bluuebunny/crossref_metadata_2025 huggingface dataset:
Eigenfunctions of the Electron Spin Operators
Authors: Secrest Don , Holm Lloyd Martin
Published: January 2005

The problem of finding linear combinations of Slater determinants which are eigenfunctions of both the S2 and Sz operators is considered. The method of projection operators is used in forming eigenfunctions of S2. A recursion relation is given for the coefficients in the linear combination. Rules are given for selecting a complete linearly independent set of eigenfunctions and an explicit formula is given for the members of the set.
Systematic Characterization of mth-Order Energy-Level Spacing Distributions
Authors: Leff Harvey S.
Published: January 2005

The mth-order energy-level spacing distributions p(m)(x) for complex spectra are defined in terms of a general joint probability distribution PN(λ1, … λN) for N consecutive eigenvalues. The precise limiting processes involved are explained, and are subsequently used to obtain two formal representations of p(m)(x). Both representations yield p(m)(x) = xm/m! exp (−x) for statistically independent eigenvalues. One of the representations, which is an extension of Dyson's method for m = 0, 1, is applied to the superposition of n independent sequences of levels. General asymptotic results are found for the mth-order distributions for (a) small x, arbitrary n, and (b) arbitrary x with n → ∞.
A Remarkable Connection Between Kemmer Algebras and Unitary Groups
Authors: Roman P.
Published: January 2005

Starting from a Kemmer algebra generated by n elements, a new algebra generated by ½(n2 + n) elements is derived. This algebra has only one irreducible representation which is of dimension (n + 1). This representation is equivalent to the first elementary representation of the infinitesimal operators of the unitary group in a space of (n + 1) dimensions. Omitting the identity element, one of course obtains the first elementary representation of the unitary unimodular group in a space of (n + 1) dimensions. The procedure is first illustrated for the case of n = 3 and explicit representations of the infinitesimal operators of U4 (SU4) are obtained. Next, the proof for arbitrary n is outlined. Finally, the degenerate case of U3 (SU3) is discussed.
An Algebraic Approach to Quantum Field Theory
Authors: Haag Rudolf , Kastler Daniel
Published: January 2005

It is shown that two quantum theories dealing, respectively, in the Hilbert spaces of state vectors ℌ1 and ℌ2 are physically equivalent whenever we have a faithful representation of the same abstract algebra of observables in both spaces, no matter whether the representations are unitarily equivalent or not. This allows a purely algebraic formulation of the theory. The framework of an algebraic version of quantum field theory is discussed and compared to the customary operator approach. It is pointed out that one reason (and possibly the only one) for the existence of unitarily inequivalent faithful, irreducible representations in quantum field theory is the (physically irrelevant) behavior of the states with respect to observations made infinitely far away. The separation between such ``global'' features and the local ones is studied. An application of this point of view to superselection rules shows that, for example, in electrodynamics the Hilbert space of states with charge zero carries already all the relevant physical information.
Zero-Mass Representations of the Proper Inhomogeneous Lorentz Group
Authors: Korff David
Published: January 2005

The representations of the proper inhomogeneous Lorentz group are investigated as a function of both real and imaginary mass, in the limit as the mass approaches zero. One obtains only the physical mass-zero representations in either limit, the infinite spin representations being unique to zero mass. It is found that there exists a superselection rule which prohibits the position and spin operators from being physically observable in the mass-zero limit.
Lower Bounds on the Lehmann Weights in Spin-Zero Meson Theory
Authors: Hagen C. R.
Published: January 2005

It is shown within the framework of conventional spin-zero meson theory that, if all renormalizations are assumed finite, the Lehmann weights of the fermion and meson Green's functions cannot decrease arbitrarily rapidly as a function of energy.
Lagrangian and Hamiltonian Formalisms with Supplementary Conditions
Authors: Schwartz Melvin
Published: January 2005

The present note generalizes a transformation due to Takahashi. The Takahashi transformation introduces a Hamiltonian formalism in the presence of a single linear supplementary condition imposed on n + 1 coordinates. The Takahashi transformation is generalized to treat the case when there are N independent linear supplementary conditions imposed on the differentials of n coordinates. It is shown that the generalized transformation leads to true coordinates when the coefficients of the differentials are coordinate-independent; otherwise the transformation generally leads to quasicoordinates. More general transformations are discussed, the relation with the method of Lagrange multipliers is established, and some problems which arise in connection with quantization are pointed out.
Renormalization of Singlet Amplitude in Intermediate-Vector-Boson Theory of Weak Interactions
Authors: Bolsterli M.
Published: January 2005

A simple prescription for renormalizing the singlet amplitude in the intermediate-vector-boson theory of weak interactions is given.
Existence of Scattering Solutions for the Schrödinger Equation
Authors: Belinfante Johan G.
Published: January 2005

The Fredholm alternative for the Schrödinger (Lippmann-Schwinger) equation is established for potentials with finite first moment, using the Ascoli selection theorem and Banach space methods.
On Trapped Trajectories in Brownian Motion
Authors: Jackson Julius L. , Coriell Sam R.
Published: January 2005

The Brownian motion of an ion in the attractive field of an infinite line charge is investigated with reference to the existence of trapped trajectories, i.e., trajectories which never depart from the neighborhood of the center of attraction. It is shown that there is a discontinuity in the nature of the Brownian motion as a function of the line charge density, marked by the appearance of trapped trajectories when the line charge density becomes greater than a critical value.
The Representation of Canonical Variables as the Limit of Infinite Space Volume: The Case of the BCS Model
Authors: Ezawa Hiroshi
Published: January 2005

We examine the possibility of obtaining the representation suitable for a given Hamiltonian through a limiting procedure. The model employed is that of the Bardeen-Cooper-Schrieffer's superconductivity theory, for which the infinite-volume limit of the Wightman functions gives the right representation. This method is found to be helpful in analyzing the so-called symmetry breaking solutions. In particular, it serves for throwing some light to the Haag's work on the gauge property of the BCS model as well as to the Nambu's dynamical model of elementary particles. What is meant by the limit of infinite volume is studied with regard to the convergence of the Hamiltonian operator.
Product Property and Cluster Property Equivalence
Authors: Sherman S.
Published: January 2005

Uhlenbeck and Ford in their presentation of the Ursell development express a sequence of symmetric functions WN in terms of another sequence of symmetric functions UN. They invert this sequence of equations and remark that from the first set of equations it follows that the product property of WN sequence is equivalent to the cluster property for the UN sequence. They ask for a simple proof of the equivalence. Here a modification of an algebra developed by Bohnenblust to prove Spitzer's formula on the fluctuations of the sums of independent, identically distributed, random variables is used systematically to invert the relations and to prove the equivalence under lighter assumptions than previously used.
On Multiple-Valued Random Functions
Authors: Lumley J. L.
Published: January 2005

A technique is developed for the statistical description of multiple-valued functions. For the special case of single-valued functions, the relations developed are shown to reduce in a natural way to the customary ones. Transformation laws are developed relating the statistical descriptions of a multiple-valued function in two coordinate systems. The relationship of the present work to the Rice formula for the zeros of a random function is shown. The statistical description of homogeneous, isotropic, multiple-valued functions in two dimensions is developed. Previously known results regarding expected arc length of random functions are obtained using the present technique.
Characters of Irreducible Representations of the Simple Groups. I. General Theory
Authors: Antoine J.-P. , Speiser D.
Published: January 2005

New formulas for the characters of irreducible representations of simple groups are presented. They yield the character directly as a sum instead of a quotient as does Weyl's formula. The procedure is purely geometrical and is based on the properties of the regular lattice associated with every simple group in the ``global'' theory of Lie groups due to Hopf and Stiefel.
Nonrelativistic Coulomb Green's Function in Momentum Space
Authors: Hostler Levere
Published: January 2005

The nonrelativistic Coulomb Green's function in momentum space is obtained in closed form by Fourier transforming the known expression for the coordinate-space Green's function. Also, an integral representation for the momentum-space Green's function is obtained which looks rather attractive from the point of view of applications.
A Model of Interacting Radiation and Matter
Authors: Willis Charles R.
Published: January 2005

We investigate the long-time behavior of a model consisting of N two-level atoms in a lossless cavity. The Hamiltonian of our system contains the radiation oscillators in addition to the matter Hamiltonian and the usual ∫j·Adv interaction term. In order to treat the system perturbatively, it would be necessary to remove the tremendous degeneracy of the system. Since this is prohibitively difficult, and since we are interested in the long-time behavior of the system, we solve the quantum mechanical Liouville equation directly for a wide class of physically important initial distribution functions. We show the effective expansion parameter is γ̃Nγ̃ where γ̃ is a dimensionless atomic dipole moment and N is the number of atoms. In the lowest order we find the self-consistent field approximation. In the next order, particle-field correlations appear. We explicitly solve the equations of motion for the particle-field correlations in terms of the average quantities that appear in the self-consistent field approximation. We show the self-consistent field approximation consists of five first-order differential equations. Next we show the equations of motion for the density matrix of the system correct to order (γ̃Nγ̃)2 are equivalent to eight first-order differential equations. The three additional equations are needed to describe the three second moments of the density matrix of the electromagnetic field that appear in second order. Our lowest-order microscopic equations are equivalent to semiphenomenological theories and our higher-order equations contain only the measurable second-order moments of the electromagnetic field in addition to the variables that appear in semi-phenomenological theories.
On Some Topological Properties of Feynman Graphs and Their Application to Formulas Related to the Feynman Amplitudes
Authors: Chow Yutze
Published: January 2005

The purpose of this note is to demonstrate the usefulness of the topological concept of the tree sets introduced into any Feynman graph. We demonstrate here a relationship between certain functions appearing in two different forms of parametrized Feynman amplitudes by using the properties of the tree sets and some purely determinantal manipulations. As another exposition, we also present an alternative proof of a theorem due to Nakanishi by using only the concept of tree sets with almost no algebraic manipulation involved; the proof in this case is seen to be particularly simple and lucid.
Scattering of Electromagnetic Waves by a Ferrite in a Waveguide
Authors: Kalikstein Kalman , Spruch Larry
Published: January 2005

As for any multichannel scattering problem, variational techniques can be utilized in the determination of the elements of the scattering matrix or of the equivalent network elements for a gyromagnetic obstacle in a waveguide. As always, however, it can be quite difficult to interpret numerical results which in general are neither upper nor lower bounds. A variational bound originally developed for the determination of the phase shift for a given angular momentum in a quantum mechanical central potential scattering problem is here adapted to the solution of a transversally magnetized, lossless ferrite slab in a rectangular waveguide propagating only one mode, the TE10 mode. With a simple trial function and with the aid of a comparison scattering problem which need not be tensor in character (so that the determination of upper and lower bounds is not really difficult), close bounds are obtained on cot ηe and cot η0, the cotangent of the real uncoupled phase shifts associated with the even and odd standing waves, respectively. The bounds obtained on cot ηe and cot η0 determine bounds on the equivalent π network. A second variational bound, which can be simpler to apply and which can be applied to a wider class of problems, is also developed. This too is an adaption of a formalism originally introduced in quantum mechanical scattering problems, and depends upon a consideration of the spectrum of the fundamental operator of the theory, the Hamiltonian in the quantum mechanical case and an analogue thereof in the electromagnetic case.
Kinetic Theory of a Weakly Coupled Gas
Authors: Su C. H.
Published: January 2005

Using the multiple-time-scale method on the BBGKY hierarchy, the weak coupling expansion is carried out to higher orders. It is found that there are two local breakdowns of the expansion. One occurs at small relative velocities between particles. The correct asymptotic representation for the small relative velocity region is given. The second breakdown occurs for particles having a large separation at t with their relative velocity oriented in such a way that they were in collision at t = 0. Such a breakdown indicates that in contrast to the Bogoliubov functional assumption, the higher-order correlation functions should vary on the kinetic time scale in their own right. A sufficient condition on the smoothness of the initial correlation functions is given such that one obtains the Fokker-Planck equation at the lowest-order approximation in the expansion. The connection between irreversibility and the requirement of nonsecularity in the multiple-time-scale formulation is also indicated.
The Uniqueness of Weak Solutions of the One-Dimensional Scalar Analog to the Navier-Stokes Equation
Authors: Kraut Edgar A.
Published: January 2005

A uniqueness theorem for solutions of the one-dimensional scalar analog to the Navier-Stokes equation is stated and rigorously established.
Spin and Statistics with an Electromagnetic Field
Authors: Yao York-Peng
Published: January 2005

Because of the impossibility of simultaneously satisfying the requirements of manifest Lorentz covariance and positive-definite (finite) norm in the Hilbert space, no simple proof of the connection between spin and statistics with an electromagnetic field has been given. This note is to point out that it is indeed not necessary to have manifest Lorentz covariance in the full 3 + 1 space to show such a connection. Using the axial gauge A3 = 0, we have succeeded in constructing a simple straightforward proof.
The Clebsch-Gordan Series for SU(3)
Authors: Coleman Sidney
Published: January 2005

A method is explained for the calculation of the reduction of direct products of irreducible representations of SU(3). The method is believed to be simpler and more efficient than the usual algorithms.
Scalar Invariants of a Rotational System in a Lie Algebra
Authors: Pease M. C.
Published: January 2005

We consider matrix equations of the form dW/dz = [S, W], where S is a matrix function of z that is embedded in a given Lie algebra L-i.e., it is a curve in L. If the initial condition on W is in L, then W describes a curve in L. If the Killing form is used as a metric on L, then the behavior of the system is a pure rotation about an axis that is a function of z. A set of scalar invariants of such a system is obtained. These invariants form a set of conservation laws that the system obeys regardless of the detailed behavior of S(z).

If S describes a curve in some L1, which is a semisimple subalgebra of the algebra L in which the whole system is embedded, then we can split the initial condition into two parts, one of which is in L1 and generates a solution in L1, the other in a subspace that is orthogonal to L1 and that generates a curve that remains in the subspace. We can, then, obtain conservation laws that apply separately to the two parts.

The results have application to quantum mechanics since the density matrix obeys this type of equation. They also have application to coupled mode theory if we use, instead of the vector the corresponding power density spectrum matrix or the like.
Perturbation Theory of Microwave Interaction with Gyroelectric Plasmas
Authors: Nagelberg Elliott R.
Published: January 2005

As a means of determining the effects of a uniform but arbitrarily directed magnetic field on cylindrical and spherical wave propagation in a cold, homogeneous plasma, we regard the magnetic field as a small perturbation. Assuming an expansion for the electric and magnetic fields in powers of the parameter iωg/ω, where ωg is the static gyrofrequency of the electron, we solve for the linear terms. This solution is carried out under the assumption that the fields are known for the limit of vanishing static magnetic field.

The first-order theory is then applied to cylindrical and spherical systems. When the approximate solution for the axially magnetized column is compared with the exact result, agreement is obtained provided that the static magnetic field is weak, as expected.
Summation of Partial Waves
Authors: Brysk Henry
Published: January 2005

For high-energy potential scattering, when the partial wave expansion converges too slowly to be directly useful, a technique is proposed for extracting the differential cross section more effectively from the phase shifts by means of a weight function and orthogonal polynomial approach.
The Structure of Space and the Formalism of Relativistic Quantum Theory. I
Authors: Ahmavaara Yrjö
Published: January 2005

The structure of a Euclidean space can be approached, with an unlimited accuracy, by a part of a maximally ordered finite linear space. Accordingly, all the physical theories based on the space-time continuum can also be considered in such a finite space-time. The finiteness of the underlying space makes also some new kinds of theories possible. Among them is a purely group theoretical formalism of relativistic quantum theory, including a free-particle theory as well as a group formalism of interaction of particles. The free-particle formalism of a finite space-time is considered here (Part I). An essential difference in comparison with the formalism of continuous space-time is that there is, as a consequence of the relations of Euclidicity to be imposed on observable 4-vectors, a nontrivial spectrum of momentum, mass, and energy in a finite geometry.
Random Walks on Lattices. II
Authors: Montroll Elliott W. , Weiss George H.
Published: January 2005

Formulas are obtained for the mean first passage times (as well as their dispersion) in random walks from the origin to an arbitrary lattice point on a periodic space lattice with periodic boundary conditions. Generally this time is proportional to the number of lattice points.

The number of distinct points visited after n steps on a k-dimensional lattice (with k ≥ 3) when n is large is a1n + a2n½ + a3 + a4n−½ + …. The constants a1 − a4 have been obtained for walks on a simple cubic lattice when k = 3 and a1 and a2 are given for simple and face-centered cubic lattices. Formulas have also been obtained for the number of points visited r times in n steps as well as the average number of times a given point has been visited.

The probability F(c) that a walker on a one-dimensional lattice returns to his starting point before being trapped on a lattice of trap concentration c is F(c) = 1 + [c/(1 − c)] log c.

Most of the results in this paper have been derived by the method of Green's functions.
Exponential Ensemble for Random Matrices
Authors: Bronk Burt V.
Published: January 2005

Using the ideas of information theory, it is pointed out that the Gaussian ensemble for random Hermitian matrices can be characterized as the ``most random'' ensemble of these matrices. Pursuing the same characterization for positive matrices, we are led in a natural way to the definition of the exponential ensemble. Transforming to a representation of eigenvectors and eigenvalues, the joint distribution function for the eigenvalues of positive Hermitian matrices is found for this ensemble. The asymptotic single-level density formula is derived, using a semiclassical model. It is found that the density is convex from below over most of the domain of eigenvalues. Since this is similar to the exponential dependence expected for nuclear spectra, the density is examined in a region near the level taken to correspond with the lowest nuclear level. It turns out, however, that the density is concave from below near this level, and that a large number of levels are contained in this concave region. Hence the exponential ensemble does not fairly represent nuclear energy levels, at least in this region. Various changes are made in the measure on the matrix ensemble to determine to what extent the level density depends on this measure. It is seen that the level density graph retains a characteristic shape for a wide variety of measures. The relationship of the limiting behavior of the level density for positive matrices to the semicircle law is noted.
Derivation of Low-Temperature Expansions for the Ising Model of a Ferromagnet and an Antiferromagnet
Authors: Sykes M. F. , Essam J. W. , Gaunt D. S.
Published: January 2005

Low-temperature expansions for the free energy of the Ising model of a ferromagnet and an antiferromagnet are derived for the more usual two- and three-dimensional lattices. The underlying enumerative problem is studied and a new method described that makes it possible to obtain more terms than available previously without undue labor.
Special Class of Feynman Integrals in Two-Dimensional Space-Time
Authors: Källén G. , Toll J.
Published: January 2005

Contributions of Feynman diagrams consisting of a single loop with an arbitrary number of vertices are explicitly evaluated in two-dimensional space-time. The result can be written as a sum of logarithms multiplied by algebraic expressions. Each logarithm is characteristic of a simple diagram of one loop with two external lines, while the coefficients can be obtained from rules analogous to the rules of residue calculus.
Adiabatic Particle Orbits in Discontinuous Fields
Authors: Schindler Karl
Published: January 2005

The invariance properties of the action integral J = ∮ p dq are studied for the motion of charged particles in one-dimensional electromagnetic fields. Attention is concentrated on those situations where the field gradients become large. Whereas in the case of smooth fields and finite gradients the asymptotic theory of the one-dimensional oscillator as developed by Gardner and by Lenard applies, the presence of large gradients, requires a special treatment. The present considerations are restricted to cases where the strong field variation is confined to a small region such that the transition can be approximated by a discontinuity. It is shown that for time intervals of order 1/ε, J is an adiabatic invariant of at least order ε½, ε measuring the time variation of the fields. As an example, the motion of high-speed particles through a plane discontinuity is shown to be adiabatic in that sense. Consequently, the initial and the final magnetic moments differ only slightly.
The Dirac Equation and the Unitary Representations of the Inhomogeneous Lorentz Group
Authors: Redei L. B.
Published: January 2005

The Dirac equation in its quantized form is discussed in order to deduce the structure of the unitary representation of the Poincaré group in Fock space. There is a short account of discrete symmetries in Fock space and a simple method of calculating spin matrix elements is outlined.
Quantum Statistics of Multicomponent Systems
Authors: Mohling Franz , Grandy W. T.
Published: January 2005

Methods of quantum statistics are applied to the general multicomponent system and explicit rules for the calculation of thermodynamic quantities and distribution functions in terms of the basic particle interactions are derived. Particular attention is given to the nonrelativistic system of charged particles, and several inherent difficulties are investigated and treated. Included are the renormalization, by the Λ transformation, of charged-particle energy contributions due to the emission and reabsorption of photons and the formal sum over all long-range Coulomb effects wherever (diagrammatically) they give divergent expressions. The latter sum is shown to result in the familiar Debye-Hückel theory in the high-temperature, low-density limit.
Quantum Mechanics of a Many-Boson System and the Representation of Canonical Variables
Authors: Ezawa Hiroshi
Published: January 2005

We present a method of treating the assembly of interacting bosons under Bose-Einstein condensation. Without applying the Bogoliubov approximation in which the creation and the annihilation operators of zero-momentum particles are replaced by a c number, we keep the quantum nature of these operators—thus the title, ``Quantum Mechanics.'' The method is a quantum mechanical adaptation of the theory of small oscillation. The oscillation means the fluctuation of the number of condensed particles. The interaction between particles determines the stability of this oscillation. When it is stable and its amplitude is not macroscopic, the Bogoliubov approximation is valid. In this way, our method provides a validity criterion for the Bogoliubov approximation as well as an estimation of the errors thereby committed. We have to note that the excitations associated with the fluctuation of condensed particles can never be obtained within that approximation. Our method is applied to the Huang model, the assembly of bosons interacting through a hard core plus weak attractive potential. Having found that, within the physically accessible range of the particle density, the above-mentioned oscillation is stable, we can conclude that Huang's treatment is well founded. We have discussed the mathematical background of our approximation by invoking the representation theory of canonical variables of an infinitely large system.
Dynamics of a Simple Many-Body System of Hard Rods
Authors: Jepsen D. W.
Published: January 2005

General formulas are given for the exact calculation of the nonequilibrium properties of the one-dimensional system of equal-mass hard rods both for a finite but large system and in the limit of infinite size. Only properties which depend upon labeling one or more of the particles are nontrivial in this system. Various results are obtained on Poincaré cycles, delocalization of a particle with time and electrical conductivity when one particle is charged.
Statistical Ensembles of Complex, Quaternion, and Real Matrices
Authors: Ginibre Jean
Published: January 2005

Statistical ensembles of complex, quaternion, and real matrices with Gaussian probability distribution, are studied. We determine the over-all eigenvalue distribution in these three cases (in the real case, under the restriction that all eigenvalues are real). We also determine, in the complex case, all the correlation functions of the eigenvalues, as well as their limits when the order N of the matrices becomes infinite. In particular, the limit of the eigenvalue density as N → ∞ is constant over the whole complex plane.
Some Recent Developments in Invariant Imbedding with Applications
Authors: Bailey Paul B. , Wing G. Milton
Published: January 2005

A summary of some recent developments in the theory of invariant imbedding is presented. Applications to some simple problems in wave propagation, diffusion theory, and transport theory demonstrate some of the advantages of the new and almost ``mechanistic'' approach. In addition, a slightly different attack is applied to nonlinear difference equations and to the classical phase-shift problem.
Construction of the Charge Operator for Higher Symmetry Schemes
Authors: Gerstein I. S.
Published: January 2005

The problem of the construction of an additively conserved operator with integral eigenvalues, to be identified with the electric charge is solved in complete generality for the groups locally isomorphic to U(1)⊗ SU(n). It is found that the representations fall into classes on which different charge operators may be defined. Several results previously obtained for particular classes of representations are found here as special cases. We specialize the results to N = 4 and discuss several models presently in the literature.
Fourier Series Expansion for the General Power of the Distance between Two Points
Authors: Ashour Attia A.
Published: January 2005

The general power rn = (r12 − 2r1 r2 cos ω + r22)n/2 of the distance between two points is expressed as a Fourier series Σ Rn, l(r1, r2) cos lω. Following Sack's method, the radial functions Rn, l are obtained as power series in r. Symmetrical expressions in r1 and r2 and recurrence relations are found for Rn, l.
Statistical Mechanics of Assemblies of Coupled Oscillators
Authors: Ford G. W. , Kac M. , Mazur P.
Published: January 2005

It is shown that a system of coupled harmonic oscillators can be made a model of a heat bath. Thus a particle coupled harmonically to the bath and by an arbitrary force to a fixed center will (in an appropriate limit) exhibit Brownian motion. Both classical and quantum mechanical treatments are given.
Classification of Three-Particle States According to SU3
Authors: Dragt A. J.
Published: January 2005

It is shown that the set of states for three noninteracting particles can be put into a one-to-one correspondence with the set of irreducible representations of SU3. This classification leads to three-particle angular momentum states which treat all particles on an equal footing. The states exhibit the maximum localization compatible with a given total energy, momentum, and angular momentum. Only nonrelativistic particles are treated.
Diagram Renormalization, Variational Principles, and the Infinite-Dimensional Ising Model
Authors: Bloch Claude , Langer James S.
Published: January 2005

The Ising model has been employed in a study of the variational principles which are associated with renormalized diagram expansions in statistical mechanics. For this model the variational functional is expressed in terms of the renormalized semi-invariants, which play the role of the one-particle density or the one-particle Green's function in quantum statistics. We review the derivation of this functional and discuss some of its properties. In order to examine the mathematical content of the variational principle in more detail, we specialize to the exactly soluble infinite-dimensional (infinite-range) model. We find that, by virtue of the Dyson relation, the variational theorem is not valid over all possible values of the renormalized semi-invariants but applies only within a restricted domain. Within this domain the variational functional is multiple valued. The renormalized expansion is asymptotically convergent to the branch of this functional which describes a single phase with uniform magnetization.
Note on Positronium
Authors: Tiktopoulos George
Published: January 2005

By means of an inequality providing an upper bound for the norm of integral operators, it is shown that the Bethe-Salpeter equation for bound states of the electron-positron system (in the ladder approximation) admits solutions associated with a discrete spectrum of binding energies. It is found that in the weak coupling limit the spectrum B(α) approaches asymptotically the Coulomb spectrum in the sense lim lim α→0(B(α)/mα2)={(1/4n2) | n=1, 2, 3, … ∞},where α is the fine-structure constant and m the electron rest mass.
Uniform Asymptotic Estimates for Wave Packets in the Quantum Theory of Scattering
Authors: Wilcox Calvin H.
Published: January 2005

The theory of the scattering of particles by a potential field V(x) according to the classical wave mechanics of Schrödinger is considered. Various conditions on V(x) are known which imply that the wave packets ψ(x, t) describing scattered particles are asymptotically equal in L2(Rn) to wave packets ψ0(x, t) for free particles when t → ∞ or − ∞. In this paper, additional conditions on V(x) and the initial values of the wave packets are given which imply that ψ(x, t) and ψ0(x, t) have square-integrable (or continuous) partial derivatives of a prescribed order whose difference tends to zero in L2(Rn) (or uniformly in Rn) when t → ∞ or − ∞.
Uniqueness of Steady-State Solutions to the Fokker-Planck Equation
Authors: Gray A. H.
Published: January 2005

The uniqueness of steady-state probability densities in certain continuous Markov processes is proven when the Fokker-Planck or Kolmogorov equation satisfied by the transition probability is of a variety called ``steady.'' The tendency of other probability densities to approach such steady-state densities is formally demonstrated.
Vertex Modification for Coulombic Interactions by Exact Summation of Ladder Diagrams
Authors: Pao Yoh-Han , Frisch H. L.
Published: January 2005

We have been able to sum a class of ladder diagrams for Coulombic interactions to infinite order exactly and are able therefore to replace a ``bare'' interaction with a self-consistent effective interaction or equivalently we are able to modify the vertex operator in a manner which may be of particular significance for systems of low particle density in analogy with what can be done for nuclear matter and liquid helium. The method is only valid for nearly free particles, the exact energy being negative but the zero-order description being that of free particles. Effective attractive interactions between two electrons or interaction with an external Coulombic field constitute perturbations which fall within the range of validity of the scheme. Despite the limitation of the specific results, the method is probably of more general use and interest and both the approach and specific results are described in this note.
Internal Symmetry and the Poincaré Group
Authors: Tomozawa Yukio
Published: January 2005

Theorems concerning the interrelation between the Poincaré group and the symmetry group of elementary-particle interactions have been proved under weaker conditions than that of McGlinn.
Relativistic Formulation of the Lifetime Matrix in the Potential Theory of Collision
Authors: Gien Tran Trong
Published: January 2005

The lifetime matrices for collision processes as described by the Klein-Gordon and Dirac equations are formulated. The results obtained are exactly the same as in the case of a collision described by the Schrödinger equation. This gives us a consistent operator theory for the collision lifetime in the case of potential scattering.
Symbolic Calculus of the Wiener Process and Wiener-Hermite Functionals
Authors: Imamura Tsutomu , Meecham William C. , Siegel Armand
Published: January 2005

A new definition is given for the ``ideal random function'' (derivative of the Wiener function), which separates out infinite factors by fullest exploitation of the possibilities of the Dirac delta function. By allowing all integrals to be written formally as sums, this facilitates the definition and manipulation of the Wiener-Hermite functionals, especially for vector random processes of multiple argument. Expansion of a random function in Wiener-Hermite functionals is discussed. An expression is derived for the expectation value of the product of any number of Wiener-Hermite functionals; this is all that is needed in principle to obtain full statistical information from the Wiener-Hermite functional expansion of a random function. The method is illustrated by the calculation of the first correction to the flatness factor (measure of Gaussianity) of a nearly-Gaussian random function.
Wiener-Hermite Expansion in Model Turbulence in the Late Decay Stage
Authors: Siegel Armand , Imamura Tsutomu , Meecham William C.
Published: January 2005

The Wiener-Hermite functional expansion, which is the expansion of a random function about a Gaussian function, is here substituted into the Burgers one-dimensional model equation of turbulence. The result is a hierarchy of equations which (along with initial conditions) determine the kernel functions which play the role of expansion coefficients in the series. Initial conditions are postulated, based on physical reasoning, criteria of simplicity, and the assumption that the series is to represent the late decay stage (in which the Gaussian correction is small and also decreasing with time). These are shown to justify an iterative solution to the equations. The first correction to the Gaussian approximation is calculated. This is then tested by evaluating the correction to the flatness factor, which for an exactly Gaussian function has the value 3, but which has been found by experiment (in real three-dimensional fluids, of course) to have a value which deviates from the Gaussian value increasingly rapidly with the order of the derivative. We utilize this effect as a test of the inherent ability of the Wiener-Hermite expansion to bring to realization the physical properties implicit in the Navier-Stokes or Burgers equations. The various contributions to the flatness-factor deviation, when computed, do show a potential capability of providing a theoretical basis for the effect.
On ``Diagonal'' Coherent-State Representations for Quantum-Mechanical Density Matrices
Authors: Klauder John R. , McKenna James , Currie Douglas G.
Published: January 2005

It is proved that every density matrix is the limit, in the sense of weak operator convergence, of a sequence of operators each of which may be represented as an integral over projection operators onto coherent states (in the sense of Glauber) with a square-integrable weight function. This result is a special case of one that holds for all operators with trace and for overcomplete families of states other than just the coherent states. We prove our more general result, at no cost of complexity, within the more general framework of continuous-representation theory. The significance of our results for representing traces of operators is indicated.
S-Operator Theory. I. Formulation
Authors: Pugh Robert E.
Published: January 2005

It is postulated that the S operator is dynamically independent of the wavefunctions of the asymptotic states. From this postulate a functional differential equation for the S operator is developed. Its solutions include both the renormalized and unrenormalized Feynman-Dyson S operators; appropriate boundary conditions distinguish the two solutions. Interpolating quantum fields are defined in terms of the S operator and are only of secondary importance in this theory. Calculations in this theory are not appreciably more difficult than the corresponding calculations with the unrenormalized S operator.
Compactification of Minkowski Space and SU(3) Symmetry
Authors: Hermann Robert
Published: January 2005

Several ideas from the geometric side of Lie group theory are presented that may be relevant to the search for groups containing the internal and Poincaré group symmetries.
Unitarity and On-Mass-Shell Analyticity as a Basis for S-Matrix Theories. II
Authors: Gunson J.
Published: January 2005

Sets of postulates for S-matrix theories are given and used to construct potentially complete dynamical theories for scattering of strongly interacting particles, based essentially on the unitarity relations and on-mass-shell analyticity of scattering amplitudes. In one formulation, the unitarity relations for stable particles are used to derive the corresponding relations for cuts associated with unstable particle states. In a second formulation, an attempt is made to treat stable and unstable particles on a more equal footing. Crossing symmetries and decomposition laws for single-particle poles appear as consequences of the postulates.
Unitarity and On-Mass-Shell Analyticity as a Basis for S-Matrix Theories. III
Authors: Gunson J.
Published: January 2005

The singularity structure of many-particle scattering amplitudes in momentum transfer variables is investigated in terms of the many-particle unitarity conditions. Cauchy-type kernels for analytic functions defined on complex rotation group in three dimensions are constructed and related to a theory of local representations of the rotation group, corresponding to complex angular momenta.
Projected Angular Momentum States
Authors: Karayianis N. , Morrison C. A.
Published: January 2005

Known results regarding the projection of total spin states from product wavefunctions involving spin-½ particles are generalized. It is shown that total spin states projected from product wavefunctions involving particles with arbitrary spin, but otherwise restricted to having either maximum or minimum z component of spin, are no more complicated in form than for the spin-½ case.
Transformation from a Linear Momentum to an Angular Momentum Basis for Particles of Zero Mass and Finite Spin
Authors: Moses H. E.
Published: January 2005

The infinitesimal generators of the inhomogeneous Lorentz group have been given in a basis in which the components of the linear momentum operators are diagonal and in another basis in which the square of the angular momentum is diagonal for all unitary irreducible ray representations of the group. In the present paper we show how the two bases are related for representations corresponding to zero mass and any (finite) spin. It will be shown how this relation enables one to integrate the infinitesimal generators in the angular momentum basis and thereby permits one to show how the angular momentum of a particle changes under the inhomogeneous Lorentz group. In particular, we study the way that the angular momentum of a massless particle of any spin appears in translated and moving frames of reference.
Calculation of Some Homology Groups Relevant to Sixth-Order Feynman Diagrams
Authors: Federbush Paul
Published: January 2005

A homology group that determines an upper bound for the number of linearly independent analytic functions connected with the sixth-order ladder diagram is here computed. The formalism is that of Fotiadi, Froissart, Lascoux, and Pham. Calculations use the standard methods of homology theory. We find that there are at most 127 such functions in general.
Solution of the Källén-Pauli Equation
Authors: Pagnamenta Antonio
Published: January 2005

We give the relevant solution to the integral equation which Källén and Pauli have derived for elastic VΘ-scattering in the Lee model. As an application of this result the exact Tamm-Dancoff solution for the entire VΘ-sector is obtained. This includes the amplitudes for VΘ- and N2Θ elastic scattering and VΘ-N2Θ production, as well as their extensions off the mass shell.
Momentum Transfer Cross-Section Theorem
Authors: Gerjuoy E.
Published: January 2005

The momentum transfer cross section is expressed in terms of a matrix element of grad V, where V is the potential which need not be spherically symmetric. The result may be useful for estimating the momentum transfer cross section in circumstances where the usual expansion in partial waves is inconvenient or inapplicable as, e.g., when V is noncentral.
Note on Bég's Approach To Peratization
Authors: Leader Elliot
Published: January 2005

It is shown that Bég's elegant formulation of the peratization result of Feinberg and Pais can be rigorously justified.
Disconnected Groups as Higher Symmetry Groups
Authors: Klink W. H.
Published: January 2005

Disconnected groups are investigated to see whether they can be used as higher symmetry groups. Some disconnected groups are given which satisfy certain minimal physical conditions. Two disconnected groups are analyzed in detail using little group techniques, and it is shown how a doubling of certain multiplets results, so that particles and their antiparticles are in the same multiplets—rather than being in separate, though equivalent multiplets as in the usual SU2 or SU3 schemes.
Algebraic Difficulties of Preserving Dynamical Relations When Forming Quantum-Mechanical Operators
Authors: Arens Richard , Babbitt Donald
Published: January 2005

Proofs are presented showing impossibility of assigning differential operators (quantum observables) to classical mechanical observables in such a way as to preserve the usual bracket formalism. Difficulty is shown to arise even if we limit ourselves to preserving brackets between the Hamiltonian and a rather limited set of observables. Some other algebraic difficulties inherent in the operator assignment problem are also discussed.
Time-Dependent One-Speed Albedo Problem for a Semi-Infinite Medium
Authors: Kuščer I. , Zweifel P. F.
Published: January 2005

A Laplace transformation technique is used to determine the neutron distribution in a semi-infinite medium which has been irradiated by a neutron pulse. The result is given in terms of known solutions of Milne's problem and of the steady-state albedo problem, which in turn are expressed by aid of Case's X-function. Simple asymptotic approximations, valid for t ≫ 1, are deduced from the exact result.
Microcanonical Ensemble in Quantum Statistical Mechanics
Authors: Griffiths Robert B.
Published: January 2005

The infinite-volume limit of thermodynamic functions calculated in the quantum microcanonical ensemble is shown to exist for a fairly wide class of spin systems and quantum gases. The entropy is set equal to the logarithm of the number of eigenstates in an energy interval which increases linearly with the size of the system, but is otherwise arbitrary. The limiting entropy per unit volume agrees with that calculated in the canonical formalism, and possesses certain convexity properties required for thermodynamic stability. A precise criterion, in terms of the energy spectra of large systems, is given for determining the limit of the thermodynamic entropy as the temperature approaches zero. This is not determined by the degeneracy of the ground state, contrary to the discussion of the ``third law of thermodynamics'' found in some textbooks.
On Some Unitary Representations of the Galilei Group I. Irreducible Representations
Authors: Voisin J.
Published: January 2005

The true irreducible unitary representations of central extensions GM of the Galilei universal covering group G and hence the physical representations of G are constructed by Mackey's method of induced representations. The elements of the representation space ℋ are obtained from functions defined on GM and restricted to their values at one representative of each left coset of GM modulo K where K is the induction subgroup. The physical interpretation of these functions is in terms of wave functions and comes from the definition of a basis in ℋ. This interpretation depends on the choice of a fundamental frame of reference in space-time and on the physical meaning given to a fundamental state. To a change of the representatives corresponds a change of basis in ℋ. By a suitable choice of these representatives, we obtain in particular the momentum-spin representation and the momentum-helicity representation. The zero mass case named class II by Inönü and Wigner is then obtained by the limit process M → 0 applied to the helicity representation.