bluuebunny/crossref_metadata_2025 huggingface dataset:
New Approach to the Ising Problem
Authors: Hurst C. A.
Published: January 2005

The partition function for the Ising model on a two-dimensional rectangular lattice is cast into a form which closely resembles the vacuum expectation value of the S-matrix in quantum field theory. The standard Onsager expression is obtained very simply in this formalism. It is further shown how the nonsoluble models can be expressed as field theories with quartic interactions, thereby resembling the standard many-body fermion theories.
Numerical Solution of Non-Fredholm (Singular) Integral Equations by Matrix Inversion
Authors: Jones C. Edward , Tiktopoulos George
Published: January 2005

It is shown that the simple matrix-inversion techniques often used in numerically solving linear integral equations with a Fredholm-Schmidt (i.e., square-integrable) kernel can also be employed for a wide class of non-Fredholm (``singular'') equations. This class includes equations the kernel of which is the sum of a Fredholm-Schmidt kernel and a kernel whose norm (in the operator sense) is less than one. In particular, the integral equations of the so-called ``new strip approximation'' in particle dynamics belong to this class.
Methods of Quadrature for Euler Transform Integrals
Authors: Livingston Peter M.
Published: January 2005

Several methods of treating Euler transform integrals exist. One such method follows from the expression of the Euler transform kernel as a bilinear series of independent solutions to the Jacobi equation valid for the integration variable in the real interval −1 to 1 and the transform variable outside. The transform function then is expressed as a series of solutions of the second kind to the Jacobi equation whose coefficients are the expansion coefficients of the function to be transformed in the complete set of Jacobi polynomials, provided the latter exist. Such a series is absolutely convergent for the transform variable not on the real interval cited above. Another method, due to MacRobert, permits quadrature of the Euler transform integral directly for certain integrands. Finally, the expansion of the Euler kernel in a bilinear series of Bessel functions and Neumann polynomials valid for the integration variable on the finite interval, 0 to a, is mentioned, and applied to several integrals. Examples of all three methods are given.
Some Spatially Homogeneous Anisotropic Relativistic Cosmological Models
Authors: Kantowski R. , Sachs R. K.
Published: January 2005

Some solutions of the Einstein field equations for a dust source are given in explicit form. They are spatially homogeneous, irrotational, and anisotropic. They can be characterized as those spatially homogeneous expanding models that do not permit a simply transitive three-parameter group of motions. The models are compared in detail with observations and with the Friedmann models. In a few instances slightly longer time scales are obtained with the present models than from the corresponding Friedmann models.
Geometrization of a Complex Scalar Field. I. Algebra
Authors: Penney R.
Published: January 2005

The necessary and sufficient conditions that a Ricci tensor should algebraically represent the energy tensor of a complex scalar field are given.
Two-Center Coulomb Integrals between Atomic Orbitals
Authors: O-Ohata Kiyosi , Ruedenberg Klaus
Published: January 2005

General analytical formulas are developed for the two-center Coulomb integrals between Slater-type atomic orbitals of arbitrary integral quantum numbers (nlm). The Coulomb integral C is obtained by integrating the Poisson equation ΔC = —4πS, where S is the corresponding overlap integral. The Coulomb integrals are expressed as trilinear forms of powers and certain auxiliary functions Gαβγ, Hαβγ, Jμνλ, which are related to confluent hypergeometric functions and Jacobi polynomials respectively. Stable recurrence procedures for their evaluation are given, and use with an electronic computer program showed the loss in numerical accuracy to be small for all argument values as well as for high quantum numbers.
Functional Formulation of the Lee Model
Authors: Fried H. M.
Published: January 2005

The Lee model is defined as a particular member of a class of approximations to a relativistic, local, nontrivial field theory. Functional methods are applied to obtain generalizations of familiar properties, and to suggest variants of the model.
Asymptotic Renormalizability Conditions on Field Propagators
Authors: Boyce J. F.
Published: January 2005

High-energy boundary conditions upon the basic Green's functions which limit the types of divergence arising in any approximational method of solution of a field theory are derived and shown to be related to the recently defined stability criterion.
Empty Space-Times of Embedding Class Two
Authors: Collinson C. D.
Published: January 2005

The null tetrad notation of Newman and Penrose is used to investigate empty space-times of embedding class two. Necessary conditions are found for algebraically special empty space-times to have this property.
Continuity of Bound and Unbound States in a Fermi Gas: A Soluble Example
Authors: Majumdar Chanchal K.
Published: January 2005

Considering a gas of independent fermions in the presence of an attractive localized potential, one can show that the properties of the system as a whole are smooth (analytic) functions of the strength parameter of the potential, even at those values where new single-particle bound states appear. Thus for the system as a whole, the transition from ``unbound'' to ``bound'' states is continuous and the concept of a bound state cannot be made precise. This is illustrated here for a simple mathematically soluble model—noninteracting spinless fermions moving in the presence of a delta-function potential in one dimension. Some related physical ideas are also presented.
Asymptotic Correlations of the Hamiltonian Matrix Elements
Authors: Ullah Nazakat
Published: January 2005

The statistical correlations of the matrix elements of real symmetric Hamiltonians are studied using the assumption of representation invariance and the limit of large dimension N. The diagonal-diagonal correlation coefficient is expressed in terms of a parameter which gives the ratio of the dispersion of off-diagonal element to that of the diagonal element. It is shown that for a certain class of real-symmetric Hamiltonian ensembles in the limit N → ∞, the diagonal-diagonal correlation coefficient goes as λN−1, where λ is some positive constant independent of N and the correlation coefficient of two different eigenvalues is the same as the one obtained using the weak assumption of independent probabilities.
q-Equivalent Particle Hamiltonians. I. The Classical One-Dimensional Case
Authors: Currie Douglas G. , Saletan Eugene J.
Published: January 2005

The classes of equivalent Lagrangians in one-dimensional particle dynamics are found. These classes contain not only Lagrangians yielding the same equations of motion (Lagrangians differing by a total time derivative), but also those implying each other's equations of motion. The corresponding classes of Hamiltonians, all of which give the same orbits in configuration space, but in general different orbits in phase space, are also found. Some specific examples are presented.
Kinetic Equation for the Electron-Phonon Gas
Authors: Chappell Willard R.
Published: January 2005

The random phase approximation equations of motion for the electron-phonon system are solved to obtain long-time expressions for the basic operators that appear in the calculation of correlation functions. These operator expressions are used to obtain a kinetic equation for the electrons and an expression for the phonon spectral function. The kinetic equation has the same form as the Balescu-Lenard equation. Expressions for the density autocorrelation function and the two-particle correlation function are also obtained. The latter is shown to have the same form as the test-particle result for a classical plasma.
On the Poles of the S Matrix for Long-Range Potentials
Authors: Ferreira Erasmo M. , Teixeira Antonio F. F.
Published: January 2005

The analytic properties in the complex k plane of the S matrix for scattering by a screened Coulomb potential are studied. Particular attention is given to the limit as the screening radius tends to infinity, so as to show, in an explicit example, the effect of the tail of the potential on the properties of the analytically extended S matrix. It is shown that the pole configuration obtained in this way is different from that obtained in the usual description of the analytic properties of the Coulomb S matrix.
One-Dimensional Impenetrable Bosons in Thermal Equilibrium
Authors: Lenard A.
Published: January 2005

The simple relationship between the wavefunctions of a system of impenetrable one-dimensional bosons and impenetrable one-dimensional fermions is exploited to derive an expansion of the boson reduced density matrices in terms of the fermion reduced density matrices and vice versa. This expansion is independent of the statistical ensemble used, and of the interparticle potential (subject to the impenetrability condition). The special case of zero impenetrability radius with no other forces is treated in detail, using the grand canonical ensemble. This leads to a natural generalization of a formula previously found for zero temperature only, for the one-particle reduced density matrix.
Global Covariant Conservation Laws in Riemannian Spaces. II
Authors: Shepelavey Bohdan
Published: January 2005

Using the idea of tensor integration, the vector field developed in Part I of this report, and the full Bianchi identities, it is shown that in a general Riemannian space there are four global covariantly-conserved tensors. The ranks of these tensors are three, four, five, and six. The traces of the first two of these tensors yield the generally covariant equivalent of the familiar linear and angular momentum. The remaining four traceless tensors describe, residually, the gravitational field. With each covariantly conserved tensor one can associate a number of independent invariants. Such invariants are conserved in the ordinary sense. Among these are two types of rest energies and two types of angular momentum magnitudes obtained from the trace and traceless tensors. Examples of global, conserved tensors are derived for a Schwarzschild metric with the electron mass, and a metric of a point electron. It is shown that the rest energy of the Coulomb field diverges as ln(1 /r) at the origin and the second rest energy, that is, the rest energy of the gravitational field diverges as ln r as r approaches infinity. When cutoffs are introduced at the Schwarzschild radius r0, at the classical electron radius r1, and at the radius of the visible universe r2, the rest energy of the gravitational field contained in the shell of thickness r1 − r0 is approximately 100 times that of the electron rest energy. It is twice this value in the entire visible universe. Since the gravitational field is described by the traceless tensors and the former forms a heavy, compact cloud around the point particle, it is conjectured that the traceless tensors represent the internal degrees of freedom of the elementary particles.
A Note on the Quantum Theory of Electrical Conductivity
Authors: Tjon J. A.
Published: January 2005

A formal method is given for calculating the electrical conductivity tensor in ascending powers of the interaction for a system of electrons interacting with a random distribution of scattering centers.
Laws like Newton's
Authors: Arens Richard
Published: January 2005

Instantaneous interaction for n particles is defined in terms of kinematic concepts only. A set of ``laws,'' reminiscent of, but much weaker than, Newton's three laws, is formulated in solely kinematic terms. Invariance under the Euclidean group, the Galilean group, and the special Galilean group is defined, and the most general interactions satisfying the laws, and invariant under these groups, are found. It is shown that they all satisfy Newton's laws. It is shown that the interaction between moving charges cannot be instantaneous and Galilean-invariant.
Degeneracy of Cyclotron Motion
Authors: Dulock Victor A. , McIntosh Harold V.
Published: January 2005

The classical problem of planar cyclotron motion of a charged particle in a uniform magnetic field possesses symmetries which account for the ``accidental'' degeneracies of the analogous nonrelativistic Schrödinger equation, as found by Johnson and Lippman. The essentially quadratic nature of the Hamiltonian is not changed by considering the particle moving in a harmonic oscillator potential, a ``Zeeman effect'' for the harmonic oscillator. The transitions to the limiting cases of a weak magnetic field (pure harmonic oscillator) or a strong field (pure cyclotron motion) involve the contraction of the corresponding symmetry groups, yielding Larmor precession of the oscillator orbits in the first case, and the drift of the cyclotron orbit in the second. The constants of the motion generate the unitary unimodular group SU2 in all cases except for pure cyclotron motion, in which case one obtains the commutation rules of creation and annihilation operators. Only for certain ratios of magnetic field strength to the oscillator frequency does one obtain bounded closed orbits, and presumably only in these cases do degeneracies exist quantum-mechanically. A transition to a rotating coordinate system reduces the problem to that of a plane harmonic oscillator; however, the time dependencies of the transformation must be allowed for interpreting the constants thereby arising. Moreover, the velocity-dependent forces introduce gauge transformations which also affect the interpretation of the symmetries. There are two kinds of symmetries—inner symmetries involving the canonical coordinates and governing the shape of the orbits, and outer symmetries involving the mechanical coordinates and governing the location of the orbits.
Probability and Entropy of Macroscopic Fluctuations
Authors: Zwanzig Robert
Published: January 2005

A macroscopic fluctuation is a fluctuation of order N in an N-particle system. This article contains a calculation of the probability of a macroscopic fluctuation, and its associated entropy, which is asymptotically correct in the limit of large N. Sufficiently small macroscopic fluctuations are shown to obey the same Gaussian distribution law as spontaneous microscopic fluctuations (of order N½). The modifications necessary to describe large macroscopic fluctuations are found. The entropy of a macroscopic nonequilibrium state is expressed by means of various moments calculated at equilibrium. The nonlinear thermodynamic force for a nonequilibruim state far from equilibrium is found. The calculation is based on a cumulant expansion of the characteristic function of the probability distribution and a stationary phase estimate of its Fourier transform.
Lattice Constant Systems and Graph Theory
Authors: Sykes M. F. , Essam J. W. , Heap B. R. , Hiley B. J.
Published: January 2005

The two principal systems of lattice constants that have arisen in the study of cooperative phenomena and related problems on crystal lattices are the strong (low-temperature) and the weak (high-temperature) systems. The two systems are defined in terms of the concepts of graph theory, and a general theorem relevant to cluster expansions is stated. The interrelation of the two systems is studied and exploited to derive configurational data for the face-centered cubic lattice. All star graphs with up to seven points (vertices) or nine lines (edges) that are embeddable on the face-centered cubic are described. A general classification of stars with cyclomatic index 3 is given.
N Degenerate Modes
Authors: Bruch L. W.
Published: January 2005

The effect of a bilinear coupling in a system of degenerate modes is discussed. The normal mode frequencies are obtained for some simple couplings. If each mode is coupled to all the other degenerate modes, the maximum shift in normal mode frequency due to the coupling increases with the number of degenerate modes. Using the same methods, a many-boson problem and a truncated version of it can be compared. The energy eigenvalues have the same relationship as the exact and first-order perturbation excitation energies of the Bogoliubov model Hamiltonian for bosons.
A Formalism for Generalized Quantum Mechanics
Authors: Prugovečki Eduard
Published: January 2005

A generalization of a non-Hilbert-space formalism, developed in two previous papers, is presented. In this generalization observables with a many-dimensional spectrum are considered. Furthermore, the fundamental mathematical tools are generalized by introducing generalized functions instead of measures. It is then proved that a physical equivalence of the present formalism with the Hilbert-space formalism can still be established.
Ordering Theorems
Authors: Rohrlich F. , Wray J. G.
Published: January 2005

Various theorems related especially to time-ordered products are proven. Applications to quantum field theory include particularly simple derivations of the off-mass-shell extensions of the S-operator and of the interpolating field. The latter are a generalization of the results of Glaser, Lehmann, and Zimmermann.
Numerical Solutions of Singular Fredholm Equations
Authors: Ullman Nelly , Ullman Robert
Published: January 2005

A singular Fredholm equation of the second kind is solved numerically by a Fourier series analysis in which the singularity is removed naturally, and by a Gaussian quadrature procedure in which the singularity was eliminated by an approximation using the law of the mean. In addition, an analytic solution of an associated Fredholm equation of the first kind was used for comparison with the numerical results. Some properties of these numerical solutions are indicated, and a brief discussion of the errors is given.
Canonical Root Vectors of SUn
Authors: Wenger Daniel L.
Published: January 2005

The present paper extends the study of SUn to the determination of particular root vectors and fundamental weights. The root vectors are chosen so as to show explicitly the canonical nesting of subgroups SUn⊃ SUn−1 × U1, SUn−1⊃ SUn−2 × U1, etc. Explicit matrix representations of the fundamental representations are given. The results are all dependent upon the use of n vectors which define the fundamental region of SUn. Eigenvalues of fundamental invariant operators are also given in terms of these n vectors.
Representation Functions of the Group of Motions of Clifford Space
Authors: Wenger Daniel L.
Published: January 2005

We find the representation functions of the group of motions of the three-dimensional Clifford space and of the three-dimensional Einstein space. These functions are generalizations of spherical and cylindrical waves of three-dimensional Euclidian space and reduce to these familiar functions in the Euclidian limit. The generalization of the plane wave is also found.
General Coupling Coefficients for the Group SU(3)
Authors: Resnikoff M.
Published: January 2005

A Hilbert space method, previously applied to the group SU(2), is employed to examine the representations 𝒟λμ and the reduction of the direct-product representation of the group SU(3). The base vectors |λμ;α>, an orthogonal Hilbert space of homogeneous polynomials, are transformed to the base vectors |λμ;α>c, and are associated with the complex conjugate representation by an explicit R- conjugation operation. For the general direct-product representation 𝒟λ1μ1 ⊗ 𝒟λ2μ2, explicit expressions are derived for the vector coupling coefficients and the number of times the irreducible representation 𝒟λ3μ3 is contained in the direct product. Two methods of labeling the degenerate states are given, the reduction of the direct product is shown to be complete, and the symmetry relations of the 3(λμ) coefficients are discussed.
Phase-Integral Approximation in Momentum Space and the Bound States of an Atom
Authors: Gutzwiller Martin C.
Published: January 2005

The phase integral approximation of the Green's function in momentum space is investigated for an electron of negative energy (corresponding to a bound state) which moves in a spherically symmetric potential. If the propagator rather than the wavefunction is considered, all classical orbits enter into the formulas, rather than only the ones which satisfy certain quantum conditions, and the separation of variables can be avoided. The distinction between classically accessible and classically inaccessible regions does not arise in momentum space, because any two momenta can be connected by a classical trajectory of given negative energy for a typical atomic potential. Three approaches are discussed: the Fourier transform of the phase integral approximation in coordinate space, the approximate solution of Schrödinger's equation in momentum space by a WKB ansatz, and taking the limit of small Planck's quantum in the Feynman-type functional integral which was recently proposed by Garrod for the energy-momentum representation. In particular, the last procedure is used to obtain the phase jumps of π/2 which occur every time neighboring classical trajectories cross one another. These extra phase factors are directly related to the signature of the second variation for the action function, and provide a physical application of Morse's calculus of variation in the large. The phase integral approximation in momentum space is then applied to the Coulomb potential. The location of the poles on the negative energy axis gives the Bohr formula for the bound-state energies, and the residues of the approximate Green's function are shown to yield all the exact wavefunctions for the bound states of the hydrogen atom.
Upper Bounds on Holomorphy Envelopes for Wightman Functions
Authors: Manoharan A. C.
Published: January 2005

As a consequence of Lorentz invariance, reasonable mass spectrum and local commutativity, the holomorphy envelopes H(SnP) for the analytic functions arising from vacuum expectation values of products of n field operators are related in such a way that H(SnP) furnishes upper bounds (in the coordinates) for all H(𝔖N) with N > n.
Invariants of Nearly Periodic Hamiltonian Systems
Authors: McNamara B. , Whiteman K. J.
Published: January 2005

A new and simple method of finding an invariant J of a nearly periodic dynamical system is presented. The Hamiltonian is written as H = p1 + εΩ(qipi), where Ω is periodic in q1 and ε « 1. The first four terms of the invariant series are found explicitly in terms of Ω using Poisson bracket and averaging operators. This invariant is related to the adiabatic invariant and to various constants of motion discussed in celestial mechanics, such as Whittaker's adelphic integral. J is shown to be an asymptotic constant by using the rigorous methods of Kruskal to calculate the adiabatic invariant K; it is found that K/τ = H - εJ, where τ is the period in q1. The adelphic integral has different functional forms depending on the presence of resonant denominators, but is shown to be always a function of H and J. The present method provides a single functional form which is applicable even when Ω is only almost periodic in q1. It is also much simpler than the methods of adiabatic invariant theory.
Solution of the Differential Equation (∂2∂x∂y+ax∂∂x+by∂∂y+cxy+∂∂t)P=0
Authors: Lambropoulos P.
Published: January 2005

A series solution of the above differential equation is presented.
Internal Multiplicity Structure and Clebsch-Gordan Series for the Exceptional Group G(2)
Authors: Radhakrishnan D. , Santhanam T. S.
Published: January 2005

An explicit algebraic formula is obtained for the multiplicity M̄(γ) of a vector γ belonging to the fundamental domain of the group G(2). Using this, the internal multiplicity Mm(m′) of a weight m′ of the irreducible representation D(m) with the highest weight m is calculated through Kostant's formula for the dominant weights. The Clebsch-Gordan decomposition of the direct product of two irreducible representations is then obtained.
Connection between the Marchenko Formalism and N/D Equations: Regular Interactions. I
Authors: Cornille Henri
Published: January 2005

In this paper and in the following, we study, in potential scattering, the existence and meaning of the solutions of the N/D equations in the equivalent formulation f/f. For S waves, considering only regular discontinuity μΔ(x), such that the resulting integral equation is of the Fredholm type, we study the corresponding Fredholm determinant 𝒟(μ). We remark that Marchenko formalism gives exactly the same resulting equation and then we have the possibility to interpret in terms of local potentials. We show the connection between the nth trace of the kernel of the resulting integral equation and the nth term of the potential reconstructed from the discontinuity. The connection between dispersion relation and the corresponding potential reconstructed from the discontinuity is given by the relation D(μ)=exp−12 ∫ 0∞ ∫ r∞V(t, μ) dt dr. In the present paper we limit our study to |μ| less than the smallest modulus root of 𝒟(μ) where a perturbation expansion of the solution exists and we show that V(r, μ) is regular at r = 0. On the other hand, for Yukawa-type potentials where the inverse Laplace transform is λC(α) the Fredholm determinant is exp (−∫m∞λC(α)/α2 dα) and cannot vanish such that the corresponding solutions of the resulting integral equation exist always.
On the Determination of the Many-Channel Potential Matrix and S Matrix from a Single Function
Authors: Cox Joseph R.
Published: January 2005

It is shown that, for a certain model of many-channel potential scattering, there exists, under rather general conditions, a single function from which both the potential matrix V and the corresponding S matrix can be constructed. This function is the Fredholm determinant of the Lippmann-Schwinger equation for the physical wavefunction of a system having a potential matrix which is identical with V in the interval from the origin out to a distance r, but which vanishes identically beyond this distance. A one-channel and a two-channel example are discussed.
Solution of the Dimer Problem by the Transfer Matrix Method
Authors: Lieb Elliott H.
Published: January 2005

It is shown how the monomer-dimer problem can be formulated in terms of a transfer matrix, and hence in terms of simple spin operators as was originally done for the Ising problem. Thus, we rederive the solution to the pure dimer problem without using Pfaffians. The solution is extremely simple once one sees how to formulate the transfer matrix.
Properties of the Wavefunction for Singular Potentials
Authors: Spector Richard M.
Published: January 2005

For potentials more singular than the inverse square at the origin and with complex strength, the nature of the solutions to the Schrödinger equation is investigated. The difficulties occurring with attractive real potentials in the three-dimensional equation are discussed, and it is shown that no solutions exist in a domain including the origin. For complex or repulsive singular potentials with radial form varying as the inverse fourth or sixth power, a relatively simple series solution exists. This is also true for the singular Yukawa with inverse fourth power. These series are shown to be asymptotic by means of general theorems from the theory of ordinary differential equations. Other inverse powers have much more complicated solutions. In the Dirac case, all inverse powers and singular Yukawa forms have asymptotic series solutions, and the series is given explicitly for the singular powers. A form of the S-wave scattering length for the singular inverse fourth-power Yukawa V = g2e−iΔe−μr/r4 is derived which is valid for small values of μg.
Dynamical Content of Differential Geometry
Authors: Meckler Alvin
Published: January 2005

A formulation of the notion of a Cartesian basis for a vector space suitably adapted to the local differential vector space at a space-time point leads to this description of and derivation for the law of motion of a mass point: There exists a class of equivalent 4-dimensional Cartesian frames in which the trajectory is a straight line, the velocity being constant along the trajectory. The lived-in coordinate system, in which the trajectory is in general not a straight line, is in general not a Cartesian coordinate system. The equation for the motion of the point is purely a statement of differential geometry—an integrability condition—and the forces which are responsible for the deviation from constant velocity are immediately derivable from the metric tensor of the non-Cartesian system. The equation of motion and the relations among dynamical variables are more relativistic than Newtonian, but the metric tensor is positive definite; no indefinite metric is needed. A Newtonian equation of motion is an approximation based on small velocity and it covers the general case of motion in a noninertial system in an external field of force. Intrinsic mass is identified as a constant of the motion and, although the relevancy of mass is somewhat limited in this description of the motion of a single point in an immutable force field, it is seen that the effective mass of the point contains a contribution by the potential energy.
Internal Dynamics of Particlelike Solutions to Nonlinear Field Theories
Authors: Rosen Gerald
Published: January 2005

We set up a Rayleigh-Ritz procedure for determining the approximate classical and quantum internal motion of a singularity-free solution to a nonlinear field theory. The dynamical approximation procedure is applied to the particlelike solutions of a nonlinear model scalar field theory and to the particlelike solutions of a class of nonlinear model scalar-spinor theories. Dynamically stable, metastable, periodic, and unstable particlelike solutions all follow for certain ranges of the physical parameters from the classical theory, but only corresponding metastable and unstable associated quantum stationary states are obtained for the nonlinear model theories considered here. The functional dependence of the decay constant for the metastable quantum stationary states is such that very long-lived, practically stable, states are admissible. Group-theoretic techniques for the systematic derivation of rigorous particlelike solutions are also described and illustrated with examples.
Selection Rules and the Decomposition of the Kronecker Square of Irreducible Representations
Authors: Smith P. R. , Wybourne B. G.
Published: January 2005

The irreducible representations occurring in the decomposition of the Kronecker squares of irreducible representations of finite and continuous groups are shown to be readily separable into symmetric and antisymmetric parts using Littlewood's method of plethysm. Particular applications of the rotation and symplectic groups, together with selection rules for isoscalar factors, are given.
Discrete Series for the Universal Covering Group of the 3 + 2 de Sitter Group
Authors: Evans N. T.
Published: January 2005

A classification is given of the irreducible unitary representations of the universal covering group of the 3 + 2 de Sitter group which contract to the usual physical representations of the Poincaré group. These representations include the discrete series for the 3 + 2 de Sitter group. The classification problem is reduced from one for the group to the corresponding one for the Lie algebra. The method used by Thomas for the representations of the 4 + 1 de Sitter group is then followed, except that a representation is reduced out with respect to the irreducible unitary representations of a noncompact 2 + 2 subalgebra. It is conjectured that each representation of this subalgebra occurs at most once. The action on the representation spaces of a basis for the Lie algebra is given. The contractions of the representations to those of the Poincaré, oscillator and the Galilei groups are briefly considered.
Twistor Algebra
Authors: Penrose R.
Published: January 2005

A new type of algebra for Minkowski space-time is described, in terms of which it is possible to express any conformally covariant or Poincaré covariant operation. The elements of the algebra (twistors) are combined according to tensor-type rules, but they differ from tensors or spinors in that they describe locational properties in addition to directional ones. The representation of a null line by a pair of two-component spinors, one of which defines the direction of the line and the other, its moment about the origin, gives the simplest type of twistor, with four complex components. The rules for generating other types of twistor are then determined by the geometry. One-index twistors define a four-dimensional, four-valued (``spinor'') representation of the (restricted) conformal group. For the Poincaré group a skew-symmetric metric twistor is introduced. Twistor space defines a complex projective three-space C, which gives an alternative picture equivalent to the Minkowski space-time M (which must be completed by a null cone at infinity). Points in C represent null lines or ``complexified'' null lines in M; lines in C represent real or complex points in M (so M, when complexified, is the Klein representation of C. Conformal transformations of M, including space and time reversals (and complex conjugation) are discussed in detail in twistor terms. A theorem of Kerr is described which shows that the complex analytic surfaces in C define the shear-free null congruences in the real space M. Twistors are used to derive new theorems about the real geometry of M. The general twistor description of physical fields is left to a later paper.
Local Weights Which Determine Area, and the Ising Model
Authors: Baxter Glen
Published: January 2005

A group G of local weights is constructed which assigns to closed paths in the square lattice the enclosed area and the number of turns of the tangent vector (mod 2) to the path. Special cases of this group have been used previously in explicit evaluations of the partition function for the Ising model in 2-dimensions. Properties of G are examined to cast light on the combinatorial approach to the Ising problem developed by Kac and Ward, Feynman, and Sherman. It is shown that their method breaks down in the general case.
Variational Theorem for Reduced Density Matrices
Authors: Kiang Hsiao S.
Published: January 2005

This paper discusses a theorem concerning the variational description of the eigenfunctions and eigenvalues of the two complementary reduced density matrices for a many-particle system in a bound state.
Calculation of Intersect Distribution Functions in Small Angle X-Ray Scattering
Authors: Schmidt Paul W.
Published: January 2005

As has been recently suggested, many subjects in small angle x-ray scattering theory can be discussed by using a function called the intersect distribution function G(M), which gives an average value of the distribution of lines with length M which pass through a point in a particle and which also have both ends lying on the boundary of the particle. Some properties of the intersect distribution function for a plane lamina with a convex boundary are investigated. The calculation is found to require the use of a weighting factor which is expressible in terms of the function generating the boundary of the lamina. The relation between G(M) and the two-dimensional characteristic function is given. The exact intersect distribution function is found for a circle, and an approximate calculation of G(M) is carried out for small M for an arbitrary plane lamina with a convex boundary.
Padé Approximant and Partial-Wave Integral Equation
Authors: Masson D.
Published: January 2005

The Padé approximant is applied to the partial-wave integral equation and is shown to yield approximate solutions which satisfy exact two-body unitarity and which converge to the N/D solution.
Eigenvalue Problem for Lagrangian Systems
Authors: Barston E. M.
Published: January 2005

The problem of small oscillations about a state of steady motion of a Lagrangian system is considered. Upper and lower bounds for the growth rates of unstable systems are obtained; sufficient conditions for instability are given for finite dimensional systems; an existence theorem for stable modes for systems with an infinite number of degrees of freedom is presented (valid when the operators are completely continuous in Hilbert space); and finally the orthogonality and completeness properties of the modes of stable finite dimensional systems are discussed.
Chapman-Enskog-Hilbert Expansion for a Class of Solutions of the Telegraph Equation
Authors: McKean H. P.
Published: January 2005

The Chapman-Enskog-Hilbert expansion is a method for describing a gas in the ``hydrodynamical stage,'' beginning from the Boltzmann equation. The present paper is devoted to the analog of this expansion for the problem ∂p/∂t + e∂p/∂x = p(−e) − p(+e), where e = ±1 and x ∈ R1. Though the situation is vastly simpler than in the Boltzmann case, new and amusing mathematical phenomena are encountered. One studies solutions of ∂p/∂t + e∂p/∂x = ε−1[p(−e) − p(+e)] which are (formal) power series in ε (Hilbert solutions): such a solution solves ∂p/∂t = ε−1[(1 + ε2∂2/ ∂x2)½ − 1]p (hydrodynamical equation) and is completely determined by the initial value of p(−e) + p(+e) (Hilbert paradox). Also, every solution of the original problem comes very rapidly close to a Hilbert solution which is actually convergent (hydrodynamical stage).
Short Simple Evaluation of Expressions of the Debye-Waller Form Valid for Anharmonic Modes not in Thermal Equilibrium
Authors: Schultz T. D.
Published: January 2005

A one-sentence nonalgebraic evaluation of averages, such as occur in the theory of the Debye-Waller factor, is given, which is valid for certain types of anharmonic modes and also for ensembles more general than the canonical ensemble.
Quaternionic Representations of Compact Metric Groups
Authors: Natarajan S. , Viswanath K.
Published: January 2005

Representations of compact metric groups in Hilbert spaces over the quaternions are studied. A generalization of the Peter-Weyl theorem is formulated and proved. The problem of finding all the irreducible quaternionic representations of an arbitrary compact metric group is solved, and a rule is given for computing the ``Q-characters'' of all the irreducible quaternionic representations once the characters of all the irreducible complex representations are known. For the Abelian case, it is shown that every irreducible quaternionic representation is equivalent to a complex representation and hence one dimensional. An example is given of a non-Abelian group whose irreducible quaternionic representations are all one dimensional.
Representations of the Lie Algebra of the Homogeneous Galilei Group and Their Relation to the Representations of the Lorentz Algebra
Authors: Voisin Jacques
Published: January 2005

The irreducible Hermitian representations of the Lie algebra of the homogeneous Galilei group were first constructed by using the method of the maximal compact subgroup. The same representations were then obtained by contracting the irreducible representations of the homogeneous Lorentz group. In the Appendix, the faithful representations are given with the diagonalized generators of the velocity transformations.
Relation between the Relativistic and Nonrelativistic S-Matrices
Authors: Halpern Francis R.
Published: January 2005

It is shown that, given any Lorentz invariant S-matrix, it is possible to calculate from it, by purely algebraic operations, a Galilean invariant S-matrix, and vice versa.
Tensor Operators and Mass Formula in the Minimal Extension of U3 by Charge Conjugation
Authors: Barut A. O. , Kleinert H.
Published: January 2005

The transformation property of the mass (or mass square) M is specified in the extended group. The most general mass formula, M=M0+a(α)[I(I+1)−14Y2−K(α)]+b1(α)NY+b2(α)Q2Y, is derived where a(α), bi(α) are constants depending on a charge-even set α, specified in the text, and Q2 is the second invariant operator of SU3.
Canonical Definition of Wigner Coefficients in Un
Authors: Biedenharn L. C. , Giovannini A. , Louck J. D.
Published: January 2005

Two general results applicable to the problem of a canonical definition of the Wigner coefficient in Un are demonstrated: (1) the existence of a canonical imbedding of Un × Un into Un2 and (2) a general factorization lemma for operators defined in the boson calculus. Using these results, a resolution of the multiplicity problem for U3 is demonstrated, in which all degenerate operators are shown to split completely upon projection into U2.
Double Representations of Space Groups
Authors: Glück M. , Gur Y. , Zak J.
Published: January 2005

It is shown that, for calculating characters of double representations of space groups, the same methods can be used as for single representations. Two methods are reviewed: the ray representation method and the induction method. Examples are presented for both methods.
Application of Houston's Method to the Sum of Plane Waves over the Brillouin Zone. II. Body-Centered Cubic Lattice
Authors: Miąsek Maria , Dodziuk Helena
Published: January 2005

In a previous paper we considered the function χ(r) ≡ (1/N) Σk eik·r, where the sum runs over the first Brillouin zone of a crystal, and its expansion into series of Cubic Harmonics Kj:χ(r)=∑j=0∞gj(r)×Ki(θ,φ). Houston's method was used in order to find the radial functions gj(r) for several values of j, for χ(r) given for the simple cubic and the face-centered cubic lattices. In this paper, the same considerations are applied to χ(r) given for the body-centered lattice. gj(r), with j = 0, 2, 3, are calculated in the region of small r which is assumed as 0 ≤ r ≤ 2a, where a is the lattice constant. In most of the problems of solid-state physics, where the function χ(r) occurs, it is satisfactory to know its values only for small r, usually not larger than 2a. The function g0(r) is calculated using 3-, 6-, and 9-term expansion formulas, g2(r) and g3(r) using only 3- and 6-term formulas. Comparing gj(r) obtained from the formulas with different number of terms it is established that, for r in the region 〈0, 2a〉, the 6-term approximation is very good.
Reduction of Reducible Representations of the Infinitesimal Generators of the Proper, Orthochronous, Inhomogeneous Lorentz Group
Authors: Lomont J. S. , Moses H. E.
Published: January 2005

In the present paper we show explicitly how to reduce reducible representations of the infinitesimal generators of the proper, orthochronous, inhomogeneous Lorentz group. We first construct a basis in which all the reducible representations are expressed as integrals of representations over masses and in which the infinitesimal generators act as in the Foldy-Shirokov realization for nonzero masses and in the Lomont-Moses realization for zero masses. However, the spin operators which appear in the Foldy-Shirokov realizations and the generators of the Euclidean group which appear in the Lomont-Moses realizations are reducible in general. Thus reducible representations are only partially reduced in this basis. On carrying out the reduction of the spin operators and the generators of the Euclidean group, we introduce a second basis such that the reducible representations are completely reduced. By changing the emphasis slightly, the methods of the present paper can be used to obtain the irreducible representations of the generators. One of us (H. E. M.) has already used the methods of the present paper to reduce the electromagnetic vector potential and, in papers which follow the present one, will show how to reduce wavefunctions in general and will also derive the Clebsch-Gordan expansion for the direct product of two massless representations of finite spin and the same sign of energy. From the work of Mautner and Mackey it is known that every reducible unitary ray representation of the proper, orthochronous, inhomogeneous Lorentz group can be reduced to a direct integral of the irreducible representations. The techniques of the present paper thus enable us to carry out the reduction explicitly.
Necessary Conditions on Radial Distribution Functions
Authors: Wu F. Y. , Tan H. T. , Feenberg E.
Published: January 2005

Certain necessary conditions must be met by any function introduced to serve as a radial distribution function of a uniform N-particle system. One large class of necessary conditions is based on the statement that the expectation value of a potential energy (for an arbitrary potential function between pairs of particles) cannot fall below the classical minimum potential energy of the system. To convert this statement into a family of useful inequalities, we have evaluated the classical potential energies of close-packed crystals for a linear combination of Yukawa and Coulomb two-particle interactions. The numerical evaluation of lattice sums is performed by two procedures: (1) direct summation over the lattice (suitable for short-range potentials), and (2) an adaptation of the Ewald summation procedure suitable for long-range potentials. Results are given for the Coulomb and Yukawa potentials and also for the potentials 1/r(r + a) and 1/r(r + a)2. Two simple approximate forms are developed both giving close lower bounds on the classical potential energy of interacting particles forming a regular crystalline lattice.
Analytic Continuation of the Off-Energy-Shell Scattering Amplitude
Authors: Nuttall J.
Published: January 2005

It is shown that the two-body scattering amplitude 〈k| T(E) |k′〉 may be analytically continued in E through the physical cut into a region of meromorphy, provided that the potential satisfies certain requirements. The residue at a pole is a separable operator whose form agrees with previous work. The proof also demonstrates the existence of a region of meromorphy of 〈k| T(E) |k′〉 as a function of seven complex variables.
Exponential Operators and Parameter Differentiation in Quantum Physics
Authors: Wilcox R. M.
Published: January 2005

Elementary parameter-differentiation techniques are developed to systematically derive a wide variety of operator identities, expansions, and solutions to differential equations of interest to quantum physics. The treatment is largely centered around a general closed formula for the derivative of an exponential operator with respect to a parameter. Derivations are given of the Baker-Campbell-Hausdorff formula and its dual, the Zassenhaus formula. The continuous analogs of these formulas which solve the differential equation dY(t)/dt = A(t) Y(t), the solutions of Magnus and Fer, respectively, are similarly derived in a recursive manner which manifestly displays the general repeated-commutator nature of these expansions and which is quite suitable for computer programming. An expansion recently obtained by Kumar and another new expansion are shown to be derivable from the Fer and Magnus solutions, respectively, in the same way. Useful similarity transformations involving linear combinations of elements of a Lie algebra are obtained. Some cases where the product eAeB can be written as a closed-form single exponential are considered which generalize results of Sack and of Weiss and Maradudin. Closed-form single-exponential solutions to the differential equation dY(t)/dt = A(t) Y(t) are obtained for two cases and compared with the corresponding multiple-exponential solutions of Wei and Norman. Normal ordering of operators is also treated and derivations, corollaries, or generalization of a number of known results are efficiently obtained. Higher derivatives of exponential and general operators are discussed by means of a formula due to Poincaré which is the operator analog of the Cauchy integral formula of complex variable theory. It is shown how results obtained by Aizu for matrix elements and traces of derivatives may be readily derived from the Poincaré formula. Some applications of the results of this paper to quantum statistics and to the Weyl prescription for converting a classical function to a quantum operator are given. A corollary to a theorem of Bloch is obtained which permits one to obtain harmonic-oscillator canonical-ensemble averages of general operators defined by the Weyl prescription. Solutions of the density-matrix equation are also discussed. It is shown that an initially canonical ensemble behaves as though its temperature remains constant with a ``canonical distribution'' determined by a certain fictitious Hamiltonian.