bluuebunny/crossref_metadata_2025 huggingface dataset:
Computation of Matrix Elements of Integrable Nuclear Two-Particle Interaction Operators
Authors: Nesbet R. K.
Published: January 2005

A general procedure is described for computing matrix elements of operators that occur in a realistic nuclear Schrödinger Hamiltonian, in a basis of orbitals with radial factors of the form Nar2na+la exp (−γar2), where na is an arbitrary nonnegative integer and γa is an arbitrary positive number. The method is suitable for efficient large-scale computation of these matrix elements, needed when orbitals of physical interest (such as Hartree-Fock orbitals) are expressed as linear combinations of basis functions of the kind indicated. The analysis of matrix elements of a tensor operator provides a new method of reduction to linearly independent reduced matrix elements, the number of which is smaller than it is in the usual analysis. Thus, the number of independent parameters in the corresponding empirical theory of complex nuclear spectra is reduced. The specific operator forms considered here are those present in the asymptotic one-pion exchange potential, except that the functional forms of the potentials V(r) are not prescribed beyond the requirement that they should be integrable.
Invariant Operators of the Unitary Unimodular Group in n Dimensions
Authors: Klein Abraham
Published: January 2005

An elementary derivation is given of Biedenharn's construction of a complete set of independent invariants for the group SU(n). The basic tool is the mapping of the adjoint representation onto the linear space of generators in the defining representation. The trace of any algebraic function of the matrix thus associated is seen to constitute an invariant of the adjoint representation and yields by substitution an invariant operator. The independent invariants are recognized by their isomorphy to the invariant forms under the permutation group.
Computer Studies of Energy Sharing and Ergodicity for Nonlinear Oscillator Systems
Authors: Ford Joseph , Waters John
Published: January 2005

Weakly coupled systems of N oscillators are investigated using Hamiltonians of the form H=12 ∑ k=1N(pk2+ωk2qk2)+α ∑ j,k,l=1NAjklqjqkql,where the Ajkl are constants and where α is chosen to be sufficiently small that the coupling energy never exceeds some small fraction of the total energy. Starting from selected initial conditions, a computer is used to provide exact solutions to the equations of motion for systems of 2, 3, 5, and 15 oscillators. Various perturbation schemes are used to predict, interpret, and extend these computer results. In particular, it is demonstrated that these systems can share energy only if the uncoupled frequencies ωk satisfy resonance conditions of the form ∑ nkωk ≲αfor certain integers nk determined by the particular coupling. It is shown that these systems have N normal modes, where a normal mode is defined as motion for which each oscillator moves with essentially constant amplitude and at a given frequency or some harmonic of this frequency. These systems are shown to have, at least, one constant of the motion, analytic in q, p, and α, other than the total energy. Finally, it is demonstrated that the single-oscillator energy distribution density for a 5-oscillator linear and nonlinear system has the Boltzmann form predicted by statistical mechanics. Thus, these nonlinear systems are shown to have many features in common with linear systems. In particular, it is unlikely that they are ergodic. From the standpoint of statistical mechanics, it is argued that this lack of ergodicity may be a welcome feature.
Remarks on the Combinatorial Approach to the Ising Problem
Authors: Burgoyne P. N.
Published: January 2005

A new proof is given of a certain conjecture due to Feynman. This conjecture relates graphs and paths on a lattice and was first proved by Sherman. It is the key step in a particular method of obtaining Onsager's formula for the partition function of the two-dimensional Ising model.
Approximate Evaluation of Feynman Functional Integrals
Authors: Rosen Gerald
Published: January 2005

By considering a new set of functions for the integration domain, Feynman's functional-integral representation of the propagation kernel is evaluated approximately for values of the time that are not too large. The approximation theory is applied to systems of uncoupled and strongly coupled oscillators.
Method for Construction of Unitary Operators in Quantum Field Theory
Authors: Moravek Paul H. , Joseph David W.
Published: January 2005

A method for the construction of explicit representations of unitary Hillbert-space operators in the particle-number representation is presented and illustrated by application to several examples.
Comparison of Two Generalizations of Maxwell's Equations Involving Creation of Charge
Authors: Chambers Ll. G.
Published: January 2005

A comparison is made of the Lyttleton-Bondi, and the Watson theories for the electromagnetic fields produced when charge is created. It is shown that, except when dimensions of the order of the radius of the Universe are involved, the difference is negligible and that consequently the Watson theory, being mathematically more simple, is the better.
Fundamental Properties of Perturbation-Theoretical Integral Representations. II
Authors: Nakanishi Noboru
Published: January 2005

In the first part of this paper, it is investigated, apart from the perturbation-theoretical basis, under what conditions the perturbation-theoretical integral representations can be derived, and two theorems are given concerning this problem. In the second part, the asymptotic behavior of the weight function in the integral representation is investigated in perturbation theory. It is proved that the weight function vanishes at infinity for an infinite sum over certain graphs which are much more general than the ladderlike graphs. This result gives the analyticity in the right half-plane of complex angular momentum.
High-Energy Behavior in Perturbation Theory. II
Authors: Polkinghorne J. C.
Published: January 2005

Contributions to the asymptotic behavior of Feynman integrals are evaluated which correspond to pinches in the interior of the hypercontour of integration. It is shown that they give the Gribov-Pomeranchuk phenomenon and Regge cuts. A set of diagrams is investigated which gives a Regge cut on the physical sheet.
Coupled Schrödinger Equations and Statistical Boundary Conditions
Authors: Dullemond C.
Published: January 2005

The possibility of introducing distribution functions for the phase of a radial wavefunction at a boundary is studied for the case of elastic scattering. The properties of a simple class of distribution functions are discussed. The results are generalized to the solutions of an arbitrary number of coupled Schrödinger equations which satisfy time-reversal invariance, provided there are no closed channels.
Canonical Variables for the Interacting Gravitational and Dirac Fields
Authors: Kibble T. W. B.
Published: January 2005

The problem of reducing the Lagrangian for the interacting gravitational and Dirac fields to canonical form is discussed, using the vierbein formalism. The arbitrary gauge variables corresponding to local Lorentz transformations of the vierbein are removed by imposing Schwinger's ``time-guage'' condition, and a further condition that the spatial part of the vierbein be symmetric. It is shown that in this guage the Lagrangian can be expressed in a canonical form involving essentially the same gravitational-field variables as in the absence of matter, and that the generators of spatial translations and rotations have the expected form.
A Class of Null Flat-Space Coordinate Systems
Authors: Newman Ezra T. , Unti T. W. J.
Published: January 2005

A new coordinate system, intrinsically attached to an arbitrary timelike world line, is investigated in flat-space time. The Maxwell field tensor associated with the field of an arbitrarily moving charged particle assumes a particularly simple form in this, its intrinsic coordinate system. This reference frame is expected to be useful in General Relativity, in asymptotic studies of radiation, and equations of motion.
Theory of the Soret Effect in Electrolytic Solutions
Authors: Helfand E. , Bearman R. J. , Vaidhyanathan V. S.
Published: January 2005

The limiting law for the square-root concentration dependence of the heat of transport of a simple electrolyte is calculated by considering the Soret effect. The calculation is accomplished by relating the intermolecular forces arising from the temperature gradient in the nonuniform Soret stationary state to the equilibrium gradients of chemical potential required to maintain the same concentration gradients. The contribution arising from ion-ion interactions is identical with that determined by Helfand and Kirkwood from a consideration of heat flow accompanying diffusion. Hence, the present work provides a verification of the heat-matter reciprocal relation without explicitly invoking time-reversal invariance.
Structure of the Three-Particle Scattering Operator in Classical Gases
Authors: Résibois P.
Published: January 2005

It is shown that in all cases where the finite duration of the collision may be neglected, there is a complete equivalence between the 3-particle scattering operator obtained by Prigogine and his co-workers and the corresponding expression derived by Choh and Uhlenbeck using Bogolubov's method. Great emphasis is put on the ``irreducible character'' of the three-body collision: Only situations in which the three particles are simultaneously interacting play a role in both theories.
On the Kinetic Theory of Dense Gases
Authors: Cohen E. G. D.
Published: January 2005

A statistical mechanical theory of a dense gas that is not in equilibrium is presented, which is completely analogous to the well known theory of a dense gas in equilibrium. In particular, an expansion of the pair distribution function in powers of the density for a gas not in equilibrium is given, corresponding with that in equilibrium to all orders in the density, that can be represented by the same diagrams. The expansion can be reduced to that derived by Bogolubov, Uhlenbeck, and Choh from a solution of the B-B-G-K-Y hierarchy. The conditions for the validity of the expansion are, for an infinite system at not too high density, and after the lapse of some time after t = 0: (1) a statistical assumption at t = 0; (2) some conditions on the interaction potential; (3) coarse-grained distribution functions. A simple generalization of the Boltzmann equation to general order in the density is included. Also, the connection with a Master equation for a spatially homogeneous system is discussed.
Variational Statistical Mechanics in Terms of ``Observables'' for Normal and Superfluid Systems
Authors: De Dominicis C.
Published: January 2005

Thermodynamical functions are expressed by perturbation theory as stationary functionals of the 1-, and 2-body potentials; the stationary conditions yield the 1-, and 2-body correlation functions in terms of the potentials. Through the use of diagramatic methods, it is possible to explicitly invert these relations and express the potentials in terms of the correlation functions; in turn the thermo-dynamical functions become stationary functionals of the correlation functions. This procedure is carried out for normal (quantum and classical) systems. For superfluid systems, 12−,32− body potentials, and correlation functions need also be considered and the above procedure becomes imperative to eliminate the nonphysical 12−,32− body potentials. Its first two steps are illustrated, and various features of this formulation and its usefulness are discussed.
The Master Equation with Special Transition Probabilities
Authors: Verboven E.
Published: January 2005

The Van Hove master equation to general order is solved, making a special ansatz for the function W11′(kk′). Approximate solutions are found for strong and weak coupling. For λ = 1, a comparison is given with the exact numerical solution.
Dimer Statistics and Phase Transitions
Authors: Kasteleyn P. W.
Published: January 2005

After the introduction of the concept of lattice graph and a brief discussion of its role in the theory of the Ising model, a related combinatorial problem is discussed, namely that of the statistics of non-overlapping dimers, each occupying two neighboring sites of a lattice graph. It is shown that the configurational partition function of this system can be expressed in terms of a Pfaffian, and hence calculated explicitly, if the lattice graph is planar and if the dimers occupy all lattice sites. By the examples of the quadratic and the hexagonal lattice, it is found that the dimer system may show a phase transition similar to that of a two-dimensional Ising model, or one of a different nature, or no transition at all, depending on the activities of various classes of bonds. The Ising problem is then shown to be equivalent to a generalized dimer problem, and a rederivation, of Onsager's expression for the Ising partition function of a rectangular lattice graph is sketched on the basis of this equivalence.
Mechanical Model for Quantum Field Theory
Authors: Peres Asher
Published: January 2005

We consider two harmonic oscillators, coupled during a finite time. Initial and final states can be defined unambiguously, and if the duration of the coupling is sufficiently short, the S matrix can be computed explicitly. The coupling gx2y is investigated in detail, for complex values of g. It is found that, along the real axis, the S matrix behaves smoothly as a function of g and tends to the unit matrix for g → 0, as it should. However, the S matrix has a line of essential singularities along the imaginary g axis, including the origin, so that it cannot be expanded into powers of g. If such an expansion is sought by means of a perturbation procedure, it is found that each term of the series is finite (no need of renormalization), but the series as a whole diverges.
On a New Method in the Theory of Irreversible Processes
Authors: Frieman Edward A.
Published: January 2005

A new method is presented for obtaining irreversible equations describing the approach to equilibrium in systems of many particles. The basic idea is the removal of secular terms arising in a perturbation expansion by the technique used in nonlinear mechanics. The irreversible equations then appear as consistency conditions for the existence of a well behaved expansion. The method relies heavily on the existence of the natural fine-scale mixing occurring in the dynamics.
Symmetry Functions of the Cube
Authors: Trammell George T.
Published: January 2005

A simple method is presented for reducing the 2J + 1 state vectors of good total angular momentum J to cubic symmetry types (basic vectors of the irreducible representations of the cubic rotation group). The cubic symmetry functions are written explicitly in terms of the eigenfunctions of Jz, Jx, and Jy, and formulas and tables which facilitate their use in obtaining eigenfunctions and energy splittings of ions in fields of cubic symmetry are presented and applied.
Symmetry in Quantum Theory
Authors: Emch Gérard , Piron Constantin
Published: January 2005

The aim of this note is to give a precise definition of symmetry in quantum theory in order to generalize Wigner's representation theorem, in the framework of lattice theory and projective geometry. We do not require the concept of physical state; the results are valid for any field (division ring) used in the realization of the lattice of propositions. Physical systems with the most general superselection rules are included in the theory.
Derivation of the Gell-Mann-Okubo Mass Formula
Authors: Goldberg H. , Lehrer-Ilamed Y.
Published: January 2005

A short proof is given for a mathematical identity which is used to derive the Gell-Mann-Okubo mass formula.
Transition Matrix for Nucleon-Nucleon Scattering. II
Authors: Kowalski K. L. , Feldman D.
Published: January 2005

The Fredholm reduction of the singular integral equation satisfied by the reactance matrix, which was developed in a previous paper, is extended so as to constitute a complete and unified Fredholm formalism for the various integral equations which occur in the momentum-space formulation of two-body (potential) scattering problems. The essential simplification which permits this unified treatment is the demonstration of the formal similarity of the scattering integral equations, whether or not part of the interaction includes a hard core. The principal result consists in an apparatus which is useful in performing those two-body calculations which occur in investigations of high-energy nucleon-nucleus scattering; in particular, the technique is used to obtain the solutions of integral equations satisfied by transition matrices which appear typically in multiple-scattering theories. The relation between the uniqueness of solutions of the scattering integral equations and the validity of the Fredholm formalism is also discussed. Finally, some methods which were previously considered for obtaining approximate solutions of the Fredholm equations are generalized.
Nonlinear Coupled Oscillators. I. Perturbation Theory; Ergodic Problem
Authors: Jackson E. Atlee
Published: January 2005

A study is made of some of the problems which arise in determining the long-time behavior of a system of coupled oscillators. Standard perturbation methods are examined in the light of certain classic results due to Poincaré and Whittaker concerning the construction of constants of motion which are analytic in the coupling constant, λ. These considerations lead to the study of perturbation methods which are not ordered in powers of λ. An examination of the various advantages of these methods leads to a method which removes secular terms in such a way as to mitigate the classic problem of small divisors. The significance of studies which attempt to relate the existence of analytic constants of the motion with the ergodic behavior of a system is examined.
Lyapunov's Stability Criteria for Plasmas
Authors: Fowler T. K.
Published: January 2005

The orbit stability theory of Lyapunov has been adapted to the Vlasov-Boltzmann equation governing plasmas. Both linear and nonlinear stability are considered. The theory is characterized by a search for Lyapunov functions, whose existence implies stability in analogy with particles trapped in a potential well, as in the energy principle. The most important result is an existence theorem for Lyapunov functions quadratic in perturbations in all linearly stable cases in which perturbations damp asymptotically (sufficiently fast). As a corollary, without damping, the existence of a quadratic Lyapunov function is necessary and sufficient to prevent exponential growth of perturbations. A prescription is given for finding Lyapunov functions which are constants of motion. An example is treated. The implication of nonlinear stability from linear stability with damping is discussed, and Dr. C. S. Gardner's direct proof of nonlinear stability of a Maxwellian plasma by Lyapunov's method is reported.
Partition Function for Certain Simple Lie Algebras
Authors: Tarski Jan
Published: January 2005

The partition function `which yields the multiplicities of weights in representations' is computed for the following Lie algebras: A2, B2, G2, and A3.
Note on the One-Dimensional Gas of Impenetrable Point-Particle Bosons
Authors: Schultz T. D.
Published: January 2005

A one-dimensional gas of impenetrable point-particle bosons is considered. An exact expression for the one-particle density matrix is derived in terms of the Fredholm determinant and resolvent of a certain simple kernel. It is proved that for large r, the density matrix is bounded by const × r−4/π2, and that this implies the absence of a generalized Bose-Einstein condensation, contrary to a recent approximate calculation by Girardeau, who first defined such a condensation.
Statistical Theory of the Energy Levels of Complex Systems. V
Authors: Mehta Madan Lal , Dyson Freeman J.
Published: January 2005

This paper is divided into three disconnected parts. (i) An identity is proved which establishes an intimate connection between the statistical behavior of eigenvalues of random unitary matrices over the real field and over the quaternion field. It is proved that the joint distribution function of all the eigenvalues of a random unitary self-dual quaternion matrix of order N is identical with the joint distribution function of a set of N alternate eigenvalues of a random unitary symmetric matrix of order 2N. A corollary of this result is the following: the distribution of spacings between next-nearest-neighbor eigenvalues in a real symmetric matrix of large order is identical with the distribution of nearest-neighbor spacings in a self-dual Hermitian quaternion matrix of large order. (ii) A conjecture is made which gives an exact analytic formula for the partition function of a finite gas of N point charges free to move on an infinite straight line under the influence of an external harmonic potential. This conjecture is at the same time a statement about the statistical properties of the eigenvalues of Hermitian matrices whose elements are Gaussian random variables. (iii) A list is made of several other problems which remain unsolved in the statistical theory of eigenvalues of random matrices.
Acnodes and Cusps and the Mandelstam Representation
Authors: Islam Jamal N.
Published: January 2005

Complex singularities on the physical sheet associated with acnodes (isolated real points) and cusps are found for a particular Feynman diagram. These singularities emerge before the appearance of anomalous thresholds, and cause a breakdown of the Mandelstam representation.
Density Matrices Arising from Incomplete Measurements
Authors: Wichmann Eyvind H.
Published: January 2005

The problem of how to associate a statistical ensemble of a quantum mechanical system with an incomplete set of measured ensemble averages is discussed. The view adhered to in this note is that the most chaotic ensemble consistent with the measured ensemble averages is a reasonable choice of ensemble. The properties of this ensemble are studied rigorously for the case when the quantum mechanical system is associated with a finite-dimensional Hilbert space. The results are extended heuristically to some particular ensembles of many-boson and many-fermion systems.
A Remarkable Representation of the 3 + 2 de Sitter Group
Authors: Dirac P. A. M.
Published: January 2005

Among the infinitesimal operators of the 3 + 2 de Sitter group, there are four independent cyclic ones, one of which is separate from the other three. A representation is obtained for which this one has integral eigenvalues while the other three have half-odd eigenvalues, or vice versa. The representation is of a specially simple kind, with the wavefunctions involving only two variables.
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.
Initial-Value Problem on Einstein-Rosen Manifolds
Authors: Lindquist Richard W.
Published: January 2005

A class of solutions of the time-symmetric initial-value equations for gravitation and electromagnetism is obtained on a two-sheeted manifold containing N Einstein-Rosen bridges. The initial metric tensor and electric field are expressed in terms of a pair of harmonic functions, called ``metric potentials,'' which are required to be analytic and asymptotically flat and to satisfy certain match-up conditions; these potentials are then determined by applying the method of inversion images. The particular case of two identical Einstein-Rosen bridges is examined in detail, and is shown to correspond to a wormhole in an asymptotically flat (one-sheeted) universe. A charge and ``renormalized mass'' are defined for each Einstein-Rosen bridge, as well as an interaction energy (gravitational plus electrostatic potential energy) for the system as a whole. Expressions for these and other physical characteristics are evaluated explicitly in the two-body case.
New Method for Solving Eigenvalue Problems
Authors: Sasakawa Tatuya
Published: January 2005

A new method of solving eigenvalue problems is proposed. The secular equation is represented in the form of a dispersion relation. Three methods of solving the dispersion relation are given. Several examples including both nondegenerate cases and degenerate cases are presented. All these examples demonstrate that the present treatment is better than other conventional perturbation treatments. The transformation coefficients between the unperturbed wavefunctions and the perturbed ones are given in terms of the eigenvalues thus obtained. The wavefunctions of the perturbed states form a complete orthonormal set if those of the unperturbed one do.
Solution of the Cauchy Problem for Laplace's Equation in Axially Symmetric Systems
Authors: Harker Kenneth J.
Published: January 2005

The determination of the electrodes for an electron gun reduces to solving the Cauchy problem for Laplace's equation. Using Green's Theorem, an analytic solution for the case of axial symmetry is obtained in terms of a line integral involving the product of the analytically continued values of the voltage and normal electric fields at the beam edge by an appropriate weighting function. Since an explicit solution is given in terms of line integrals, numerical results should be obtained far more accurately and quickly than those obtained by integrating Laplace's equation directly by finite-difference methods.
Kinetic Equation for an Unstable Plasma
Authors: Balescu R.
Published: January 2005

A kinetic equation is derived for the description of the evolution in time of the distribution of velocities in a spatially homogeneous ionized gas which, at the initial time, is able to sustain exponentially growing oscillations. This equation is expressed in terms of a functional of the distribution function which obeys the same integral equation as in the stable case. Although the method of solution used in the stable case breaks down, the equation can still be solved in closed form under unstable conditions, and hence an explicit form of the kinetic equation is obtained. The latter contains the ``normal'' collision term and a new additional term describing the stabilization of the plasma. The latter acts through friction and diffusion and brings the plasma into a state of neutral stability. From there on the system evolves towards thermal equilibrium under the action of the normal collision term as well as of an additional Fokker-Planck-like term with time-dependent coefficients, which however becomes less and less efficient as the plasma approaches equilibrium.
A Thermodynamical Limitation on Compressibility
Authors: Bragg Lincoln E. , Coleman Bernard D.
Published: January 2005

In their theory of thermostatics, Coleman and Noll have obtained a convexity inequality which places restrictions on admissible stress-strain functions for elastic materials. Here we show that for an arbitrary elastic material in an arbitrary state of strain F, the general convexity inequality implies that the modulus of compression k obeys the inequality k(F)≥23p̄(F) where p̄ is the mean pressure, i.e. minus one-third the sum of the principal stresses. Here k is defined to be the derivative of p̄ with respect to the mass density along a deformation process representing a uniform expansion from the state F.
Study of Several Lattice Systems with Long-Range Forces
Authors: Kac M. , Helfand E.
Published: January 2005

A number of one- and two-dimensional Ising lattice systems with long-range ferromagnetic interactions are studied. The theory introduces as basic variables stochastic fields acting at each site, but goes beyond Weiss mean-field theory (or the Bragg-Williams approximation) in giving a complete account of the statistics of these fields. A transition is manifest in these systems by a shift in the values of the stochastic fields which are important for the calculation of the partition function. Particular attention is devoted to the critical region where the range of significant stochastic fields broadens. The equation of state for the lattice gas corresponding to this model is of the van der Waals type. Comparison is frequently made between these results and the properties of an analogous one-dimensional continuum system studied by Kac, Uhlenbeck, and Hemmer.
Bases for the Irreducible Representations of the Unitary Groups and Some Applications
Authors: Moshinsky Marcos
Published: January 2005

In this paper we show that sets of polynomials in the components of (2j + 1)-dimensional vectors, solutions of certain invariant partial differential equations, form bases for all the irreducible representations of the unitary group U2j+1. These polynomials will play, for the group U2j+1, the same role that the solid spherical harmonics (themselves polynomials in the components of a three-dimensional vector) play for the rotation group R3. With the help of these polynomials we define and determine the reduced Wigner coefficients for the unitary groups, which we then use to derive the Wigner coefficients of U2j+1 by a factorization procedure. An ambiguity remains in the explicit expression for the Wigner coefficients as the Kronecker product of two irreducible representations of U2j+1 is not, in general, multiplicity-free. We show how to eliminate this ambiguity with the help of operators that serve to characterize completely the rows of representations of unitary groups for a particular chain of subgroups. The procedure developed to determine the polynomial bases of U2j+1 seems, in principle, generalizable to arbitrary semisimple compact Lie groups.
Spin-Component Analysis of Single-Determinant Wavefunctions
Authors: Sasaki Fukashi , Ohno Kimio
Published: January 2005

The weights of spin components involved in a single-determinant wavefunction are obtained. The behavior of the weights for a system of a large number of electrons is discussed.
Lower Bounds for Energy Levels of Molecular Systems
Authors: Bazley Norman W. , Fox David W.
Published: January 2005

It is shown that the Hamiltonian H for a molecule in which the nuclei are regarded as fixed has the form H = Σ Hα + H′, where H′ is positive and each Hα has known eigenvalues and eigenvectors. A procedure is developed for the calculation of lower bounds to the eigenvalues of H. The optimization of the procedure and the existence of the eigenvalues are discussed.
Kinetic Equations for Electrons and Phonons
Authors: Ron Amiram
Published: January 2005

A method is presented for deriving a set of kinetic equations for a system of electrons and phonons in a simplified model of a metal. By employing the second quantization representation for the creation and annihilation operators of the electrons and the phonons, an hierarchy of equations for the distribution functions and correlation functions is introduced. This hierarchy is studied, using an approach originally developed by Bogoliubov, where both truncation of the hierarchy and irreversibility are achieved under general assumptions. A set of kinetic equations is obtained for an homogeneous system, where the electrons dynamically shield both each other and the phonons.
Poles in Coupled Scattering Amplitudes
Authors: Dragt A. J. , Karplus R.
Published: January 2005

Elastic scattering amplitudes which have the analytic structure of the Mandelstam representation and which satisfy the unitarity condition and substitution law are severely restricted. Because of these restrictions, poles cannot occur independently and arbitrarily in amplitudes coupled by the unitarity condition. Instead, the locations and residues of poles in coupled amplitudes must satisfy the same relations as do poles in perturbation theory amplitudes.
Approximate Calculations of Scattering Phase Shifts
Authors: y Walter R.
Published: January 2005

The nonrelativistic scattering problem is reformulated by transforming the partial-wave equation into similarity with the corresponding equation for a known or soluble problem. It is shown that if the potentials for the two cases are sufficiently close (within about 10% typically), a particularly simple and accurate expression for the desired phase shifts can be written. Numerical examples for a Gaussian potential are given. The method is also extended to the calculation of bound-state energy levels, but no numerical examples are given.
An Integral Equation for the Associated Legendre Function of the First Kind
Authors: Ashour Attia A.
Published: January 2005

The solution of the Fredholm homogeneous equationψ(ξ)=λ ∫ 01M(ξ,ξ′)ψ(ξ′)dξ′,where M(ξ,ξ′)= ∫ 0πcosmφ dφ(x2−2xx′cosφ+x′2)12and x = (1 − ξ2)½ is found to be the associated Legendre function Pnm(ξ),n+m even, and the characteristic numbers of this kernel are obtained. The solution of the corresponding equation of the second kind is also found. The kernel of the homogeneous equation whose solution is Pnm(ξ), n + m odd, is obtained.
Duality Invariance and Riemannian Geometry
Authors: Penney R.
Published: January 2005

It is shown that the postulate of indistinguishability of the Maxwell field tensor from its dual leads to the concept of the electromagnetic field tensor as a spinor component in dual space. The demand for algebraic consistency dictates a unique connection with the gravitational field. The Maxwell field must be viewed as a set of potentials, and the necessity for a duality gauge condition excludes the existence of magnetic monopoles.
Reduction of the N-Particle Variational Problem
Authors: Garrod Claude , Percus Jerome K.
Published: January 2005

A variational method is presented which is applicable to N-particle boson or fermion systems with two-body interactions. For these systems the energy may be expressed in terms of the two-particle density matrix: Γ(1, 2 | 1′, 2′)=(Ψ |a2+a1+a1′a2′| Ψ). In order to have the variational equation: δE/δΓ = 0 yield the correct ground-state density matrix one must restrict Γ to the set of density matrices which are actually derivable from N-particle boson (or fermion) systems. Subsidiary conditions are presented which are necessary and sufficient to insure that Γ is so derivable. These conditions are of a form which render them unsuited for practical application. However the following necessary (but not sufficient) conditions are shown by some applications to yield good results: It is proven that if Γ(1, 2 | 1′, 2′) and γ(1 | 1′) are the two-particle and one-particle density matrices of an N-particle system [normalized by tr Γ = N(N − 1) and trγ = N] then the associated operator: G(1,2 | 1′,2′)=δ(1−1′)γ(2 | 2′)+σ Γ(1′,2 | 1,2′)−γ(2 | 1)γ(1′ | 2′) is a nonnegative operator. [Here σ is + 1 or − 1 for bosons or fermions respectively.]
Continuous-Representation Theory. III. On Functional Quantization of Classical Systems
Authors: Klauder John R.
Published: January 2005

The form of Schrödinger's equation in a continuous representation is indicated for general systems and analyzed in detail for elementary Bose and Fermi systems for which illustrative solutions are given. For any system, a natural continuous representation exists in which state vectors are expressed as continuous, bounded functions of the corresponding classical variables. The natural continuous representation is generated by a suitable set 𝔖 of unit vectors labeled by classical variables for which, for the system in question, the quantum action functional restricted to the domain 𝔖 is equivalent to the classical action. When a classical action is viewed in this manner it contains considerable information about the quantum system. Augmenting the classical action with some physical significance of its variables, we prove that the classical theory virtually determines the quantum theory for the Bose system, while it uniquely determines the quantum theory for the Fermi system.
Separation of Angular Variables in the Bethe-Salpeter Equation for Two Spin-½ Particles
Authors: Günther Marian
Published: January 2005

The Bethe-Salpeter equation for two fermions, in its usual differential form, is reduced here to one which involves a single spatial and a single temporal variable. Concomitantly, the original underlying 16-dimensional spinor space is reducible to one of 8 dimensions for angular momentum l ≠ 0 and of 4 dimensions for l = 0. The possibility of this reduction derives from the existence of one (or two) normal divisors of the matrices operating on the original spinor space, in these respective cases. The procedure is illustrated with the positronium problem, but may easily be seen to be quite general.
Regge Behavior of Forward Elastic Scattering Amplitudes
Authors: Bjorken James D.
Published: January 2005

A theorem concerning the asymptotic behavior of forward elastic scattering amplitudes in relativistic theories is stated and proved. The assumptions made are (1) identical spinless particles interact via Gφ3 and λφ4 couplings; (2) a cutoff of the propagators is introduced; (3) the forward scattering amplitude satisfies a Bethe-Salpeter equation in the crossed channel; (4) the kernel of the equation is an arbitrary finite subset of the Feynman graphs which compose the exact kernel. The theorem states that under these assumptions, the forward scattering amplitude exhibits Regge behavior, i. e., A(s, 0) → sα + O (1) as s → ∞.
Accuracy of the Semicircle Approximation for the Density of Eigenvalues of Random Matrices
Authors: Bronk Burt V.
Published: January 2005

Certain statistical properties of the energy levels of complex physical systems have been found to coincide with those for distributions of eigenvalues derived from ensembles of random matrices. However, if ensembles of random matrices give a fair representation for the Hamiltonian of a complex physical system, the density of the characteristic values at the lower end of the spectrum should show some similarity with the exponential dependence found in nuclear spectra. The limiting distribution of the density for very high-dimensional random matrices is a semicircle, i.e., concave from below if plotted against the characteristic value which represents, in this case, the energy. Hence the deviations from the limiting distribution are investigated and it is shown that there is a region, at the very lowest part of the spectrum, where the density is convex from below, similar to an exponential function. The region of convexity is called the tail of the distribution. It is shown, however, that the avergae number of roots in the tail is very small, of the order of 1. It is concluded that those ensembles of random matrices which have been studied up to now, do not give a fair representation of Hamiltonians of complex systems.
Theory of Vibrational Structure in Optical Spectra of Impurities in Solids. I. Singlets
Authors: McCumber D. E.
Published: January 2005

General expressions are derived describing the modifications of the optical spectra of isolated impurities induced by the subclass of interactions with host-lattice phonons which shift the energy eigen-values of the internal impurity state but which do not mix those internal states. The results apply to localized phonon modes as well as to extended modes (whose individual coupling decreases with increasing lattice volume), to anharmonic- as well as harmonic-phonon systems, and to phonon-impurity couplings which need not be linear in the dynamic local strain field. They take the form of ``linked-cluster'' expansions which, in the linear-coupling harmonic-phonon case, terminate after the second term. The relation of these expansions to moment methods is indicated. As a simple application of the linked-cluster expansions, we compute the dependence of the sharp ``no-phonon'' spectral line on impurity mass. We conclude that the no-phonon line should not exhibit isotopic-mass displacements in those impurity sites for which the lattice is inversion symmetric.
Stroboscopic-Perturbation Procedure for Treating a Class of Nonlinear Wave Equations
Authors: Kruskal Martin D. , Zabusky Norman J.
Published: January 2005

A new perturbation procedure is presented for treating initial-value problems of nonlinear hyperbolic partial differential equations. The characteristic variables of the partial differential equation and the functions of these variables are expanded in powers of ε, and the formal solution is uniformly valid over time intervals O(1/ε). The uniform first-order solution is evaluated for the equationytt=(1+εyx)yxx,subject to the standing-wave initial conditions: y(x, 0) = a sin πx, yt(x, 0) = 0. This equation is the lowest continuum limit of an equation for which numerical computations are available. The uniform zero-order solution breaks down after a time tB = 4/εaπ. A detailed study of the solution is made in the vicinity of the breakdown region of the (x, t) plane, and it demonstrates that the formal solution for yx and yt goes from a single-valued to a triple-valued function while yxx and ytt become singular. To compare the solutions with the available numerical computations, the yx and yt waveforms are decomposed into spatial Fourier modes. The effect of breakdown is manifest in the modal amplitudes ∝ Jn(nT)/nT. The modal amplitudes change their asymptotic behavior, from exponentially decreasing as n → ∞, to algebraically decreasing when t goes from smaller to larger than tB. In the time interval up to breakdown, t < tB, the modal energies are in excellent agreement with the modal energies of the numerical computations, whereas for t > tB they diverge. For t < tB, the total energy calculated from the uniform zero-order solution is conserved and equal to the initial value, EC(0)=2a2π2 ∑ n=1∞[Jn(nT)(nT)]2=12a2π2=EC |t=0,t≤tB. Thus, the lowest-continuum-limit equations describe the dynamics of a discrete model for a finite time. A heuristic discussion is given which suggests that the time of description can be extended beyond tB by including higher spatial derivatives in the continuum model.
Explicit Formal Construction of Nonlinear Quantum Fields
Authors: Segal I. E.
Published: January 2005

It is shown that there can be associated with any given nonlinear relativistic partial differential equation an operator field satisfying the canonical commutation relations, transforming appropriately under the action of the Lorentz group, and propagated in accordance with the given differential equation. This quantization procedure is unique, apart from the scale of the commutators. The treatment is intuitive, but is capable of rigorization in terms of the mathematical theory of analysis in function space (functional integration). The results are in harmony with crucial conventional ones, so far as has been determined, and provide in principle a possible means of computing the collision matrix for particular systems without recourse to perturbation theory.
Some Properties of Triangular Representations of SU(3)
Authors: Rosen S. P.
Published: January 2005

The occurrence of an equal-spacing mass rule in the unitary decuplet can be explained either by observing that the isotopic spin and hypercharge of each particle are related by T = 1 + ½Y, or by making use of a theorem due to Diu and Ginibre. This theorem states that, in all triangular representations of SU(3), the matrix elements of an arbitrary tensor operator depend upon one reduced matrix element instead of two. Here we present a new proof of the Diu-Ginibre theorem and use it show that relations of the form T = λ ± ½Y exist for all triangular representations. We also show that the L and K spins are related to their corresponding hypercharges by L = λ ± ½YL and K = λ ± ½YK. One consequence is that the masses and magnetic moments of particles in a triangular multiplet are equally spaced. Other consequences are also discussed.
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.
Functional Differential Calculus of Operators
Authors: Rohrlich F.
Published: January 2005

The functional derivative with respect to operators of operator functionals is defined for operators which satisfy certain commutation relations of interest in quantum field theory. From this definition, a functional differential calculus is developed for functionals of tensor as well as spinor fields. It is noted that an implicit definition of the functional derivative can always be given while an explicit one seems to exist only for those operator fields which need not be restricted by supplementary operator conditions, and for which not more than one derivative occurs in the commutation relations.
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.