bluuebunny/crossref_metadata_2025_split huggingface dataset:
Bounds on Elements of the S Matrix for Elastic Scattering: One-Dimensional Scattering
Authors: Bartram Ralph , Spruch Larry 
Published: January 2005

For the quantum-mechanical scattering of a particle by a static potential, a method due to Kato determines upper and lower bounds on cotη, where η is the phase shift for a given angular momentum. The method is readily generalizable to the case of elastic multi-channel scattering if one can a priori diagonalize the scattering matrix S to determine the standing-wave eigenmodes (by symmetry considerations, for example). We here consider the case when this can not be done. Both bounds are obtained on the independent elements Bij of B, where B=i(l+S)(l−S)−1, which then determine bounds on the elements of S, and on the eigenphase shifts and mixing parameters. One-dimensional scattering by V(x)≠V(−x) provides a concrete two-channel example. The numerical results obtained are very encouraging.
The Kato method is useful only if one can obtain solutions of related scattering problems. The most significant feature of the present approach is that the related scattering problems can be single-channel scattering problems.
Traveling-Wave Technique of Cyclotron Radiation
Authors: Parzen P. , Nomicos G. 
Published: January 2005

A traveling-wave technique of computing the radiation from accelerated electrons is described. In contrast to the usual method which expands the field due to the moving electrons in spherical waves, an expansion in traveling waves along a preferred direction is utilized.
This method is especially suitable in those cases where the radiation is confined by cylindrical conducting boundaries or plasmas. The technique is illustrated by considering the problem of the radiation from an electron in circular motion both in free space and within a coaxial conducting cylinder, and from an electron in a variable-pitch helical orbit.
On the Restricted Lorentz Group and Groups Homomorphically Related to It
Authors: Macfarlane A. J. 
Published: January 2005

A study is made of the real restricted Lorentz group, L, and of its relationship
(a) to the group, SL(2C), of complex unimodular two-dimensional matrices, and
(b) to the group, O3, of orthogonal transformations in a complex space of three dimensions.
The discussion of case (a) is an improved version of the treatment by Wightman. Its notable features are, firstly, that it gives important formulas in new concise forms and their proofs in an elegant and economical manner, and, secondly, that it deals with the nontrivial matter of proving the internal consistency of the formalism. To illustrate the practical utility of the theory, the product of two nonparallel pure Lorentz transformations is studied. In the discussion of case (b), explicit formulas realizing the isomorphism of O3 and L are obtained. These formulas are new and have been applied, for illustrative purposes, to the derivation of the transformation properties under L of the electromagnetic field vectors, regarded as a complex three-vector (E + iH). A result analagous to the factorization of the general element of L into a spatial rotation and a pure Lorentz transformation, and to the polar decomposition of the general element of SL(2C), is derived for O3. Insight into the relationship of O3 to L is provided by considering the unimodular matrix description of the complex Lorentz group, and the contrasting specializations of it that lead to the unimodular matrix descriptions of its subgroups, O3 and L.
On a Functional Averaging Method in the Classical Ensemble Theory
Authors: Galindo A. 
Published: January 2005

Albertoni, Bocchieri, and Loinger (ABL) have given a general prescription to calculate the average E(F) of functionals F[ρ] defined on the space L of the initial states (Liouville density functions) ρ(x) of an arbitrary (finite) dynamical system (S, Σ, μ, H). Then they have proved that E(BH) = 0 for any B ∈ Σ, where BH[ρ]≡lim lim T→∞1/T× ∫ 0T| ∫ Bρ(x,t) dμ(x)−μ−1(S)μ(B)|2dt[ABL theorem]. On the other hand, these authors claim that E gives the same weight to each ρ∈L and therefore, they contend that their theorem is sufficient to justify the classical statistical mechanics.
In this note, however, the following statements are proved: (A) If (S, Σ, μ, H) is ergodic with discrete spectrum, then every probability measure ν in L for which BH[ρ] = 0, ν-almost everywhere (for any B ∈ Σ) must be concentrated at e [e(x) ≡ μ−1(S) on S]. (B) Given (S, Σ, μ, Ho) with Ho = 0, every (normally defined) functional average A(F) such that A(BHo) = 0 for any B ∈ Σ must define a probability measure ν in L such that ν({e}) = 1. From (B) it will follow that: (1) The ABL average E(F) does not give the same weight to each ρ∈L; actually, E defines a probability measure in L concentrated at e; consequently, no significant nontrivial conclusion can be naturally drawn from the ABL theorem regarding the foundations of the classical statistical mechanics; (2) it is impossible to restore in general the validity of the ABL theorem by choosing a different normal functional averaging method. To illustrate (2) a new (nonconcentrated) functional average E*(F) is introduced and it is proved that E*(BH) ≠ 0 for every nonweakly mixing system (S, Σ, μ, H) and some B ∈ H.
Evaluation of the Quantum-Mechanical Ring Sum with Boltzmann Statistics
Authors: DeWitt Hugh E. 
Published: January 2005

A procedure is given for the evaluation of the quantum-mechanical ring sum at finite temperature. The method is used for the evaluation of the quantum corrections to the classical Debye-Hückel free energy for an electron gas obeying Boltzmann statistics. The ring sum is shown to be of the form (βe2/πλD)P(γ), where γ=ƛ/λD,ƛ=ℏ/(2mkT)1/2, and λD is the Debye screening length. The quantum effects for finite γ are due only to the operation of the uncertainty principle. The function P(γ) decreases monotonically from the classical value π/3, and the form is shown to be P(γ)=(π/3)[1+ ∑ n=2anγ2(n−1)]3/2− ∑ n=2bnγ2n−3. The coefficients an and bn are evaluated exactly for small n and asymptotically for large n. The two series converge for γ2 &lt; γc2 = 2.042 ….
For γ ≫ γc the function P(γ) is also evaluated as an asymptotic expansion in inverse powers of γ1/2. Thus the low-temperature correlation energy of distinguishable electrons is obtained in random phase approximation. At zero temperature, the result is the same as the correlation energy obtained by Foldy for charged bosons. The correlation pressure in this approximation is negative and diverges as ρ1/4 at high density.
Particle Charge Symmetries as R8 Subgroups
Authors: Peaslee D. C. 
Published: January 2005

It is shown how the formalism of 8 × 8 Dirac matrices can be extended to include all the groups so far proposed as relevant to elementary-particle charge symmetries: R4, R7, G2, SU3. These are all treated as subgroups of R8, which appears to determine the eightfold structure of the particle families, even before the particle interactions are ``switched on''. Since these subgroups of R8 are incompatible, they will lead to a ``clash of symmetries'', as observed experimentally. It is pointed out that if the plausible association is made between real and charge-space statistics in representations of R8, the group SU3 satisfies charge-conjugation invariance for 3-boson interactions but not for two-fermion-one-boson interactions.
An argument is given that the representations 8 and 8̄ of SU3 plus the representations (7 + 1) of G2 and R7 completely span the symmetries obeying the limited invariance implied by conservation of isotopic spin I and hypercharge Y.
Quantum Electrodynamics of the Stimulated Emission of Radiation. I
Authors: Bevensee R. M. 
Published: January 2005

The quantum electrodynamic formalism of Senitzky is developed so as to yield long-time differential equations for the time-average expectation values of electric field energy and excess molecular population in a system composed of one resonant cavity mode, two-level molecules, and a dissipation mechanism. The cavity-mode resonant frequency is essentially the mean molecular transition frequency of either a Gaussian or Lorentzian distribution of transition frequencies. The intermolecular and lattice T1 and T2 time constants are assumed to be much longer than the stimulated emission period. The field is coupled to either the electric or the magnetic dipole moments of the molecules.
Senitzky's papers are summarized and the relevant expressions for the Heisenberg field and molecular operators are listed. Modified time-average expressions are obtained for the presence of a transient coherent driving field and perhaps an ``off-resonance'' molecular distribution of transition frequencies. Time averages of the derivatives of various terms in the expected field energy are compared to derivatives of time averages of those terms. Well-behaved differential equations are obtained for the time-average expectation values of field energy and molecular excess population, justified on a long-time basis by the application of an intermittent similarity transformation. A differential equation for the time-average dispersion (second moment) of electric field energy is obtained, which indicates that the relative dispersion tends to decrease during the pumping interval and increase back to the thermal value during the emission interval. Energy transfer from pumping field to molecules to the resonant field during the pumping interval is described qualitatively. The direct-product form for the density matrix ρ = ρ(field) × ρ(molecules) × ρ(dissipation mechanism) is justified by maximum-entropy inference. In conclusion, the equations of motion for the time-average expectation values of field energy and molecular population are interpreted so as to explain the envelope modulation of a solid-state laser beam during the emission period.
Integration of the Partial Differential Equations for the Hypergeometric Functions F1 and FD of Two and More Variables
Authors: Olsson Per O. M. 
Published: January 2005

From Kummer's 24 solutions of the ordinary hypergeometric equation and their connections, the general solution of the partial differential equations for the Appell function F1 in the neighborhood of the singular points is found. Connections between the solutions are given to the extent required to continue the F1 function to the neighborhood of any of its singular points.
The corresponding problem for the hypergeometric function FD of more than two variables is briefly indicated.
Effects of Finite Sample Size on the Measurement of Transport Quantities
Authors: Nossal Ralph 
Published: January 2005

A determination is made of the temporal evolution of the momentum autocorrelation function for a noninteracting gas constrained by walls. A frequency-dependent dielectric constant is calculated from the latter by applying an appropriate Kubo relation. Deviations from the Drude conductivity are obtained which, for high frequency and/ or large systems, are found to be ∼ o(102/βmω2L2), where L is the spatial dimension of the assembly and ω the frequency of the applied field.
A similar calculation is performed to determine the effects of boundaries upon measurements of scattering cross sections for slow neutrons. It is found that, for realistic experimental conditions, wall perturbations are unlikely to be of importance.
Proof that Scattering Implies Production in Quantum Field Theory
Authors: Aks Stanley O/. 
Published: January 2005

A proof of the necessity of production processes in quantum field theory is carried out in the axiomatic framework. It is shown that a field theory that is assumed to have a nontrivial scattering amplitude violates crossing symmetry, if production processes are absent. The proof is based on the rigorous analytic properties of the scattering amplitude, particularly the analyticity in the invariant-scattering variables s, t, and u. In the case of a scalar theory with pairing symmetry, the scattering amplitude is known to be analytic within the domain |stu| &lt; 7168m6, except for the usual cuts. Under the working assumption that production processes are null, it is shown that this domain can be enlarged by applying the elastic unitarity conditions beyond the (usual) elastic region. The domain is enlarged sufficiently to include the first Landau singularity of the absorptive part of the scattering amplitude. This singularity is not symmetric in s and t within the extended domain, and this is incompatible with the crossing symmetry of the scattering amplitude. In order to avoid a contradiction, the discontinuity across this Landau singularity must be null. It follows that the scattering amplitude must itself be null.
In the course of the proof it is shown that the conclusion is valid for a scattering amplitude satisfying the requirements of an S-matrix theory embodied in the Mandelstam representation.
On a Special Class of Type-I Gravitational Fields
Authors: Trümper Manfred 
Published: January 2005

This paper contains an investigation of the algebraic structure and the analytic properties of a class of normal hyperbolic Riemannian 4-spaces restricted by the following condition: There exists a timelike unit vector ua such that the Riemann tensor satisfies *Rabcdubud = 0. This condition is shown to be equivalent to the statement that the conform tensor is Petrov type I with real eigenvalues, ua being a principal vector and an eigenvector of the Ricci tensor. This means that there is no flux of nongravitational energy relative to an observer travelling with 4-velocity ua.
The eigen null directions (Debever vectors) of the conform tensor lie in a timelike hyperplane spanned by ua and the two eigenvectors of εac ≡ −Cabcdubud belonging to the eigenvalues with largest absolute value. The conform tensor is degenerate (type D) if and only if a Debever direction projected into the rest space of an observer ua is an eigendirection of εab.
The complete set of Bianchi identities is examined. It yields an expression for the covariant eigentime derivative of εab and an algebraic relation linking the rotation and shear of ua to the curvature tensor of the Riemannian space.
The general results are applied to special Einstein spaces (Rab = 0) admitting a congruence of timelike curves without shear and rotation. We get a new simplified proof of the theorem that in the case of nondegeneracy such spaces are static, the curves of the congruence being paths of an isometric motion.
N-Particle Kinematics and Group-Theoretical Treatment of Phase Space I. Nonrelativistic
Authors: Lévy-Leblond Jean-Marc , Lurçat François 
Published: January 2005

This paper is a group-theoretical study of the kinematics of n nonrelativistic particles. A systematic method is given to construct a new complete set of commuting observables. The method is based on the existence of a group (the ``great group'') which acts transitively on the phase-space manifold and preserves the phase-space volume element; the observables are then Casimir operators of the great group and of some of its subgroups, including the usual three-dimensional rotation group.
Among these collective observables, in addition to the total angular momentum, the most interesting is the ``togetherness operator'' which describes the simultaneous localization of the n particles. This operator is a generalization to n &gt; 2 of the square of orbital angular momentum; its use allows to generalize to n particles the familiar centrifugal barrier arguments.
Rigorous Treatment of the Van Der Waals-Maxwell Theory of the Liquid-Vapor Transition
Authors: Lebowitz J. L. , Penrose O. 
Published: January 2005

Rigorous upper and lower bounds are obtained for the thermodynamic free-energy density a(ρ, γ) of a classical system of particles with two-body interaction potential q(r) + γνφ(γr) where ν is the number of space dimensions and ρ the density, in terms of the free-energy density a0(ρ) for the corresponding system with φ(x) ≡ 0. When φ(x) belongs to a class of functions, which includes those which are nonpositive and those whose ν-dimensional Fourier transforms are nonnegative, the upper and lower bounds coincide in the limit γ → 0 and limγ → 0 a(ρ, γ) is the maximal convex function of ρ not exceeding a0(ρ) + ½αρ2, where α ≡ ∫ φ(x) dx. The corresponding equation of state is given by Maxwell's equal-area rule applied to the function p0(ρ) + ½αρ2 where p0(ρ) is the pressure for φ(x) ≡ 0. If a0(ρ) + ½αρ2 is not convex the behavior of the limiting free energy indicates a first-order phase transition. These results are easily generalized to lattice gases and thus apply also to Ising spin systems.
The two-body distribution function is found, in the limit γ → 0, to be normally identical with that for φ(x) ≡ 0, but if the system has a phase transition it has the form appropriate to a two-phase system.
Some of the upper and lower bounds on a(ρ, γ) are simple enough to be useful for finite γ. Also, some of our results remain valid for quantum systems.
Permutation-Algebraic Formulation of Spin-Free Transition Density Matrices
Authors: Poshusta R. D. , Matsen F. A. 
Published: January 2005

Spin-free transition density matrices are derived from spin-free kets which are symmetry-adapted to the symmetric group and its algebra. The Dirac identity establishes that these spin-free density matrices are identical to those obtained by integrating the spin from the full-spin density martices. Derivations are first given for arbitrary primitive kets which may be geminals of higher polymals, after which we consider products of orbitals, either orthonormal or nonorthonormal.
Correlation in the spin-free space is discussed and we show the influence of permutational symmetry on the probability of coincidence of pairs. A special case of this correlation is the well-known Fermi hole.
Generalizations of the Virial and Wall Theorems in Classical Statistical Mechanics
Authors: Siegert Arnold J. F. , Meeron Emmanuel 
Published: January 2005

A generalized virial theorem which expresses inverse compressibility in terms of integrals of virials and canonical distribution functions through the four-particle distribution is transformed to the grand canonical ensemble and becomes an expression for compressibility in terms of the same integrals formed with grand canonical distribution functions. The integrals are of a mixed (virial and fluctuation) type.
While the thermodynamic functions expressed by the same integrals with canonical and grand canonical distribution functions are quite different, the two formulas agree in the thermodynamic limit because of the different asymptotic behavior of canonical and grand canonical distribution functions. We also derived an alternative form of the second virial theorem which expresses compressibility in terms of integrals over virials and grand canonical distribution functions through the three-particle distribution function only. It is shown that this form and its generalizations to higher derivatives of the density, as well as the hierarchy of fluctuation theorems and the fugacity expansions of distribution and correlation functions can all be very simply derived from a set of integro-differential equations satisfied by the grand canonical distribution functions. A generalization of the wall theorem (P/kT = ρwall) is derived and shown to be equivalent to the generalized virial theorem (canonical form).
A Closed Form for Elsasser Integrals
Authors: Wolfson Seymour J. 
Published: January 2005

An Elsasser integral is defined as the definite integral of a particular product of three spherical harmonics, namely, ∫ 02π ∫ 0πYn1*m1[∂Yn2m2∂θ∂Yn3m3∂φ−∂Yn2m2∂φ∂Yn3m3∂θ]dθ dφ.
This integral occurs in connection with the ``dynamo theory'' of the magnetic field of the earth and thus it is desirable to have a value of the integral involving only the indices of the spherical harmonics. Such a closed form is developed and some properties of the integral are discussed.
n-Representability Problem for Reduced Density Matrices
Authors: Kummer Hans 
Published: January 2005

In this paper we prove some theorems about the n-representability problem for reduced density operators. The first theorem (Theorem 6) sharpens a theorem proved by Garrod and Percus. Let Tnp be the set of all n-representable p-density operators. Then a density operator Dp belongs to Tnp¯ (the bar indicates the closure with respect to a certain topology) if and only if Tr (DpBp) ≥ 0 for all bounded self-adjoint p-particle operators Bp, such that their n-expansion (pn)ΓpnBp≡ ∑ i1&lt;…&lt;ipBp(i1…ip)is a positive operator in n-particle space. Moreover, it is shown that Tnp¯ is the closed convex hull of the exposed points of Tnp of finite one-rank (Theorem 9). A more practical version of this theorem may be formulated in the following manner (cf. Theorem 8).
Consider the set γp of subspaces of the n-particle space, occurring as an eigenspace to the deepest eigenvalue of a bounded n-particle operator which is the n expansion of some p-particle operator. Choose from every element of γp one (and only one) vector (function) and form the corresponding reduced p-particle operator. Tnp¯ is the closed convex hull of all these p-density operators (cf. Theorem 9). For p = 1, this theorem reduces to Coleman's theorem about the n representability of the 1 matrix.
Foundations for a Treatment of the Scattering of Light by the Hydrodynamical and Statistical Atom Model
Authors: Wakano Masami 
Published: January 2005

A study has been made on the scattering of light by the hydrodynamical and statistical atom model. Bloch's treatment of the hydrodynamical equations of motion for this model is supplemented here by inclusion of the interaction with the electromagnetic field. We limited attention to oscillations of small amplitude. By correspondence principle arguments, general expressions were derived for the cross sections for absorption, coherent and incoherent scattering.
The energy can be expressed—following Bloch—as the energy of a Thomas-Fermi atom plus a Hamiltonian which is associated with departures from the Thomas-Fermi distribution. Using Bloch's quantization of this Hamiltonian and applying the method of quantum field theory, we rederived the correspondence principle results for elementary cross sections.
Then applying the correspondence theoretical argument to the matrix element, we rederived Heisenberg's result for the total intensity of the Compton scattering. We also apply the method of stationary phase to the hydrodynamical treatment and show that this method gives the same result as does the linear term in the momentum transfer in Heisenberg's expression, except for a numerical factor 3½/2—a point that was discussed by Bloch many years ago.
Application of the general formulas given here for angular distribution of Rayleigh and Compton scattering will require electronic machine calculations of higher modes of oscillation of the gas model of the atom analogous to those made by Wheeler and Fireman for l=1.
Renormalization of a Classical Gravitational Field Interacting with Quantized Matter Fields
Authors: Utiyama Ryoyu , DeWitt Bryce S. 
Published: January 2005

The behavior of the mean value of the energy-momentum tensor of a set of quantized matter fields interacting with a classical gravitational field which is, in turn, produced by this mean value, is investigated. Singularities appear in the energy-momentum tensor corresponding to divergences of three different orders: ∞4, ∞2, and log ∞. These can be removed by the introduction of counter terms into Einstein's equation. The ∞4 singularity is removed by a ``cosmological term,'' the ∞2 singularity by a renormalization of the gravitation constant, and the log ∞ singularity by a counter term derivable from a Lagrangian which is quadratic in the Riemann tensor.
The gravitational Green's function corresponding to this semiclassical approximation to the fully quantized theory is found to have the asymptotic behavior 1/p4 instead of 1/p2, and thus to have a much weaker singularity in the coordinate representation than the Green's function of the ``bare'' linearized theory.