bluuebunny/crossref_metadata_2025 huggingface dataset:
Wigner Coefficients for the R4 Group and Some Applications
Authors: Biedenharn L. C.
Published: January 2005

The local isomorphism of the R4 group to the group R3×R3 is utilized to obtain R4 Wigner coefficients for those representations in which the subgroup R3 is diagonal. The R4 Wigner coefficients so defined are then used to obtain recursion relations and differential equations for the representation coefficients, when the group is parametrized appropriately. The R4 spherical harmonics, and their properties, are explicitly obtained as specializations of the general formulas. Physical application to the problem of geometrizing the Coulomb field is briefly discussed.
On the Interpretation of the Einstein-Schrödinger Unified Field Theory
Authors: Sciama D. W.
Published: January 2005

Vierbeins are used to analyze the Einstein-Schrödinger unified field theory, in both its real nonsymmetric and complex Hermitian versions. It appears that the skew-symmetric part of the real affine connection is related to spin rather than to electromagnetism. In its complex form the theory does appear to describe the electromagnetic field, since the vierbein analysis shows that gauge transformations arise in a natural way (the spin is then related to the skew-Hermitian part of the affine connection). The resulting identification of the electromagnetic field tensor implies that Einstein's choice of Lagrangian is physically unsatisfactory and an alternative Lagrangian is proposed. Although the electromagnetic field is thus geometrized, neither Lagrangian represents a unification of electromagnetism and gravitation.
Application of a Singular Wave-Function Operator in Scattering Theory
Authors: Mapleton Robert A.
Published: January 2005

A function which is defined in terms of the two-body singular wave-function matrix of Mo/ller acting on a plane-wave state is calculated. In the paper by T. Pradhan, the function in question was obtained as the solution of a differential equation. Here, it is found that the function has a normalization which is different from the one assumed by Pradhan, and this behavior is attributed to the long-range character of the Coulomb potential that is in the Mo/ller operator. An intermediate result of this paper agrees with the momentum-space solution of the nonrelativistic Coulomb two-body scattering problem which recently appeared in a paper published by S. Okubo and D. Feldman. As they show, the wave function must be renormalized to obtain the correct cross section, and this view is adopted in this paper. Consequently, the normalization assumed by Pradhan is reinstated. The Legendre expansion coefficients are calculated for the function,(a−b cosθ)−c.
Multiple Scattering of Waves
Authors: Waterman P. C. , Truell Rohn
Published: January 2005

Multiple scattering effects due to a random array of obstacles are considered. Employing a ``configurational averaging'' procedure, a criterion is obtained for the validity of approximate integral equations describing the various field quantities of interest. The extinction theorem is obtained and shown to give rise to the forward-amplitude theorem of multiple scattering. In the limit of vanishing correlations in position, the complex propagation constant κ of the scattering medium is obtained. Under appropriate restrictions, the expression for κ is shown to include both the square-root law of isotropic scatterers and the additive rule for cross sections valid for sufficiently low densities of anisotropic obstacles. Some specific examples from acoustics and electromagnetic theory then indicate that at least in the simplest cases the results remain valid for physically allowable densities of obstacles.
Tschebyscheff Polynomial Approximation Method of the Neutron-Transport Equation
Authors: Yabushita Shin
Published: January 2005

The Tschebyscheff polynomial approximation method of the neutron-transport equation is developed. The relations between the relaxation constants in this approximation and the positive roots of TN+1(μ)=0 are derived. Using these relations, the necessary condition for reactor criticality is discussed. Application to Milne's problem leads to an explicit expression for extrapolated end point which is formally the same in the spherical-harmonics method. Numerical comparison of this method with the spherical-harmonics method verifies Conkie's conclusion [W. R. Conkie, Nuclear Sci. and Eng. 6, 260 (1959)] that, for weak absorbers, the spherical-harmonics method gives the values for extrapolated end point which is closer to the exact value than the TN method does while, for strong absorbers, the TN method gives closer value than the PL method does.
Critical Probabilities for Cluster Size and Percolation Problems
Authors: Fisher Michael E.
Published: January 2005

When particles occupy the sites or bonds of a lattice at random with probability p, there is a critical probability pc above which an infinite connected cluster of particles forms. Rigorous bounds and inequalities are obtained for pc on a variety of lattices and compared briefly with previous numerical estimates. In particular, by extending Harris' work, it is proved that pc(s,L2)⩾12 for the site problem on a plane lattice L2 (without crossing bonds), while for the bond problem pc(b,L2)+pc(b,L2D)⩾1 where L2D is the dual lattice to L2. Simple arguments demonstrate that the bond problem is a special case of the site problem and that the critical probabilities for the bond problem on the plane square and triangular lattice cannot exceed those for the corresponding site problems.
Properties of 2-Semi-Isomorphic Graphs and Their Applications: Active Network Analysis
Authors: Frisch Ivan T. , Kim Wan H.
Published: January 2005

Two-semi-isomorphic graphs are defined by extending Whitney's concept of 2-isomorphic graphs. The properties of these newly defined graphs are then investigated. The discussion leads to the derivation of a simple formula for the determination of the sign of a tree product which is common to two 2-semi-isomorphic graphs. The formula is then applied to the topological analysis of mutually coupled active networks.
Electrical Conduction in a Stretched and Twisted Tube
Authors: Pipkin A. C. , Rivlin R. S.
Published: January 2005

The constitutive equation derived in a previous paper relating electric current and field in an isotropic material which is subjected to a time-independent deformation is applied to the solution of the problem of electrical conduction in a twisted tube of circular cross section to which a longitudinal electric field is applied. It is shown that the current follows a helical path and that an axial magnetic field is produced.
A Problem of Analytic Completion Related to the Jost-Lehmann-Dyson Formula
Authors: Bros J. , Messiah A. , Stora R.
Published: January 2005

One considers functions of a complex four-vector Q which are analytic in the domain formed by: (a) The future tube: ImQ in the future light-cone. (b) The past tube: ImQ in the past light-cone. (c) A complex neighborhood of a domain R of real Q space limited by two space-like surfaces.

It is shown, by using techniques pertaining to the theory of analytic functions of several complex variables, that all such functions can be analytically continued in a larger domain which coincides with the one predicted by the Jost-Lehmann-Dyson formula.
Role of the Asymptotic Condition in a Lagrangian Field Theory
Authors: Wellner M. , Curtis R. B.
Published: January 2005

The scattering operator of Feynman and Dyson for a self-interacting neutral scalar field is derived from a Lagrangian without the use of a canonical transformation between the Heisenberg and interaction pictures.
Coordinate Conditions
Authors: Plebanski J. , Ryten J.
Published: January 2005

Some extremal properties of coordinates imposed by coordinate conditions are studied. In particular, variational principles leading to the de Donder condition, the generalized de Donder condition, [(−g)wgαβ]|β=0; Einstein's old condition (−g)½=1; and the Einstein-Infeld conditions [(−g)12g0ρ]|ρ=0, [(−g)12gab]|b=0 are given. The class of generalized de Donder conditions is examined in the case of Schwarzschild's solution. The values of ω=0, ½, ⅔, ¾, 1 are here distinguished. For an arbitrary ω the mathematical problem is reduced to the study of certain solutions of the hypergeometric equation.
General Spherical Harmonics Formulation of Plasma Boltzmann Equation
Authors: Carpenter Chester P. , Mezger Fritz W.
Published: January 2005

The Boltzmann equation for the phase space distribution of electrons in the presence of ions is reduced to an infinite set of differential equations which do not involve angle variables. The usual method of expanding the electron phase space distribution function in terms of spherical harmonics is employed and it is assumed, in analyzing the scattering process, that the ion velocities can be neglected in comparison with the electron velocities. The expansion includes both polar and azimuthal angles obviating the assumption of symmetry about a polar axis made in previous work. The differential equation for the general component of the spherical harmonics expansion is derived and explicit equations for the first few components are presented. The component equations are seen to be considerably more tractable for cases which involve electric and/or magnetic fields along a single axis.
Comparison of Atom and Bond Percolation Processes
Authors: Hammersley J. M.
Published: January 2005

Various inequalities, some of them strict, are proved concerning probabilities associated with percolation processes. In particular, it is shown that the critical probability of an atom percolation process is not less than the critical probability of the corresponding bond percolation process.
Eigenvalue Problems in Matrix Mechanics
Authors: Chasman Richard R.
Published: January 2005

The techniques of the Heisenberg matrix mechanics are extended to treat all potentials of the form Q2n, for n a positive integer. The square well (Q∞) and the potential Q4 are treated in detail.
Remarks on the Continued Fraction Calculation of Eigenvalues and Eigenvectors
Authors: Swalen Jerome D. , Pierce Louis
Published: January 2005

For eigenvalue problems in which the secular determinant has tridiagonal form, e.g., the rigid asymmetric rotor; the secular equation may be written in the form f(λ′)=0, where f(λ′) is a continued fraction and λ′ an eigenvalue. Furthermore, if the secular problem is of nth order, then the continued fraction f(λ′) may be developed in n different ways. Since the eigenvalues are roots of a function f(λ), it is convenient to find the eigenvalues by means of the Newton-Raphson iterative procedure. This requires that the derivative of f(λ) with respect to λ(f′(λ)) be determined. An exact expression for f′(λ) is derived and it is shown that f′(λ′) is in fact the norm of the eigenvector belonging to the eigenvalue λ′. A simple recursion formula, in continued fraction form, for the eigenvector elements is also derived. The Newton-Raphson procedure is further shown to be equivalent to the variational method for iterative calculation of eigenvalues. The former procedure has, however, the advantage of bypassing the necessity of solving a set of simultaneous equations. Advantage is taken of the relation between f′(λ′) and the eigenvector of λ′ to formulate a reasonable criterion for choosing the best possible development of f(λ) is order to avoid convergence to an undesired root of f(λ).
Statistical Theory of the Energy Levels of Complex Systems. I
Authors: Dyson Freeman J.
Published: January 2005

New kinds of statistical ensemble are defined, representing a mathematical idealization of the notion of ``all physical systems with equal probability.'' Three such ensembles are studied in detail, based mathematically upon the orthogonal, unitary, and symplectic groups. The orthogonal ensemble is relevant in most practical circumstances, the unitary ensemble applies only when time-reversal invariance is violated, and the symplectic ensemble applies only to odd-spin systems without rotational symmetry. The probability-distributions for the energy levels are calculated in the three cases. Repulsion between neighboring levels is strongest in the symplectic ensemble and weakest in the orthogonal ensemble. An exact mathematical correspondence is found between these eigenvalue distributions and the statistical mechanics of a one-dimensional classical Coulomb gas at three different temperatures. An unproved conjecture is put forward, expressing the thermodynamic variables of the Coulomb gas in closed analytic form as functions of temperature. By means of general group-theoretical arguments, the conjecture is proved for the three temperatures which are directly relevant to the eigenvalue distribution problem. The electrostatic analog is exploited in order to deduce precise statements concerning the entropy, or degree of irregularity, of the eigenvalue distributions. Comparison of the theory with experimental data will be made in a subsequent paper.
Cluster Sums for the Ising Model
Authors: Rushbrooke G. S. , Scoins H. I.
Published: January 2005

Expressions are derived for the first seven irreducible cluster sums, β1, ⋯, β7, for plane-square, simple-cubic, and body-centered cubic Ising models. The connection between these expressions and high-temperature expansions of the zero-field partition function and zero-field inverse susceptibility is indicated. Comparison is made with the work of other authors.
High-Temperature Ising Partition Function and Related Noncrossing Polygons for the Simple Cubic Lattice
Authors: Rushbrooke G. S. , Eve J.
Published: January 2005

We have found the high-temperature expansion of the partition function for the simple-cubic lattice Ising problem up to the term involving u14, where u is the high-temperature variable tanhw/2kT. Comment is passed on an odd feature of the coefficients in this expansion and the corresponding expressions for the specific heat in terms of both u and w/2kT are presented. The calculations involve machine counts of the numbers pn of non-self-crossing lattice polygons; the method of obtaining these is described, and the value of p16 reported. The paper ends with a brief discussion of the trend of the numbers pn and of the closely related noncrossing chain numbers cn (using the data of Sykes).
Some Properties of the Five-Point Function in Perturbation Theory
Authors: Cook L. F. , Tarski Jan
Published: January 2005

A preliminary investigation of the five-point function in its dependence on two complex variables is presented. Only single-loop diagrams are examined. The approach involves a determination of the singularity curves and of their regular and singular parts. The geometrical properties of singularity curves are described in detail; in particular, a method for determining the tangency of two curves is given. The following general conclusions are drawn: First, real and complex vertex singularities are near the physical regions and, therefore, can produce significant experimental effects. On the other hand, scattering singularities and five-point poles seem to be further removed from the physical regions. Second, it is not likely that a simple scheme can be found for the description of the analytic properties of the five-point function. A short discussion of scattering singularities involving unstable masses is also given.
Hamiltonian Formulation of Action-at-a-Distance in Electrodynamics
Authors: Kerner Edward H.
Published: January 2005

The Wheeler-Feynman scheme of classical electrodynamics, using half-retarded+half-advanced fields between pairs of interacting charges, is written in a differential form which expresses the motion of the charges at any inertial observer's single present time through infinite-order differential equations of motion. The conversion of field variables to mechanical ones in this way does not imply an infinite number of degrees of freedom; rather it is shown how to construct a Hamiltonian, depending solely on ordinary particle-position coordinates and certain canonically conjugate momenta, which is both the energy and the generator of the ``correct'' motions. The latter are taken to be those continuously developable from free-particle motions as the strength of interaction increases from the value zero. It is just this criterion that establishes particle-position+conjugate momentum to be sufficient for the description of motion. The highness of the order of the equations of motion apparently gets to be expressed through high powers of momenta in the Hamiltonian. An example is sketched, illustrating a general algorithm, in which the particles are treated nonrelativistically and their interactions are treated to order e4 and c−4, corresponding to fourth-order equations of motion. The general form of the Hamiltonian is a double power series expansion in e2 and c−1; when the c−1 expansion is terminated covariance is sacrificed, but the possibility of collecting all powers of e2 is gained.
Field-Theory Analogs of the Lagrange and Poincaré Invariants
Authors: Sturrock P. A.
Published: January 2005

The Lagrange differential invariant and the Poincaré integral invariant of classical dynamics have as their analogs in Lagrangian field theory a ``differential divergence-free vector'' and an ``integral divergence-free vector.''

The former, which is expressible as a divergence-free vector-bracket expression, may be used to derive conservation relations associated with the transformation properties of a given system. It is not necessary that these transformations should be infinitesimal; by way of example, conservation theorems are established for systems which are periodic and for systems which are invariant under spatial inversion. The differential divergence-free vector may also be used to establish reciprocity and orthogonality relations: simple examples which are here discussed are Betti's reciprocal theorem of elasticity and Lorentz's reciprocal relation of electromagnetic theory. An extended form of the differential divergence-free vector allows for variation not only of the dependent variables but also of the independent variables.

The integral divergence-free vector associates a conserved quantity with any closed one-parameter family of solutions of the field equations. As examples, we derive the ``equation of conservation of probability'' of quantum mechanics, and a classical form of the relation between the momentum and wave vectors for a plane wave in a propagating medium.

The theorem of classical dynamics relating a complete set of Poisson brackets to a complete set of Lagrange brackets cannot be extended to the present formalism. The formula which represents the obvious extension of the classical formula for the Poisson bracket is of no interest since it can be shown not to be canonically invariant.
Absorption of Light by Light
Authors: Reiss Howard R.
Published: January 2005

The mutual absorption of two nonparallel classical photon fields through the production of electron pairs is calculated. One of the photon fields is treated accurately in the sense that wave functions are employed which are solutions of the Dirac equation containing the potential of this photon field in the Hamiltonian. The second photon field is introduced as a first-order perturbation to the system containing the first field. An interesting feature is the appearance of an index related to the number of photons in the first field, though this field is not second-quantized. It is shown that the calculation of the absorption process reduces to known results in the limit of small field strength and in the limit of small energy of the photons in the first field. The general results of the theory can be applied to examine the validity of a perturbation expansion.
Some Physical Solutions of Dirac-Type Equations
Authors: Hale Jack K. , Stokes Arnold P.
Published: January 2005

The Dirac equation of classical electrodynamics is a third-order differential equation. The purpose of the present paper is to give some sufficient conditions on the force field which will insure that there are solutions of Dirac's equation which approach a motion with constant velocity as time increases.
Multiple Scattering by Arbitrary Configurations in Three Dimensions
Authors: Twersky Victor
Published: January 2005

This paper provides the extension to three dimensions of our previous treatment of two-dimensional problems. Thus we derive a system of integral equations which specifies the multiple-scattering amplitudes for many objects in terms of corresponding functions for the isolated objects. For arbitrary configurations and large spacings, the amplitudes are expanded as series of single scattered functions and their derivatives; ``closed forms'' involving differential operators are derived for two objects. For arbitrary spacings, the amplitudes are expanded as series of spherical harmonics to obtain algebraic sets of equations relating the multiple and single scattering coefficients. Series expansions are available for arbitrary configurations, and closed forms are given for two small objects.
Analyticity Properties of the Momentum-Space Vertex Function
Authors: Brown W. S.
Published: January 2005

The problem of finding the maximal domain of analyticity of the momentum-space vertex function implied solely by Lorentz covariance, local commutativity, and mass spectrum with thresholds not all zero, is restated as a holomorphy envelope problem. If only one of the threshold masses is nonzero, the problem can be divided into two simpler problems. The triangle diagram suggests the holomorphy envelope of one, but only gives an upper bound on the holomorphy envelope of the other. It is shown that the boundary of the latter most probably consists of the three cuts and a single quasi-analytic hypersurface with certain specified properties. If two of the threshold masses are equal and the third vanishes, an appealing conjecture suggested by the triangle diagram is shown to contradict a generalization of Jost's example. In the general case some upper and lower bounds are obtained. The Källén-Toll representation and conjecture are briefly discussed. The relation of the first three terms in the representation of the position-space vertex function to the Mercedes diagram in perturbation theory is displayed, and it is shown that there is no analogous relation for the fourth term. In the single threshold case this fourth term must account for the singularities of the momentum-space vertex function on the quasi-analytic hypersurface which bounds the holomorphy envelope. The motivation for studying the analyticity properties of vertex diagrams is discussed, and the simplest totally symmetric ones are investigated.
Analytic Regularization in Relativistic Quantum Field Theory
Authors: McKenna James
Published: January 2005

The terms of a perturbation expansion of the vacuum expectation value of the time ordered product of three scalar fields are examined in x space. It is shown that these terms can be considered as the boundary values of certain analytic functions, where the variables of these functions include complex convergence parameters. Several different integral representations for these analytic functions are obtained which exhibit part of the domain of analyticity. The singular behavior of these functions as the complex convergence parameters tend to zero is discussed, and it is conjectured that the removal of the singular part of these functions is equivalent to renormalization.
Postulates of Quantum Field Theory
Authors: Haag R. , Schroer B.
Published: January 2005

We give a survey of the postulates which provide the framework of quantum field theory. Particular attention is given to the lesser known ones, namely, completeness requirements and ``primitive causality.'' Three field theoretical models are analyzed. These examples show that ``primitive causality'' is not a consequence of the commonly used postulates (which include the ``causal commutation relations''). ``Asymptotic completeness'' is not assured by any of the other postulates listed. In the formulation and analysis the concept of operator rings associated with space-time regions is used extensively.
Convergence of Perturbation Expansions in Cutoff Meson Theories
Authors: Frank William M.
Published: January 2005

It is proved that the perturbation expansion of matrix elements for a class of meson theories has a nonzero radius of convergence. Included are theories which describe bosons of nonvanishing mass, coupled linearly or bilinearly to fermions (but not antifermions) via an interaction which cuts off or attenuates sufficiently rapidly for high boson momenta. By bounding the propagators by their values for zero boson momentum one obtains a majorizing series to the perturbation expansion which converges geometrically.
Some Examples of Canonical Transformations in Quantum Mechanics
Authors: Halpern Francis R.
Published: January 2005

The set of all pairs of polynomials X(x,p) and P(x,p) in the position x and momentum p are found such that X and P have the same commutator as x and p. Several other types of representations of the commutation relations are studied. The reducibility, time, and space inversion properties of these are discussed. The unitary transformations connecting different representations are exhibited. The use of different representations to solve scattering and bound state problems is indicated. The application of one of the transformations to a partial diagonalization of a perturbed harmonic oscillator Hamiltonian is made.
Bounds on Elements of the S Matrix for Elastic Scattering: One-Dimensional Scattering
Authors: Bartram Ralph , Spruch Larry
Published: January 2005

For the quantum-mechanical scattering of a particle by a static potential, a method due to Kato determines upper and lower bounds on cotη, where η is the phase shift for a given angular momentum. The method is readily generalizable to the case of elastic multi-channel scattering if one can a priori diagonalize the scattering matrix S to determine the standing-wave eigenmodes (by symmetry considerations, for example). We here consider the case when this can not be done. Both bounds are obtained on the independent elements Bij of B, where B=i(l+S)(l−S)−1, which then determine bounds on the elements of S, and on the eigenphase shifts and mixing parameters. One-dimensional scattering by V(x)≠V(−x) provides a concrete two-channel example. The numerical results obtained are very encouraging.

The Kato method is useful only if one can obtain solutions of related scattering problems. The most significant feature of the present approach is that the related scattering problems can be single-channel scattering problems.
Symmetries of the Einstein-Maxwell Equations
Authors: Rosen Gerald
Published: January 2005

Space-time symmetries admissible according to the Einstein-Maxwell equations are analyzed from the standpoint of the groups of motions in Rainich geometry. Necessary and sufficient conditions for a motion are expressed in terms of the Ricci vierbein of principal directions. Normal Rainich geometries, for which the Ricci congruences are orthogonal to four sets of hypersurfaces, are studied in some detail. First integrals to the Rainich conditions are presented for the latter class of geometries. A new solution to the Einstein-Maxwell equations is derived in the form of a spatially homogeneous Rainich geometry.
Continuum Coulomb Wave Function
Authors: Johnson W. R. , Deck R. T.
Published: January 2005

An alternate form for the continuum-state relativistic Coulomb wave function is developed which expresses the spin dependence in terms of the usual Dirac plane wave spinor. The computational convenience of this form is indicated by a direct derivation of the Sommerfeld-Maue approximation.
Theory of Transport Coefficients. I. General Theory and Electrical Conductivity of Electron-Phonon System
Authors: Fujita S. , Abe R.
Published: January 2005

A general method is presented for the calculation of static transport coefficients based on Kubo's formulas. Techniques of perturbation expansion, diagram representation, and linked-cluster expansion are used. As an example, the electrical conductivity of an electron-phonon system is calculated with the natural introduction of irreversibility, following the ideas of Van Hove, and Prigogine and his collaborators. Under certain conditions the present method is shown to be equivalent to the conventional method by means of the Boltzmann equation. These conditions are examined and the improvement of the approximation is discussed.
Traveling-Wave Technique of Cyclotron Radiation
Authors: Parzen P. , Nomicos G.
Published: January 2005

A traveling-wave technique of computing the radiation from accelerated electrons is described. In contrast to the usual method which expands the field due to the moving electrons in spherical waves, an expansion in traveling waves along a preferred direction is utilized.

This method is especially suitable in those cases where the radiation is confined by cylindrical conducting boundaries or plasmas. The technique is illustrated by considering the problem of the radiation from an electron in circular motion both in free space and within a coaxial conducting cylinder, and from an electron in a variable-pitch helical orbit.
Modified Mandelstam Representation for Heavy Particles
Authors: Islam Jamal N.
Published: January 2005

The fourth-order Feynman amplitude ceases to satisfy the Mandelstam representation when the external masses are sufficiently large. A representation which replaces the Mandelstam one is found in the cases where the four mass invariants are equal in pairs. The physical interpretation is briefly discussed.
Axiomatic Perturbation Theory for Retarded Functions
Authors: Fried H. M.
Published: January 2005

A construction of the retarded n-point functions of perturbation theory is given within the Lehmann, Symanzik, and Zimmermann framework and without the specification of an interaction Lagrangian. An intermediate-state expansion of retarded functionals is employed to define a systematic set of equations representing approximations to the (integral) unitarity conditions; the requirement of symmetry of the n − 1 retarded coordinates of an n-point retarded function enters in an essential way. The class of solutions to these equations contains the renormalized perturbation theory retarded functions corresponding to local renormalizable Lagrangian interactions, as well as more singular functions corresponding to nonrenormalizable interactions; if the latter are excluded all the n-point functions may be successively determined to all orders in the renormalized coupling constants. The construction is explicitly performed for the first radiative corrections to the 2- and 3-point functions of a self-interacting neutral scalar boson field, yielding the finite renormalized results of perturbation theory. Similar but slightly singular results are quoted for the π-π scattering amplitude.
On the Restricted Lorentz Group and Groups Homomorphically Related to It
Authors: Macfarlane A. J.
Published: January 2005

A study is made of the real restricted Lorentz group, L, and of its relationship

(a) to the group, SL(2C), of complex unimodular two-dimensional matrices, and

(b) to the group, O3, of orthogonal transformations in a complex space of three dimensions.

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

The matrix which enters in the charge conjugation transformation of the usual spinors in 4-space is an invariant matrix and is skew symmetric. It is shown that there exists such an invariant matrix C for any number of dimensions (and independent of the number of time like dimensions). Its symmetry properties depend on the dimension number n modulo 8. With the help of the C matrix one can construct, for n = 1, 2, 7, 8 mod 8, an n-dimensional invariant bilinear in the components of a single n-dimensional spinor. Some examples are given for n = 2, 3, 7. A bilinear baryon invariant is formed for a theory with high symmetry. Its existence is closely related to the triality property of 8-space.
Studies in Perturbation Theory. V. Some Aspects on the Exact Self-Consistent Field Theory
Authors: Löwdin Per-Olov
Published: January 2005

The independent-particle model in the theory of many-particle systems is studied by means of the self-consistent-field (SCF) idea. After a review of the characteristic features of the Hartree and Hartree-Fock schemes, the extension of the SCF method developed by Brueckner is further refined by introducing the exact reaction operator containing all correlation effects. This operator is here simply defined by means of the partitioning technique, and, if the SCF potentials are derived from this operator, one obtains a formalism which is completely analogous to the Hartree scheme but which still renders the exact energy and the exact wave function. An elementary derivation of the linked-cluster theorem is given, and finally the inclusion of various symmetry properties is discussed.
On a Functional Averaging Method in the Classical Ensemble Theory
Authors: Galindo A.
Published: January 2005

Albertoni, Bocchieri, and Loinger (ABL) have given a general prescription to calculate the average E(F) of functionals F[ρ] defined on the space L of the initial states (Liouville density functions) ρ(x) of an arbitrary (finite) dynamical system (S, Σ, μ, H). Then they have proved that E(BH) = 0 for any B ∈ Σ, where BH[ρ]≡lim lim T→∞1/T× ∫ 0T| ∫ Bρ(x,t) dμ(x)−μ−1(S)μ(B)|2dt[ABL theorem]. On the other hand, these authors claim that E gives the same weight to each ρ∈L and therefore, they contend that their theorem is sufficient to justify the classical statistical mechanics.

In this note, however, the following statements are proved: (A) If (S, Σ, μ, H) is ergodic with discrete spectrum, then every probability measure ν in L for which BH[ρ] = 0, ν-almost everywhere (for any B ∈ Σ) must be concentrated at e [e(x) ≡ μ−1(S) on S]. (B) Given (S, Σ, μ, Ho) with Ho = 0, every (normally defined) functional average A(F) such that A(BHo) = 0 for any B ∈ Σ must define a probability measure ν in L such that ν({e}) = 1. From (B) it will follow that: (1) The ABL average E(F) does not give the same weight to each ρ∈L; actually, E defines a probability measure in L concentrated at e; consequently, no significant nontrivial conclusion can be naturally drawn from the ABL theorem regarding the foundations of the classical statistical mechanics; (2) it is impossible to restore in general the validity of the ABL theorem by choosing a different normal functional averaging method. To illustrate (2) a new (nonconcentrated) functional average E*(F) is introduced and it is proved that E*(BH) ≠ 0 for every nonweakly mixing system (S, Σ, μ, H) and some B ∈ H.
Evaluation of the Quantum-Mechanical Ring Sum with Boltzmann Statistics
Authors: DeWitt Hugh E.
Published: January 2005

A procedure is given for the evaluation of the quantum-mechanical ring sum at finite temperature. The method is used for the evaluation of the quantum corrections to the classical Debye-Hückel free energy for an electron gas obeying Boltzmann statistics. The ring sum is shown to be of the form (βe2/πλD)P(γ), where γ=ƛ/λD,ƛ=ℏ/(2mkT)1/2, and λD is the Debye screening length. The quantum effects for finite γ are due only to the operation of the uncertainty principle. The function P(γ) decreases monotonically from the classical value π/3, and the form is shown to be P(γ)=(π/3)[1+ ∑ n=2anγ2(n−1)]3/2− ∑ n=2bnγ2n−3. The coefficients an and bn are evaluated exactly for small n and asymptotically for large n. The two series converge for γ2 < γc2 = 2.042 ….

For γ ≫ γc the function P(γ) is also evaluated as an asymptotic expansion in inverse powers of γ1/2. Thus the low-temperature correlation energy of distinguishable electrons is obtained in random phase approximation. At zero temperature, the result is the same as the correlation energy obtained by Foldy for charged bosons. The correlation pressure in this approximation is negative and diverges as ρ1/4 at high density.
Quantum Statistics and the Boltzmann Equation
Authors: Lewis Robert M.
Published: January 2005

The system of hierarchy equations for the reduced density operators of quantum statistical mechanics is replaced by a single functional differential equation for a generating functional. A formal solution of the initial value problem for the latter equation is obtained, leading to series expansions of the reduced density operators. These expansions are used to obtain an improved derivation of the quantum-mechanical Boltzmann equation.
Theory of Transport Coefficients. III. Quantum Statistical Systems
Authors: Fujita S.
Published: January 2005

The electrical conductivity of an electron-phonon system is calculated from Kubo's formula using a perturbation method on the assumptions that the coupling is weak, the system is infinitely large, and that the electrons do not interact between them and obey the Fermi-Dirac statistics and phonons are in thermodynamic equilibrium. The result is identical with that one would obtain from the usually assumed Boltzmann-Bloch equation. The calculation is a logical extension of the previous treatment I of the present series, where the same problem is treated on the assumption that the electrons obey the Boltzmann statistics. The relation between the present calculation and the derivation of master equation is critically discussed. A brief sketch is given for the calculation of the viscosity coefficient of a dilate quantum statistical gas.
Dimers on Rectangular Lattices
Authors: Wu Tai Tsun
Published: January 2005

A simplified calculation of a known result of the dimer problem on a rectangular lattice is presented.
An Orthogonality Property of Hydrogenlike Radial Functions
Authors: Pasternack S. , Sternheimer R. M.
Published: January 2005

An integral involving r−s times the product of two hydrogenlike radial functions of the same n and different l is shown to be zero for a number of values of s.
Green's Distributions Associated with the Operator [□m − (−μ2)m]l
Authors: Bowman John Judson , Harris Joseph David
Published: January 2005

The results of a previous paper on homogeneous Green's functions for a multi-dimensional iterated Klein-Gordon operator (□ + μ2)l are extended to include homogeneous Green's functions associated with the operator [□m (− μ2)m]l in multi-dimensional spaces. The Fourier representation of the Green's functions may be expressed, after some angular integrations, as a one-dimensional infinite integral which does not in general converge. Using the concepts of distribution analysis, it is shown how this improper integral can be evaluated directly to get explicit expressions for the Green's functions. The Green's functions themselves must then be interpreted as distributions in the sense of Schwartz. Several distributions instrumental in this treatment are introduced and their properties studied. Explicit expressions for the singularities of the Green's functions on the light cone are presented. The well-known difference between even- and odd-dimensional spaces is reflected in the nature of these singularities. The singularities appearing for odd-dimensional spaces consist of a finite linear combination of derivatives of the Dirac delta function δ(s2), where s is the space-time distance. The highest derivative appearing is of order ½(n−2m−l) with n giving the number of space dimensions. The singular part for even-dimensional spaces consists of a polynomial in 1/s of degree n − 2ml + 1. No singularities appear when the order of the operator is greater than the number of space dimensions. Finally, a complete set of homogeneous Δ-function solutions is given along with their initial conditions at zero time. All of these functions would be needed in obtaining the general solution to the Cauchy problem for the operator considered.
Input Admittance of Infinitely Long Dipole Antennas Driven from Coaxial Lines
Authors: Wu Tai Tsun
Published: January 2005

For an infinitely long dipole antenna driven from a coaxial line, the reflection coefficient and hence the apparent terminal admittance are determined approximately when the radii of the coaxial line are small compared with the wavelength. This result is useful because the differences of admittances for antennas with identical geometries near the driving point are less sensitive to the driving condition and hence can be found approximately by many existing theories.
The Line Shape in a Harmonic Lattice
Authors: Englman Robert , Levi Peretz
Published: January 2005

A general formalism is developed for the treatment of lattice vibrations. The problem is reduced to the mathematics of noncommuting operators. Hausdorff's equation is then solved for a lattice interacting with linear forces. The cases treated in detail are the line shapes of a radiating atom as affected by the displacement (Franck-Condon effect) and by the nuclear recoil (Mössbauer effect). Only formal results are obtained, which are more difficult to compute than those of the method of normal modes. However, they are more general and do not require the data of vibrational frequencies.
Perpendicular Susceptibility of the Ising Model
Authors: Fisher Michael E.
Published: January 2005

A high-temperature expansion of the partition function for a lattice of N spins with Hamiltonian H=−J ∑ (ij)σizσjz−mHz ∑ iσiz−mHx ∑ iσixis derived and thence an expansion for the zero-field perpendicular susceptibility is found. By perturbation theory, χ⊥(T) is also expanded at low temperatures and seen, in general, to increase with T from the value χ⊥(O) = Nm2/q | J |, where q is the coordination number of the lattice. The perpendicular susceptibility is re-expressed in terms of near neighbor pair and higher-order spin correlation functions in zero field. This yields exact closed formulas for the linear chain, the Bethe pseudolattices, and for the plane square and honeycomb lattices. The behavior of χ⊥(T) in the critical region is investigated for these lattices and for the plane superexchange lattice.
Normalization Condition for the Bethe-Salpeter Wavefunction and a Formal Solution to the Bethe-Salpeter Equation
Authors: Sato Iwao
Published: January 2005

By the use of an inhomogeneous Bethe-Salpeter equation, a normalization condition for the Bethe-Salpeter wavefunction is obtained. This condition requires the normalization integral to be positive. A formal solution is obtained in the ladder approximation, and convergence of the normalization integral is proved by the use of this solution. This solution is also used to prove a dispersion relation for the vertex function of the compound particle and to give an approximate solution The positiveness of the normalization integral is proved in the nonrelativistic limit. The bound state of nucleon and antinucleon is studied in the ladder-chain approximation and it is found that the normalization condition gives a finite wavefunction in spite of divergency of the normalization integral.
A Unified Variational Formulation of Classical and Quantum Dynamics. I
Authors: Corson Edward M.
Published: January 2005

We have critically re-examined the role and status of Lagrangian concepts in both classical and quantal contexts. We find a new, unified variational principle which, subject to the superadded postulates of determinism or indeterminism, respectively, leads uniquely to the classical concept of state and Lagrange equations of motion or to the quantal concept of state and Schrödinger equations of motion.
The Diffraction of Waves By a Penetrable Ribbon
Authors: Yeh C.
Published: January 2005

The exact solution of the diffraction of waves by a dielectric ribbon or by an elliptical dielectric cylinder is obtained. Results are given in terms of Mathieu and modified Mathieu functions. It is found that each expansion coefficient of the scattered or transmitted wave is coupled to all coefficients of the series expansion for the incident wave, except when the elliptical cylinder degenerates to a circular one. Both polarizations of the incident wave are considered: one with the incident electric vector in the axial direction and the other with the incident magnetic vector in the axial direction. It is noted that the technique used in this paper to satisfy the boundary conditions may be applied to similar types of problems; such as the plasma-coated ribbon radiator and the corresponding acoustical problems.
Canonical Form of the Covariant Free-Particle Equations
Authors: Chakrabarti A.
Published: January 2005

The canonical form of the covariant equations for free particles of nonzero rest mass is proposed to be taken as [(p νpν)12−κ] ψ=0, instead of [γ0(p2+κ2)12−p0] ψ=0, as suggested by Foldy. The connection of our representation with the usual forms of the Dirac and the Klein-Gordon (K-G) equations are discussed, each feature being compared with the corresponding one in Foldy's case. The case of the Dirac equation is treated in some detail. A study of the infinitesimal operators of the Poincaré group and the transformation properties of the wavefunction and the polarization operator in our representation lead us to conclude that the choice of operators and the definition of spin states adopted by Iu. M. Shirokov in his study of the Poincaré group corresponds directly to our representation and the canonical form proposed, rather than that proposed by Foldy, as is sometimes supposed. It is also shown that the proposed canonical form corresponds to Wigner's unitary representation of the Poincaré group in terms of the little group of (κ, 0, 0, 0) (for κ > 0). In Appendix A, we give a brief outline of the decomposition of the direct-product representation of the Poincaré group to bring out the special features that arise in our representation. In Appendix B, we compare in detail, for the case of the Dirac equation, our transformation with the well-known Foldy-Wouthuysen transformation. The case of zero rest mass has not been considered. Also the discussion of the position operators in our representation has been left aside and is to be taken up in a following article.
Remarks on the Double Dispersion Approach to the Bethe-Salpeter Equation
Authors: Nakanishi Noboru
Published: January 2005

The following remarks are made on the applicability of the double dispersion approach to the Bethe-Salpeter equation introduced previously. (1) Any invariant solution of the Bethe-Salpeter equation in ladder approximation satisfies the double dispersion representation when the total energy-momentum is spacelike. (2) There are some exceptional invariant solutions which are not given by the previous method in the equal-mass case, but the existence of such solutions is very unlikely in the unequal-mass case. (3) In the case of the general separated kernel the previous results give the correct solutions even if the kernel does not reproduce the double dispersion representation.