Volume 20, Issue 10, October 1979
Index of content:

Matrix elements for infinitesimal operators of the groups U(p+q) and U(p,q) in a U(p) ×U(q) basis. I
View Description Hide DescriptionIn this article explicit expressions are obtained for the action of the infinitesimal operators of the principal nonunitary series representations of the groups U(p,q) in a U(p) ×U(q) basis. It is moreover shown how the finite dimensional irreducible representations of the group U(p,q) and the group U(p+q) with respect to a U(p) ×U(q) basis are obtained from the principal nonunitary series representations of the group U(p,q).

Matrix elements for infinitesimal operators of the groups U(p+q) and U(p,q) in a U(p) ×U(q) basis. II
View Description Hide DescriptionThe general expressions derived in Ref. 1 for the matrix elements of the infinitesimal operators of the groups U(p+q) and U(p,q) in a U(p) ×U(q) basis are utilized in this article to obtain e x p l i c i t expressions for the matrix elements of the infinitesimal operators of (a) the degenerate series representations of U(p,q) in a U(p) ×U (q) basis, and (b) the representations of U(p+q) with highest weight (m _{1},0, ..., 0, m _{2}) in a U(p) ×U(q) basis. The operator which unitarizes the U(p+q) representations considered is given in e x p l i c i t form.

Sum rules for the matrices of the generators of SU(3) in an SO(3) basis
View Description Hide DescriptionWe consider various sum rules for the semireduced [i.e., reduced with respect to SO(3)] matrix elements of the generators of SU(3) in a basis of an irreducible representation [p q] corresponding to the group reduction SU(3) ⊆SO(3) ⊆SO(2). We use basis states which diagonalize an additional labeling operator K, but avoid their explicit construction. We build all the needed operators from the two independent SU(3) vector operators X and V, where X= (L,Q) is made of the SU(3) generators and V= (V ^{ L }, V ^{ Q }) is defined in terms of them. First we obtain an analytical formula for the linear sum rule satisfied by the diagonal semireduced matrix elements of Q. Then, from the set of quadratic equations fulfilled by the semireduced matrix elements of Q and V ^{ Q }, we obtain explicit expressions for the quadratic sum rules satisfied by these quantities. All the above‐mentioned sum rules are independent of the selection made for K. When K is defined as the third order operator L.V ^{ L }, we show that a relation between some nondiagonal matrix elements of Q and V ^{ Q } exists enabling the determination of a k‐weighted quadratic sum rule for the semireduced matrix elements of Q. As by‐products of the preceding results we obtain general formulas for the eigenvaluesk of K for all the values of L whose multiplicity does not exceed 3, and we show that we are able to compute analytically the individual matrix elements of Q for not too high dimensionalities by working out the case of the irreducible representation [10,5].

Clebsch–Gordan coefficients of finite magnetic groups
View Description Hide DescriptionA detailed method is given for the calculation of Clebsch–Gordan coefficients of finite magnetic groups. This method is a generalization of a new method for the calculation of Clebsch–Gordan coefficients of finite nonmagnetic groups which makes use of the fact that the Clebsch–Gordan coefficients may be arranged into vectors which are eigenvectors of certain projection matrices.

Stokes multipliers for a class of ordinary differential equations
View Description Hide DescriptionA new method is presented for calculating the Stokes multipliers for a class of linear second‐order ordinary differential equations. The Stokes multipliers allow the asymptotic solutions of these equations to be continued across the Stokes lines on which they are dominant. The differential equations, of the class considered here, have an irregular singular point at infinity and a singular point at the origin, which may be either regular or irregular. The Stokes multipliers, as functions of the coefficients in the differential equation, are obtained in the form of convergent infinite series, whose terms must be obtained from the solution of recursion relations, which are derived. In the case of Whittaker’s equation (when the origin is a regular singular point), the known results are obtained analytically. When the origin is an irregular singular point, numerical evaluation of the series is necessary, but the method seems to be quite efficient for use with digital computers. In the special case of an equation, with two irregular singular points, which can be transformed to Mathieu’s equation, the numerical results for the Stokes multiplier show good agreement with available known results for the characteristic exponents of Mathieu’s equation.

Gel’fand–Levitan equations with comparison measures and comparison potentials
View Description Hide DescriptionUsing an abstract form of the Gel’fand–Levitan equation, it is shown how a solution of the equation corresponding to a given weight operator can be found in terms of a solution for the equation with a different weight operator. The resulting Gel’fand–Levitan equation is a generalization of the original one. To achieve our result, an analog of a canonical transformation for direct scattering is used. The effect of the use of the transformation is to include part of the scattering potential (the comparison potential) in the unperturbed Hamiltonian. The generalized Gel’fand–Levitan equation has the advantage that if the weight operator for a given Gel’fand–Levitan equation is close to that for an already solved Gel’fand–Levitan equation, the solution of the first can be obtained from the second by using the solution of the second as a first approximation in an iteration procedure or as a trial function in a variational procedure. The method is illustrated by considering the inverse problem for the one‐dimensional Schrödinger equation, a generalized radial Schrödinger equation, and the Marchenko equation for the zero angular momentum radial Schrödinger equation. Though the use of a comparison weight function for some of the cases above has been given by others, the work of the present paper represents a systematic approach to the problem. The role of a variational principle will also be discussed.

Noether’s theorem, time‐dependent invariants and nonlinear equations of motion
View Description Hide DescriptionNoether’s theorem is applied to a Lagrangian for a system with nonlinear equations of motion. Noether’s theorem leads to a time‐dependent constant of the motion along with an auxiliary equation of motion. Special cases of this invariant have been used to quantize the time‐dependent harmonic oscillator. We also discuss the solution of the original equations of motion in terms of the solutions to the auxiliary equation.

Gauge theory in Hamiltonian classical mechanics : The electromagnetic and gravitational fields
View Description Hide DescriptionGauge potentials are directly defined from Hamiltonian classical mechanics. Gauge transformations belong to canonical transformations and are determined by a first order development of generating functions. The electromagnetic and gravitational fields, and they only, are obtained.

Quasiprobability distributions and the analysis of the linear quantum channel with thermal noise
View Description Hide DescriptionThe quasiprobability distributions (QPD) introduced by Cahill and Glauber much simplify calculating the density operator of the output of a linear quantum channel from the density operator of the input when the channel attenuates and is corrupted by thermal noise. This channel models, for instance, an attenuator viewed as a simple harmonic oscillator in contact with a heat bath, or the relation between the radiating field mode in the aperture of a transmitter and the field mode it excites at the aperture of a distant receiver. It is shown how the attenuation and the noise in the channel modify the dependence of the QPD on the ordering parameter s. The form of the QPD at the output indicates that the greater the attenuation, the more nearly the state of the output resembles a classical state. Indeed, the addition of thermal noise contributing on the average only one photon suffices to convert an arbitrary unimodal state into a classical state. The principal result of the paper is applied to determining the output of a quantum channel whose input mode is in a generalized coherent state.

Gravitational duality and Bäcklund transformations
View Description Hide DescriptionWe analyze various forms of duality symmetry transformations occurring in the theory of axially symmetric gravitational fields and their relation to standard Bäcklund transformations. Appropriately interpreted, duality rotations provide genuine Bäcklund transformations for the fundamental SL(2,R) invariants associated with the metric, consisting of elementary algebraic substitutions; thereby the construction of the metric reduces to the solution of linear equations.

Self‐graviting fluids with cylindrical symmetry. II
View Description Hide DescriptionThe Einstein field equations for an irrotational perfect fluid with pressure p, equal to energy density ρ are studied when the space–time has cylindrical symmetry with no reflection symmetry. The coordinate transformation to comoving coordinates is discussed. The energy and the Hawking–Penrose inequalities are studied. Particular classes of solutions are exhibited.

Gaussian quantum stochastic processes on the CCR algebra
View Description Hide DescriptionWe define a stationary Gaussian quantum stochastic process (GQSP) on the C*‐algebra of the canonical commutation relations over a real symplectic Hilbert space. Physically GQSPs can describe diffusion with non‐Markovian memory effects in quantum harmonic oscillators of arbitrary dimension. We find that in analogy with the commutative case a GQSP is completely specified by a operator‐valued autocovariance function satisfying certain positive definiteness and reality conditions. The autocovariance function also determines the response of the system to a class of time‐dependent generalized forces, and it has a spectral representation in terms of a positive operator‐valued measure.

Quantum soliton and classical soliton
View Description Hide DescriptionThe interaction between quanta and a static soliton is discussed in the boson theory. As a natural consequence, there appears the quantum coordinate associated with the position of the soliton, which leads to a quantum or classical behavior of the soliton depending on its size and the experimental conditions. It will be shown that the presence of the quantum coordinate (or collective coordinate) is required by the equal‐time canonical commutation relation.

Deformations and spectral properties of merons
View Description Hide DescriptionWe consider a meron–antimeron pair located at a, b, ∈ R^{4}, and show that the spectrum of its stability operator is not bounded below [in precise mathematical terms: The stability operator defined on C ^{∞} _{0}(R^{4}−{a,b}) has a self‐adjoint extension, possibly many, all of which are unbounded below]. We regularize a single meron located at the origin by replacing it inside a sphere of radius R _{0} and outside a sphere of radius R by ’’half instantons,’’ and show that for R≫R _{0} the regularized configuration continues to be unstable. For R _{0} finite and R=∞, we show that the spectrum of the stability operator continues to extend to −∞. We employ a singular transformation to embed R^{4} into S ^{3}×R where the meron pair takes a simple form and its stability operator L becomes L=−d ^{2}/dτ^{2}+V, where τ∈R, and the potential V can be diagonalized in terms of the angular momenta, spin, and isospin of the vector field. The spectrum of L is continuous and extends from −2 to +∞. We determine the number of (generalized) zero eigenmodes of L, and calculate its spectrum explicitly.

Gauge field singularities and noninteger topological charge
View Description Hide DescriptionWe study the behavior of singular gauge field configurations under gauge transformations and we determine the relation between noninteger topological charge and the possibility of displacing singularities.

Casimir invariants and characteristic identities for generators of the general linear, special linear and orthosymplectic graded Lie algebras
View Description Hide DescriptionWe present the commutation and anticommutation relations, satisfied by the generators of the graded general linear, special linear and orthosymplectic Lie algebras, in canonical two‐index matrix form. Tensor operators are constructed in the enveloping algebra, including powers of the matrix of generators. Traces of the latter are shown to yield a sequence of Casimir invariants. The transformation properties of vector operators under these algebras are also exhibited. The eigenvalues of the quadratic Casimir invariants are given for the irreducible representations of ggl(m ‖ n), gsl(m ‖ n), and osp(m ‖ n) in terms of the highest‐weight vector. In such representations, characteristic polynomial identities of order (m+n), satisfied by the matrix of generators, are obtained in factorized form. These are used in each case to determine the number of independent Casimir invariants of the trace form.

A note on Sommerfeld’s diffraction problem
View Description Hide DescriptionA direct transform technique is found to be most suitable for attacking two‐dimensional diffraction problems. As a first example of the application of the technique, the well‐known Sommerfeld problem is reconsidered and the solution of the problem of diffraction, by a half‐plane, of a cylindrical pulse is made use of in deducing the solution of the problem of diffraction of a plane wave by a soft half‐plane.

Analysis of the dispersion function for anisotropic longitudinal plasma waves
View Description Hide DescriptionAn analysis of the zeros of the dispersion function for longitudinal plasma waves is made. In particular, the plasma equilibriumdistribution function is assumed to have two relative maxima and is not necessarily an even function. The results of this analysis are used to obtain the Wiener–Hopf factorization of the dispersion function. A brief analysis of the coupled nonlinear integral equations for the Wiener–Hopf factors is also presented.

An approximation method for electrostatic Vlasov turbulence
View Description Hide DescriptionElectrostatic Vlasov turbulence in a bounded spatial region is considered. An iterative approximation method with a proof of convergence is constructed. The method is nonlinear and applicable to strong turbulence.

First‐stage magnetization and metastable migration field in a type I superconducting slab
View Description Hide DescriptionThe penetration of the field in the edge of a type I superconducting slab, placed in a uniform magnetic field, perpendicular to its plane, is analyzed by means of a suited complex potential, derived from general methods used in Dirichlet’s type problems. The way of taking into account the singularities of the field distribution at the ends of the edge structure in the intermediate state, and the boundary conditions are discussed. A computational method is described for calculating the potential and flux profiles along the edges and thereby, the thermodynamic potential of the system. The first stage magnetization law together with the equilibrium dimensions of the edge structure and the previously defined migration threshold are deduced from the theory.