Showing papers on "Ladder operator published in 1986"

Operator Algebra and Correlation Functions in the Two-Dimensional Wess-Zumino SU(2) x SU(2) Chiral Model

Abstract: A conformally invariant solution of the Wess-Zumino SU(2)xSU(2) chiral model in two-dimensional space-time is studied. The anomalous dimensions of all fields, structure constants of the operator algebra and four-point correlation functions are calculated exactly.

A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors

Abstract: On a Riemannian spin manifold, we give a lower bound for the square of the eigenvalues of the Dirac operator by the smallest eigenvalue of the conformal Laplacian (the Yamabe operator). We prove, in the limiting case, that the eigenspinor field is a killing spinor, i.e., parallel with respect to a natural connection. In particular, if the scalar curvature is positive, the eigenspinor field is annihilated by harmonic forms and the metric is Einstein.

Phase in quantum optics

Abstract: Dirac's prescription for quantisation does not lead to a unique phase operator for the electromagnetic field. The authors consider the commonly employed phase operators due to Susskind and Glogower (1964) and their extension to unitary exponential phase operators. However, they find that phase measuring experiments respond to a different operator. They discuss the form of the measured phase operator and its properties.

Eigenvalue bounds for the Dirac operator

Abstract: A natural question in the study of geometric operators is that of how much information is needed to estimate the eigenvalues of an operator. For the square of the Dirac operator, such a question has at least peripheral physical import. When coupled to gauge fields, the lowest eigenvalue is related to chiral symmetry breaking. In the pure metric case, lower eigenvalue estimates may help to give a sharper estimate of the ADM mass of an asymptotically flat spacetime with black holes. We use three tools to estimate the eigenvalues of the square of the (purely metric) Dirac operator the conformal covariance of the operator, a patching method and a heat kernel bound.

Harmonic oscillator with complex frequency

Abstract: In the present paper we study the problem of the harmonic oscillator with complex frequency. A special case of this problem is the determination of the eigenvalues and eigenfunctions of the squeeze operator in quantum optics. The Hamilton operator of the complex harmonic oscillator is non-Hermitian and its study leads to the Lie-admissible theory. Because of the complex frequency the eigenvalues of the energy are complex numbers and the partition function of Boltzman and the free energy of Helmholtz are complex functions. Especially the imaginary part of the free energy describes the metastable states.

Hypervirial theorem and ladder operators. Recurrence relations for harmonic oscillator integrals

Abstract: A second-quantization formalism combined with a hypervirial theorem is used to derive new recurrence relations for one-dimensional harmonic oscillator matrix elements. The most general case of 〈m|f(â, â+)|n〉 is considered, and the recurrence relations forf(â, â) = Xk, exp(−βX), and exp(−X2) are given as examples. The relations obtained are considerably simpler than those derived by using only the hypervirial theorem; comparatively, the recurrence relations presented here have the advantage of avoiding the use of the quantum mechanical sum-rules when determining initial matrix elements. The proposed procedure can be used to determine the recurrence relations for other potentials as well as to evaluate the two-center integrals.

Some spectral properties of an integral operator in potential theory

Abstract: In [7] Plemelj established some fundamental results in two- and three-dimensional potential theory about the eigenvalues of both the double layer potential operator and its adjoint, the normal derivative of the single layer potential operator. In [3] Blumenfeld and Mayer established some additional results concerning the eigenvalues of these integral operators in the case of ℝ 2 . The spectral properties established by Plemelj [7] and by Blumenfeld and Mayer [3] have had a profound effect in the area of integral equation methods in scattering and potential theory in both ℝ 2 and ℝ 3 .

Consistent algebra for the constraints of quantum gravity

Abstract: In the previous paper, a proposal was advanced for the ordering of the operatorsOpen image in new windowμ that arise in Dirac's programme for the quantization of gravity. The resulting algebra, however, was found to contain an undesired anomalous operator. Here we present a minimal modification of the canonical commutation relations of gravity in order to ensure that covariance is maintained for noncommuting tensor operators. As a result of the modification, the algebra of the quantum operator constraints is found to close exactly as in the classical case.

Ladder operator relations for hypergeometric functions

Abstract: Two first-order differential operators are introduced to generate recursion formulae for hypergeometric functions. These operators, on the one hand, factorise the associated second-order operator and, on the other hand, reproduce the equations resulting from the action of the supersymmetry generators on the positive-energy solution of the Coulomb field. The results for the confluent hypergeometric functions also provide a natural basis for constructing ladder operator recursion relations for the Coulomb Green function with the outgoing wave boundary condition.

On the Discretization of Certain Operator Field Equations

Abstract: Two new explicit finite difference schemes are proposed for certain operator field equations in a lattice.

Matrix methods in discrete-time quantum mechanics.

Abstract: The operator difference equations that arise in the finite-element treatment of a quantum theory are implicit and therefore difficult to solve. By introducing a matrix formulation it is possible to circumvent the implicit character of these equations and obtain explicit closed-form solutions for arbitrary matrix elements of any operator.

Factorization-Algebraization-Path Integration and Dynamical Groups

Abstract: Schrodingerl proposed an elegant method of factorization of a quantum-mechanical second-order linear differential equation into a product of two first-order differential operators often referred to as ladder operators. These ladder operators when acting on respective eigenfunctions create new eigenfunctions with a quantum number raised or lowered by one unit. Schrodinger’s method was further systematically studied for a class of second-order linear differential equations in particular by Infeld and Hull2 who have shown that a second-order differential equation which may be brought into the form: (0.1) may be factorized into products of two first-order ladder operators such that (0.2)

Momentum projection of solitons including quantum corrections

Abstract: The method of projection is applied to a relativistic field theory of fermions interacting with a nonlinear scalar field, specifically the Friedberg-Lee soliton model. Projection is effected by operating on a localized “bag” state with the translation operator exp (iP·Z), and integrating overZ. The resulting state is an eigenstate of zero momentum. The energy and the expectation value of other physical operators can be expressed as Gaussian moments of the Hamiltonian or the physical operator times powers of the momentum operator taken with respect to the bag state. Renormalization in the one-loop approximation is discussed in detail for the boson sector, and briefly for the fermion sector. The method can be tested for convergence against nonexpansion techniques. The latter, however, cannot so easily handle distortion of the Bose modes or the distortion of the Dirac sea.

Construction of orbital angular momentum eigenfunctions for many‐particle systems by harmonic projection

Abstract: Formulas are derived which allow the direct construction of total orbital angular momentum eigenfunctions for many-particle systems without the use of Clebsch–Gordan coefficients. One of the equations is closely analogous to Dirac' identity for the total spin operator. This equation describes the action of L2 on a function of the particle coordinates in terms of a class operator of the symmetric group and a "contraction operator." A general projection operator for constructing symmetric eigenfunctions of L2 is presented.

Theory of potential scattering, taking into account spatial anisotropy

Abstract: We get new tests for the existence and completeness of wave operators under perturbation of a pseudodifferential operator with constant symbol P(ξ) by a bounded potential v(x). The term anisotropic is understood in the sense that the growth of P(ξ) as ξ→∞ and the decrease of v(x) as x→∞ can depend essentially on the direction of the vectors ξ and x respectively. This permits us to include in the sphere of applications of the abstract scattering theory of a nonelliptic unperturbed operator the D'Alembert operator, an ultrahyperbolic operator, nonstationary Schrodinger operator, etc. In view of the anisotropic character of the assumptions on the potential, the results obtained are new even in the elliptic case. As an example we consider a Schrodinger operator with potential close to the energy of a pair of interacting systems of many particles.

On the limit behavior of the largest eigenvalue of an elliptic operator with a small parameter

Abstract: The asymptotic behavior of the largest eigenvalue of a differential operator with a small parameter is studied.Figures: 1. Bibliography: 10 titles.

Operator ordering in quantum field theory. Nonlinear Lagrangian of scalar fields

Abstract: Operator ordering in the Lagrangian formalism of the relativistic quantum field theory is investigated for a system corresponding to a classical one described byLc=–1/2∂μϕcaϱab(ϕc)∂μϕcb-v(ϕc). The most general form of the Lagrangian density operator and that of current operator accommodating the freedom of the order of the field operators are taken first, and the conditions on them required for the realization of the symmetry properties in the operator formalism are investigated. The requirement of the Poincare symmetry determines the Lagrangian density operator, the field equation and the current operators up to an unknown function. The unknown function is restricted to a scalar under the transformations of the Poincare and internal symmetries.

Antisymmetry and the direct integral decomposition of unstarred operator algebras

Abstract: It is shown that the direct integral decomposition of a non-self-adjoint operator algebra Q/ has the diagonal w n s&'* of this algebra as the algebra of diagonalizable operators if and only if almost all direct integrands of _v are antisymmetric algebras. By using the antisymmetric decomposition a direct integral model of a commutative, reflexive algebra is obtained.

Optical reflection from planetary surfaces as an operator-eigenvalue problem

Abstract: The understanding of quantum mechanical phenomena has come to rely heavily on theory framed in terms of operators and their eigenvalue equations. This paper investigates the utility of that technique as related to the reciprocity principle in diffuse reflection. The reciprocity operator is shown to be unitary and Hermitian; hence, its eigenvectors form a complete orthonormal basis. The relevant eigenvalue is found to be infinitely degenerate. A superposition of the eigenfunctions found from solution by separation of variables is inadequate to form a general solution that can be fitted to a one-dimensional boundary condition, because the difficulty of resolving the reciprocity operator into a superposition of independent one-dimensional operators has yet to be overcome. A particular lunar application in the form of a failed prediction of limb-darkening of the full Moon from brightness versus phase illustrates this problem. A general solution is derived which fully exploits the determinative powers of the reciprocity operator as an unresolved two-dimensional operator. However, a solution based on a sum of one-dimensional operators, if possible, would be much more powerful. A close association is found between the reciprocity operator and the particle-exchange operator of quantum mechanics, which may indicate the direction for further successful exploitation of the approach based on the operational calculus.