL;DR: In this article, the authors apply the time-dependent projection operator approach to a small system coupled to a reservoir and derive a closed nonlinear equation of motion for the reduced statistical operator of the small system.
Abstract: The time-dependent projection operator technique according to Willis and Picard (Phys. Rev. A 9 (1974) 1343) offers a unique quantum statistical description of two interacting subsystems. The technique is used here to go beyond the standard quantum master equation (QME) for a small system coupled to a reservoir. Applying the time-dependent projection operator approach, one is able to overcome a perturbational treatment of the system–reservoir coupling and can incorporate, (i) how the dynamics of the system may drive the reservoir out of the equilibrium, and (ii) how this nonequilibrium state reacts back on the system dynamics. The case of a reservoir of harmonic oscillators coupled linearly by its coordinates to the small system is studied in detail. The derivation of a closed nonlinear equation of motion for the reduced statistical operator of the small system is demonstrated. The method is used to describe the motion of a quantum particle in a molecular system, e.g. a tunneling electron or an exciton, which interacts strongly with its environment.

Journal ArticleDOI
TL;DR: In this paper, the authors studied the self-adjointness of the Liouvillian of a symmetric operator and showed that quantum theory in Hilbert and Liouville spaces is not equivalent.
Abstract: We study the self-adjointness of the Liouvillianof a symmetric operator. We also discuss some cases ofthe spectrum of the Liouville operator of a self-adjointHamiltonian with purely continuous singular spectrum. The presence of an absolutelycontinuous part for the spectrum of Liouvillianscorresponding to Hamiltonians with purely continuoussingular spectrum shows that quantum theory in Hilbertand Liouville spaces is not equivalent.

Posted Content
TL;DR: In this paper, the Dirac operator defined over a conformal surface immersed in R^4 was determined to be the Lax operator of the nonlinear Schrodinger and the modified Novikov-Veselov equation.
Abstract: In the previous report (J. Phys. A (1997) 30 4019-4029), I showed that the Dirac operator defined over a conformal surface immersed in R^3 is identified with the Dirac operator which is generalized the Weierstrass- Enneper equation and Lax operator of the modified Novikov-Veselov (MNV) equation. In this article, I determine the Dirac operator defined over a conformal surface immersed in R^4, which is reduced to the Lax operators of the nonlinear Schrodinger and the MNV equations by taking appropriate limits. Thus the Dirac operator might be the Lax operator of (2+1)- dimensional soliton equation.

Journal ArticleDOI
TL;DR: Parastatistics, defined as triple operator algebras represented on Fock space, are unified in a simple way using the transition number operators as discussed by the authors, expressed as a normal ordered expansion of creation and annihilation operators.
Abstract: Parastatistics, defined as triple operator algebras represented on Fock space, are unified in a simple way using the transition number operators. They are expressed as a normal ordered expansion of creation and annihilation operators. We discuss several examples of parastatistics, particularly Okubo's and Palev's parastatistics connected to many-body Wigner quantum systems and relate them to the notion of extended Haldane statistics.

Journal ArticleDOI
TL;DR: In this article, the authors discuss the solvability and approximation of operator equations on the half line, and show that the approximation can be computed in polynomial time on the line.
Abstract: This paper discusses the solvability and approximation of operator equations on the half line.

Journal ArticleDOI
TL;DR: In this paper, a comparison between different lattice regularizations of the Dirac operator for massless fermions in the framework of the single and two flavor Schwinger model is performed.