bluuebunny/crossref_metadata_2025 huggingface dataset:
Twistors and induced representations of SU(2,2)
Authors: Lerner David E.
Published: March 2003

We give an explicit realization of a series of representations of SU(2,2) induced by R+⊕SL(2,C). Vectors in these representation spaces are homogeneous spinor-valued functions of two twistor variables. They may also be realized, in a frame-dependent way, either as conformally invariant fields in Minkowski space or as homogeneous spinor-valued functions on the O(2,4) null cone. The conformal invariance of the massless free fields is discussed from this point of view, and the twistor version of the field equations is derived. Finally, irreducible twistors are shown to correspond to conformally invariant fields satisfying the generalized twistor equation.
Post-Newtonian two-body and n-body problems with electric charge in general relativity
Authors: Barker B. M. , O’Connell R. F.
Published: March 2003

Starting with the Bażański two-body post-Newtonian Lagrangian with electric charge in general relativity, we construct a coordinate transformation (not involving center-of-mass coordinates) with two arbitrary parameters and obtain a Hamiltonian which is in agreement with one derived from quantum field theory. The field theory Hamiltonian corresponds to using an arbitrary parameter xp in the photon propagator as well as an arbitrary parameter xg in the graviton propagator. These results are also generalized to the case of n bodies. The condition for static balance ei=±G1/2mi is found to hold both for the exact Reissner–Nordstro/m ’’one-body’’ problem and for the post-Newtonian n-body problem. An alternate condition for static balance ei=± (Gm1m2)1/2 is found to hold for the post-Newtonian two-body problem. The precession of the perihelion for the post-Newtonian two-body problem is given along with four special cases, one of which is the two-body generalization of the ’’one-body’’ special relativity result of Sommerfeld. Post-Newtonian two-body equations of motion (in center-of-mass coordinates) with the condition of static balance are also examined.
On the bulk distribution functions and fluctuation theorems
Authors: Ziff Robert M.
Published: March 2003

A new integral relation linking the bulk distribution functions of a one phase fluid is given. It is related to the Ornstein–Zernike compressibility formula, which is also discussed and generalized. In addition, an expression is found for the probability that a region of volume V in an infinite system of given density and temperature contains exactly N particles. This enables the fluctuation–compressibility relation and its generalization to be calculated, without the use of the grand canonical ensemble.
Markov systems of imprimitivity
Authors: Chen Etang
Published: March 2003

It is shown that Nelson’s Markoff system is a Markov system of imprimitivity; and conversely, an ergodic Markov system of imprimitivity determines a Nelson’s Markoff system. Markovian dynamics are also constructed on some Markov systems of imprimitivity.
A general theory of linear finite-difference equations with a few applications
Authors: Phares Alain J.
Published: March 2003

Recently a new formalism has been developed giving the solutions in explicit form of multiterm linear homogeneous recursion relations with nonconstant coefficients. The basic idea was to relate this problem to the one of partitioning an interval into parts of given lengths. This idea is extensively used here to obtain the solutions of linear inhomogeneous difference equations. The resulting method has the advantage of being general: It does not rely on any special device and does not assume any special form for the recursion relation. Applications of these techniques to physical problems are presented elsewhere. Here we show how the method works for first and second order equations. The three-term Legendre polynomial recursion relation with an arbitrary inhomogeneous term is discussed in detail. The Legendre polynomials are then viewed as special cases of combinatorics functions, P (j,m;z), based on the partitions of an interval (j,m), 0⩽j⩽m, into parts of lengths 1 and 2. P (j,m;z) reduces to the Legendre polynomial, Pm(z), for j=0. It is also interesting to note that P (j,m;z) are not orthogonal in general. However, a new set of orthogonal functions can be constructed as the solutions of the inhomogeneous Legendre recursion relation with a suitable choice of the inhomogeneous term.
Generalization of the inversion equations and application to nonlinear partial differential equations. I
Authors: Cornille Henri
Published: March 2003

The generalization of the inversion equations corresponding to a system of n linear coupled equations is obtained in two different cases : first for the second order differential linear operator μ2(x)[∂2/∂x2−V0(x)] and second for the linear first order operator μ (x) ∂/∂x. For the latter (which is an extension of the 2×2 case of Ablowitz et al.) we consider only a twofold eigenvalue problem for the n×n system, leaving the more general case to a subsequent paper. The equations of inversion are obtained by simple algebraic method, without considering both the direct and inverse scattering problem as in the classical method and without investigating the properties of the Jost solutions in the complex eigenvalue plane. Always using pedestrian algebraic method, and assuming that the kernel of the integral equation satisfies a linear partial differential equation, we deduce, for the solution of the integral equation, the corresponding nonlinear part of the evolution equation. In this way we show that the explicit construction of the classical solvable nonlinear evolution equations can be extended in two cases : coupled equations and substitution of μ (x) ∂/∂x for ∂/∂x.
U(n) lowering operators
Authors: Bincer Adam M.
Published: March 2003

Lowering operators for the U(n) groups are given in product form.
WKB approximation for bound states by Heisenberg matrix mechanics
Authors: Klein Abraham
Published: March 2003

The WKB approximation for bound states is derived from Heisenberg matrix mechanics. It is shown that the result can be obtained from the Bohr–Sommerfeld quantization of a suitably chosen solution of the classical equations of motion. It is also derived from the commutation relation and from a quantum form of Hamilton’s variational principle.
The linear potential: A solution in terms of combinatorics functions
Authors: Antippa Adel F. , Phares Alain J.
Published: March 2003

The mathematical formalism recently developed for solving multiterm linear recursion relations is used to obtain a solution of the linear potential problem, for any value of the orbital quantum number l, in terms of combinatorics functions. The wavefunctions are given as power series expansions and the energy eigenvalues as the roots of an infinite order polynomial.
Orthocomplemented subspaces of nondegenerate partial inner product spaces
Authors: Antoine J.-P. , Grossmann A.
Published: March 2003

A vector subspace of a nondegenerate partial inner product space is orthocomplemented if and only if it satisfies a certain topological regularity property. If a nondegenerate, positive definite partial inner product space has ’’many’’ orthocomplemented subspaces, it is a Hilbert space.
Dense electron-gas response at any degeneracy
Authors: Gouedard C. , Deutsch C.
Published: March 2003

An exact expression for the linear response function of the dense electron gas valid at any temperature is worked out in the ring (RPA) approximation. The T=0 and T=∞ limits reproduce the already known results. It is used to explain the longitudinal oscillations and the screening around a test charge. The latter is either Thomas–Fermi-like or Friedel-like according to the values of the parameters.
Dynamics and phase transitions for a continuous system of quantum particles in a box
Authors: Battle Guy A.
Published: March 2003

A particular type of continuous quantum system with infinitely many particles is analyzed, and the existence of dynamics is proven in the GNS representations of certain states. The dynamics is not a group of automorphisms on the original algebra, so equilibrium states are defined in terms of the KMS condition in the representations of the states. The basic theorems about KMS states do not apply here. Nevertheless, for a special class of interactions it is proven that the central decomposition of an equilibrium state is concentrated on a Borel set of equilibrium factor states and that such factor states are precisely the extremal equilibrium states. Furthermore, the equilibrium factor states are in one-to-one correspondence with sets of functions satisfying a certain system of trace equations. This explicit correspondence is then used to show that there are no phase transitions for high temperature, and an example of a phase transition is constructed for low temperature. The phase transition also provides an example of continuous symmetry breaking.
Inverse Gaussian transforms: General properties and application to Slater-type orbitals with noninteger and integer n in the coordinate and momentum representations
Authors: Taylor William J.
Published: March 2003

The use of Gaussian-type orbitals (GTO) facilitates the evaluation of the multicenter integrals encountered in quantum chemistry by reducing all integrals of more than two centers to two-center integrals. On the other hand Slater-type orbitals (STO), while leading to more time-consuming integral evaluations, provide a better approximation to variationally determined atomic orbitals. Thus, for a basis set of given size, STO’s generally give better accuracy than GTO’s. Kikuchi proposed the representation of STO’s as integral Gaussian transforms, or in effect by continuous expansions in GTO’s, and Shavitt, Karplus, and Kern have applied this technique to the evaluation of multicenter integrals over STO’s. If these procedures are to be extended, it is desirable to develop a more systematic approach to the representation of a given basis function, ψ (r), as a Gaussian transform, ψ (r) =𝒢[f (t);r] =ℱ∞0f (t)exp(−r2t) dt; what this reduces to is the problem of calculating the inverse Gaussian transform, f (t) =𝒢−1[ψ (r);t]. In the present investigation it is pointed out that f (t) =ℒ−1[ψ (s1/2);t], where ℒ−1 represents the inverse Laplace transformation. On this basis conditions on ψ (r) necessary for the existence of a unique continuous Gaussian inverse, f (t), are formulated, and general rules for the manipulation of inverse Gaussian transforms are developed. Finally, the formulas for the inverse Gaussian transforms of STO’s obtained previously by Kikuchi and Wright are generalized to noninteger principal quantum number, and angle-dependent STO’s, in both the coordinate and momentum representations.
The three-dimensional convolution of reduced Bessel functions and other functions of physical interest
Authors: Filter E. , Steinborn E. O.
Published: March 2003

A method for evaluating convolution integrals over rather general functions is suggested, based on the analytical evaluation of convolution integrals over functions BMν,L(r) = (2/π)1/2rL+νKν (r) YML(ϑ,φ), which are products of modified Bessel functions of the second kind Kν(r), regular solid spherical harmonics rLYML(ϑ,φ), and powers rν.
Eigenvalues of S⋅Π for spins 1/2, 1, and 3/2
Authors: Weaver D. L.
Published: March 2003

The eigenvalues of the matrix operator S⋅Π for a constant magnetic field are derived in a parallel way for spins 1/2, 1, and 3/2 using only the algebra of the spin matrices and the commutation relations of the components of Π.
Spaces of positive and negative frequency solutions of field equations in curved space–times. II. The massive vector field equations in static space–times
Authors: Moreno Carlos
Published: March 2003

The space–times considered in this article are static, Vn×R, with compact space-section manifolds without boundary, Vn, and such that the trajectories of the Killing vector field are geodesics. For the physical field of spin 1 and mass m≳0 in these space–times, field equations are solved in any adapted atlas, by the one-parameter groups of unitary operators generated by scalar and vector Hamiltonians, i−1Tj−1, j=0,1, in Sobolev spaces Hjl−1(Vn) ×Hjl 1(Vn), lεR. Hilbert spaces of positive energy solutions of field equations, as well as those of reduced solutions and their canonical symplectic and complex structures, are determined. The existence and the uniqueness of Lichnerowicz’s (1−1) current on space–time are established, and the corresponding frequency-solution Hilbert spaces are constructed. Within the framework of Segal, a definition of quantum field operators is given, leading to the postulated commutator for the physical field concerned.
Forms of all spacetime metrics which admit [(11) (11)] Killing tensors with nonconstant eigenvalues
Authors: Hauser I. , Malhiot R. J.
Published: March 2003

Forms are obtained for all of the spacetime metrics which admit or are conformal to those which admit Killing tensors whose Segre characteristics are [(11) (11)] and whose eigenvalues λ(1) and λ(2) are not constants. No a priori assumptions are made concerning separability, isometries, invertibility, Petrov type, or the matter tensor. Some of our metrical forms, viz., those for which the Hamilton–Jacobi equation is separable or partially separable after multiplication by an integrating factor, are already known; in particular, they include all of Carter’s Hamilton–Jacobi separable spacetimes. Also, some of our metrical forms are new and may be applicable to finding interesting nonvacuum metrics which admit [(11) (11)] Killing tensors, but which do not necessarily have any Killing vectors. Though our results include a class of metrics for which λ(1) and/or λ(2) are constant, they do not include all such metrics; in an appendix, we prove that the set of all metrics which admit [(11) (11)] Killing tensors with constant λ(1) and λ(2) is identical except for conformal factors with the set of all metrics which admit [(11) (11)] conformal Killing tensors.
Uniqueness connection between charge conjugation and statistics
Authors: Hammer C. L. , DeFacio B.
Published: March 2003

The charge conjugation properties of bilinear quantum field theories are examined in considerable detail. It is shown that the connection between charge conjugation and statistics is unique. The relation between spin and statistics for a large class of these theories and the statistics of unusual fields such as Faddeev–Popov ghost fields and Gupta’s regularizing fields with negative norm are discussed.
Theory of vibrations of coated, thermopiezoelectric laminae
Authors: Dökmeci M. Cengiz
Published: March 2003

This study presents a theory for dynamic problems of coated laminae in which there is coupling between mechanical and electrical as well as thermal fields. The laminae is coated completely with perfectly conducting electrodes on both its faces, and it may comprise any number of bonded layers, each with a distinct but uniform thickness, curvature and electromechanical properties. First, a generalized variational theorem is derived so as to describe the complete set of the fundamental equations of thermopiezoelectricity. Next, by the use of this theorem, a system of two-dimensional, approximate governing equations of the coated laminae is constructed for the case when the mechanical displacement, electric potential, and temperature fields vary linearly across the laminae thickness. The effects of elastic stiffnesses of, and the interactions between, layers of the laminae and its electrodes are all taken into account. Also, the uniqueness of the governing equations is examined, and a theorem which includes the conditions sufficient for the uniqueness is given.
Transforming Gaussians into Wannier functions
Authors: Wannier Gregory H.
Published: March 2003

It is clear a priori that equal Gaussian functions, spread over a lattice, can be transformed into Wannier functions. The transformation is carried out here analytically in one dimension, with the help of the theory of theta functions. The results confirm and illustrate the properties commonly assigned to these functions, with one startling exception.
A characterization of Lorentz transformations
Authors: Lenard A.
Published: March 2003

If a one-to-one correspondence of Minkowski space–time onto itself is such that timelike lines, and only timelike lines, map onto timelike lines, then the correspondence is an inhomogeneous Lorentz transformation combined with a dilation.
A new approach to the eigenvalues of the Gel’fand invariants for the unitary, orthogonal, and symplectic groups
Authors: Edwards S. A.
Published: March 2003

The expressions for the Gel’fand invariants (Casimir operators) of U(n), O(n), and Sp(n) in terms of the IR labels are derived by relating them to the trace of a suitably defined operator Pk. The method has unity and simplicity, since trace (Pk) can be related directly to the IR labels using the Weyl dimension function—without having to determine the eigenvectors of P.
On the self-induced transparency effect of the three-wave resonance process
Authors: Chiu S. C.
Published: March 2003

The self-induced transparency effect of the three-wave resonance is studied by the inverse scattering method. By transforming to the characteristic coordinates of the background wave, the inverse-scattering theory becomes greatly simplified. With the presence of a constant background wave, the three-wave process is dispersive, and the solitons and continuum behave in a different way from those of spatially bounded wavepackets. The continuum decays away, depositing energy to the background. The solitons have velocities which are amplitude dependent.
On the first approximation of the K-harmonics method
Authors: Castilho Alcarás J. A.
Published: March 2003

Formulas for the calculation of two-body central interactions in the first approximation of the K-harmonics method are presented. These formulas are exact and account for diagonal and off-diagonal matrix elements as well.
Identification of the velocity operator in an irreducible unitary representation of the Poincaré group for imaginary mass or zero mass and variable helicity
Authors: Jordan Thomas F.
Published: March 2003

For an irreducible unitary representation of the Poincaré group with imaginary mass or zero mass and variable helicity it is shown, as it was previously for positive mass or zero mass and fixed helicity, that the velocity operator can be identified by its transformations under the Poincaré group together with the assumption that its different components commute with each other. In particular, there is no way to use an unconventional velocity operator to avoid the tachyon interpretation of imaginary-mass representations.
The spinor Helmholtz equation
Authors: Hillion Pierre
Published: March 2003

In this paper, we introduce the spinor Helmholtz equation, a first order partial differential system, which, on the one hand, describes the behavior of a polarized electromagnetic beam at the Fresnel approximation and which, on the other hand, can also be used, from a numerical viewpoint, instead of the scalar Helmholtz equation for solving the Dirichlet problem. We give the main properties of this equation and of its solutions.
Transformation brackets for generalized Bogolyubov–boson transformations
Authors: Tikochinsky Yoel
Published: March 2003

We consider the transformation bi=Σsj=1 (λijaj+μijaj) between two sets of boson operators, with λij and μij complex. Closed formulas are derived for the transformation brackets connecting base states of the two sets. As an application, the general quadratic Hamiltonian in one dimension is diagonalized and two examples, involving time-dependent real transformation and time-independent complex transformation, are worked out.
Solutions of the three-magnon bound state equation. III. The physical eigenstate
Authors: Majumdar Chanchal K. , Bose Indrani
Published: March 2003

The physical eigenvalue and eigenfunction are found analytically by directly solving the three-magnon bound state equation in one dimension. The structure of the equation suggests a simple ansatz for the eigenfunction, and the calculation is reduced to the solution of a finite number of linear algebraic equations. The method is elementary, but the explicit solution is found after a long calculation. The connection of the wavefunction with previous results is also established.
On the inversion of the Weyl transform
Authors: Liu Kuang Chi
Published: March 2003

An inversion theorem for the Weyl operators of the Hilbert–Schmidt class is proved by using the polar decomposition of an operator.
Killing vectors in gauge supersymmetry
Authors: Kannenberg L.
Published: March 2003

The definition of a Killing vector for spaces with both Bose and Fermi dimensions is developed. Some properties of these vectors are studied, and they are used to define spaces of maximal supersymmetry. Certain theorems on symmetric spaces are shown to have supersymmetric analogs.
General expressions for divergence relations and multipole expansions for arbitrary scalar functions
Authors: Davies K. T. R. , Becker Richard L.
Published: March 2003

In theories of the cohesion of a system whose matter density is constant (e.g., classical liquid drops) or nearly constant except in a thin surface region (e.g., nuclei or neutron stars), there occur computationally difficult single and double volume integrals of energy densities. Such integrals have been important in recent dynamical calculations of fission and heavy-ion reactions. Even if the matter density is diffuse, the integrals can be written as integrals over finite volumes by modeling the density as the convolution of a step function and a diffuseness function. If the integrands can be written as divergences of tensor fields, the integrals over finite volumes can be reduced to surface integrals by Gauss’ theorem. We have found general expressions, which we call divergence relations, for a vector field whose divergence is a given scalar function and for a second-rank tensor field whose double divergence is the scalar. The equations derived are much easier to use and apply to a wider class of functions than formulas previously obtained in the literature. The generalization to nth order for application to many-body forces is included, and for all orders the dimensionality is arbitrary. The interaction energy of two nonoverlapping systems is often most simply evaluated by using the generalized Slater multipole expansion of the two-body interaction. A new expression is derived for the radial factor Gl(r1,r2) appearing in the multipole expansion of an arbitrary scalar, two-body function. This Gl is expressed as an integral involving the product of the Fourier transform of the interaction and two Bessel functions. For some cases this integral can easily be evaluated by contour integration.
Vector constants and their algebras for classical Hamiltonians
Authors: Mitchell Thomas P.
Published: March 2003

The explicit form of a vector constant of the motion for an arbitrary relativistic spherically symmetric time independent classical Hamiltonian is obtained by showing that its construction is achieved on solving a linear second order ordinary differential equation. The solution of this equation is presented and the vector, in conjunction with the angular momentum to which it is normal, is used to generate the algebras of the Euclidean group E(3), the orthogonal rotation group O(4) and the special unitary group SU(3). The mass is assumed to be structureless and to move in an externally prescribed scalar potential field.
Coulomb-like quantization of the electromagnetic field on spacelike hyperboloids
Authors: Hostler Levere
Published: March 2003

Maxwell’s equations relative to a pseudospherical coordinate system in Minkowski space are cast in a form remarkably similar to the usual Cartesian form. The vector spherical harmonics of the unit hyperboloid derived in an earlier work are then applied to the problem of quantizing the electromagnetic field from the standpoint of the method of quantization on spacelike hyperboloids. A Coulomb-like gauge is introduced for which quantization on spacelike hyperboloids parallels that of ordinary Coulomb gauge quantization. In a kind of ’’turning around’’ of the situation which obtains for ordinary Coulomb gauge quantization, our new Coulomb-like gauge condition has manifest covariance with respect to transformations of the homogeneous Lorentz group but not with respect to translations.
The linear representations of SL̄ (3,R) on a Banach space
Authors: Angelopoulos E.
Published: March 2003

All continuous irreducible linear representations of S̄L̄ (3,R) on a Banach space are studied and classified, using successively infinitesimal methods and methods of induced representations. Irreducibility and equivalence criteria are given in terms of the values of the invariants of the representation as well as in terms of the inducing characters. The reduction into irreducible factors 𝒟j of the restriction to SU(2) is carried out and the multiplicity of each 𝒟j is determined. A necessary and sufficient condition for an irreducible representation to be equivalent to a unitary representation is also established.
Conditions on the uniqueness of the solution of the elastic unitarity equation
Authors: Sakmar I. A.
Published: March 2003

We report new conditions for the uniqueness of the solution of the unitarity integral equation. These conditions follow from the consideration of the smallest value of the integral.
Super-convergent adiabatic invariants with resonant denominators by Lie transforms
Authors: McNamara Brendan
Published: March 2003

Adiabatic invariants of motion for perturbed Hamitonian systems are very important in plasma physics. It has been shown how to deal with resonant denominators to first order in the perturbation by judicious choice of the zeroth order invariant. The method is extended to higher orders by solving the Liouville equation and by using the Lie transform technique. The resulting invariants are shown to be equivalent by using the operator algebra of the averaging method involving the Poisson bracket, integrating, and averaging operations. Partial Lie transforms are introduced to describe the internal structure of any resonance in any order. The super convergent expansions of Kolmogorov are easily illustrated in the Lie formalism and allow a quick analysis of high-order resonances to be made.
Bound states for r−2-like potentials in one and three dimensions
Authors: van Haeringen H.
Published: March 2003

We investigate the bound states for a symmetric one-dimensional potential V (x) =c (‖x‖+d)−2 for ‖x‖≳R⩾0, V (x) arbitrary for ‖x‖<R, and the bound states for its three-dimensional analog. We prove that the number of bound states is finite when c⩾−1/4 and infinite when c<−1/4. For the borderline case c=−1/4 this is a new result. For R=0 we obtain explicit approximate expressions for many binding energies. In this case, when −1/4⩽c<0, there is exactly one bound state in the one-dimensional case and no bound state in the three-dimensional case. The one-dimensional ground-state energy is continuous in c at c =−1/4, whereas the excited-state energies are not. In the case c< −1/4 the binding energies form approximately a greometric progession, both for the odd solutions and for the even solutions. The sequence of all binding energies is approximately a geometric progression when R=0 and the potential strength is sufficiently large. However, in general the complete sequence of binding energies shows an odd–even staggering.
Multiplicity free and finite multiplicity indecomposable representations of the algebra su(1,1)
Authors: Gruber Bruno , Klimyk A. U.
Published: March 2003

A classification is given for the multiplicity free indecomposable representations of the simple Lie algebra su(1,1), which are unbounded on both sides. Formulas have been obtained for the matrix elements of the generators of su(1,1) for all these representations. Representations of su(1,1) are analyzed which have the property that all their weight subspaces are infinite dimensional. Subrepresentations and representations on quotient spaces of this infinite multiplicity representations are considered and their relationship to the multiplicity free indecomposable representations is determined (both, unbounded on both sides, and bounded on one side). Finite multiplicity indecomposable representations are obtained from the infinite multiplicity representation for special values of the Casimir operator. A decomposition of the infinite multiplicity representation into a direct sum of multiplicity free representations and finite multiplicity indecomposable respresentations is given in two different ways. Finally, formulas for the matrix elements of su(1,1) are given for the finite multiplicity indecomposable representations.
Average force and force correlation formulas for momentum transfer cross section
Authors: March N. H.
Published: March 2003

Gerjuoy has related the momentum transfer cross section to the average force. The force correlation function formula of Rousseau, Stoddart, and March is here shown to be transformable into Gerjouy’s result for a spherically symmetrical scatterer.
On the computation formulas of the SO(n−1,1) representation matrix elements
Authors: Maekawa Takayoshi
Published: March 2003

The formulas for computing the boost matrix elements are found for all classes of the unitary irreducible representations of SO(n−1,1) by defining the invariant scalar product in the space consisting of the D functions of SO(n−1) and assuming that functions with some property exist for the complementary series. The normalization constants of the bases are completely determined by requiring that the boost matrix elements in the finite transformations agree with those obtained by the method of the infinitesimal operators in the infinitesimal transformations.
On invariant integration over SU(N)
Authors: Creutz Michael
Published: March 2003

We give a graphical algorithm for evaluation of invariant integrals of polynomials in SU(N) group elements. Such integrals occur in strongly coupled lattice gauge theory. The results are expressed in terms of totally antisymmetric tensors and Kronecker delta symbols.
An investigation of Dubourdieu’s list of space–times which admit holonomy groups
Authors: Harris R. A. , Zund J. D.
Published: March 2003

An investigation is made of fourteen space–times given by Dubourdieu which admit holonomy groups. It is shown that, although nine of these space–times admit only trivial vacuum gravitational fields, the remaining ones are of Bel–Petrov types III and N. Many of the latter metrics can be identified with known exact solutions of the source-free Einstein equations including the Kerr–Goldberg metric and certain type III metrics recently studied by the authors.
Continuum calculus. III. Skorohod’s weak distributions in the evaluation of a class of Feynman path integrals
Authors: Lee L. L.
Published: March 2003

Path integrals for functionals are studied from the point of view of the continuum calculus proposed earlier [J. Math. Phys. 17, 1988 (1976)]. The weak distributions of Skorohod in an infinite-dimensional Hilbert space and the p-integral method of continuum calculus are employed to derive a formula for the functional integral, which is in turn evaluated through a natural extension of the weak distribution expression. Generalizations are made to measures with density functions in the function space. As a demonstration, the formula is tested against the polynomial functionals studied by Friedrichs, and valid results are obtained for the general case.
Symmetries of the 3j coefficient
Authors: Venkatesh K.
Published: March 2003

An explicit form of the five Regge symmetries of the 3j coefficient is given. It is shown that a set of six 3F2(1) hypergeometric functions is necessary and sufficient to account for the 72 symmetries of the 3j coefficient, each accounting for 12 symmetries. The eight-element group, recently discussed by Lockwood, accounts for only eight symmetries of the 3j coefficient.
A new form of the Mayer expansion in classical statistical mechanics
Authors: Brydges David , Federbush Paul
Published: March 2003

New expressions are given for the expansion coefficients in the Mayer expansion (and thus the virial expansion). These promise to be useful in applications, as well as provide a simple rigorous proof of the convergence of the Mayer series and some of its properties.
O(3) shift operators: The general analysis
Authors: Hughes J. W. B. , Yadegar J.
Published: March 2003

O(3) shift operators are constructed in terms of tensor operators T (j,μ) and the O(3) generators. These are of type Bkl, k=−j,...,j, where Bkl raises l by k, l (l+1) being the eigenvalue of the O(3) Casimir operator. Various convenient normalizations of these operators are constructed, and their properties and uses considered.
Path integrals with a periodic constraint: Entangled strings
Authors: Inomata Akira , Singh Vijay A.
Published: March 2003

Path integrals with a periodic constraint ∫θ̇ ds =Θ̄+2πn (n=integer) are studied. In particular, the path integral for a string entangled around a singular point in two dimensions is evaluated in polar coordinates. Applications are made for the entangled polymers with and without interactions, the Aharonov–Bohm effect, and the angular momentum projection of a spinning top.
Electromagnetic and gravitational Hertz potentials
Authors: Plebański Jerzy F. , Robinson Ivor
Published: March 2003

With generality of complex relativity, the classical theory of the electromagnetic Hertz potentials is outlined in terms of spinors and forms. Particularly interesting are D (0,1) and D (1,0) null Hertz potentials. Then, a new spinorial approach to heavens (ℋ spaces) is proposed, which reveals in their structure the presence of the left null gravitational Hertz potential [of the type D (0,2)]. The relevant hints which follow from our results and concern the structure of the most general solutions of the Einstein vacuum equations (type G⊗G), are discussed, in particular on the level of the linearized theory.
Presymplectic manifolds and the Dirac–Bergmann theory of constraints
Authors: Gotay Mark J. , Nester James M. , Hinds George
Published: March 2003

We present an algorithm which enables us to state necessary and sufficient conditions for the solvability of generalized Hamilton-type equations of the form ι (X) ω=α on a presymplectic manifold (M,ω) where α is a closed 1-form. The algorithm is phrased in the context of global infinite-dimensional symplectic geometry, and generalizes as well as improves upon the local Dirac–Bergmann theory of constraints. The relation between our algorithm and the geometric constraint theory of Śniatycki, Tulczyjew, and Lichnerowicz is discussed.
A classification of four-dimensional Lie superalgebras
Authors: Backhouse Nigel
Published: March 2003

A classification is made of all Lie superalgebras (graded Lie algebras) of maximum dimension four.
Relativistic quantum kinematics on stochastic phase spaces for massive particles
Authors: Prugovečki Eduard
Published: March 2003

It is shown that to every Galilei-covariant nonrelativistic stochastic phase space representation of a system of massive particles, whose generator is rotationally invariant, corresponds a Poincaré-covariant relativistic representation sharing the same generator. The stochastic phase space probability densities of the two representations overlap in the limit of nonrelativistic velocities in the laboratory frame. The relativistic representations give rise to covariant and conserved probability currents at stochastic space–time points, in complete analogy with their nonrelativistic counterparts. This parallelism extends to the existence of a global representation of the proper Poincaré group in L2(Γ), which is reduced by each subspace of L2(Γ) spanned by the set of phase-space wavefunctions generated by some stochastic phase space representation of all pure states of the system.
Stochastic phase space kinematics of the photon
Authors: Prugovečki Eduard
Published: March 2003

A stochastic phase space description of photon states is obtained by attaching to each circular polarization mode a probability amplitude at stochastic phase space points that are frontally localized. The ensuing formalism gives rise to a probability density at such points that transforms as a scalar under proper Lorentz transformations. There also exist conserved covariant currents associated with these probability densities. The wavefunctions corresponding to all extremal stochastic phase space representations can be embedded in a single Hilbert space, and give rise there to irreducible representations of the proper Poincaré group 𝒫↑+.
Representations of the Weyl Lie algebra as models of elementary particles
Authors: Snellman Håkan
Published: March 2003

Some irreducible representations of the 11-parameter Weyl Lie algebra are suggested as models of elementary particles.
Geometrical spacetime perturbation theory: Regular first-order structures
Authors: Gowdy Robert H.
Published: March 2003

Spacetime perturbation theory is formulated in a coordinate independent way by regarding a family of spacetimes as a (4+n) -dimensional manifold with a particular standard connection and deriving analogs of the Gauss–Weingarten equations to describe the imbedding of each spacetime in the family.
Coherent states and lattice sums
Authors: Boon M. , Zak J.
Published: March 2003

The expansion of harmonic oscillator states in discrete coherent states on a von Neumann lattice leads to relationships between lattice sums and expansion coefficients of the Weierstrass σ function. It is shown that these relationships can be generalized to arbitrary lattices. Some interesting identities are obtained between infinite sums of different convergence rates.
Asymptotic graph counting techniques in ψ2N field theory
Authors: Bender Carl M. , Caswell William E.
Published: March 2003

We discuss two different techniques for obtaining asymptotic estimates of the number of n-vertex graphs in a ψ2N field theory as n→∞. The first technique relies on difference equations and the second makes use of Lipatov’s methods. We show how various topological constraints, such as connectedness and Wick ordering, affect the total number of graphs. We show, for example, that Wick ordering reduces the total number of connected graphs by a factor of eN−1/2 as n→∞.
Analytic solution of a model pulse propagation problem
Authors: Lax M.
Published: March 2003

A closed form solution is found to a model problem for pulse propagation in a medium that responds hydrodynamically to the pulse. This solution is useful for testing pulse propagation code.
The transport equation with anisotropic scattering in finite slab geometry
Authors: Gibbs Alan G. , Seto Richard
Published: March 2003

We consider one-speed neutron transport or monochromatic radiative transfer in a slab of finite width, with particles incident on both faces. The scattering kernel is represented as a finite sum of Legendre polynomials. An exact expression is obtained for the expansion coefficients of the flux in a singular eigenfunction representation. It involves the X and Y functions of Chandrasekhar and the ql and sl polynomials of Sobolev. Case’s full- and half-range expansions are obtained as limits when the slab thickness becomes zero or infinite, respectively. Orthogonality relations for the singular eigenfunctions and their adjoints are given. Also, we derive a number of orthogonality relations of the Inönü-type involving moments of the eigenfunctions. Expressions for surface densities, currents, and higher order Legendre moments of the surface fluxes are obtained. Singular integral transform relations between the Chandrasekhar φ and ψ functions and the polynomials gl(ν), are given. Also, we give the solution of slab problems with one face having diffuse or specular reflecting boundary conditions.
A new approach to the inverse problem
Authors: Siewert C. E.
Published: March 2003

The full-range orthogonality theorem concerning the elementary solutions of the equation of transfer is used to develop a solution of the inverse problem for a finite plane-parallel slab.
SU2 monopole harmonics
Authors: Yang Chen Ning
Published: March 2003

For a particle of I-spin I in an SU2 monopole field in five-dimensional Euclidean space, the monopole harmonics are found. They belong to the irreducible representation (p,q)5 where p =q+2I. They form a complete set of wave sections.
Direct-inverse problems in transport theory, the inverse albedo problem for a finite medium
Authors: Kanal Madhoo , Moses Harry E.
Published: March 2003

In this paper we deal with the inverse problem in radiative transfer, which is equally applicable to the neutron transport, for a finite homogeneous medium. We give a method for computing the scattering function and the albedo for single scattering. The solution is given in terms of Legendre expansion of the scattering function. A decomposition of the equation of transfer is also given in which the relation between the direct and the inverse problem is exhibited via the principles of invariance. A relation of this work with Case’s method of approach is outlined. This work should be of practical value to the problems associated with remote sensing of the terrestrial atmosphere and the neutron piles.
A system theoretic representation of mechanical systems. I. Decomposition of a mechanical system into a hierarchy of orthogonal stationary linear dynamical systems
Authors: Saraswati Dipak K. , Primas H.
Published: March 2003

A system theoretic representation of mechanical systems is proposed. It is shown that mechanical systems, classical or quantal, described by the Liouville equation, can be realized as a stationary linear input–output system. This state space representation of mechanical systems is then extended to decompose a mechanical system into a hierarchy of orthogonal stationary linear dynamical systems.
A system theoretic representation of mechanical systems. II. Stochastic interpretation
Authors: Saraswati Dipak K. , Primas H.
Published: March 2003

In a previous paper by the authors it was shown that mechanical systems, classical or quantal, can be realized as stationary linear dynamical systems. In this state space representation of mechanical systems, the input space, the output space, and the state space are Hilbert spaces and to make a connection with the language of a mathematical system theory, in which the corresponding spaces are topological vector spaces, it is shown here that the input, the output, and the state of the dynamical systems representing mechanical systems can be interpreted as second order stochastic processes. An orthogonal decomposition of mechanical motion into a deterministic and a purely nondeterministic motion, suggested by the stochastic interpretation, is discussed and it is pointed out that by iterating the decomposition procedure one can obtain a simple model which approximates the true mechanical motion.
Note on the entropy production in a discrete Markov system
Authors: Moreau M.
Published: March 2003

The Prigogine inequalities on the rate of entropy production are derived by information theoretic methods for a discrete open Markov system. A new inequality is proposed. A comparison is made with similar results due to Levine and co-workers.
Criticality of neutron transport in a slab with finite reflectors
Authors: Pao C. V.
Published: March 2003

The purpose of this paper is to investigate the subcriticality and the supercriticality for the neutron transport in a slab which is surrounded by two finite reflectors. The mathematical problem is to determine when the coupled boundary-value problem has or has no positive solution. It is shown under some explicit conditions on the material properties of the transport mediums and the size of the slab length that the coupled problem has a unique solution which insures the subcriticality of the system. It is also shown under some different conditions on the same physical quantities that the system cannot have a nonnegative solution when there is an external source, and it only has the trivial solution when there is no source in the system. This conclusion leads to the supercriticality of the system. Both upper and lower bounds for the critical length of the slab are explicitly given.
The point form of quantum dynamics and a 4-vector coordinate operator for a spinless particle
Authors: Bracken A. J.
Published: March 2003

We construct the analog in the quantum mechanics of a free spinless particle, of Dirac’s formula for the generators of space–time translations in his point form of classical dynamics, where one takes as fundamental variables the generators of homogeneous Lorentz transformations and the coordinate 4-vector of the point where the world line of the particle meets one sheet of a two-sheeted hyperboloid in space–time. A 4-vector coordinate operator is determined for such a particle, with commuting Hermitian components. The corresponding observable is the analog of the coordinate 4-vector of the point on the hyperboloid. This operator bears the same relation to such a surface as the Newton–Wigner operator does to an instant.
Properties of the self dual equations for an SU(n) gauge theory
Authors: Brihaye Y. , Fairlie D. B. , Nuyts J. , Yates R. G.
Published: March 2003

The Yang equations for all self dual solutions of SU(r,s) gauge theory are exhibited in simple form. Algebraic and Bäcklund transformations of the solutions of these equations are derived. The Bäcklund transformations change an SU(r,s) solution into an SU(r−1, s+1) solution.
Some results on automorphisms of affine algebraic groups
Authors: Ekong Samuel D.
Published: March 2003

We study the group of automorphisms Aut(G) of a given group G from the groups of automorphisms (supposed known) of some of its subgroups; basic results are applied to topological groups and widely developed inside the Borel–Chevalley theory of linear groups; affine algebraic complete groups are studied in detail.
Analysis of the neutron slowing down equation
Authors: Sengupta A. , Karnick H.
Published: March 2003

The infinite series solution of the elementary neutron slowing down equation is studied using the theory of entire functions of exponential type and nonharmonic Fourier series. It is shown from Muntz–Szasz and Paley–Wiener theorems, that the set of exponentials {exp(iλnu) }∞n=−∞, where {λn}∞n=−∞ are the roots of the transcendental equation in slowing down theory, is complete and forms a basis in a lethargy interval ε. This distinctive role of the maximum lethargy change per collision is due to the Fredholm character of the slowing down operator which need not be quasinilpotent. The discontinuities in the derivatives of the collision density are examined by treating the slowing down equation in its differential-difference form. The solution (Hilbert) space is the union of a countable number of subspaces L2(−ε/2, ε/2) over each of which the exponential functions are complete.
On the structure of divergence-free tensors
Authors: Anderson I. M.
Published: March 2003

Contravariant rank two tensors which are divergence-free on one index and which are constructed from the metric tensor, an auxiliary collection of arbitrary tensor fields, and the first and second partial derivatives of these quantities are classified. The results generalize existing mathematical arguments in support of the Einstein field equations.
Spectral and scattering inverse problems
Authors: Sabatier Pierre C.
Published: March 2003

The reconstruction of a differential operator form discrete spectra is reduced to its reconstruction from an S-matrix. This method makes it possible to solve the singular Sturm–Liouville problems which determine certain modes of a sphere. The results pave the way for handling studies in which information on modes and scattering results would all be taken into account. They are applied to the earth inverse problem and partial answers are given to a well-known conjecture. Finally the relevance of the JWKB approximation in this kind of problem is briefly discussed.