bluuebunny/crossref_metadata_2025 huggingface dataset:
Bases for Irreducible Representations of U2n in the Chain U2n⊃Un+̇Un
Authors: Devi V. Syamala , Venkatarayudu T.
Published: February 2004

The highest weight polynomials of irreducible representations of a U3 × U3 subgroup of U6, which occur in the reduction of a given irreducible representation of U6, are obtained, making use of Littlewood's rules for the reduction of an irreducible representation of U6 with respect to a U3 × U3 subgroup. The various irreducible representations of U3 × U3 occurring in the reduction are uniquely labeled by parameters which are obtained in a natural and obvious way from Littlewood diagrams. These polynomials could not be obtained by the algebraic methods developed so far. The method described here overcomes the division problem encountered there and makes the solution possible. It can also be directly extended to the case of U2n⊃Un+̇Un for any n.
Asymptotic Fields in Some Models of Quantum Field Theory. III
Authors: Ho/egh-Krohn Raphael
Published: February 2004

We consider the model of a relativistic boson field, interacting in a local way with a nonrelativistic nucleon field. This model was studied by Nelson, who proved that after an infinite nucleon self-energy renormalization, the total energy becomes a self-adjoint operator. In this paper we prove that the asymptotic limits of the boson field exist.
On the Relaxation to Quantum-Statistical Equilibrium of the Wigner-Weisskopf Atom in a One-Dimensional Radiation Field. I. A Study of Spontaneous Emission
Authors: Davidson Russell , Kozak John J.
Published: February 2004

A theoretical study of the phenomenon of spontaneous emission has been carried out, using as a model the Wigner-Weisskopf atom in a one-dimensional radiation field. The calculation is performed within the framework of the Prigogine theory of nonequilibrium statistical mechanics. When the model is solved exactly to first order in the coupling parameter α and the evolution in time of the diagonal elements of the density matrix ρ is studied, it is found that the relaxation to equilibrium is characterized in part by a sequence of slowly damped oscillations. This result seems to be in agreement with the observation made by Zwanzig, namely, that exponential decay in time seems not to be universal, and may, in fact, be hidden behind some other kind of time dependence. An approximate theory is developed alongside the exact one, and corresponding terms in each treatment are compared numerically. It is found that, for small values of the coupling parameter α (α ≤ 0.1) and for sufficiently large values of τ, defined as τ = αEt where E is frequency and t is time, the approximate theory gives a satisfactory representation of the exact solution to first order in α. The importance and relevance of the model introduced by Van Hove and coworkers, in which nonexponential behavior was also observed, will be noted but not stressed in this paper, as this relationship will be developed in considerable detail in a subsequent contribution. Finally, the possible relevance of the theory to a problem of interest in magnetic resonance is mentioned.
Frequency Spectrum of a One-Dimensional Lattice with an Isotopic Impurity
Authors: Kinzer Earl T.
Published: February 2004

The frequency spectrum of a one-dimensional lattice with a substitutional isotopic impurity is calculated directly and compared with previous calculations for the correction to the spectrum of a perfect lattice. The calculation makes use of certain properties of the delta function to facilitate the derivation of the frequency spectrum. Minor corrections to previous results are presented.
Exact Solution of the Faddeev Equations for a One-Dimensional System
Authors: Dodd L. R.
Published: February 2004

The Faddeev equations for the one-dimensional system consisting of three identical particles interacting through δ-function potentials are shown to be exactly solvable. The scattering solutions in the form of half-off-shell rearrangement and elastic amplitudes are constructed explicitly and the ground state of the system is derived.
Renormalized Perturbation Theory for the Weakly Nonlinear Oscillator
Authors: Aks S. O/. , Carhart R. A.
Published: February 2004

The problem of finding solutions for the weakly nonlinear quantum oscillator is investigated in the Heisenberg representation in one dimension. The perturbation method developed makes allowance for the nonisochronous nature of nonlinear oscillations and avoids at any level of approximation the secularity—terms increasing without bound as t → ∞—intrinsic to the usual type of iteration scheme. The treatment of the general quantum equation (d2/dt2 + ω2)x(t) = εf(x), for the Heisenberg position operator x(t), is first motivated by the classical analog. The iteration equations for the quantum case are derived, and the case f(x) = x3 is studied fully to order ε, and partially to order ε2.
Spherically Symmetric Boundary-Value Problems in One-Speed Transport Theory
Authors: Case K. M. , Zelazny R. , Kanal M.
Published: February 2004

A Green's function technique is applied to one-velocity neutron problems with spherical symmetry. The main advantage of the approach is that it bypasses the need to construct explicitly the appropriate spherical eigenfunctions. Indeed, these can be directly deduced from this new formulation, if one so desires.
On Crossing Relations for Helicity Amplitudes
Authors: Hara Yasuo
Published: February 2004

Crossing relations between s-channel helicity amplitudes and u-channel helicity amplitudes are derived, following the derivation of the crossing relations between s-channel helicity amplitudes and t-channel helicity amplitudes by Trueman and Wick [T. L. Trueman and G.-C. Wick, Ann. Phys. (N.Y.) 26, 322 (1964)]. The over-all phase factor of crossing relations, which was not given by Trueman and Wick, is also derived by specifying the path of analytic continuation and by applying these relations in the static approximation.
Considerations on the Lorentz-Dirac Equation
Authors: Cohn Jack
Published: February 2004

Charged particles whose motions are described by the Lorentz-Dirac equation are considered. It is proved that, for an arbitrary collection of such classical charges moving solely under mutual electromagnetic interaction, the position of each charge and the force acting on each are infinitely differentiable functions of the proper time throughout any time interval for which the particle separations are all nonzero. A corollary of this result is then shown to be that discontinuous-, step-, or δ-function-type external boundary forces which might act on nonzero-size charged particles are not permissible.
On the Translation Invariance of Local Internal Symmetries
Authors: Landau Lawrence J. , Wichmann Eyvind H.
Published: February 2004

The notion of a local internal symmetry is discussed with reference to local physical theories and tentative definitions are presented. It is shown that under fairly general assumptions commonly made about local theories a local internal symmetry commutes with all translations.
Wavepacket Volkov Solutions
Authors: Meyer James W.
Published: February 2004

The problem of a scalar particle interacting with a planewave external field is solved in terms of the complete set of normalized in-states which are present in the asymptotic past. By expressing the field operator in terms of the annihilation and creation operators for the in-states, we show that the normal LSZ asymptotic condition holds allowing an unambiguous interpretation of the wavefunctions, i.e., in the sharp-momentum limit of the wavepacket Volkov solutions, the momentum parameter appearing in the solution is the momentum of the particle before the particle entered the beam. A brief discussion on how to include other interactions is also presented.
Six Integral Theorems for Vector Spherical Harmonics
Authors: Clapp Roger E. , Li H. Tung
Published: February 2004

With the aid of a generalized addition theorem for spherical harmonics [J. Math. Phys. 11, 1 (1970), preceding paper], six integral theorems for vector spherical harmonics are proved. A source-point direction (θ1, φ1) is first expressed in terms of a field-point direction (θ, φ) and a polar-coordinate angle-pair (θ′, φ′), which has as its polar axis the field-point direction. For a particular choice of n and m, all the components of the vector spherical harmonics for the source, expressed in terms of (θ1, φ1), are integrated over the relative azimuth angle φ′ while the field-point direction (θ, φ) and the relative polar angle θ′ are held fixed. The result in each case is a spherical harmonic or vector spherical harmonic of the field-point direction, with the same n and m but now depending on (θ, φ) instead of (θ1, φ1), multiplied by an explicit function of the relative polar angle θ′.
Field-Theoretic Description of Massless Particles with Higher Spin and Definite Parity. I. Integer Spin
Authors: Cukierda T. , Lukierski J.
Published: February 2004

It is known that there exist S + 1 equivalent covariant formulations of the theory of free massive particles with integer spin S and definite parity P. From these S + 1 equivalent theories one gets by putting m = 0 the description of S + 1 different massless particles, with helicities (k, −k), k = 0, 1, …, S. This property is already known in the case S = 1. In this paper we demonstrate explicitly how to obtain three kinds of massless particles, with different helicities, from three equivalent formulations of free massive theory with S = 2. Further, we outline the general argument for arbitrary integer value of S.
Connection between Marchenko Formalism and N/D Equations: Regular Interactions. II
Authors: Cornille Henri
Published: February 2004

In a previous paper we have shown, for S-waves, that the resulting integral equations of the f/f equations (equivalent to the N/D approach) can be obtained from the Marchenko formalism. The potential V(μ, r) reconstructed from the discontinuity μΔ(x) is −2(d/dr)[(dD/dr)/D], where D(μ,r) is the Fredholm denominator of the Jost solution and D(μ,0) that of the resulting integral equation. For ``regular discontinuities'' we find different classes of V(μ, r). First, if D(μ,0)=0, then V(μ, r) is not ``regular at the origin'' [in general, we find that V(μ, r) becomes marginally singular: repulsive and singular like r−2]. Secondly, if D(μ,0)≠0, then V(μ, r) is ``regular'' at the origin and we obtain the following: (i) If |μ| is less than the smallest modulus root of D(μ,0), then V(μ, r) has no poles for r ≥ 0. This range of |μ|-values where the iteration series of the resulting integral equations converge is limited by the smallest |μ| value where a real or complex ghost can appear or where a bound state can appear at zero energy. (ii) For |μ| larger than this smallest modulus root but μ inside the interval given by the first positive and negative roots, V(μ, r) has no second-order poles for r ≥ 0. These results (i) and (ii) are obtained with the restriction that in the considered interval there do not exist (μ, r)-values such that D(±μ,r)=0, and from our study we cannot conclude that this is always true. (iii) For μ outside the above interval, V(μ, r) has poles of the second order for r > 0, the ``bound states'' being, in general, real or complex ghosts, or ``bound states'' corresponding to badly behaved potentials. We find also that the Jost solutions for energy equal to zero are D(−μ,r)/D(μ,r). This gives the connection between ghosts and, in general, possible bound states appearing at zero energy; this gives also the relations between poles of V(μ, r) corresponding to opposite μ values. These results for the Jost function correspond to a normalization at infinity, so we have considered the problem of subtractions with normalization at an arbitrary point. Then the new Fredholm determinant is the product of the old one by the value of the Jost function at the subtracted point. It follows that, if the first μ-greater-than-zero and the first μ-lessthan-zero roots of D(μ,r) are not opposite (r ≥ 0), the |μ| interval of convergence of the iteration series is enlarged for the subtracted equation.
Diffraction of Waves by a Conducting Cylinder Coated with a Moving Plasma Sheath
Authors: Yeh C.
Published: February 2004

The scattering of plane electromagnetic waves by a perfectly conducting cylinder coated with a moving dielectric or plasma sheath is investigated theoretically. The homogeneous sheath is assumed to be moving in the axial direction with a uniform velocity vz with respect to the conducting cylinder. Solutions of this problem are obtained by making use of the special theory of relativity, the covariance of Maxwell's equations, and the Lorentz transformations. Results are given in terms of the radiation patterns of the scattered fields. A rather unique feature concerning mode coupling between the incident wave and the scattered wave is found. Even at normal incidence for vz ≠ 0, an incident E wave or H wave will produce a scattered wave which contains both E and H waves. Detailed discussions are presented.
Reduction of a Class of O(4, 2) Representations with Respect to SO(4, 1) and SO(3, 2)
Authors: Barut A. O. , Böhm A.
Published: February 2004

A complete classification of representations of SO(4, 2) with infinitesimal generators SAB, characterized by the representation relation {SAB, SAC} = −2agBC, and their extension by parity have been determined. The possible values of a are a = 1 − S2, S = 0, 12, 1, 32, 2, ⋯. These representations have then been reduced according to the two chains SO(4,2)⊃SO(4,1)⊃SO(4)⊃SO(3) and SO(4,2)⊃SO(3,2)⊃SO(3)⊗SO(2). The equivalence of these representations with the oscillator representations is established.
Approach to Equilibrium in a Simple Model
Authors: Radin Charles
Published: February 2004

The time evolution of a class of generalized quantum Ising models (with various long-range interactions, including Dyson's 1/rα) has been studied from the C*-algebraic point of view. We establish that: (1) All 〈A〉t are weakly almost periodic in time; (2) there exists a unique averaging procedure over time; (3) the time evolution in the thermodynamical limit can be locally implemented by effective Hamiltonians in the algebra of quasilocal observables; (4) there exists a specific connection between the spectral properties of the time evolution of the initial state and the approach to equilibrium; (5) there are examples in which the time evolution is not G-Abelian.
Classical Thermodynamics Simplified
Authors: Rastall Peter
Published: February 2004

Classical thermodynamics is developed in a rigorous and quite general form. The approach is similar to Carathéodory's in that entropy and temperature are defined in terms of quantities which are more directly measurable, but Pfaffian forms and quasistatic processes do not appear. The mathematics used is elementary, apart from a small amount of symbolic logic and a very little topology.
Coulomb Green's Function in f-Dimensional Space
Authors: Hostler Levere C.
Published: February 2004

It is shown that the f-dimensional nonrelativistic Coulomb Green's function and the associated reduced Green's functions can be obtained by differentiation of the corresponding functions in the 1-dimensional (f odd) or 2-dimensional (f even) case. A new expansion of the 3-dimensional coordinate-space Coulomb Green's function and a new sum formula for a product of two Laguerre polynomials with different arguments are derived.
Lorentz Invariance in a Gravitational Field
Authors: Kirkwood Robert L.
Published: February 2004

In any theory of gravity in which free particles move along the geodesics of a 4-dimensional metric tensor, a particular class of metrics can be defined which correspond to the fields of Newton's theory of gravity. In these Newtonian fields the metric coefficients which describe intrinsic properties of space and time are clearly separated from those that describe the gravitational field. This separation suggests an invariance in the gravitational field which is quite similar to the usual Lorentz invariance of electro-magnetism. The infinitesimal form of the generalized Lorentz transformation is determined by the fact that the 3-dimensional geometry remains Euclidean under the transformation. The finite form is determined so that the transformations form a group, and the group is found to be the usual Lorentz group. The transformation is then applied to fields that are not necessarily Newtonian.
Time Translations in the Algebraic Formulation of Statistical Mechanics
Authors: Dubin D. A. , Sewell G. L.
Published: February 2004

We present a new description of time translations in the C*-algebraic formulation of statistical mechanics. This description is based on weaker assumptions than the hitherto accepted ones, due to Haag, Hugenholtz, and Winnink (HHW) [Commun. Math. Phys. 5, 215 (1967)]. It is shown that these weaker assumptions still lead to the principal results of HHW for Gibbs states and, further, that our assumptions, unlike those of HHW, are valid for the ideal Bose gas and strong-coupling BCS models.
Integral Transformations in Momentum Space and Conformal Invariance
Authors: Castell L.
Published: February 2004

A certain element Z of the identity component of the conformal group together with the Poincaré subgroup generate the whole conformal group. In order to prove the conformal invariance of an S-matrix, only the invariance under Z has to be checked, once relativistic invariance has been established. The explicit form of Z for certain physically important representations of the different covering groups of the conformal group will be derived. The transformation Z turns out to be an integral transformation.
Derivatives of Phase Shifts and Binding Energies by Use of Variational Principles
Authors: Tikochinsky Y.
Published: February 2004

The Kohn-Hulthén variational principle for the phase shifts, as well as the Rayleigh-Ritz principle for the binding energies, are used to determine the derivatives of δl = δ(V, E, l, m, ℏ) and E = E(V, l, m, ℏ) with respect to the listed parameters. A similar treatment utilizing Hamilton's variational principle leads to the corresponding classical results. The relation between the quantum mechanical and the classical expressions is examined. In particular, it is found that the quantum-mechanical binding energy corresponds to a certain path average of the classical energy. Some applications of resulting formulas are briefly reviewed. This work is an extension of ideas originated by Fock and Demkov.
Rough-Surface Scattering: Shadowing, Multiple Scatter, and Energy Conservation
Authors: Lynch P. J. , Wagner R. J.
Published: February 2004

Multiple-scatter and shadowing effects are included in an extended theory of high-frequency scattering from a surface rough in one dimension. The single-scatter probability of slopes relation, corrected for shadowing, is an immediate consequence for any stationary random process. The double-scatter contribution (shadow corrected) is derived as well, and it provides a significant correction for surfaces with appreciable rms slope. The total power scattered by a perfectly reflecting rough surface is numerically evaluated as a test of energy conservation; the results show that the double-scatter formulation is substantially more accurate than the conventional single-scatter, unshadowed theory, particularly in the cases of large angles of incidence or very rough surfaces.
General Expansion of the Determinant of the Maxwell Dyadic
Authors: Gelman Harry
Published: February 2004

Electromagnetic propagation, influenced by arbitrary tensor constitutive functions in an unbounded medium, is considered. The general expansion of the determinantal eigenvalue equation for the dispersion relations is obtained, exhibiting, for the first time, the functional dependence of the eigenvalue equation on the constitutive tensors.
New Solutions of the Kinematic Dynamo Problem
Authors: Childress Stephen
Published: February 2004

The steady-state kinematic dynamo problem in a homogeneous 3-dimensional core is studied. The existence of a class of smooth solenoidal dynamos, satisfying a no-slip condition on the core boundary, is proved using perturbation theory. The dynamos are of the form q = q(1) + q(2) + q(3), where q(1) is spatially periodic on a sufficiently small scale of length, q(2) is zero except near the core boundary, and q(3) is an arbitrary sufficiently small motion. The term q(1) is also a spatially periodic dynamo in an appropriate sense for an infinite core. The last property allows a simple characterization of the bounded dynamos in terms of the admissible q(1).
Multiplicities in the Classical Groups. I
Authors: Gruber B.
Published: February 2004

A unified treatment of all three multiplicities, valid for all classical (compact, connected, simple Lie) groups, is described. The general theory is given and then applied to the rank-2 groups.
Linear Integral Equations for a Certain Class of H-Functions Applicable to the Theory of Neutron Transport and Radiative Transfer
Authors: Burniston E. E. , Siewert C. E.
Published: February 2004

A matrix version of the classical Riemann-Hilbert problem defined on an open contour is discussed. The problem is reduced to a quasiregular integral equation for cases where the sufficient Hölder continuity condition is satisfied and the component indices are nonnegative. As an illustration of this procedure, linear integral equations (rather than the usual nonlinear forms), for Chandrasekhar's functions Hl(μ) and Hr(μ) are established in a form amenable to solution by numerical iteration.
Exact Recursion Relation for 2 × N Arrays of Dumbbells
Authors: McQuistan R. B. , Lichtman S. J.
Published: February 2004

It is shown that A(q, N), the number of ways of arranging q indistinguishable dumbbells on a 2 × N rectangular array of compartments, is exactly described by the recursion relationA(q,N)=A(q,N−1)+2A(q−1,N−1)+A(q−1,N−2)−A(q−3,N−3). For large values of N the normalization of the distribution generated by this recursion relation is found to be 0.665(3.214)N and the maximum number of arrangements occurs when the array is approximately 61% occupied.
The Lorentz Group and the Sphere
Authors: Held A. , Newman E. T. , Posadas R.
Published: February 2004

A direct connection between the spin and conformally weighted functions on the sphere and geometric objects in Minkowski space is established through the isomorphism of the conformal group of the sphere to the restricted Lorentz group. It is shown that with the use of these functions one can duplicate all the standard work on the representations of the Lorentz group. It is shown further that these functions can be used to obtain a generalization of the classical equations of motion in which internal degrees of freedom arise naturally.
Model of Phase Transitions for Interacting Fermi Systems
Authors: Scott H. L.
Published: February 2004

The behavior of a previously discussed model of phase transitions for Fermi systems is analyzed in the region very near the critical point. It is shown that for a very general class of 2-body interactions the chemical potential is analytic in the temperature and density at the critical point, so that the model is in this sense equivalent to the classical theory of phase transitions. An extension of the model to include certain 3-body and higher interactions leaves this conclusion unchanged.
Calculation of the Inner Multiplicity of Weights by Means of the Branching-Law Method and by Racah's Recurrence Relation
Authors: Wong M. K. F. , Gruber B.
Published: February 2004

We discuss here in detail the validity of the branching-law method suggested in a previous paper [M. K. F. Wong, J. Math. Phys. 11, 1489 (1970)] for the calculation of the inner multiplicity of weights in an irreducible representation of a classical group. It is found that the method works for all the irreducible representations of SU(n) and SO(2n + 1), but that in the case of SO(2n) and Sp(2n) the method does not always give complete solutions except in some simple cases. It is then suggested that Racah's recurrence relation be used in these cases so that complete solutions may be obtained. It is also noted that Racah's recurrence relation alone is sufficient to obtain the inner multiplicity of all weights. This fact is utilized in the calculation of inner multiplicities in another paper [B. Gruber, J. Math. Phys. 11, 3077 (1970)]. The method suggested in this paper is illustrated through the calculation of some typical examples of the inner multiplicity of weights in the two classical groups SO(2n) and Sp(2n).
SU(4)⊃SU(2)⊗SU(2)
Projection Techniques
Authors: Draayer J. P.
Published: February 2004

The twofold multiplicity problem associated with the Wigner supermultiplet reduction SU(4)⊃SU(2)⊗SU(2) is resolved by spin-isospin projection techniques analogous to the angular momentum projection technique introduced by Elliott to resolve the SU(3)⊃R(3) multiplicity problem. The projection quantum numbers, which furnish either an integer or half-integer characterization of the multiplicity, are assigned according to an (ST)-multiplicity formula derived from a consideration of the symmetry properties of spin-isospin degeneracy diagrams. An expression is obtained for the coefficients which relate the SU(4)⊃SU(2)⊗SU(2) projected basis states to states labeled according to the natural U(4)⊃U(3)⊃U(2)⊃U(1) chain. General expressions for SU(4)⊃SU(2)⊗SU(2) coupling coefficients and tensorial matrix elements are given in terms of the corresponding U(4)⊃U(3)⊃U(2)⊃U(1) quantities.
Theorem of Uniqueness and Local Stability for Liouville-Einstein Equations
Authors: Choquet-Bruhat Yvonne
Published: February 2004

We prove, by use of energy inequalities, a theorem of uniqueness and local (i.e., for finite time) stability for the solution of Cauchy problem relative to the integro-differential system of Einstein and Liouville. A global theorem of geometrical uniqueness follows from a general method, previously given. We will prove elsewhere an existence theorem.
Hamiltonian Operators via Feynman Path Integrals
Authors: Cohen Leon
Published: February 2004

It is shown that Kerner and Sutcliffe's derivation leading to a unique quantum mechanical Hamiltonian from its classical counterpart is indeed not unique. This is done by using the same method as Kerner and Sutcliffe to derive different quantum Hamiltonians from the same classical Hamiltonian. It is also shown that other ordering rules besides that of Born and Jordan can be derived with the same technique.
Adiabatic Response to an Oscillatory Field
Authors: Young Ralph H. , Deal Walter J.
Published: February 2004

The behavior of a quantum-mechanical system with a slowly modulated oscillatory Hamiltonian is characterized by an adiabatic theorem similar to that for a system with a slowly changing ``static'' Hamiltonian. Quasiperiodic states—solutions of the instantaneous Schrödinger equation with an oscillatory Hamiltonian—play the same role as eigenfunctions of the instantaneous Hamiltonian do in an adiabatic theorem for a nearly static Hamiltonian. As an example, the theorem is used to establish the correct wavefunction to be used in computing the refractive index of atomic hydrogen.
Finding Hidden Symmetry by Numerical Methods
Authors: Heilmann Ole J.
Published: February 2004

Given a Hermitian matrix that depends upon some parameter, one often wants to be able to find all the parameter independent symmetries of the matrix. A method for accomplishing this is given here. The only numerical procedure involved is the diagonalization of Hermitian matrices.
Ground State Functional of the Linearized Gravitational Field
Authors: Kuchař Karel
Published: February 2004

The ground state functional of the linearized Einstein theory of gravitation is given as a functional of the gauge invariant Ricci tensor, and compared with the corresponding electromagnetic expression. The connection of the canonically quantized nonlinear theory of gravitation with the linearized theory is exhibited. Time is treated as a momentum variable rather than as a superspace coordinate, which leads to an ``extrinsic time representation'' hTTik, hi, t = −½Δ−1πT. The state functional of the linearized theory is shown to be the initial value of the state functional of the canonical theory on a constant extrinsic time hypersurface in the lowest order of a perturbation expansion. By means of the Einstein-Schrödinger equation, this functional can be integrated off this initial hypersurface.
Einstein Tensor and 3-Parameter Groups of Isometries with 2-Dimensional Orbits
Authors: Goenner Hubert , Stachel John
Published: February 2004

The algebraic classification of the Weyl and Ricci tensors and the relation between them in a Riemann space with an isometry group possessing a nontrivial isotropy group are reviewed. All metrics with Minkowski signature, invariant under a 3-parameter isometry group with 2-dimensional orbits having nondegenerate metrics, are constructed from the group properties and are shown to have Ricci tensors with a double eigenvalue, and the orbits are shown to be surfaces of constant curvature. The null orbits are shown to have a triply degenerate eigenvalue of the Ricci tensor. The various additionally degenerate metrics are classified in further detail, extending the work of Plebański and Stachel.
Perturbation Theory for Damped Nonlinear Oscillations
Authors: Mendelson Kenneth S.
Published: February 2004

A perturbation theory has been worked out for the decay of autonomous, nonlinear oscillations in the case where there is large linear damping. The solution reduces to a solution obtained by Kryloff and Bogoliuboff for small damping and to the perturbation solution for periodic oscillations for vanishing damping. The solution is applied to the decay of oscillations in Duffing's equation. In this case it shows good agreement with a solution obtained by numerical integration.
Half-Range Expansion Theorems in Studies of Polarized Light
Authors: Burniston E. E. , Siewert C. E.
Published: February 2004

The established normal modes of the vector equation of transfer describing the transport of polarized light are used to construct solutions to typical half-space problems. The half-range completeness theorem required by this method is discussed in the context of systems of singular integral equations. Although the Riemann-Hilbert problem encountered here is defined in terms of continuous rather than Hölder-continuous functions, the existence of a canonical solution is established, and the developed properties of this canonical matrix are used to complete the proof of the necessary half-range expansion theorem.
Group Representations and Geometry
Authors: Robinson G. de B.
Published: February 2004

We confine our attention here to simply reducible groups and show how six of the seven points of a finite projective plane PG(2, 2) constitute a ``Pasch'' diagram representing a 6j symbol. The class of all equivalent symbols may thus be represented by the seventh point in the plane. Analyzing the symmetries of such configurations, we derive two theorems, the first of which is the geometrical analog of Regge's result while the second gives the geometrical analog of the multiplication of two 6j symbols. In these terms the analogs of Eqs. (I1), (I2), and (I3) of Appendix I of Irreducible Tensorial Sets by Fano and Racah are very simply expressed. In particular, the Biedenharn identity (I3) becomes a vector equation (mod 2), and the relation with Desargues' theorem is clarified. The advantage of this geometrical model is that the structure alone survives and all summations and complicated coefficients disappear.
Generalized Operators
Authors: Svetlichny George
Published: February 2004

A formal expression T in creation and annihilation operators (e.g., the Hamiltonian for a field theory model) is generally not a densely defined bona fide Hilbert space operator but is usually a densely defined sesquilinear form; as such it is convenient to consider it as a linear map from a dense domain Φ− of a Hilbert space Φ0 to a still larger space Φ+ of antilinear functionals on Φ−; that is, T:Φ−→Φ+⊃Φ0. We give here the basis of a mathematical structure theory of such generalized operators. The idea which we explore is that, associated with T, there is a (not necessarily unique) analytic family Rλ of generalized operators called the resolvent of T. Formally, Rλ = (λ − T)−1, an equation to which we give more precise interpretations. The ambiguities in determining Rλ are associated with the arbitrary adjustments that are characteristic of renormalization programs. When appropriate conditions are met, we can construct from Rλ a new Hilbert space Ψ0 and a bona fide operator TR (the renormalized T) which is related to T by a formal intertwining equation TRΔ = ΔT, where Δ maps Φ− into a space containing Ψ0. Given several generalized operators, we outline a procedure by which a subset of these can be renormalized to bona fide operators while the rest are reinterpreted as new generalized operators in the new Hilbert space. These are the rudiments of a multiplicity theory. Numerous examples illustrate the methods; in particular, the Nθ sector of the Lee model with arbitrary cutoff (including none) is treated in detail.
S Matrix in the Heisenberg Representation
Authors: Borie Edith
Published: February 2004

The elements of the S matrix are calculated directly from an operator formalism, using the method of Yang and Feldman. This method has the advantage of providing a simple and direct justification of the Feynman rules for gauge fields, which express the contribution to the S matrix from a diagram containing closed loops in terms of sums over lowest-order physical amplitudes (tree amplitudes) in which all external lines are on the mass shell and have physical polarizations. This guarantees unitarity. A condensed notation due to DeWitt is used. First, the one- and two-loop contributions to the amplitude for production of a single quantum, and the amplitudes for pair production and scattering of a single particle by a classical background field are calculated in the absence of an invariance group. Noncausal chains (loops of cylically connected advanced or retarded Green's functions) never appear at any stage of the calculation, thus giving the decomposition into sums over tree amplitudes directly. This result is then generalized in an obvious way to the case in which an invariance group is present. The amplitudes are expressed in terms of a noncovariant propagator which propagates only physical (transverse) quanta. Rewriting these expressions in terms of covariant propagators leads to the formal appearance of ``fictitious quanta,'' which compensate the nonphysical modes carried by these propagators. All results are in agreement with those obtained by other methods.
On the Accuracy of the Adiabatic Separation Method
Authors: Caswell Randall S. , Danos Michael
Published: February 2004

Numerical experiments performed for a model of two strongly coupled oscillators indicate that the adiabatic separation method yields accurate results even where the condition of adiabaticity is violated to a very high degree, except in those cases where two levels are degenerate in the adiabatic approximation. An accurate solution for those cases can be obtained by diagonalizing the 2 × 2 Hamiltonian submatrix built on the two degenerate adiabatic states. It is conjectured that the adiabatic separation method can be expected quite generally to yield highly accurate results, at least for states belonging to the discrete spectrum.
Approximations for the Frequency Spectrum of a Simple Lattice
Authors: Baroody E. M.
Published: February 2004

The simple cubic lattice with harmonic forces between nearest neighbors only is considered. Starting with the expression for the spectrum of squared frequency G(x) as a Fourier transform of a product of Bessel functions, the asymptotic expansions about singular points are studied. For a range of values of the ratio of noncentral to central force constants σ, simple approximations are obtained which give quantitative information on the spectrum, not only very near the singular points, but over a substantial part of the frequency range. For σ = ½, it has been found that a cusp at x = ½ has been overlooked in previous work. In this case, essentially the entire spectrum is dominated by five singularities (at x = 0, ¼, ½, ¾, 1), and can be represented accurately by simple expressions. The approximations developed break down for very small σ, but appear to work well for ¼ ≤ σ ≤ 1.
Ising-Model Spin Correlations on the Triangular Lattice. III. Isotropic Antiferromagnetic Lattice
Authors: Stephenson John
Published: February 2004

The asymptotic behavior of the pair correlation ω2(r) = 〈σ0σr〉 between two spins at sites 0 and r on an axis of an isotropic antiferromagnetic triangular lattice is investigated with the aid of the theory of Toeplitz determinants as developed by Wu. The leading terms in the asymptotic expansion are obtained for large spin separation at fixed nonzero temperature. Evidence is presented that the zero-point behavior of the correlation is of the form ω2(r) ∼ ε0r−½ cos ⅔πr, where r = |r| is the spin separation and ε0=212(E0T)2=0.632226080…,E0T being the decay amplitude of the pair correlation at the Curie point (critical point) of an isotropic ferromagnetic triangular lattice. A special class of fourth-order correlations ω4(r) = 〈σ0σδσr σr+δ〉 − 〈σ0σδ〉 〈σrσr+δ〉 between the four spins at sites 0, δ, r, and r + δ on the same lattice axis, where δ is a lattice vector, is reconsidered. The asymptotic form of the correlation for large separation of pairs of spins r = |r| is obtained for all fixed temperatures.
Evaluation of a Unitary Integral. II. Mandelstam Iteration
Authors: Dilley James
Published: February 2004

The methods of a previous communication are used to derive a general expression for the first Mandelstam iteration of inputs of the form At(s,t)= ∑ jgj(s,t)δ(t−tRj),where the sum is over terms representing t-channel resonances in narrow-width approximation. The coefficients gj(s, t) can be either analytic in s, as for an expansion in t-channel partial waves, or meromorphic in s, as in a Padé- or Schlessinger-type expansion.
On Some Relations between the Solutions and the Parameters of Second-Order Linear Differential Equations
Authors: Goel Narendra S. , Buff Frank P.
Published: February 2004

The calculation of the Green's function for a point charge near an anisotropic planar diffuse layer leads to the differential equation U″ − k2D2(u)U = −2kδ(u − u′). Solutions for small and large k are obtained readily by a procedure equivalent to the principle of invariant imbedding. It is shown that the additional terms introduced in the solution for large k, due to the lack of smoothness in D(u) specifying the dielectric properties of the layer, may lead to singularities in the image potential. Similar considerations apply in the case of 1-dimensional wave propagation through a stratified medium.
Born Approximations in Statistical Mechanics
Authors: Marchand Jean-Paul
Published: February 2004

There are various possibilities in defining Born approximations for the evolution of density matrices. They differ, however, from the ordinary Born approximation for pure states by the fact that they do not, in general, conserve the fundamental properties of physical states. We shall describe these possibilities geometrically in the so-called Liouville space of Hilbert-Schmidt operators and investigate their significance in nonequilibrium dynamics. The results admit of an analogous interpretation in classical statistical mechanics.
Exact Stationary States in Local Relativistic Field Theories
Authors: Rosen Gerald
Published: February 2004

For theories featuring n scalar fields and an interaction energy density that converges rapidly for large field magnitudes to the form representative of a linear theory, we present exact physical solutions to the Schrödinger stationary state equation. These solutions show that such quantum field theories are effectively linear with no interaction and scattering between the quanta.
Exponential Hilbert Space: Fock Space Revisited
Authors: Klauder John R.
Published: February 2004

An exponential Hilbert space, which is an abstraction of the familiar Fock space for bosons, provides a natural framework to discuss a wide class of field-operator representations. This framework is especially convenient when wide invariance groups, such as a unique translationally invariant state, are involved. In this paper, we develop the theory of exponential Hilbert spaces in a functional fashion suitable to discuss representations of field operators enjoying such invariance features. Representations of both current algebras and canonical field operators are discussed, and it is shown that these representations are natural generalizations of those characterizing infinitely divisible random processes. Questions of reducibility and equivalence are treated, and we prove that our construction gives rise to infinitely many unitarily inequivalent representations. Nevertheless, an extremely simple expression, bilinear in annihilation and creation operators, abstractly characterizes the operators of both the current algebras and canonical fields. Dynamical applications to quantum field theory will be treated in subsequent papers.
Variational Methods in Potential Scattering
Authors: Morawitz Hans
Published: February 2004

A comparison of a variety of standard variational methods with a new method due to Harris is made with a view towards selection of the optimal method to be used in realistic many-body scattering calculations. Numerical results for two short-range potentials, the attractive exponential and the attractive Yukawa potential, are given and compared with exact results obtained analytically or by direct numerical integration. It is demonstrated by calculation that the source of an anomaly observed in earlier studies by Schwartz is not due to the attributed reason and, furthermore, we find the Kohn method significantly more accurate at the Harris eigenvalues than any of the other methods. Therefore, we propose the use of the Kohn method at the eigenvalues of the matrix Hamiltonian in the trial function subspace as the optimal method to be used in variational scattering calculations.
Many-Body Point Transforms
Authors: Witriol Norman M.
Published: February 2004

The method of point transforms is extended to the consideration of the truly many-body terms of many-body systems. The point transformation is taken to be regional (one body isolated, two bodies interacting, three bodies interacting, …, N bodies interacting) and continuous in the 3N-dimensional transformed space. The transformed Hamiltonian and wavefunction of an N-body system can then be easily approximated by one-body, two-body, three-body, …, or N-body additivity. As an example of the use of this method, a set of regional point transforms is developed for the short range, strongly repulsive, or hard-core many-body problem. Applying these transforms to the many-body Hamiltonian, we obtain a set of equivalent Hermitian Hamiltonians which are Fourier analyzable and amenable to ordinary perturbation and variational techniques. Assuming two-body additivity, this set is shown to be the set of pairwise point transformations developed by Eger and Gross.
Formulation of the Many-Body Problem for Composite Particles. II. Equivalence of Physical and Ideal State Spaces
Authors: Girardeau M. D.
Published: February 2004

It is shown that all inner products are preserved by the correspondence between the physical and ideal state spaces.
On the Green's Functions for the Bethe-Salpeter Equation
Authors: Huang Justin C. , DeFacio Brian
Published: February 2004

We have studied four possible Green's functions of the Bethe-Salpeter equation for equal mass scalar bosons by considering contours in an off-the-mass-shell complex energy (Ω) plane and an on-the-mass-shell magnitude of 3-momenta (k) or energy (ωk) complex plane. We find that three of the Green's functions—causal, retarded, and advanced Green's function—cannot lead to scattering solutions. Only a new Green's function, which we have called the scattering Green's function, with appropriate restrictions on the interaction V(x), can lead to scattering states.
Green's Functions for Multidimensional Neutron Transport in a Slab
Authors: Garrettson G. , Leonard A.
Published: February 2004

The integral form of the one-speed, steady-state Boltzmann transport equation is solved for a point source in a homogeneous, isotropically scattering slab. In addition, solutions are obtained for line sources and plane sources in the slab, both normal and parallel to the slab faces. Using Fourier and Laplace transforms, the problem is reduced to that of solving a 1-dimensional integral equation with a difference kernel. This equation is transformed into a singular integral equation which is solved using standard methods. The Green's functions are subsequently obtained as generalized eigenfunction expansions over the spectrum of the 1-dimensional integral operator. This form yields a simple solution far from the source, and alternate expressions are obtained to facilitate evaluation near the source. In a thick slab the exact solutions are shown to reduce to simple closed expressions plus correction terms that decrease exponentially as the slab thickness increases. Most of the work previously done in multidimensional transport in slabs is shown to be easily reproduced using this theory in the thick-slab approximation. Also, virtually all other problems of this type can be solved using the theory presented here. In particular, the density from a pencil beam of particles normally incident to the slab is obtained.
A Superposition Principle in Physics
Authors: Gudder Stanley P.
Published: February 2004

A superposition principle is considered both in classical mechanics and in the quantum logic approach to quantum mechanics. It is shown, roughly speaking, that in classical mechanics the only type of superposition of states is a mixture and that no pure state is a nontrivial superposition of other pure states. In quantum mechanics it is shown that, if a superposition principle holds, then the quantum logic is a complete atomic lattice.
Clebsch-Gordan Coefficients for the Coupling of SL(2, C) Principal-Series Representations
Authors: Anderson R. L. , Rączka R. , Rashid M. A. , Winternitz P.
Published: February 2004

An explicit expression is obtained for the Clebsch-Gordan coefficients which appear in the specification of a basis corresponding to the decomposition of the tensor product of two arbitrary SL(2, C) principal-series representations. Special cases of physical interest are treated in detail.
Radiation from a Dipole Imbedded in an Inhomogeneous Media with Critical Coupling Regions
Authors: Bahar E.
Published: February 2004

Maxwell's equations for inhomogeneous media are transformed into loosely coupled first-order differential equations by taking into account the local features of the refractive index profile. A set of generalized WKB type solutions are derived for the electromagnetic fields which are valid for regions of critical coupling. These solutions also merge with the familiar WKB solutions for vertically polarized waves propagating in slowly varying media.
Electromagnetic Form Factors in SL(n, C)-Calculation of Ultraspherical Polynomials
Authors: Thornber N. S.
Published: February 2004

We derive a simple expression for the functions encountered in electromagnetic form factor calculations done with infinite-component fields. Our expression holds for the general case of SL(n, C). We relate our functions to Gegenbauer polynomials.