bluuebunny/crossref_metadata_2025 huggingface dataset:
Unified Variational Formulation of Classical and Quantum Dynamics. II
Authors: Corson Edward M.
Published: January 2005

In a previous article in this series [E. M. Corson, J. Math. Phys. 4, 42(1963); herein referred to as I], we showed that a new variational principle, of Lagrangian structure, includes Hamilton's principle as a unique derived consequent and encompasses both the classical and quantal definitions of ``state'' and the corresponding Lagrangian and Schrödinger equations of motion, respectively. In this theory, the essential difference between classical and quantal domains was shown to arise solely from the superadded postulates of determinism vs indeterminism; other than this, the two basic postulates may be reduced to (a) Newton's equations and (b) the new variation principle.

It was shown that in the quantal domain, Schrödinger's equations obtain for the transition amplitudes (propagators) between pure eigenstates of homologous variables in configuration space, at different times, thus clearly implying all the results of conventional Hamiltonian-operator quantum formalism. In the present article we show that our variational formulation, under the familiar classical canonical transformation to momentum space, also automatically leads to the corresponding Schrödinger equations for the transition amplitudes between pure eigenstates of homologous variables in momentum space, at different times, the unitarity of the formalism and the reciprocal Fourier integral relations between coordinate and momentum representations. Thus we bridge a long-standing gap in theory and derive, uniquely and apparently for the first time, the basic quantum commutators (P.B.'s) which are postulated in the more familiar Hamiltonian theory.
Lie Algebraic Solution of Linear Differential Equations
Authors: Wei James , Norman Edward
Published: January 2005

The solution U(t) to the linear differential equation dU/dt = h(t)U can be represented by a finite product of exponential operators; In many interesting cases the representation is global. U(t) = exp[g1(t)H1] exp [g2(t)H2] … exp[gn(t)Hn] where gi(t) are scalar functions and Hi are constant operators. The number, n, of terms in this expansion is equal to the dimension of the Lie algebra generated by H(t). Each term in this product has time-independent eigenvectors. Some applications of this solution to physical problems are given.
Possible Schemes for Global Symmetry
Authors: Speiser David R. , Tarski Jan
Published: January 2005

An attempt was made to determine all possible schemes for global symmetry as representations of groups. All possible schemes which correspond to connected groups were found, but the conclusions about nonconnected groups are incomplete. Physical interpretation of the schemes is discussed, and a detailed summary of group-theoretic methods is included.
Note on Uniqueness of Canonical Commutation Relations
Authors: Arnowitt R. , Deser S.
Published: January 2005

It has been pointed out by Wigner that the consistency requirement between the Lagrange and Heisenberg equations of motion does not uniquely determine the canonical commutation relations, at least for one-dimensional systems. It is shown here that this ambiguity does not arise in local field theory whose basic equal-time commutators commute with the translation operator.
Asymptotic Form of the Proper Self-Energy Function in Many-Particle Systems
Authors: Fredkin Donald R.
Published: January 2005

The asymptotic form at high frequencies of the reciprocal of a two-time thermodynamic Green's function is shown to be a constant. This constant is expressed as the exact expectation value of a second (functional) derivative of the Hamiltonian.
Representations of the Canonical Commutation Relations Describing a Nonrelativistic Infinite Free Bose Gas
Authors: Araki H. , Woods E. J.
Published: January 2005

The existence of inequivalent representations of the canonical commutation relations which describe a nonrelativistic infinite free Bose gas of uniform density is investigated, with a view to possible applications to statistical mechanics. The functional E(f, g)=(Ψ, eiφ(f) eiπ(g)Ψ) is used to describe the inequivalent representations. This functional is calculated for the free Bose gas in a box of volume V, and the limit is then taken as V → ∞. In this way we construct cyclic representations describing an infinite system of particles with a density distribution ρ(k) in momentum space. For a given ρ(k) the operator algebra generated by the φ(f), π(g) is reducible. For the ground-state representation (all particles in the zero-momentum state), the representation is a direct integral of irreducible representations (analogous to BCS theory). For finite temperatures the situation is complicated by the occurrence of representations which are not type I. The physical significance of the reducibility of the representations is discussed.

It is argued that the thermal ensemble for the infinite system is a pure state at zero temperature, although there is some ambiguity as to which operators should belong to the algebra of observables. For finite temperatures, the thermal ensemble seems to be a mixture. The case of an interacting Bose gas is considered briefly.
Statistical Theory of the Energy Levels of Complex Systems. IV
Authors: Dyson Freeman J. , Mehta Madan Lal
Published: January 2005

In previous papers of this series, a theory was constructed for the description of the statistical properties of the eigenvalues of a random matrix of high order. This paper deduces from the same theoretical model the statistical behavior to be expected for a finite stretch of observed eigenvalues chosen out of a much longer stretch of unobserved ones. In comparing the model with experimental data, we have always to deal with such a finite stretch of observed levels. Three ``statistics'' are investigated; these are quantities which can be computed directly from the observed data, and for which the expectation values and statistical variances can be calculated from the theory. One statistic gives only a precise way of measuring the average level spacing. One provides a test of the model by measuring the degree of long-range order in the level series. The third provides an independent test of the model by measuring the degree of short-range order. In Sec. V these methods are applied to a preliminary analysis of some experimental data on neutron capture levels in heavy nuclei. The results are inconclusive. Discrepancies between theory and observation are large, but the discrepancies might be produced either by incompleteness of the data or by incorrectness of the theoretical model.
Wigner-Eckart Theorem and Simple Lie Groups
Authors: Ginibre Jean
Published: January 2005

For simple Lie groups, matrix elements of vector operators (i.e. operators which transform according to the adjoint representation of the group) within an irreducible (finite-dimensional) representation, are studied. By use of the Wigner-Eckart theorem, they are shown to be linear combinations of the matrix elements of a finite number of operators. The number of linearly independent terms is calculated and shown to be at most equal to the rank l of the group. l vector operators are constructed explicitly, among which a basis can be chosen for this decomposition. Okubo's mass formula arises as a consequence.
More Integral Representations for Scattering Amplitudes with Complex Singularities
Authors: Fronsdal C. , Norton R. E. , Mahanthappa K. T.
Published: January 2005

Previous work is generalized in order to achieve a better understanding of the role of complex singularities in connection with integral representations. The most general conditions under which the box-diagram contribution to a scattering amplitude satisfies a representation with real integration contours are derived. Explicit representations are derived in several special cases. It is frequently found possible to obtain representations that are essentially of the Mandelstam type, although the more general Bergman-Oka-Weil representation must be invoked in general. One example of a three-dimensional representation is given, which exhibits the analytic structure in one of the external masses in addition to the kinematical variables. The significance of the ``physical sheet'' is discussed.
N-Dimensional Total Orbital Angular-Momentum Operator
Authors: Granzow Kenneth D.
Published: January 2005

A set of generalized polar coordinate systems are defined in N-dimensional space. The total orbital angular-momentum operator as defined by Louck is found to be a tensor invariant on the (N − 1)-dimensional unit sphere; hence it is easily explicitly determined in any of the possible generalized polar coordinate systems. Commuting operators can be found and the eigenvalue problem solved in many coordinate systems. Two examples are given: (1) a coordinate system of 3M dimensions where M is an integer, and (2) a coordinate system of 8 dimensions.
The Flatter Regions of Newman, Unti, and Tamburino's Generalized Schwarzschild Space
Authors: Misner Charles W.
Published: January 2005

The ``generalized Schwarzschild'' metric discovered by Newman, Unti, and Tamburino, which is stationary and spherically symmetric, is investigated. We find that the orbit of a point under the group of time translations is a circle, rather than a line as in the Schwarzschild case. The time-like hypersurfaces r = const which are left invariant by the group of motions are topologically three-spheres S3, in contrast to the topology S2 × R (or S2 × S1) for the r = const surfaces in the Schwarzschild case. In the Schwarzschild case, the intersection of a spacelike surface t = const and an r = const surface is a sphere S2. If σ is any spacelike hypersurface in the generalized metric, then its (two-dimensional) intersection with an r = const surface is not any closed two-dimensional manifold, that is, the generalized metric admits no reasonable spacelike surfaces. Thus, even though all curvature invariants vanish as r → ∞, in fact Rμναβ = O(1/r3) as in the Schwarzschild case, this metric is not asymptotically flat in the sense that coordinates can be introduced for which gμν − ημν = O(1/r). An apparent singularity in the metric at small values of r, which appears to be similar to the spurious Schwarzschild singularity at r = 2m, has not been studied. If this singularity should again be spurious, then the ``generalized Schwarzschild'' space would represent a terminal phase in the evolution of an entirely nonsingular cosmological model which, in other phases, contains closed spacelike hypersurfaces but no matter.
Undulatory Riemannian Spaces
Authors: Lanczos C.
Published: January 2005

A demonstration is given that Riemannian spaces of very high curvature in submicroscopic domains do not contradict the existence of a macroscopic line element which is nearly Minkowskian. The signature of the microscopic line element is positive definite and the wave property of the metric in macroscopic domains comes about by a peculiar ``wave-guide action'' of a strongly curved, two-dimensional line element, in harmony with the particlelike behavior of the photon. The four-dimensional lattice structure of the metrical vacuum field does not establish an absolute frame of reference and can be harmonized with the macroscopic validity of the Lorentz transformations.
Regularity of the T Matrix in the Case of Dirac Potential Scattering
Authors: Pagnamenta Antonio
Published: January 2005

Applying Hunziker's method to the case of Dirac potential scattering, we prove analyticity of the T matrix in energy and momentum transfer. Our conditions on the potential are somewhat weaker, and the domains of analyticity contain the known ones as special cases.
The Nature of the Axioms of Relativistic Quantum Field Theory. II
Authors: Sudarshan E. C. G.
Published: January 2005

This paper continues the study of the nature and interdependence of the axioms of relativistic field theory; attention is focused on the notion of relativistic invariance. The central result of the present paper is the derivation of necessary and sufficient conditions for a representation of the covariant free field to admit a unitary representation of the inhomogeneous Lorentz group associated with the field operator. It is shown that only the standard Fock-Cook representation has this property. The relevance of the requirement that the Lorentz group is represented by a unitary family associated with the field operator is exhibited by an analysis of the covariant representations of Shale and Segal. These representations involve extremal states which are not pure, and group representations by intertwining operators.
Random Matrix Diagonalization—Some Numerical Computations
Authors: Porter C. E.
Published: January 2005

Numerical results of Monte-Carlo calculations of spacing and eigenvalue distributions for the invariant and independent Gaussian orthogonal ensemble of Hamiltonian matrices are presented. Many of the histograms should be useful for comparison with experimental data. A table of the first few moments of each distribution is given. For the spacing distributions, such moments are equivalent to spacing correlation coefficients, and hence these are also made available indirectly.
Real Unitary Representation of the Many-Channel S Matrix for Complex l and E
Authors: Chan Hong-Mo
Published: January 2005

A representation is established for the multichannel S matrix in terms of a matrix function A(l, E) and a scalar function B(l, E), both holomorphic in the domain formed by the product of the whole finite l plane with the finite E plane, cut only along the left-hand dynamical cut. The representation satisfies the known analytic properties of S(l, E), and also all the generalized unitarity conditions of Peierls, LeCouteur, and Newton. The reality condition on S for complex l and E is guaranteed by a simple condition on A and B.
Continuous-Representation Theory. I. Postulates of Continuous-Representation Theory
Authors: Klauder John R.
Published: January 2005

In a continuous representation of Hilbert space, each vector ψ is represented by a complex, continuous, bounded function ψ(φ) ≡ (φ, ψ) defined on a set S of continuously many, nonindependent unit vectors φ having rather special properties: Each vector in S possesses an arbitrarily close neighboring vector, and the identity operator is expressable as an integral over projections onto individual vectors in S. In particular cases it is convenient to introduce labels for the vectors in S whereupon each ψ is represented by a complex, continuous, bounded, label-space function. Basic properties common to all continuous representations are presented, and some applications of the general formalism are indicated.
Irreducible Representations of Generalized Oscillator Operators
Authors: Jordan T. F. , Mukunda N. , Pepper S. V.
Published: January 2005

All of the irreducible representations are found for a single pair of creation and annihilation operators which together with the symmetric or antisymmetric number operator satisfy the generalized commutation relation characteristic of para-Bose or para-Fermi field quantization. The procedure is simply to identify certain combinations of these three operators with the three generators of the three-dimensional rotation group in the para-Fermi case, and with the three generators of the three-dimensional Lorentz group in the para-Bose case. The irreducible representations are then easily obtained by the usual raising and lowering operator techniques. The applicability of these techniques is demonstrated by a simple argument which shows that the commutation relations require that the generator to be diagonalized have a discrete spectrum.
Expansions Associated with a Pair of Singular First-Order Differential Equations
Authors: Roos B. W. , Sangren W. C.
Published: January 2005

This paper reports on the continuation of a study of the spectral properties of a Dirac operator. The analytical methods developed by Weyl and Titchmarsh for the analysis of the Sturm-Liouville equation are extended to the investigation of a system of two singular first-order differential equations. Expansions associated with the system are established and a convergence theorem is presented.
Relativistic Kinetic Theory of a Simple Gas
Authors: Israel Werner
Published: January 2005

A consistent relativistic theory of transport processes in a simple gas is developed. The approach is the four-dimensional geometric one due to Synge. Scalar and vector eigenfunctions of the linearized collision operator are derived when the scattering cross section is a simple separable function of scattering angle and relative velocity (Maxwellian particles). The bulk viscosity and thermal conductivity are computed explicitly for this case.
Observable Properties of Large Relativistic Masses
Authors: Metzner A. W. K.
Published: January 2005

Interstellar space may be full of very dense and very faint stars (supernova remnants, very old white dwarfs, etc.), on whose surfaces the gravitational field intensity is very high. What, according to general relativity, are their observable properties, and what are the maximum gravitational effects as a matter of principle? To answer this, the exact test particle orbits (geodesics) of the Schwarzchild metric are derived, classified and described, on the basis of an exact classical model. The latter is of considerable help in making the properties of the orbits immediately evident, and represents the principal advantage of the present derivation over previous ones. Also, as radial coordinate, instead of the usual largely arbitrary r, the metric coefficient g00 = A2(r) is used. A2 has immediate physical significance as the red shift ratio v(r)/v(∞), and its use also simplifies the formulas. The test-particle scattering angle (generalizing the results of Darwin), and the differential scattering cross section, as well as the capture cross section of a sphere of radius A2(r) = A2(R) = AR2, are calculated as a function of test-particle energy, and presented graphically. The observed, augmented, angular diameter of a sphere is calculated in terms of AR2, and some peculiar lenslike effects of masses, first discussed by Einstein, are reviewed. It is then pointed out that an important result of Curtis for the interior field, implies AR2 > 0.514, so that the exterior field extends at most down to A2 = 0.514, and the region A2 < 0.514—which would be the most pathological region of the exterior—cannot in fact exist, if Curtis' arguments are assumed valid. As a result, quasihyperbolic test-particle orbits exist with pericentrum equal to the radius of any conceivable spherical mass, and the radius can therefore in principle always be determined by an asymptotic scattering experiment. As a further result, the maximum photon-scattering angle can be no more than about 110° (and larger for slower particles), providing a cutoff at this point in the photon-scattering cross section, and allowing a massive star to produce at most one secondary image of another star. As a practical matter, none of the effects discussed here seems large enough to be measurable today, with the sole well-known exception of the red shift, which for the Curtis limiting sphere would be large enough to shift a visible spectrum into the near infrared.
Mathematical Analysis on the Effect of a Prolate Spheroidal Core in a Magnetic Dipole Field
Authors: Islam Mohammed Azizul
Published: January 2005

The magnetic vector potential has been obtained for the case of a circular loop of current surrounding a material core of a prolate spheroidal shape, by solving Maxwell's equations and suitable boundary conditions. It is shown that this vector potential consists of two parts; the first part is that due to the loop alone, the second part being due to the presence of the core.
Addendum: Combinatorial Aspects of the Ising Model for Ferromagnetism. I. A Conjecture of Feynman on Paths and Graphs
Authors: Sherman S.
Published: January 2005

A gap in the proof of Theorem 1 of the cited paper is filled.
Excitation Spectrum of an Impurity in the Hard-Sphere Bose Gas
Authors: Soda Toshio
Published: January 2005

The impurity excitation spectrum is calculated for the hard-sphere Bose gas with dilute impurity concentration, using the pseudopotential method of Lee, Huang, and Yang. The spectrum for the ideal case, in which the mass and interaction of impurity are the same with those of the bosons in the gas, is k2 in the unit of 2ma = h/ = 1 (ma being the mass of boson, and k the momentum of impurity), with an effective mass correction of −6445[(ρa3)12/π12]k2+O[(ρa3)k2,k4], where ρ is the density of the boson gas, and a is the hard-sphere diameter of the boson. The general impurity case is also calculated and found to have an impurity spectrum rk2, with a similar effective mass correction, where r is the mass ratio of the boson to the impurity.
Stationary Entropy Principle and Renormalization in Normal and Superfluid Systems. I. Algebraic Formulation
Authors: De Dominicis Cyrano , Martin Paul C.
Published: January 2005

This paper shows how the dynamical and thermodynamical properties of an interacting quantum mechanical system with many degrees of freedom may be expressed and calculated solely in terms of renormalized propagators and renormalized vertices or interactions. The formulation employed is sufficiently general to encompass systems which have several components, with Fermi or Bose statistics, whether or not they exhibit superfluidity or superconductivity. The process of renormalization is the functional generalization of the thermodynamic transformation from the chemical potential and temperature to the energy and matter densities. With each set of variables (here, functions) is associated a natural thermodynamic function (here a functional). The natural functional for the unrenormalized potentials which occur in the Hamiltonian is the logarithm of the grand partition function; the natural functional for the fully renormalized variables, the distribution functions, is the entropy. In particular, a stationarity principle for a functional F(2) of distribution functions subject to constraints is shown to provide a fully renormalized description of the system. The numerical value of this functional, at the stationarity point at which the distribution functions take their actual value, is the entropy of the system. The equations of stationarity are expressions for the unrenormalized ν-body potentials vν in terms of the ν′-body distribution functions Gν′. The functionals F(2) and vν (of the distribution functions Gν′) are expressed as the solutions of closed functional differential equations which may be used to generate their power-series expansions. For a superfluid Bose system, as for the electromagnetic field interacting with matter, it is necessary to consider expectation values of odd, as well as even, numbers of field operators. In particular it is necessary to employ the expectation values Gν for 2ν = 1, 2, 3, 4 field operators. For a fermion system, even if it is superconducting, only the functions Gν for 2ν = 2, 4 are required. In contrast to other thermodynamical functionals, the entropy functional F(2) makes no reference to equilibrium parameters such as temperature and chemical potential.
Lorentz Invariant Multichannel Scattering Formalism
Authors: Jordan Thomas F.
Published: January 2005

A manifestly Lorentz invariant Hamiltonian formalism for multichannel scattering and production processes is developed by making two simple and natural extensions of the ordinary quantum mechanical formalism. The first is the asymptotic covariance postulate of Fong and Sucher which is essentially a necessary and sufficient condition for Lorentz invariance of the scattering amplitudes. The second is a Lorentz invariant extension of the asymptotic condition. It is shown that the latter is in fact no extension at all in a case where the total momentum operators of the asymptotic (unperturbed) systems are the same as the total momentum operators of the interacting system. In such a case the ordinary multichannel scattering formalism is completely Lorentz invariant whenever asymptotic covariance is satisfied.
The Graviton as a Spin-2 Particle
Authors: Brulin Olof , Hjalmars Stig
Published: January 2005

It is proved that the linearized gravitational field can be described by means of a first-order relativistic wave equation with matrix coefficients, obtained in a simple way from the generators of the full linear group in five dimensions.
The Hose Instability Dispersion Relation
Authors: Weinberg Steven
Published: January 2005

The dispersion relation is obtained for the ``hose'' instability in a modulated beam of charged particles traveling through a finite ohmic plasma channel. For a relativistic beam, the low-frequency mode obeys a dispersion relation of the same form as for the unmodulated beam, but it involves a constant which will have to be obtained by machine computation. For large enough modulation frequency the dispersion relation is identical to that for an unmodulated beam.
Complex Angular Momenta and Many-Particle States. I. Properties of Local Representations of the Rotation Group
Authors: Andrews M. , Gunson J.
Published: January 2005

The properties of the ``local representations'' of the rotation group corresponding to complex angular momentum are further developed. Completeness and bi-orthogonality relations are derived and a reduction of products is carried out, giving a generalization of the Clebsch-Gordan reduction. The connection with the representation theory of the group SL(2,R) is considered and a generalization of Regge's use of the Sommerfeld-Watson transform is made to the case where three momentum transfer variables occur in the description of scattering amplitudes.
Statistical Mechanics of Quenched Solid Solutions with Application to Magnetically Dilute Alloys
Authors: Morita Tohru
Published: January 2005

The arrangement of atoms in solid solutions and alloys, prepared at high temperatures and cooled nonadiabatically, is not the one which is thermodynamically most stable. In establishing theories of phenomena related to the internal degrees of freedom of such a system, such as magnetism, one must be careful to account for this nonequilibrium distribution of atoms. In this paper, systems are treated with the aid of a fictitious equilibrium system. This fictitious system is constructed such that its thermal equilibrium properties are the same as the properties of the non-thermal-equilibrium system. Thus one can treat nonequilibrium systems by applying well known thermal equilibrium techniques to the fictitious system.

The method is illustrated via the example of a magnetically dilute alloy. Brout's result for a very dilute Ising system is obtained with the aid of the theory of classical fluids, without collecting diagrams. A method for applying the higher approximations developed for classical fluids to the present problem is suggested; calculations and discussions of which are retained for a forthcoming paper.
Representations of the Inhomogeneous Lorentz Group in Terms of an Angular Momentum Basis: Derivation for the Cases of Nonzero Mass and Zero Mass, Discrete Spin
Authors: Lomont J. S. , Moses H. E.
Published: January 2005

In a previous paper the authors showed how the infinitesimal generators of the proper, orthochronous, inhomogeneous Lorentz group acted in a basis in which the square of the angular momentum, the z component of the angular momentum, the helicity and the energy were diagonal for the irreducible representations which correspond to the cases of nonzero and zero mass, discrete spin. In that paper no derivation of the results were given. It was possible, however, to verify them directly. In the present paper we carry out the derivation.
If Microns Were Fermis
Authors: Corben H. C.
Published: January 2005

Evidence is rapidly accumulating that elementary particle states are simply the rotational levels of a symmetrical top, as derived from relativistic quantum mechanics. Some of this evidence is presented for the case of zero-strangeness bosons. An analogy is drawn between this point of view and the nonrelativistic theory of the rotational levels of a rigid molecule.
Electromagnetic Radiation in the Presence of Moving Simple Media
Authors: Lee K. S. H. , Papas C. H.
Published: January 2005

The radiation pattern of an arbitrary source immersed in a moving simple medium is calculated by deducing the differential equation for the potential 4-vector in the rest frame of the source and then solving the equation in terms of a Green's function. As an illustrative example, the case where the source is an oscillating dipole is worked out in detail.
High-Field Magnetoresistance of Inhomogeneous Semiconductors and Plasmas. II. Two-Dimensional Inhomogeneity Distributions
Authors: Morrison J. A. , Frisch H. L.
Published: January 2005

We extend in this paper the treatment of the high field magnetoresistance of a previously described classical model of a semiconductor (plasma) containing a two-dimensional distribution of inhomogeneities. The basic assumptions on the classical model are that the scale of the inhomogeneities is large compared to the mean thermal wavelength of an electron and the Landau level spacing is large compared to kT. The magnetic field ℋ is taken parallel to the z coordinate and the inhomogeneity distribution is characterized by a sufficiently smooth potential φ(x, z). The 4-moment equations are solved asymptotically for large ℋ, and an equivalent asymptotic solution is obtained, subject to certain mathematical assumptions, for the transport equation. The magnetoresistance is shown, in general, not to saturate, but to increase, as ℋ2, with increasing ℋ.
Generalization of the Variational Method of Kahan, Rideau, and Roussopoulos and Its Application to Neutron Transport Theory
Authors: Kostin Morton D. , Brooks Harvey
Published: January 2005

The variational method of Kahan, Rideau, and Roussopoulos (KRR) frequently used in neutron transport theory to estimate weighted averages is extended and generalized. In the KRR variational method a first variation in the trial functions produces a second variation in the estimate of the weighted average. Two generalized variational functionals which depend on trial operators instead of trial functions are given. A first variation in the trial operators produces an nth variation in the estimate of the weighted average when an nth order generalized variational functional is used. Both perturbation theory and the KRR variational method are derived as special cases of the generalized variational method. Several examples including calculations of transport equation spatial moments using diffusion equation solutions as trial operators are studied with good results.
On the Representations of the Semisimple Lie Groups. III. The Explicit Conjugation Operation for SUn
Authors: Baird G. E. , Biedenharn L. C.
Published: January 2005

The analog of the SU2 (1 − j) symbol is defined and discussed in detail for SUn. An appropriate generalization to SUn of the Condon-Shortley phase convention is explicitly given.
On the Representations of the Semisimple Lie Groups. IV. A Canonical Classification for Tensor Operators in SU3
Authors: Baird G. E. , Biedenharn L. C.
Published: January 2005

It is shown that the multiplicity structure of the general SUn operators may be put in a one-to-one correspondence with the multiplicity structure of the corresponding states. This result allows a convenient labeling scheme to be devised for the general SUn Wigner operator and leads in a natural way to the concept of a reduced Wigner operator. The problem of multiplicity in tensor operators is shown to have a canonical resolution in the conjugation classification which is discussed in detail for the SU3 case.
Applications of the Lorentz Transformation Properties of Canonical Spin Tensors
Authors: Chakrabarti A.
Published: January 2005

Some applications of the Lorentz transformations of relativistic spin tensors in the canonical representation are discussed. The problem of precession of polarization is discussed in Sec. 2. It is shown that the kinematical equation, obtained quite simply, already contains the ``Thomas factor.'' In Sec. 3, applications to polarization analysis of decay products are considered. The canonical form of S-matrix elements are used and multipole parameters for successive decays of the type a → b + c are obtained in an arbitrary frame in a relatively simple way. The exact relativistic way in which the multipole parameters depend, in an arbitrary frame, on the particle momenta are discussed for decays of the type a → b + c + d. While the canonical representation is used mainly, the corresponding technique in the spinor representation is discussed.
Effect of Small Irregularities on Electromagnetic Scattering from an Interface of Arbitrary Shape
Authors: Mitzner K. M.
Published: January 2005

A perturbation analysis is used to determine the effect of small irregularities on electromagnetic scattering from an interface between two media. The interface irregularities are replaced by approximate equivalent surface currents, and the field in space can then be found using the dyadic Green's function of the unperturbed problem. The approach is valid when the irregularity has small slope and amplitude small compared to the wavelengths and local radii of curvature. To facilitate applications, the theory of dyadic Green's functions is developed, and the necessary functions are given for some important geometries.
Comparison of Two Methods for Lower Bounds to Eigenvalues
Authors: Börsch-Supan Wolfgang
Published: January 2005

It is shown that two procedures given by Bazley and Fox for the calculation of lower bounds to eigenvalues of self-adjoint operators are essentially equivalent.
S Theorem and Construction of the Invariants of the Semisimple Compact Lie Algebras
Authors: Gruber B. , O'Raifeartaigh L.
Published: January 2005

An infinitesimal proof of the S theorem, which states that the invariants of a compact semisimple Lie algebra are symmetric with respect to the discrete Weyl group of the algebra, is given. The complete set of invariants of the various compact semisimple Lie algebras found by Racah are rederived in a somewhat different and explicit way, the S theorem being used to establish their completeness.
On the Derivation of Statistical Thermodynamics from Purely Phenomenological Principles
Authors: Mandelbrot Benoit
Published: January 2005

Szilard pointed out in 1925 that it is possible to base the foundations of statistical thermodynamics upon ``phenomenological'' principles, analogous to those of the non-statistical ``classical'' theory. This approach is discussed and developed.
Asymptotic Properties of the Electromagnetic Field
Authors: Goldberg J. N. , Kerr R. P.
Published: January 2005

From the integral form of the general solution for the retarded electromagnetic field of a localized charge-current distribution, the asymptotic field is shown to have the behavior Fμν = Nμν/R + IIIμν/R2 + 2Jμν/R3, where the coefficients satisfy Nμνkν = 0, IIIμνkν = Akμ, and kμkν = 0. The remainder 2Jμν is shown to be bounded by using the second-mean-value theorem. Thus the algebraically special character of the asymptotic electromagnetic field is exhibited.
Generalization of Laplace's Expansion to Arbitrary Powers and Functions of the Distance between Two Points
Authors: Sack R. A.
Published: January 2005

In analogy to Laplace's expansion, an arbitrary power rn of the distance r between two points (r1, ϑ1, φ1) and (r2, ϑ2, φ2) is expanded in terms of Legendre polynomials of cos ϑ12. The coefficients are homogeneous functions of r1 and r2 of degree n satisfying simple differential equations; they are solved in terms of Gauss' hypergeometric functions of the variable (r)2. The transformation theory of hypergeometric functions is applied to describe the nature of the singularities as r1 tends to r2 and of the analytic continuation of the functions past these singularities. Expressions symmetric in r1 and r2 are obtained by quadratic transformations; for n = −1 and n = −2; one of these has previously been given by Fontana. Some three-term recurrence relations between the radial functions are established, and the expressions for the logarithm and the inverse square of the distance are discussed in detail. For arbitrary analytic functions f(r), three analogous expansions are derived; the radial dependence involves spherical Bessel functions of (r<∂/∂r>) of of related operators acting on f(r>), f(r1 + r2) or f[(r12 + r22)½].
The Representations of the Inhomogeneous Lorentz Group in Terms of an Angular Momentum Basis
Authors: Lomont J. S. , Moses H. E.
Published: January 2005

The irreducible ray representations of the proper, orthochronous, inhomogeneous Lorentz group were originally given by Wigner in terms of a basis in which the energy and linear momenta are diagonal. In the present paper we show how the infinitesimal generators of the irreducible representations act on a basis in which the energy, the square of the angular momentum, the component of the angular momentum along the z axis, and the helicity (or circular polarization) are diagonal.

We consider representations corresponding to particles of nonzero mass, and any spin and of zero mass and finite spin. The continuous-spin case is to be treated in a later paper.
Feynman Integrals and the Schrödinger Equation
Authors: Nelson Edward
Published: January 2005

Feynman integrals, in the context of the Schrödinger equation with a scalar potential, are defined by means of an analytic continuation in the mass parameter from the corresponding Wiener integrals. The method yields a new interpretation of highly singular attractive potentials in quantum mechanics. For the example of the attractive 1/r2 potential, this interpretation agrees with classical mechanics in the correspondence limit.
An Application of Sommerfeld's Complex-Order Wavefunctions to an Antenna Problem
Authors: Yeh C.
Published: January 2005

Using the orthogonality relations of Sommerfeld's complex-order wavefunctions, the exact solution for the problem of electromagnetic radiation from a circularly symmetric slot on the conducting surface of a dielectric-coated cone is obtained. The results are valid for the near-zone region as well as for the far-zone region, and they are applicable for arbitrary-angle cones. It is noted that the technique used to solve this problem may be applied to similar types of problems involving conical structure, such as the diffraction of waves by a dielectric-coated, spherically tipped cone.
The Connection between Conservation Laws and Laws of Motion in Affine Spaces
Authors: Havas Peter
Published: January 2005

It was pointed out recently that, for any theory describing matter as a collection of mass points in a metric space and subject to a covariant conservation law for a symmetric tensor density Pμ ν, the geodesic law of motion as well as the form of Pμ ν follow from the conservation law alone, independent of any equations obeyed by the metric. This result is shown to be valid in any affine space, independent of any equations obeyed by the affine connection; conversely, the geodesic law implies a conservation law for a singular symmetric tensor density. Similarly, the existence in any affine space of a covariant conservation law for a vector density Fν describing a collection of point charges is shown to imply the constancy of charge, and the form of Fν; conversely, the constancy of charge implies a conservation law for a singular vector density. Some applications of these results are presented. An Appendix contains a discussion of the laws of motion for particles with an intrinsic dipole moment.
The Complete High-Energy Behavior of Ladder Diagrams in Perturbation Theory
Authors: Polkinghorne J. C.
Published: January 2005

The Mellin-transform method for obtaining the high-energy behavior of Feynman integrals is modified and applied to the set of ladder diagrams. The complete set of terms of the form s−1(ln s)n is summed, and gives an equation for the trajectory function which is analogous to that obtained by Fredholm methods for a Yukawa potential. A perturbation expansion for α(t) valid for t large is given, and the threshold behavior investigated. The results confirm the reliability of the perturbation-theory method of investigation. They also exhibit directly the connection between high-energy behavior and the poles of the scattering amplitude.
Multiplicative Symmetries in Axiomatic Quantum Field Theory
Authors: Prugovečki Eduard
Published: January 2005

The group of all multiplicative symmetries defined for a finite number of interacting fields is studied in detail. Various theorems are proved connecting the existence of multiplicative symmetries with properties of the Wightman distributions of the fields.
Field Operators for Bosons with Impenetrable Cores. II. Equations of Motion and General Operator Formalism
Authors: Aizu Ko , Dell'Antonio Gianfausto , Siegert Arnold J. F.
Published: January 2005

Continuing the investigation started in part I, we derive the equations of motion in the Fock representation and a nonrelativistic field theory for bosons interacting as hard spheres.
Representations for the Nonrelativistic Coulomb Green's Function
Authors: Mano Koichi
Published: January 2005

Derivation of the representations for the nonrelativistic Coulomb Green's function is discussed. It is shown that the recently published representations, both in integral and closed forms, can be obtained from an expression for the difference of the diverging and converging wave solutions of the Green's function which is given in terms of the wavefunctions summed by Gordon.
Elastic, Electromagnetic, and Other Waves in a Random Medium
Authors: Karal Frank C. , Keller Joseph B.
Published: January 2005

Propagation of any type of wave in a random medium is analyzed on the assumption that the medium differs slightly from a homogeneous medium. An equation satisfied by the average wave is deduced which is correct through terms of order ε2, where ε measures the deviation of the medium from homogeneity. From this equation, the propagation constant of the medium is determined. The general formulation applies to any type of linear differential or integral equation with random coefficients. It is applied to time-harmonic waves satisfying the reduced wave equation, to the equations of elasticity and to Maxwell's equations. The propagation constant for the average or coherent wave is complex even for a nondissipative medium, because the coherent wave is continually scattered by the inhomogeneities and converted into the incoherent wave. The propagation velocity of the average wave is also diminished by the inhomogeneities. This propagation constant depends upon certain trigonometric integrals of the auto- and cross-correlation functions of the coefficients in the original equations, i.e., of the various coefficients characterizing the medium. To illustrate the results, media with particular random variations are considered and the propagation constants are determined for them.
A Theorem on the Conductivity of a Composite Medium
Authors: Keller Joseph B.
Published: January 2005

A composite medium consisting of a rectangular lattice of identical parallel cylinders of arbitrary cross section is considered. The cylinders have conductivity σ2 and are imbedded in a medium of conductivity σ1. Simple properties of the conductivity tensor of the composite medium are deduced from the theory of harmonic functions.
Scattering of Electromagnetic Waves by a Composite Cylinder
Authors: Islam Mohammed Azizul
Published: January 2005

The scattering of an obliquely incident electromagnetic wave by a composite cylinder has been obtained using fundamental electromagnetic principles. The general result has been reduced to simpler forms for certain special cases.
Application of Operational Methods to the Analysis of Uniform Plasmas
Authors: Silven Saul
Published: January 2005

Various aspects of plasma theory are derived by the application of operational methods to the equation of electron motion. Discussed herein are the complex tensor dielectric constant, the tensor permittivity and conductivity, plasma transient behavior, and plasma oscillations.
Expansions of Integrals of Bessel Functions of Large Order
Authors: Curtiss C. F.
Published: January 2005

Series expansions in powers of 1/v of integrals over x of the product of the Bessel function J v(vx/y), [J v(vx/y)]2, or J v(vx/y) J v+1(vx/y) with an arbitrary function of a restricted class are developed.
Matrix Elements of the Octet Operator of SU3
Authors: Lurié D. , Macfarlane A. J.
Published: January 2005

All the nonvanishing matrix elements of all the components of the tensor operator which belongs to the regular representation (the octet) of SU3 have been evaluated. Of special interest is the component 𝒴, for it is usual in the broken unitary symmetry theory of strong interactions to assume that the interactions which break exact SU3 invariance have the same transformation properties as 𝒴. Previously, matrix elements of 𝒴 connecting states of the same irreducible representation of SU3 have been given by Okubo in the form of the mass formula. Knowledge of all the matrix elements of 𝒴 is essential however if one is to do more than evaluate one-particle matrix elements in the broken unitary symmetry theory. Our method provides such knowledge for all components of the octet tensor operator with little more effort than is needed to treat 𝒴 alone.
Application of Nonlocal Field Operators to a System of Hard-Sphere Bose Gas
Authors: Wong K. W.
Published: January 2005

Some relationships of the nonlocal field operators developed by Siegert et al. and the usual free-field operators are obtained. The interaction Hamiltonian derived with the help of these relationships is compared with the pseudopotential recently obtained by Liu and Wong. A study of the fluid dynamical equations at extreme low temperature has been made.
One-Speed Neutron Transport in Two Adjacent Half-Spaces
Authors: Mendelson M. R. , Summerfield G. C.
Published: January 2005

Using Case's method for solving the one-speed transport equation with isotropic scattering, the Milne problem solution, the solution for a constant source in one half-space, and the Green's function solution are obtained for two adjacent half-spaces. These problems have been solved previously by other methods. Here the derivations are greatly simplified by using Case's method.
The Necessary and Sufficient Condition in Terms of Wightman Functions for a Field to be a Generalized Free Field
Authors: Garczynski W.
Published: January 2005

The description of the generalized free field in terms of the Wightman functions is given. The necessary and sufficient condition for a field to be a generalized free field is established.
On Dirac's Wave Equation in a Gravitational Field
Authors: Peres Asher
Published: January 2005

The Dirac equation is investigated in the combined electric and gravitational field of a point charge in General Relativity. The wavefunctions are weakly singular at the origin, but still normalizable for a continuous range of the energy. The Hamiltonian, however, is not self-adjoint over the manifold of its own ``eigenstates,'' though it can be made self-adjoint by a suitable choice of its domain of definition. The theory, however, is unable to decide how the Hamiltonian should be defined, and what are the bound states.
Properties of a Class of Nonlocal Solvable Interactions
Authors: Ghirardi G. C. , Rimini A.
Published: January 2005

A generalization of factorable interactions is taken into account. The direct and inverse problems for the relative Schrödinger equation are investigated and they turn out to be workable. It is shown that such interactions can produce several bound states. The solution of the direct problem is given. A class of interactions of the considered type is constructed which produces an arbitrarily assigned finite set of bound states.
Asymptotic Convergence and the Coulomb Interaction
Authors: Dollard John D.
Published: January 2005

A definition of asymptotic convergence is given for nonrelativistic time-dependent scattering problems involving Coulomb potentials. Convergence proofs have been found both for potential and for n-body multichannel scattering. For pure Coulomb potential scattering, the Mo/ller wave matrix is computed explicitly and found to have its usual meaning.