bluuebunny/crossref_metadata_2025 huggingface dataset:
Combinatorial Approach to the N-Representability of P-Density Matrices
Authors: Yoseloff Mark L. , Kuhn Harold W.
Published: February 2004

This paper considers the determination of N-representability (for diagonal elements) of p-density matrices restricted to certain finite-dimensional subspaces of l2 of the configuration space of N identical antisymmetric particles. In particular, an arbitrary set of N + p spin orbitals is selected and one considers the (N+pp)-dimensional subspace generated by all possible Slater determinants of the spin orbitals being considered. Applying a combinatorial approach to the problem, a necessary and sufficient set of conditions is determined; previous work has dealt only with necessary conditions, except in the 1-matrix case. The paper concludes by presenting a probabilistic interpretation of these conditions which seems of particular interest for the 2-matrix case. The conditions presented here in combination with the Pauli principle give a probabilistic view of the expected occupation of p-tuples of spin orbitals in terms of the expected occupations of lower-order-tuples of spin orbitals.
Level Density of a Fermi System: Nonperiodic Perturbations of the Energy-Level Scheme
Authors: Kahn Peter B. , Rosenzweig N.
Published: February 2004

The level density of a degenerate Fermi system is modified by the perturbation in the position of a single-particle level. To study this effect, miscellaneous exact relations between the level densities of the perturbed and unperturbed system are derived. For the special case of the perturbed uniform model, these connections become a set of recursion relations which lead to a complete solution of the problem. Results are also obtained in the saddle-point approximation, and these have a simple interpretation in terms of the usual Fermi occupation probabilities. If a single-particle level is deleted from the scheme, the resultant diminution in the level density persists indefinitely with increasing excitation energy. Information about the adequacy of the saddle-point approximation is obtained by comparison with some exact solutions.
Linear Inequalities for Density Matrices
Authors: Davidson Ernest R.
Published: February 2004

A set of linear inequalities for the reduced two-body density matrix is presented. The general problem of finding all such inequalities is discussed. It is shown that all of the linear inequalities found so far for ensemble representability may be written as a single family of inequalities.
Algebraic Theory of Ray Representations of Finite Groups
Authors: Harter William G.
Published: February 2004

A theory of characters of ray representations of finite groups, that does not use any reference to a covering group, is derived by defining two generalizations of the concept of a group's class. Orthogonality relations are obtained over one of these generalized classes. This theory is used to discuss subduction and induction of ray representations while the Frobenius reciprocity theorem and generalizations thereof are proved. The theory provides a more efficient method of deriving and treating ray representations of finite groups for a given factor system than has previously been made available.
Algebraic Classification of Four-Dimensional Riemann Spaces
Authors: Thompson James , Schrank Glen
Published: February 2004

An algebraic classification of four-dimensional Riemann spaces with signature -2 is given. The possible Riemann and Ricci tensors are listed and inversion formulas from one to the other are obtained. Classifications are given for the two special spaces Rij = 0 and RklRmk=0.
On the Isospin Crossing Matrix
Authors: Lee Huan
Published: February 2004

New properties and relations to Clebsch-Gordan coefficients have been found and proved for a certain subset of isospin crossing matrix elements. These properties, coupled with the well-known fact that the crossing matrix is simply related to a real orthogonal matrix, provide a quick method to calculate crossing matrices of rank ≤ 4. Procedures of construction and tables are given for all crossing matrices involving I ≤ 32. For the sake of completeness as well as clarification of the confusing point about phases, we include a systematic discussion of crossing relations and a general expression for crossing matrices applicable to arbitrary phase conventions. For the crossing matrix in which both direct and crossed reactions are elastic, some interesting inequalities among elements in the first and last columns are noticed and their physical implication is briefly discussed.
Classical Fields on Spacelike Mass Shells
Authors: Rühl W.
Published: February 2004

We define classical fields, corresponding to unitary representations of the inhomogeneous Lorentz group with M2 < 0, which belong to the discrete series. These fields satisfy Bargmann-Wigner equations which are given in explicit matrix form.
Nonsaturation of Gravitational Forces
Authors: Lévy-Leblond Jean-Marc
Published: February 2004

Rigorous inequalities are derived for the ground-state energy of a nonrelativistic quantum-mechanical system of N particles in gravitational interaction. It is shown that gravitational forces do not saturate, the binding energy per particle increasing with N, like N2 for a Bose system, like (N4/3) for a Fermi system. As a by-product, we obtain a generally valid Heisenberg-like inequality for N-fermion systems, expressing very simply the effect of the Pauli exclusion principle. These results are extended to the case of a system of oppositely charged particles which is shown to behave, with respect to gravitational forces, as a Fermi system as soon as particles with one sign of charge only are identical fermions. This explains quantitatively how and when gravitational forces finally predominate over Coulomb forces for large enough bodies (planets). A further extension to the case where relativistic effects enter only at the kinematical level permits us to derive rigorously from first principles the existence and an estimate of the Chandrasekhar mass limit, above which no collection of cold matter is stable (white dwarf stars).
Zero-Mass Representation of Poincaré Group and Conformal Invariance
Authors: Bose S. K. , Parker R.
Published: February 2004

By explicit construction it is shown how to extend zero-mass, discrete spin representations of the Poincaré group to corresponding representations of the conformal group of Minkowski space.
Best Error Bounds for Padé Approximants to Convergent Series of Stieltjes
Authors: Baker George A.
Published: February 2004

We consider series of Stieltjes with a nonzero radius of convergence R. We establish by way of Padé approximants the allowed range of values for such functions at any point in the cut (−∞ < z ≤ − R) complex plane when a finite number of Taylor-series coefficients are known. The previous results for z real and positive are sharpened. We investigate the fitting problem and again give error bounds throughout the cut complex plane and we give necessary and sufficient conditions that the set of values fitted be values of a series of Stieltjes with radius of convergence at least R.
Application of the Kirillov Theory to the Representations of O(2, 1)
Authors: Dunne S. A.
Published: February 2004

The Kirillov construction is applied to the semisimple Lie group O(2, 1). All the unitary irreducible representations (except the supplementary series) are found, provided that complex subalgebras and complex points on orbits are admitted. The characters of the representations are calculated and the relation of their Fourier transforms to the orbits is examined.
Measure-Theoretical Description of Klein-Gordon Multiparticle States
Authors: Prugovečki Eduard
Published: February 2004

It is shown that a multiparticle state constructed from solutions of the Klein-Gordon equation can be also described by a family of complex measures having simple properties. These measures are the expectation values for that state of products of projectors corresponding to the spectral decompositions of certain self-adjoint operators. The results obtained are used in discussing the problem of defining quantized free fields at a point.
Orear Behavior in Potential Scattering. II
Authors: Johnson Porter W.
Published: February 2004

Here we will extend certain results obtained in a previous paper [J. Math. Phys. 9, 712 (1968)] concerning high-energy large-angle scattering upon nonsingular potentials V(r) which are even in r and analytic in a finite strip about the real r axis. (In the previous paper, it was required that the potentials have a larger domain of analyticity than is required here.) The scattering amplitude is compared with the first Born approximation for this wider class of potentials.
Next-Nearest-Neighbor Ising Model
Authors: Gibberd R. W.
Published: February 2004

The next-nearest-neighbor two-dimensional Ising model is cast into a form which resembles a one-dimensional interacting many-fermion system. An approximation, which has previously been shown to be successful in giving the critical properties of the two-dimensional ferroelectric problem, is used. From the approximate expression obtained, the critical indices are found to be α = α′ = 0, β = ⅛, which agrees with the results obtained from series expansions and plausible physical arguments. The critical temperature obtained agrees to within 6% of the series expansion results.
Geometrization of the Brans-Dicke Scalar Field
Authors: Peters P. C.
Published: February 2004

The Brans-Dicke gravitational scalar field is geometrized in the spirit of the Rainich-Misner-Wheeler geometrization of electromagnetism. Geometric equations are derived which imply that the Brans-Dicke field is present and an explicit expression is given for this field in terms of geometrical quantities.
Algebraic Matrices for the Configurations (d + s)n
p
Authors: Roth C.
Published: February 2004

The construction of the algebraic matrices for the configurations (d + s)np in L–S coupling is discussed. Particular emphasis is given to configurations with a half-filled d shell. From a total of 42 parameters specifying the various interactions for the configurations (d + s)np only the matrices of 8 parameters need to be calculated explicitly for the complementary configurations (d + s)np, n > 6. The matrices of the other 34 parameters can be obtained from the corresponding algebraic matrices of (d + s)np, n ≤ 6, either directly or by simple changes in sign.
Quadratic Action Principle of Relativity
Authors: Lanczos C.
Published: February 2004

By an improved mathematical technique, the field equations derivable from a Lagrangian which is quadratic in the curvature components can now be studied in greater detail. The intercalation of a high-frequency metrical plateau between the flat Minkowskian metric and the macroscopic physical world has the consequence that the resulting perturbation equations are no longer of fourth but only of second order, thus making a comparison with Einstein's equations possible. The principal difference is that the free vector of Einstein's theory is now restricted by a divergence condition, with the result that the equations of the electromagnetic field, expressed in terms of the vector potential, become solutions of the macroscopic field equations. The cases of fourfold symmetry with imaginary time and 3 + 1 symmetry with real time are discussed. The gravitational phenomena and the second-order interaction terms, needed for the construction of material particles, remain outside the limits of the present investigation.
Sufficient Conditions for Stability of the Faddeev Equations
Authors: Zachary W. W.
Published: February 2004

Functional analysis techniques are used to obtain stability theorems for the Faddeev equations in the quantum-mechanical three-particle nonrelativistic scattering theory. Sufficient conditions are obtained in order that the solutions of these equations not be sensitive to small variations of the off-shell two-particle amplitudes. These conditions provide criteria for the validity of some of the previous formal investigations of the Faddeev equations.
Magnetization of Ising Model in Nonzero Magnetic Field
Authors: Mattis Daniel C. , Plischke Michael
Published: February 2004

Knowing only the zero-field magnetization (e.g., Yang's result) of the Ising model in any number of dimensions, one can construct a lower bound on m(h), the magnetization in finite field. Knowledge of u, the internal energy per bond, enables a more efficient lower bound to be constructed. Both are applications of the Griffiths inequality, as recently generalized by Kelly and Sherman, and should prove useful in the lattice gas problem where it is essential to know m(h).
Comments on Kinetic Equation for Autocorrelation Functions
Authors: Résibois P. , Brocas J. , Decan G.
Published: February 2004

The non-Markoffian kinetic equation for the one-particle momentum autocorrelation function, derived by Zwanzig and studied in great detail recently by Berne, Boon, and Rice, is analyzed in the weak coupling limit. It is shown that, in this limit, this kinetic equation remains non-Markoffian because the kernel which determines the memory effects only decays very slowly. More precisely, it tends to zero over times of the order of the relaxation time itself and not, as could be expected, over the much shorter collision time. The comparison with the more traditional approach, based on the solution of a transport equation, is also discussed.
Broken Symmetry and Generalized Bose Condensation in Restricted Geometries
Authors: Girardeau M. D.
Published: February 2004

It is suggested, on the basis of soluble models, that generalized Bose condensation results from a broken symmetry associated with a nonvanishing pairing amplitude with very large low-momentum components, leading to a nonzero ``pairing density'' pc as well as a nonzero generalized condensate density ρc. For systems of interacting bosons it is proved that (1) nonzero pc implies nonzero ρc, and (2) pc = 0 in one or two dimensions and, more generally, for geometries finite in one dimension and infinite in the other two (films) or finite in two dimensions and infinite in one (pores). It is pointed out that this does not exclude superfluidity in such geometries, but does show the need of a new mechanism to explain it.
Nonlinear Perturbations
Authors: Coffey Timothy P. , Ford G. W.
Published: February 2004

The perturbation theory of Bogoliubov and Mitropolsky for systems having a single rapid phase is generalized to systems having several rapid phases. It is shown that one can avoid the classic problem of small divisors to all orders in the perturbation theory. The method has the advantage of providing a single approach to many problems conventionally treated by a variety of specialized techniques.
Thermodynamics of a One-Dimensional System of Bosons with Repulsive Delta-Function Interaction
Authors: Yang C. N. , Yang C. P.
Published: February 2004

The equilibrium thermodynamics of a one-dimensional system of bosons with repulsive delta-function interaction is shown to be derivable from the solution of a simple integral equation. The excitation spectrum at any temperature T is also found.
On the Growth of the Number of Bound States with Increase in Potential Strength
Authors: Simon Barry
Published: February 2004

For a wide class of potentials, it is shown that N(λ), the number of bound states (including multiplicity) of −Δ + λV, obeys the conditions Aλ32 < N(λ) < Bλ32for λ sufficiently large. A and B are positive finite numbers. In the centrally symmetric cases, a related growth condition on lmax(λ), the largest l channel with bound states, is also obtained, namely, aλ12 < lmax(λ) < bλ12. Finally, we discuss analogous results for a larger class of central potentials and for the many-body case.
Super Hilbert Space and the Quantum-Mechanical Time Operators
Authors: Rosenbaum David M.
Published: February 2004

The basic idea of super Hilbert space is to represent physical states by continuous linear functionals on a space of good functions, rather than by functions in a Hilbert space. Since L2 is in one-to-one correspondence with a subset of super Hilbert space, everything that can be done in L2 can be done in super Hilbert space. In addition, however, it is possible to have a time operator and thus to base relativistic quantum mechanics on covariant four-dimensional commutation relations.
Exact Robertson-Walker Cosmological Solutions Containing Relativistic Fluids
Authors: Vajk J. Peter
Published: February 2004

A new derivation of the Robertson-Walker metrics is presented which elucidates the relationship between these spatially homogeneous and isotropic models and certain spatially homogeneous but anisotropic models (``three-cylinder universes''). The evolutionary equations for Robertson-Walker models containing as many as four distinct noninteracting relativistic fluids (each obeying a gamma-law equation of state) are examined, and 25 exact closed-form solutions of this type are presented explicitly.
Analysis of Electromagnetic Radiation in the Presence of a Uniformly Moving, Uniaxially Anisotropic Medium
Authors: Besieris Ioannis M.
Published: February 2004

The change in the character of electromagnetic radiation in the presence of a nondispersive, electrically and magnetically uniaxially anisotropic medium moving uniformly in the direction of its axis of symmetry is investigated. Because of the existing symmetry, it is possible to obtain explicit time-dependent analytic solutions for a longitudinally—and a transversely—oriented magnetic dipole current distribution density using a spectral representation in the space-time Fourier domain. The supports of the resulting fields are found to be oblate spheroidal wavefronts which enclose the source point if v < v1, where v and v1 signify respectively the speed of the medium and the phase speed of a wave, as measured by an observer in the rest frame of the material, and move inside circular conical regions—a phenomenon known as the Čerenkov effect—for v > v1.
Generalized Bose Operators in the Fock Space of a Single Bose Operator
Authors: Brandt Richard A. , Greenberg O. W.
Published: February 2004

Generalized Bose operators b which reduce by two the number of quanta of a Bose operator a are studied in the Fock space of a. All representations of the b's as normal-ordered (infinite degree) power series of the a's are found. The unitary operators relating the irreducible components of b to a are also exhibited. The analogous result for b(k)'s which reduce the number of a quanta by k is given and the limit k → ∞ is discussed.
Existence of Solutions for Certain Nonlinear Boundary-Value Problems
Authors: Shampine L. F.
Published: February 2004

The existence of solutions of a class of nonlinear boundary-value problems is characterized in terms of a linear eigenvalue problem. Bounds and comparison results for solutions are derived.
Algorithm for the Calculation of the Classical Equations of Motion of an N-Body System
Authors: Harrison Don E. , Gay Warren L. , Effron H. M.
Published: February 2004

The equations of motion for N-body systems are usually integrated by means of a central-difference algorithm. An alternative average force algorithm is described in this paper. The new method is shown to have both theoretical and practical advantages when compared to the central-difference method.
On the Width Distribution for a Complex System Whose Hamiltonian Contains a Small Interaction That Is Odd under Time-Reversal
Authors: McDonald J. F.
Published: February 2004

An approximate expression for the width distribution of a complex system with a small odd term is given. The expression is compared with available Monte-Carlo calculations and seems to be a good approximation for large N (10 ≤ N ≤ 100).
Type D Vacuum Metrics
Authors: Kinnersley William
Published: February 2004

Using the Newman-Penrose formalism, the vacuum field equations are solved for Petrov type D. An exhaustive set of ten metrics is obtained, including among them a new rotating solution closely related to the Ehlers-Kundt ``C'' metric. They all possess at least two Killing vectors and depend only on a small number of arbitrary constants.
F Model on a Triangular Lattice
Authors: Baxter R. J.
Published: February 2004

The Rys F model is formulated on a triangular lattice and solved for certain values of the vertex configuration probabilities (including those corresponding to the ``ice model''). As with the square lattice, it is found that the system undergoes a phase transition which is of infinite order.
Hyperplane Helicity States
Authors: McDonald S. C.
Published: February 2004

The hyperplane formalism of Fleming is developed to include a discussion of the operations of the Poincaré group, as seen by an arbitrary hyperplane observer. Basis states for the m > 0 irreducible representations of the Poincaré group are re-expressed within the framework of generalized covariance provided by the hyperplane formalism and are seen to be related to the conventional helicity states by a special Lorentz transformation.
Proof of the Symmetrization Postulate
Authors: Girardeau M. D.
Published: February 2004

It is proved within the framework of nonrelativistic quantum mechanics that identical particles are either boson or fermions. The starting assumptions are: (a) if ψ(x1 … xn) is in the space H of allowed states, then so is Pψ for every permutation P; (b) |Pψ|2 = |ψ|2 for all ψ∈H, all allowed configurations (x1 … xn), and all ψ∈H; (c) H is a vector space (principle of superposition); (d) every ψ∈H is continuous along every path in the n-particle configuration space C; and (e) there is at least one physical observable connecting each pair of irreducible components of C.
Singularities of Conformal-Invariant Scattering Amplitudes
Authors: Coon Darryl D.
Published: February 2004

It is shown that conformal invariance alone, with no specific dynamics, provides severe constraints on the location of singularities of the off-mass-shell amplitude for elastic scattering of spinless particles. Kinematical and dynamical singularities are correlated in an interesting fashion with the singular and general solutions of differential equations which follow from the momentum-space requirements of conformal invariance. Explicit expressions are found for the location of singularities which are independent of t, the square of the momentum transfer. The possibility of having an asymptotic behavior of tα for large t together with singularities specified by the function α is discussed.
Exact Static-Model Bootstrap Solutions for Arbitrary 2 × 2 Crossing Matrices
Authors: Cushing James T.
Published: February 2004

Explicit, exact solutions, satisfying a bootstrap criterion in the form of Levinson's theorem and having real coupling constants, are exhibited for the two-channel Low equation for arbitrary real values of the parameter appearing in the 2 × 2 crossing matrix. This disproves a claim made recently in the literature that the bootstrap criterion will restrict values of the crossing-matrix parameter to those corresponding to the internal-symmetry group SU2. It is demonstrated that the previous alleged proof is inconclusive.
Asymptotic Fields of the Lee Model
Authors: Jernow Stanley , Kazes Emil
Published: February 2004

The field equations for the Lee model are solved by constructing eigenfields from the most general possible combination of bare fermion and boson operators. These solutions are found to require an infinite number of boson terms, the coefficients of which obey the integral equations of scattering theory. It is discovered that the algebra of the Lee-model fermion eigenfields is not that of free particle operators. The anticommutator of the ith fermion eigenfield Ai+ has the form {Ai, Ai+}+=1−Σj≠i Aj+ Aj, where the sum is over all other fermion eigenfields. An algebra of this type is not peculiar to the Lee model. It will hold for all fields which obey a general Pauli exclusion principle and which are orthogonal to one another. The explanation for this algebra lies in the fact that these eigenfields represent operators which create states and not particles. Several other models exhibiting this same behavior (including the harmonic oscillator) are presented; for these models boson fields may also be constructed which have the above algebra. The asymptotic convergence of these eigenfields is not examined. However, it is found that all such Lee-model fermion fields, including those constructed by the Yang-Feldman method, must satisfy the above algebra and do not enjoy a free-fermion canonical algebra.
Instantaneous Action-at-a-Distance Formulation of Classical Electrodynamics
Authors: Kennedy Frederick James
Published: February 2004

The possibility of formally translating the interaction of charges from charge ↔ field ↔ charge to charge ↔ charge, where the orbits satisfy Newtonian (second order in t), yet covariant, equations of motion, is exploited for the Wheeler-Feynman interaction. A method for computing the forces on the charges correct to second order in the coupling constant e2 is presented, and ten constants of the motion correct to e2 are found. The integration is effected via the Noether theorem with the inhomogeneous Lorentz group as symmetry transformations. An important result is that a well-known correction to the Coulomb interaction which accounts for the uniform motion of charges is revealed to be, to first order in e2, a frame-invariant expression. The consequent corrected Coulomb dynamics admits first-order integrals identical to those of the Wheeler-Feynman dynamics.
Perturbation Method for a Nonlinear Wave Modulation. I
Authors: Taniuti Tosiya , Yajima Nobuo
Published: February 2004

In this paper we consider a system of nonlinear wave equations which admits, in a linear approximation, a planewave solution with high-frequency oscillation. Then, for the wave of small but finite amplitude, we investigate how slowly varying parts of the wave such as the amplitude are modulated by nonlinear self-interactions. A stretching transformation shows that, in the lowest order of an asymptotic expansion, the original system of equations can be reduced to a tractable, single, nonlinear equation to determine the amplitude modulation.
On Perturbation Expansions for Real-Time Green's Functions
Authors: Korenman Victor
Published: February 2004

A proof by Craig that the perturbation expansion for the real-time self-energy of a particle in a many-particle system has the same form in any statistical state is shown to be invalid and the stated theorem is shown to be untrue.
Phase Transition in Zero Dimensions: A Remark on the Spherical Model
Authors: Lieb Elliott H. , Thompson Colin J.
Published: February 2004

It is shown that the spherical model consisting of N spins with nonzero interaction between two spins only has a phase transition in the limit N → ∞. This is a counterexample to a suggestion of Kac which states that an Ising model will have a transition if the corresponding spherical model has a transition. Possible modifications of Kac's conjecture are suggested and discussed.
Dynamics in the Diagonal Coherent-State Representation
Authors: Miller Marvin M.
Published: February 2004

The problem of following the dynamical behavior of a quantum-mechanical system in the diagonal coherent-state representation is examined for those systems whose time evolution is specified by equations of motion for the coherent-state weight functional which resemble Fokker-Planck equations but have non-positive-definite diffusion matrices. A particular equation of this type describing a linear parametric process is considered in detail and several proposed generalizations of the diagonal representation, which include dynamical effects of the atomic system coupled to the electromagnetic field in simple models of a unimodal laser, are also briefly discussed.
On the Growth of the Ground-State Binding Energy with Increase in Potential Strength
Authors: Simon Barry
Published: February 2004

We study the asymptotic behavior of the ground-state binding energy G(λ) of −Δ + λV as λ → ∞. Unlike the number of bound states, G(λ) does not have a universal power growth as λ → ∞. It is shown, however, that as λ → ∞ for Kato potentials Aλ<G(λ)<Bλ4. Examples are presented for which G ∼ λβ for any 1 < β < 4. Other examples are presented which obey no power growth. We also prove theorems which reflect the close connection between the large λ behavior of G and the small r behavior of V for potentials with a single attractive singularity at r = 0. These can be roughly phrased as follows: If V ∼ −r−α for r → 0, then G(λ) ∼ λβ with β = 2/(2 − α) as λ → ∞.
Summation over Feynman Histories in Polar Coordinates
Authors: Peak D. , Inomata A.
Published: February 2004

Use of polar coordinates is examined in performing summation over all Feynman histories. Several relationships for the Lagrangian path integral and the Hamiltonian path integral are derived in the central-force problem. Applications are made for a harmonic oscillator, a charged particle in a uniform magnetic field, a particle in an inverse-square potential, and a rigid rotator. Transformations from Cartesian to polar coordinates in path integrals are rather different from those in ordinary calculus and this complicates evaluation of path integrals in polars. However, it is observed that for systems of central symmetry use of polars is often advantageous over Cartesians.
Group Theory and Mixed Atomic Configurations
Authors: Morrison J. C.
Published: February 2004

A group-theoretical scheme is introduced to classify the states of an atomic system having two open shells. States labeled according to this scheme may be written |(lA + lB)αQA(SALA)JA × βQB(SBLB)JB, QMQJMJ〉,where the quasispins QA and QB are coupled together to form a total quasispin Q. Although these states are, in general, mixtures of different configurations lAklBq, it is found that they serve as a convenient basis for the calculation of matrix elements in (lA + lB)N. The matrix elements of operators between the states of two configurations are obtained from these matrix elements by means of a unitary transformation. As an example matrix elements of the Coulomb interaction within (f + p)N are calculated.
On a Condition for Completeness
Authors: Wong Jack
Published: February 2004

It is shown that a completeness relation for the eigensolutions of a non-Hermitian operator H can be derived even if the resolvent operator R(H) of H is allowed to have poles of higher order than just simple poles, as required by Fonda, Ghirardi, Weber, and Rimini. A class of operators satisfying the requirements of this note is cited.
Partial Sums of Vacuum Polarization Graphs
Authors: Fink James P.
Published: February 2004

This paper contains results on the logarithmic asymptotic behavior of certain partial sums of photon self-energy graphs.
On Fluids of Particles with Short-Range Repulsion and Weak Long-Range Attractive Interaction
Authors: Jalickee John B. , Siegert Arnold J. F. , Vezzetti David J.
Published: February 2004

We have obtained an appreciably simplified derivation of the expansion in inverse range of the thermodynamic functions of a system of particles with short-range repulsion and weak long-range attractive interaction. We assume all properties of the system with only the short-range repulsive interaction to be known and consider this system as a reference system for the purpose of a perturbation expansion. We derive corrections to the van der Waals equation and to the Maxwell construction. The terms in our expansion are integrals over products of the modified Ursell functions of the reference system only and factors which are the transformed long-range potential. A simple rule is given to determine the order, in the inverse range of the attractive potential, of each term in the expansion. The simplification of the derivation is achieved by avoiding the expansion and subsequent resummation of any functions pertaining to the reference system.
Exact Quantization Conditions. II
Authors: Krieger J. B.
Published: February 2004

By employing coordinate transformations, a generalized WKB quantization condition is derived which includes a modified WKB quantization rule and the related higher-order integrals. It is observed that a necessary condition for the first-order integral to give an exact quantization rule is the vanishing of the higher-order integrals. When this condition is satisfied, the resulting quantization condition is of the form of all previously known exact-quantization rules. The higher-order integrals are shown to vanish for certain cases of interest. An error in Paper I [C. Rosenzweig and J. B. Krieger, J. Math. Phys. 9, 849 (1968)] is noted and an examination of the higher-order integrals shows that a proposed quantization rule given there is not exact.
An Exact Quantum Theory of the Time-Dependent Harmonic Oscillator and of a Charged Particle in a Time-Dependent Electromagnetic Field
Authors: Lewis H. R. , Riesenfeld W. B.
Published: February 2004

The theory of explicitly time-dependent invariants is developed for quantum systems whose Hamiltonians are explicitly time dependent. The central feature of the discussion is the derivation of a simple relation between eigenstates of such an invariant and solutions of the Schrödinger equation. As a specific well-posed application of the general theory, the case of a general Hamiltonian which settles into constant operators in the sufficiently remote past and future is treated and, in particular, the transition amplitude connecting any initial state in the remote past to any final state in the remote future is calculated in terms of eigenstates of the invariant. Two special physical systems are treated in detail: an arbitrarily time-dependent harmonic oscillator and a charged particle moving in the classical, axially symmetric electromagnetic field consisting of an arbitrarily time-dependent, uniform magnetic field, the associated induced electric field, and the electric field due to an arbitrarily time-dependent uniform charge distribution. A class of explicitly time-dependent invariants is derived for both of these systems, and the eigenvalues and eigenstates of the invariants are calculated explicitly by operator methods. The explicit connection between these eigenstates and solutions of the Schrödinger equation is also calculated. The results for the oscillator are used to obtain explicit formulas for the transition amplitude. The usual sudden and adiabatic approximations are deduced as limiting cases of the exact formulas.
Clebsch-Gordan Coefficients and Functions over Cosets
Authors: Klink William H.
Published: February 2004

Functions over cosets are used to obtain Clebsch-Gordan coefficients of induced unitary representations of noncompact groups. In particular, the Clebsch-Gordan coefficients of the Poincaré group are obtained using functions over cosets. It is also shown how the Clebsch-Gordan coefficients of other groups having induced irreducible representations, such as the Lorentz group, may be obtained.
Possible Instability of the Cut-Contributions to the Ring Equations
Authors: Baus M.
Published: February 2004

The specifically relativistic cut-contributions to the ring equations of a relativistic plasma as discussed in a previous paper are considered in more detail. The question of principle—whether or not these contributions can give rise to specifically relativistic instabilities—is investigated with the aid of a simple nonequilibrium velocity distribution which leads to unstable cut-contributions, but not necessarily to unstable Landau-pole contributions.
Schwarzschild Singularity
Authors: Bel Luis
Published: February 2004

A new point of view is presented for which the Schwarzschild singularity becomes a real point singularity on which the sources of Schwarzschild's exterior solution are localized.
Generalized Triangular Ising Lattice
Authors: Mills R. G. J. , Hurst C. A.
Published: February 2004

The partition function and critical equations for the generalized triangular Ising lattice are determined in terms of weight factors associated with the decorating lattice. As an example, a lattice which incorporates the Kagome, hexagonal, triangular, and rectangular lattices is solved by the method developed.
Rigorous Results for Ising Ferromagnets of Arbitrary Spin
Authors: Griffiths Robert B.
Published: February 2004

The following results for spin-½ Ising ferromagnets are extended to the case of arbitrary spin: (1) the theorem of Lee and Yang, that the zeros of the partition function lie on the unit circle in the complex fugacity plane; (2) inequalities of the form  ≥ , where A and B are products of spin operators; (3) the existence of spontaneous magnetization on suitable lattices. Results (2) and (3) are also extended to the infinite-spin limit in which the spin variable is continuous on the interval −1 ≤ x ≤ 1.
On the Combinatorial Structure of State Vectors in U(n). II. The Generalization of Hypergeometric Functions on U(n) States
Authors: Ciftan Mikael
Published: February 2004

The derivation of the explicit algebraic expressions of the SU(n) state vectors in the boson-operator realization is shown to lead to a generalization of hypergeometric functions. The SU(3) state vectors are rederived by the combinatorial method-propounded in Paper I [J. Math. Phys. 10, 221 (1969)] of this series of papers-and are shown to be represented by a hypergeometric distribution function and an associated generalization of the Young tableaux calculus. The SU(4) state vectors are also derived to demonstrate the main features of the general U(n) state vectors. The SU(4) state vectors are expressed in terms of the constituents of Radon transforms.
Critical-Point Singularities of the Perturbation Series for the Ground State of a Many-Fermion System
Authors: Baker George A. , Kahane Joseph
Published: February 2004

We show that the many-fermion ground-state energy with an attractive potential has a critical singularity. This singularity destroys the validity of ``low-density'' approximations. We also find that the K-matrix formalism is, in principle, not applicable to attractive potentials because of the presence of ``Emery singularities.'' We introduce an R-matrix formalism which is numerically very close to the K matrix and free from manifest ``Emery singularities.'' A model calculation is performed on the lattice gas to try to anticipate what quality of results can be expected from summing an R-matrix expansion with fixed density.
Kernel Integral Formulas for the Canonical Commutation Relations of Quantum Fields. I. Representations with Cyclic Field
Authors: Hegerfeldt Gerhard C.
Published: February 2004

We investigate the kernel or group integral for the canonical commutation relations introduced by Klauder and McKenna and its generalizations. For the finite case the kernel integral formula has been proven by means of the Schrödinger representation. Motivated by the close similarity of the Schrödinger representation to the form of a general representation with cyclic field, we examine these representations with respect to kernel integral formulas. A general criterion is derived in which the dimensionality of the test function space does not enter, i.e., it is independent of the number of degrees of freedom. In this way the finite and infinite case can be treated on equal footing. The criterion contains as special cases the kernel integral formulas of Klauder and McKenna for finitely many degrees of freedom and for direct-(or tensor-) product representations of fields. For partial tensor-product representations we obtain a somewhat modified formula. After these applications, a considerably sharpened form of the criterion is derived in which only the vacuum expectation functional enters. Under a certain cyclicity assumption it is shown that the validity of a kernel integral for just some cyclic vector implies its validity for all vectors. It is further shown that the basis-independent integral defined by a supremum over all bases of the test function space U can be replaced by an ordinary limit over a kind of diagonal sequence through finite-dimensional subspaces of U. In the last section a representation is constructed which possesses a cyclic field but does not fulfil a kernel integral formula; this is an instructive illustration of a general theorem to be proved in II of this series of papers.
Group Analysis of Maxwell's Equations
Authors: Carmeli Moshe
Published: February 2004

In this paper we write down and solve Maxwell's equations without sources when the field variables are considered as functions over the group SU2. A Hilbert space is then constructed out of the field functions. An expansion of the field functions in terms of the matrix elements of the irreducible representation of SU2 is shown to reduce the problem of solving Maxwell's equations to that of solving one partial differential equation with two variables. A Fourier transform reduces this equation into an ordinary differential equation which is identical to the partial-wave equation obtained from the Schrödinger equation with zero potential. The analogy between the mathematical method used in this paper in relation to the group SU2 and the Fourier transform in relation to the additive group of real numbers is pointed out.
Generating Functions of Classical Groups and Evaluation of Partition Functions
Authors: Santhanam T. S.
Published: February 2004

The generating functions of classical groups are used to set up recursion relations for their partition functions. These are then used to find the internal multiplicity structure of the weights using Kostant's formula.
Nonlinear Electrodynamics and General Relativity
Authors: Pellicer R. , Torrence R. J.
Published: February 2004

A generalization of Born-Infeld nonlinear electrodynamics, due to Plebanski, is reformulated in the context of general relativity theory. A class of nonsingular, static, spherically symmetric solutions of the modified Einstein—Maxwell equations are given, corresponding to a point-charge source. The metric tensors of these solutions are shown to approach the Riesner-Nordstrom metric tensor at large distances from the source, if one makes the proper identification of mass.
General Form of the Einstein Equations for a Bianchi Type IX Universe
Authors: Ryan Michael P.
Published: February 2004

The Einstein equations for a general Bianchi type IX universe are presented in a form suitable for numerical solution. As an example, the complete equations for a cosmology with a pure fluid stress tensor Tμν = εuμuν + P(gμν + uμuν) are also given.
One-Dimensional Bonded Fluids
Authors: Bell G. M.
Published: February 2004

A study is made of a linear lattice fluid in which bonds can form between pairs of molecules on second-neighbor sites with an empty site between them. At low temperatures and pressures there is, thus, competition between local open configurations and more close-packed local configurations of higher energy. By using a matrix expression for the grand partition function, the density and effective coordination number are found as functions of the absolute temperature T and one-dimensional pressure p. Equivalent results are shown to follow from a constant-pressure partition function based on an explicit expression for the configuration number. It is found that a pressure p0 exists such that for any p < p0 the open configuration is stable at T = 0, while the density passes through a maximum as T increases. For any p > p0, on the other hand, the close-packed configuration is stable at T = 0 and the density decreases monotonically with T. Using a constant-pressure partition function, we also consider continuous models with the ``bonding'' represented by a potential well separated from the hard core. With a well of parabolic shape, similar results to those of the lattice model are obtained, while with a square well the curve of density against T displays a minimum as well as a maximum for any p < p0.
Reduced Density Matrices of Energy Eigenstates
Authors: Rosina Mitja , Percus Jerome K. , Kijewski Louis J. , Garrod Claude
Published: February 2004

The following question is considered: What special properties are possessed by those reduced density matrices which come from energy eigenstates? Using the fact that 〈φ[H, A]|φ〉 = 0, where A is any operator and |φ〉 an energy eigenstate, it is shown that the elements of the two-particle density matrix are severely restricted by homogeneous linear relations. Their full content is expressed in terms of an auxiliary one-particle density which possesses additional positivity properties in the ground state.
Model for the Derivation of Kinetic Theory
Authors: Ramanathan G. V. , Sandri G.
Published: February 2004

A soluble model for the derivation of kinetic theory is discussed. It is shown that Bogoliubov's adiabatic assumption is not valid even to leading order in the kinetic regime. Furthermore, the adiabatic assumption gives rise to a singular behavior in the correlation function which is sufficiently strong to violate the ordering assumed in the expansion. The exact result for the model correlation is neither adiabatic nor singular throughout the kinetic regime. We first show that the singularity is removed by properly reordering the correlation in the relevant domains; then a method for obtaining uniformly valid asymptotic solutions for a wide class of integrodifferential problems (that includes our model) is presented. We apply our technique to the BBGKY hierarchy and show that for a weakly coupled gas (i) the familiar Landau equation is satisfied in lowest order by the velocity distribution function and (ii) the two-particle correlation function can be calculated consistently to leading order, yielding a nonadiabatic and nonsingular behavior.
Properties of Overcomplete and Nonorthogonal Basis Vectors
Authors: Rowe D. J.
Published: February 2004

An alternative to the Schmidt procedure is presented for the orthogonalization of a set of functions or state vectors. The method is particularly simple and ideally suited to the eigenvalue problem in a space spanned by nonorthonormal, and even overcomplete, basis vectors. The method is applied to the solution of the equation of motion with a non-unit-metric matrix and to the problem of eliminating spurious (or any other) states from a given vector space.
Self-Consistent Approximations in Many-Body Systems
Authors: Revzen M. , Trainor L. E. H.
Published: February 2004

A new stationary property of the grand canonical potential is presented. This property is used to define a class of self-consistent approximations which is particularly appropriate for the so-called ladder approximation. The conditions which guarantee self-consistency are only sufficient. A comparison is made with the already-known stationary properties related to Hartree-Fock and random-phase approximations. Remarks are made on the possibility of a collective excitation associated with the formalism.
Symmetrized Tensor Geometry
Authors: Tompkins Donald R.
Published: February 2004

For Weyl symmetrized tensors, the inner product structure has previously been proven to be identical within equivalent multiplets. Here the inner product structure between a set of equivalent multiplets is proven to be essentially the same as that within the equivalent multiplets. This property is important because it separates the problem of orthogonalization within equivalent multiplets from the problem of orthogonalization between equivalent multiplets. Thus, full orthogonalization can be achieved by, first, identically orthogonalizing each of a set of equivalent multiplets and, then, recoupling the equivalent multiplets with coefficients which do not depend on individual vectors within the multiplets. These statements apply to symmetrized tensor multiplets of both the permutation group and an underlying group (such as Un).
Legendre Polynomial Expansions of Hypergeometric Functions with Applications
Authors: Holdeman Jonas T.
Published: February 2004

The expansion of a class of hypergeometric functions in a series of Legendre polynomials is derived. The range of validity and the meaning to be attached to the sums is investigated. Several applications to the problem of the scattering of charged particles are presented.