bluuebunny/crossref_metadata_2025 huggingface dataset:
Recursion Relations for the Wigner Coefficients of Unitary Groups
Authors: Brody T. A. , Moshinsky M. , Renero I.
Published: January 2005

The polynomials in the components of a set of n-dimensional vectors that form a basis for an irreducible representation of S𝒰n are shown to be part of the basis of the group Unr, in which the subgroup 𝒰n × Ur is explicitly reduced and r ≥ n - 1. Using this result, the concept of auxiliary Wigner coefficient is introduced, for which the problem of multiplicity does not arise and the phase convention is related to that of Gel'fand and Zetlin; recursion relations for this auxiliary coefficient are obtained in a straightforward way, and the connection between it and the ordinary Wigner coefficient is shown to be simple. The recursion relations are being programmed for an electronic computer to allow the systematic evaluation of the Wigner coefficients of S𝒰3 and S𝒰4.
Algebraic Tabulation of the Clebsch-Gordan Coefficients for Reduction of the Product (λ, μ) ⊕ (3, 0) of Irreducible Representations of SU(3)
Authors: Pandit L. K. , Mukunda N.
Published: January 2005

Tables of Clebsch-Gordan coefficients of SU (3) for the reduction of the product (λ, μ)⊗ (3, 0) of representations of SU(3) are constructed by use of the tensor method. Derivations of some crossing and symmetry relations for the SU3 Clebsch-Gordan coefficients are given, and the Clebsch-Gordan coefficients for the product (μ, λ)⊗ (0, 3) are related to those mentioned above. The phase convention used in compiling the tables is stated and explained.
Racah Algebra for an Arbitrary Group
Authors: Derome Jean-Robert , Sharp W. T.
Published: January 2005

3j- and 6j-symbols are studied for a general group without assuming that the group is ambivalent or multiplicity-free. The choice of multiplicity label r that distinguishes the equivalent irreducible representations that may arise in a Kronecker product of irreducible representations is left arbitary and no special choices of phase are made. It is found that the 3j- and 6j-symbols obtained have essentially the same properties as the familiar 3j- and 6j-symbols for the rotation group in three dimensions.
On the Dimensionality of the Real World
Authors: Penney R.
Published: January 2005

The ``strength'' of systems of field equations is defined following a procedure due to Einstein. For an arbitrary dimension number, the strength of Einstein's field equations and of Maxwell's equations are calculated, and it is shown that only in four dimensions do the two systems of field equations determine their respective fields equally strongly. In addition, in four dimensions, Weyl's equation exhibits the same strength as the Einstein or Maxwell fields. The results are used to show that a necessary condition for a unified field-theoretic description of gravity and electromagnetism is that the world be four dimensional. If one also demands that the neutrino field be determined as strongly as the Einstein and Maxwell fields, one must have a two-component neutrino, rather than a four-component neutrino.
Analytic Properties of the Elastic Unitarity Integral
Authors: Aks S. O/. , Gilbert R. P. , Howard H. C.
Published: January 2005

In this paper, certain results concerning the singularities of holomorphic functions with integral representations are used to investigate the analytic properties of the elastic unitarity integral of the quantum theory of fields. In particular, it is shown that each of the Landau singularities of the scattering amplitude is an actual singularity. It is also shown that if the scattering amplitude is not identically zero it must have a natural boundary on the unphysical sheet.
Alternative Formulations of Scattering in Model Field Theories
Authors: Gelman Harry , Haller Kurt
Published: January 2005

A comparison is made between the expression for the boson-fermion scattering amplitude, in a model theory, obtained by renormalizing Feynman diagrams, and the one that originates from the dressed-particle picture. A proof is given of a surmise, previously verified to sixth order, that the two expressions for the transition amplitude are identical, although the former stems from a theory that leads to paradoxes from which the latter is exempt. The proof of the identity of the two representations is extended to transition amplitudes in which a weak interaction, considered to first order, is modified by a strong one considered to all orders. It is shown how the operations in the dressed-particle picture directly lead to iterations in terms of the physical mass and coupling constant, whereas the expressions obtained from the Feynman diagrams require renormalization before they have this form.
Singular Logarithmic Potentials in Coordinate Space
Authors: Cornille Henri , Predazzi Enrico
Published: January 2005

In previous works we have studied the problem of the determination of the Jost function for singular repulsive potentials behaving near the origin like inverse powers. The key was to define ``new Jost solutions'' which, still being asymptotically ingoing (or outgoing) waves, tend to constant (Jost functions) near the origin. It was shown from the perturbation expansion in coordinate space of these ``new Jost solutions'' that we can construct the Jost functions by connecting the radial coordinate r and the order of the perturbation expansion p. More precisely, if we introduce an r(p) dependence, the Jost function is the limit of convergent sequences provided r(p) goes to zero less rapidly than a given limiting dependence rL(p). In this paper, working in coordinate space, the same method is extended for two other families of singular potentials: firstly, we consider the case in which the most singular part of the potentials behaves like G2(log r−1)βr−2n, (n ≥ 1, β arbitrary) near the origin; secondly, we study exponentially singular repulsive potentials of finite range. It is found that the more singular is supposed to be the potential, the higher becomes the available limiting dependence.
Lowest-Order Contribution to the Lattice Thermal Conductivity
Authors: Hardy Robert J.
Published: January 2005

In a previous paper the correlation function formula for the thermal conductivity was used to derive equations for the lowest-order contribution to the lattice thermal conductivity Kij. An explicit but formal solution of these equations is obtained here, and it is shown how this solution simplifies in the limit of infinite volume. Transport equations equivalent to the familiar Boltzmann equation are derived for perturbations describing both anharmonic forces and lattice imperfections. No approximations are made beyond the restriction to lowest order in the perturbation. It is demonstrated that Kij is symmetric.
One-Particle Singularities of the S-Matrix in Quantum Field Theory
Authors: Hepp Klaus
Published: January 2005

In the framework of the Haag-Ruelle collision theory one-particle singularities are proved to exist in the physical region of any connected scattering amplitude. They occur with the causal propagator (p2 − m2 + iε)−1 in the dominant term and have a residue, which factors into the product of two connected amplitudes for subprocesses. The remainder of the amplitude is infinitely often differentiable in the critical variable. These results rely essentially on short-range forces and the one-particle spectrum, but neither depend on analyticity nor on unitarity.
On the Structure of the Finite Parts of the Generating Functional of Propagators in Quantumelectrodynamics
Authors: Le Gaillard Y. , Visconti A.
Published: January 2005

Starting from their generating functional, we investigate the structure of the finite and divergent parts of propagators in quantum electrodynamics. A renormalizability condition is expressed by means of a set of recursion formulas between the finite parts of radiative corrections of the generating functional; this condition is the algebraic formulation of Salam's rule for the extraction of finite parts from divergent integrals. The former sets of equation should finally allow the actual computation of the radiative correction of higher orders.
Quantum Electrodynamics of the Stimulated Emission of Radiation. II
Authors: Bevensee R. M.
Published: January 2005

The equations of motion for the resonant modes in a lossy cavity, which couples them through the nonresonant thermal loss mechanism described by Senitzky, are examined. The second-order perturbation treatment of dissipation is justified for long time intervals by reference to an intermittent similarity transformation. Self-consistent expressions for commutators such as 〈[Pn(t1), Pm(t2)]〉d, Pn being the operator-amplitude of the electric field in the nth mode, as averaged over the loss mechanism d with respect to its diagonal density matrix, are obtained from Laplace transform expressions for these operators. These averaged commutators describe correlations of fluctuations in different modes, in a way compatible with the ordinary commutation relations for lossless modes. The self-consistent anticommutator expressions, averaged over the loss mechanism, imply the Bose-Einstein formula for thermal energy in each mode at time t2 = t1. This is shown from direct examination of anticommutators and also from an integration of the correlation function. Commutators for the total electric and magnetic fields at two space-time points in the cavity have the proper causal properties. The dissipation terms in the field equations of motion cannot be derived from an equivalent Hamiltonian unless a modified analysis such as that of Kemeny is undertaken. Finally, the approximate equations of motion for the mode operator-amplitudes of a parametric amplifier are derived in the creation-annihilation operator framework and the spectral density of noise obtained at the signal frequency. The equilibrium noise present due to coupling with the idle modes is proportional to the number of photons in the idle frequency mode instead of that number plus one, in contrast to the conclusion of Wagner and Hellwarth. In this system the loss-averaged operators evolve from initial thermal equilibrium values to parametric equilibrium values according to nonlinear equations of motion containing time dependent commutators.
A New Phase-Space Distribution Function in the Statistical Theory of the Electromagnetic Field
Authors: Kano Yutaka
Published: January 2005

In a previous paper a certain new probability distribution function q(z) relating to blackbody radiation was introduced. In the present paper the properties of this function for a general radiation field are studied. Unlike the phase-space distribution function of Sudarshan (1963), this function is nonnegative and is an ordinary function. A series expansion for q(z) is given, and it is shown that the series is absolutely convergent for all eigenvalues z of the destruction operator. It is also shown that the density matrix in the Fock representation can be uniquely determined from this probability distribution function, and vice versa. The relation between q(z) and the Sudarshan's phase-space distribution function is discussed.
Half-Space Neutron Transport with Linearly Anisotropic Scattering
Authors: McCormick N. J. , Kuščer I.
Published: January 2005

The method developed by Case is used to solve four time-independent, one-speed problems for neutron transport in a homogeneous medium where the scattering function is linear in the cosine of the scattering angle. The solutions to the albedo, Milne, Green's function, and constant isotropic source problems for a half-space are facilitated by the use of half-range bi-orthogonality relations between the eigenfunctions of the homogeneous transport equation. Expressions are also derived for the emerging angular densities and the densities and net currents on the surface of the half-space.
An Expansion Method for Treating Singular Perturbation Problems
Authors: Nayfeh Ali Hasan
Published: January 2005

Cochran's method for treating singular perturbation problems is shown to give, in some cases, expansions which are not uniformly valid. This method is modified and extended to give uniformly valid expansions. The new method is applied to the problem of heat transfer in a duct to give a solution in agreement with that obtained by the WKBJ method.
Perturbation Method in the Diffraction of Electromagnetic Waves by Arbitrarily Shaped Penetrable Obstacles
Authors: Yeh C.
Published: January 2005

The perturbation technique which is based on a Taylor series expansion of the boundary conditions at the perturbed boundary is extended to consider the problem of the diffraction of waves by a dielectric object with perturbed boundary. Since this approach attacks the complete boundary-value problem, the result is valid for the near zone as well as for the far zone and is valid for all frequencies. In a way of illustration, the problem of the diffraction of electromagnetic waves by a dielectric cylinder with perturbed boundary is treated. A specific example on the scattering of plane waves by a dielectric elliptic cylinder with small eccentricity is given. Numerical results are also computed for this specific example and are compared with those obtained from the exact solution.
On the Algebraic Structure of the Cluster Expansion in Statistical Mechanics
Authors: Arf Cahït , Ïmre Kaya , Özïzmïr Ercüment
Published: January 2005

The structure of cluster expansion which is widely used in statistical mechanics is studied from an algebraic point of view. In doing this, a commutative algebra is constructed which is generated by partitions of a finite set by regarding them as operators which divide the set into disjoint parts. Physically, these operators correspond to operations which remove interaction among certain clusters of particles. It is shown that the cluster expansion stems from the relation between two basis sets of this algebra; the first set is the set of all partitions and the second is the set of pairwise orthogonal minimal idempotents. This property enables one to demonstrate the equivalence of the product versus cluster properties of the distribution and correlation functions, respectively, in general terms. This is done by constructing a simple representation space for the partition algebra corresponding to the distribution functions. A second application of the partition algebra is considered in the case when correlations are well-ordered with respect to the interaction strength λ, so that to a given order in λ the distribution functions are not all independent and can be expressed in terms of a finite irreducible set of functions involving smaller numbers of particles. The combinatorial problem of calculating the expansion coefficients is carried out explicitly using a graded representation space with respect to the order in λ. It is concluded that the partition algebra can be used as a mathematical tool in handling problems involving cluster expansion.
Decomposition of the Lorentz Transformation Matrix into Skew-Symmetric Tensors
Authors: Bazanski Stanislaw L.
Published: January 2005

Any matrix describing a finite proper orthochronous Lorentz transformation of the null tetrad in Minkowski space may be written as a polynomial of the second order in skew-symmetric tensors. From a geometrical point of view these tensors describe two-dimensional planes which are mapped by the Lorentz transformation into themselves.
Separation of the Interaction Potential into Two Parts in Treating Many-Body Systems. I. General Theory and Applications to Simple Fluids with Short-Range and Long-Range Forces
Authors: Lebowitz J. L. , Stell G. , Baer S.
Published: January 2005

Systematic methods are developed for investigating the correlation functions and thermodynamic properties of a classical system of particles interacting via a pair potential v(r) = q(r) + w(r). The method is then applied to the case in which w(r) is a ``Kac potential'' w(r, γ) = γvφ(γr) (v the dimensionality of the space) whose range γ−1 is very long compared to the range of q(r). Our work is related closely to the work of Kac, Uhlenbeck, and Hemmer. The main new feature of our method is the separation of the correlations, e. g., the two-particle Ursell function ℱ(r), into a short-range part ℱs(r, γ) and a long-range part ℱL(y, γ), y ≡ γ r; r the distance between the particles. The two parts of ℱ are defined in terms of their representation by graphs with density (or fugacity) vertices and K-and φ-bonds, K(r) = e−-βq − 1, Φ = −βw. A resummation of these graphs then yields a simple graphical representation for the long-range part of the correlation functions in terms of graphs with φ-bonds and ``hypervertices'' made up of the short-range part of the correlations. This representation is then used in this paper to make separate expansions of ℱs(r, γ) and ℱL(y, γ) and through them of the thermodynamic parameters in powers of γ. Explicit calculations of the Helmboltz free energy is carried out to a higher order in γ than done previously by Hemmer and it is shown how to carry out the calculation, in principle, to any order. The general method is further applied (in separate articles) to lattice gases, plasmas, and to the special problem of critical phenomena.
Scalar Invariants of a Rotational System in a Lie Algebra. II. Casimir Operators
Authors: Pease M. C.
Published: January 2005

We consider matrix equations of the form dW/dz = [S, W] where S(z) is a matrix-valued function of z, embedded in a given semisimple Lie algbra, L. If the set Xi is a basis for L, we expand W as wiXi. We obtain general conditions on an n-index form pijk… such that p = (uivjwk…pijk…) is a constant if ui, vj, wk, etc., are solutions of the differential equation. It is shown that these forms are those that determine the vector invariants of the corresponding group.

We also ask what transformations will map any solution of the differential equation onto another, regardless of the detailed behavior of S(z). It is shown that the transformations that do this involve these same pijk….

With this identification, we are able to obtain procedures that can be used to calculate at least the two-index forms, pij. The existence of two-index forms, other than the Killing metric gij is shown to depend on the reducibility of the adjoint representation.

Finally, we show that these same forms generate Casimir operators—i.e., operators that commute with all elements of the algebra, and hence with the group associated with it. The Casimir operator so obtained is given explicitly by (pijk…XiXjXk …), where the set, Xi, is the basis of the dual algebra.
Reduction of Representations of SUmn with respect to the Subgroup SUm ⊗ SUn
Authors: Hagen† C. R. , Macfarlane A. J.
Published: January 2005

With a view to application to theories such as the Wigner supermultiplet theory of nuclear structure and its recent extension into the domain of particle physics, the reduction of representations of the special unitary group SUmn into representations of its SUm ⊗ SUn subgroup is considered. Proof is given of the result, that, in the case of a representation of SUmn of plurality type pmn, this reduction yields only those irreducible representations of SUm ⊗ SUn, whose SUm and SUn parts have plurality types pm and pn equal to pmn modulo m and modulo n, respectively. A practical method of obtaining reductions is described and a tabulation of results in the case of SU6 and SU2 ⊗ SU3 is made. The method depends on the use of a very concise version of the Weyl character formula for unitary groups which does not seem to have been used elsewhere.
Momentum-Transfer Theorem for Inelastic Processes
Authors: Gerjuoy E.
Published: January 2005

Recently it has been shown that for potential scattering, the well-known optical theorem—relating the total cross section to the imaginary part of the forward scattering amplitude—can be generalized to yield a ``momentum-transfer cross-section theorem.'' The present paper further generalizes the previous potential scattering result. Specifically, it appears that the momentum-transfer cross-section theorem is valid also for many-particle systems, wherein inelastic processes occur. Although this last assertion probably holds quite generally, a proof is given only for the collisions of electrons with atomic hydrogen. The proof takes into account electron indistinguishability, as well as the possibility that the incident electron ionizes the atom, but assumes the forces are not spin-dependent.
Representation of Discrete Symmetry Operators
Authors: Mariwalla K. H.
Published: January 2005

Representations of discrete symmetry operators (DSO's) connected with space (𝒫), time (T), and generalized charge (𝒞) are considered. It is shown that if one writes a DSO as exp (iπΩ) × a phase transformation, then (under certain conditions on Ωs) to each DSO there corresponds a set of Ωs which is closed with respect a Lie algebra, which is isomorphic to the Lie algebra of generators of rotation in an n-dimensional Euclidean space; where n is the number of commuting observables that changes sign under this DSO in a given representation (e.g. linear momentum representation). In the particular case of the (T𝒞𝒫) operation, there are six Ωs, of which two are diagonal, viz. the generalized charge Q, and spin projection along the z axis Sz; corresponding Euclidean group is four-dimensional. For the sake of completeness, the representations are also given for the following cases: (i) nonrelativistic quantum mechanics, (ii) quantum theory of free fields, in terms of field operators.
Some Theorems on the Unimodular Complex Degree of Optical Coherence
Authors: Mehta C. L. , Wolf E. , Balachandran A. P.
Published: January 2005

The complex degree of coherence γ(r1, r2, t1 − t2) of a stationary optical field is defined as the normalized cross-correlation function (|γ| ≤ 1) of the light disturbances at two space-time points (r1, t1), (r2, t2), the disturbances being represented by means of Gabor's analytic signals.

In the present paper the general form of γ is examined under the condition that |γ| takes on the extreme value unity for all possible time differences τ = t1 − t2 (− ∞ < τ < ∞). Several cases are distinguished, depending on whether this condition is satisfied for some points or for all points in some fixed domain of space. It is pointed out, that the previously published derivations of the relevant theorems contain serious errors. The methods employed in the present paper make use of the property of nonnegative definiteness which the complex degree of coherence is shown to obey. The results have a bearing on the important but as yet unsolved ``phase problem'' of optical coherence theory.
Space-Time Theory of Elementary Particles. I. Born Reciprocity Principle
Authors: Shin Ernest E. H.
Published: January 2005

The use of the Born reciprocity principle as a postulate applied to ``internal'' Minkowski space leads to a theory of elementary particles based on a purely ``space-time'' point of view. The reciprocity transformations of Born, (xμ, pμ) → (±pμ, ∓xμ), form a discrete subgroup of the x-p rotation group SU(2), which we associate with ``isotopic spin'' symmetry. The full dynamic symmetry of the eight-dimensional phase space with Minkowski metric is a 36-parameter group which contains the x-p rotation group SU(2) and the homogeneous Lorentz group ℒ as disjoint subgroups. This gives the symmetry of ``subquantic'' matter which has oscillator-rotator properties. Some physical aspects of the generators and possible forms of mass formulas for both integer and odd-half integer spins are discussed. Finally, it is suggested that the nonrelativistic counterpart of the present formalism may explain the symmetry of atomic nuclei.

``And when we have built an altar to the Invisible Light, we may set thereon the little lights for which our bodily vision is made.''

from ``The Rock'' by T. S. Eliot
Scattering by the Singular Potential r−4
Authors: Aly H. H. , Müller H. J. W.
Published: January 2005

The potential 1/r4 is the only known potential more singular than the centrifugal term, for which the Schrödinger equation can be solved exactly. In the present investigation, we consider the potential in more detail than has been done so far. In particular the physical S-matrix is obtained, shown to be unitary, and compared with expressions of other derivations given in the literature. The eigenvalues of the Mathieu equation are finally discussed, and the behavior of the Regge trajectories is indicated.
Approximation by Sequences of Padé Approximants in Regions of Meromorphy
Authors: Chisholm J. S. R.
Published: January 2005

It is shown that sequences {Pmn(z)} of Padé approximants with m → ∞ and n → ∞ approximate a function f(z) inside a certain circle of meromorphy with centre at z = 0. Extensions of this result are discussed.
Complex Space-Time and Classical Field Theory. I
Authors: Das A.
Published: January 2005

This is the first of the series of three papers which introduces complex space-time to describe physical phenomena. The objective of this generalization is twofold: firstly, to geometrize gauge transformations and electromagnetic fields, and secondly, to quantize space-time in order to remove serious divergences from the field theory. In this paper classical fields are discussed in complex space-time with a view of subsequent generalization to quantum field theory in quantized space-time.
The Quantized Complex Space-Time and Quantum Theory of Free Fields. II
Authors: Das A.
Published: January 2005

In this paper a covariant quantization of complex space-time is proposed. As a consequence of this quantization each of the four real coordinates can take discrete values n½l, and furthermore, measurements of these coordinates are noninterfering with each other. Next the general theory of quantized free fields is developed in the background of quantized space-time. As an example the case of complex scalar field has been dealt with and it is found that the resulting Green's functions are nonsingular.
The Existence of Closed Magnetic Surfaces
Authors: Faulkner James E.
Published: January 2005

A toroidal vacuum magnetic field, which has closed magnetic surfaces in the neighborhood of the magnetic axis, is mathematically constructed. The existence of these surfaces is demonstrated by the theorem of Moser on the stability of mappings which are perturbations of rotations whose rotation angle is a function of the radius (twist mapping).
Classification of Irreducible Unitary Representations of Compact Simple Lie Groups. II
Authors: Mehta M. L. , Srivastava P. K.
Published: January 2005

In continuation to a previous article, the classification of the irreducible unitary representations of the groups E6, E7, and E8 into complex, potentially real, and pseudoreal catagories is completed.
Three-Dimensional Formulation of Gravitational Null Fields. I
Authors: Misra R. M. , Singh R. A.
Published: January 2005

The four-dimensional normal hyperbolic Riemannian space is represented as a direct product of a three-dimensional space and a timelike line. The null gravitational field is defined in a manner analogous to that of electromagnetic field. It is shown that in this way three types of gravitational null fields can be characterized. We call them gravitational null fields of types A, B, and C. We find, as necessary and sufficient conditions, that the gravitational field be null field of types A and B, respectively. It is also shown that these null fields admit null vectors in accordance with the properties of gravitational radiation fields.
Some Remarks on the Construction of Invariants of Semisimple Local Lie Groups
Authors: Santhanam T. S.
Published: January 2005

A general form of the l invariants of compact semisimple local Lie groups or rank l, as the traces of the powers of the ``velocity potential'' operator is suggested. The connection of this form of the invariants with those of Ref. 3 is described. The possible generalization beyond those of adjoint group and its connection with that of Biedenharn is discussed.
Transformation of Proca-Type System of Partial Differential Equations
Authors: Abraham C.
Published: January 2005

The Proca-type system of partial differential equations [−curl curl A + kA = τ grad φ; ▿2φ − Kφ = τ div A, with k, K, and τ constants] is transformed into a new system such that the vector potential A and the scalar potential φ satisfy separate differential equations. It is also shown that, for the solution of the system, there are only two different modes, div A = 0 with φ = 0, and curl A = 0 with φ ≠ 0, respectively.
The Validity of Perturbation Series with Zero Radius of Convergence
Authors: Baker George A. , Chisholm Roy
Published: January 2005

Although the perturbation expansion for the S-matrix of the Peres-model field theory has zero radius of convergence, it uniquely defines the S-matrix and is easily summable by the method of Padé approximants.
The Asymptotic Theory of Cerenkov Radiation
Authors: Handelsman R. , Lewis R. M.
Published: January 2005

Recently developed methods of asymptotic analysis are applied to the problem of Cerenkov radiation. The mathematical description of this physical phenomenon is given by the integro-differential system of equations for the electromagnetic field in a dispersive medium. The parameter λ is introduced into these equations, where λ is a characteristic frequency of the medium. It is for large λ that the asymptotic expansion of the electromagnetic field is sought. Isotropic, uniaxial crystalline and gyrotropic media are treated in detailed. The source function which appears in the field equations is taken to be quite general, e.g., it may be used to represent the current associated with any moving ``multipole'' source. By applying the method of stationary phase to an integral representation of the solution, the leading term of the asymptotic expansion of the electromagnetic field is obtained. More precisely, a parametric representation of the expansion is found in which certain space-time curves called ``rays'' play a key role. An expression for the total energy of the radiation is then determined.
The Discontinuities of the Triangle Graph as a Function of an Internal Mass. II
Authors: Kacser C.
Published: January 2005

We consider the discontinuities of the triangle-graph amplitude as a function of an internal mass variable. These discontinuities are important, since they form the kernel of the Aitchison-Anisovich integral equation, which is derived from the Khuri-Treiman three-body final-state-interaction dispersion relation. We evaluate the discontinuities by explicitly performing the Feynman α integrations. We also discuss their analytic continuations. Finally we consider the applicability of the Cutkosky rules to such an internal mass variable discontinuity. It is argued that these rules must be modified in two ways. One of these is straightforward, having to do with the appearance of spacelike masses. The other is more involved and is a consequence of the results of homology theory. We apply the modified Cutkosky rules to the triangle-graph discontinuities and obtain the same results as found by the direct method, so confirming the modifications which we have made.
Continuous Degenerate Representations of Noncompact Rotation Groups. II
Authors: Limić N. , Niederle J. , Raczka R.
Published: January 2005

Three principal continuous series of most degenerate unitary irreducible representations of an arbitrary noncompact rotation group SO(p, q) have been derived and their properties discussed in detail. The corresponding harmonic functions have been constructed.
Integral Forms for Quantum-Mechanical Momentum Operators
Authors: Robinson Peter D.
Published: January 2005

The usual differential form P0 for the quantum-mechanical momentum operator P which is conjugate to a generalized coordinate q (α ≤ q ≤ β) is, in atomic units, P0 = − i(g−½) ∂/∂q (g½), where g is the Jacobian of the transformation from Cartesian to generalized coordinates. However, P0 is not always self-adjoint on the domain 𝒟 of physically acceptable bound-state wavefunctions, as a proper quantum-mechanical operator should be. An integral form is proposed for P, defined by Pf(q)=(2π)−12g−12(q)  ∫ −∞∞exp (ikq)kF(k) dk,  α≤q≤β,where F(k)=(2π)−12  ∫ αβexp (−ikξ)f(ξ)g12(ξ) dξ,  f∈D. The effect of this integral operator (which is suggested by the ideas of Fourier transforms) differs from that of P0 only at the end-points of the range of q. In a sense, it is formally equivalent to an operator (suggested by Robinson and Hirschfelder) which is obtained by adding certain delta-function terms to P0, but it suffers from none of the defects, since delta-functions do not appear explicitly. Various properties of the integral operator are derived. Some discussion of the domain 𝒟 is presented as an appendix.
Existence of Particlelike Solutions to Nonlinear Field Theories
Authors: Rosen Gerald
Published: January 2005

A pseudovirial theorem is derived for time-independent particlelike solutions of finite energy (singularity-free and spatially localized time-independent solutions) to field theories associated with an action principle. It is shown that a useful necessary condition for the existence of such particlelike solutions is generally obtainable as a corollary to the pseudovirial theorem. This necessary condition is in fact sufficient to preclude existence of any well-localized particlelike solution for all but special field theories with the more common forms of algebraic interaction. On the other hand, strong satisfaction of the necessary condition can lead to model field theories with rigorous closed-form particlelike solutions, as shown by example for a class of Lorentz-covariant theories which feature a real scalar field in interaction with a two-component complex Weyl spinor field. Some of the latter particlelike solutions to the scalar-spinor theory are energetically stable with respect to spatial dilatations, hence likely to be stable in the dynamical sense. A counter example to the more general sufficiency of the strong satisfaction condition is presented, showing that strong satisfaction of the pseudovirial theorem's corollary does not always guarantee the existence of singularity-free particlelike solutions.
Analytic Properties of a Class of Potentials and the Corresponding Jost Functions
Authors: Sabatier Pierre C.
Published: January 2005

Analytical properties of the potential obtained by Newton's method in the inverse scattering problem at fixed energy are thoroughly investigated. It is found that r V(r) is analytic in the neighborhood of r = 0 and can be continued in the r complex plane as a meromorphic function, the poles of which are of order at least equal to 2. An explicit formula is given to yield the Jost functions for any complex value of the angular momentum ν. They appear to be meromorphic functions with poles located at the negative integers on the real axis. The zeros of f2(ν), for large |ν| and for Re ν and Im ν > 0, are located on a curve which is the boundary of a domain previously shown to contain no Regge pole. The interpolated scattering amplitude is unitary. It behaves for ν → −i ∞ as eπ|ν|. An important result of this paper is that the class of potentials obtained by Newton's method is much more restricted than one might think. This led the author to look for a more general approach to the inverse scattering problem, which is the subject of a forthcoming publication. In order to illustrate the method of this paper, a detailed study of an example previously introduced by R. G. Newton is given. In the last Appendix, a very remarkable property of the potentials involved in this example is given: the scattering amplitude corresponding to these potentials can be given an exact closed form. Since the corresponding potentials are strongly energy-dependent, it is very likely that this is only a mathematical curiosity.
Propagation of Correlations in a Boltzmann Gas
Authors: Frieman Edward A. , Goldman Robert
Published: January 2005

New results are obtained on the propagation of correlations in a Boltzmann gas on the scale of the mean free path and the collisional time scale which appear to support M. Green's conjecture on this subject.
Structure of Space and the Formalism of Relativistic Quantum Theory. III
Authors: Ahmavaara Yrjö
Published: January 2005

The elastic scattering of two scalar particles having equal masses is considered in the formalism of relativistic quantum theory over a Galois field GF(q). The scattering function σ determining the cross section is introduced. It is determined by the geometrical relations of Euclidicity, to be imposed on observable 4-momenta in a finite geometry. Thus the requirement of Euclidicity of observable 4-momenta can be considered as the counterpart, in a finite geometry, of the requirement of the analyticity of the invariant amplitude used in conventional S-matrix theory for the determination of the cross section.
Faddeev-Type Equations for the Four-Body Problem
Authors: Alessandrini Victor A.
Published: January 2005

The technique used by Faddeev to obtain connected equations for the nonrelativistic three-body T matrix is generalized for four particles. It is shown that the four-body equations are completely determined by the solutions of all the possible two-body subsystems, as is the case in the three-body problem. This approach can be extended to more complicated multiparticle systems.
Cartan Frames and the General Relativistic Dirac Equation
Authors: Brill Dieter R. , Cohen Jeffrey M.
Published: January 2005

The Dirac equation for spin-½ particles in curved space-time is formulated using Cartan calculus. Unlike previous formulations, this method is easy to use because it expresses the Dirac equation in terms of well known objects like partial derivatives and special relativistic Dirac matrices. It allows a simple and direct treatment of neutrinos in homogeneous nonisotropic universes and in plane-wave geometries. These solutions are compared and contrasted with the corresponding solutions containing electromagnetic radiation.
Expansion Theorem for Functions of Operators
Authors: Cohen Leon
Published: January 2005

A method is given for expanding operator functions of q and p, where p = ℏ/i(∂/∂q), such that all q factors are to the left of the p factors. The method is applicable to the rearrangement of creation and annihilation operators.
The Generalization of Choh-Uhlenbeck's Method in the Kinetic Theory of Dense Gases
Authors: García-Colín Leopoldo S. , Flores Asdrúbal
Published: January 2005

The method proposed by Choh and Uhlenbeck to deal with kinetic phenomena in dense gases is generalized to all orders in the density. The set of integral equations for the functions defining the transport coefficients is derived. It is shown that the thermal conductivity and the shear viscosity are independent of the way in which the local temperature is introduced, namely, through the kinetic energy and through the total energy density. However, the bulk viscosity does depend on the particular definition of temperature. The relationship between the corresponding bulk viscosities is explicitly obtained.
First-Order Solutions of the One-Speed Transport Equation in Plane Multiregion Geometries
Authors: Mendelson M. R.
Published: January 2005

Case's method is applied to multiregion problems in plane geometry. Half-range and ``two-media'' orthogonality relations are used to obtain a system of coupled integral equations for the appropriate expansion coefficients. The system is explicitly solved in a first-order approximation, which is valid for wide regions. Numerical analysis and comparison with transport computer codes indicates that such solutions are essentially exact for region thicknesses of five mean free paths or more.
S-Operator Theory. II. Quantum Electrodynamics
Authors: Pugh Robert E.
Published: January 2005

The formalism of S-operator theory is generalized to apply to all low-spin particles with particular emphasis on quantum electrodynamics. It is shown that only direct (nongradient) interactions are allowed, when spinor fields are involved. The Pauli interaction is explicitly shown to be forbidden. Problems of gauge invariance are discussed and commutation rules between the interpolating quantum fields and ∂μAμin(X) are derived. Finite S-matrix elements are easily calculated with complete convergence at each stage of the calculation.
On the Complex Structure of the Universe
Authors: Brown Edmund H.
Published: January 2005

A complex space—time is constructed from physical axioms. Both gravitational and electromagnetic fields are approximations to parts of a geometric object defined on this complex space, while other parts may represent strong and weak interactions. The intersections of the singularities of the three independent geometric invariants are identified as elementary particles. This identification leads to geometric definitions of mass and momentum and suggests the geometric significance of internal quantum numbers.
On the L—S Basis of the Poincaré Group for the Case of Nonzero Rest Mass
Authors: Chakrabarti A.
Published: January 2005

The action of the generators of the Poincaré group on the basis functions spanning an irreducible representation [m2, s2] (m > 0) and labeled by the eigenvalues of the energy, orbital angular momentum and spin, and the projections on a fixed axis of the two latter are considered. The explicit canonical representation of the generators (1.1) are used. The formulas are derived first for the zero-spin case (Sec. 2) and then for the general case (Sec. 3). It is shown that our L—S basis permits a much more compact and systematic derivation of the formulas than the basis involving total angular momentum and helicity, which is considered by Lomont and Moses. The separation of the orbital and spin parts allows us also to display explicitly the role played by the ``canonical'' definition of spin.
Multipole Moments in Einstein's Gravitational Theory
Authors: Lamb D. J.
Published: January 2005

Janis and Newman have proposed definitions of multipole moments of axially symmetric gravitational fields in terms of the initial data on a characteristic surface. This paper extends their definitions to source distributions without symmetry by considering the electromagnetic and linear gravitational fields. The global definition of four-momentum agrees, for sandwich waves, with the objects constructed by Mo/ller and Cornish in their attempts to provide a local definition of four-momentum.
Functional Differential Calculus of Operators. II
Authors: Rohrlich F. , Wilner M.
Published: January 2005

The calculus of operators developed in a previous paper is here carried through in momentum space. The differential quotient, i.e., the derivative of an operator with respect to a free-field operator is expressed algebraically by means of generalized functions and certain commutators. No recourse is made to configuration space but the resultant calculus in p-space is related to the previously developed one in x-space by means of Fourier transforms. The calculus is presented for spins 0, ½, and 1. The difficulties which were encountered before in the calculus for charged vector fields are resolved by working with a field equation which incorporates the supplementary condition needed to eliminate the spin-zero components.
Theorem on Gravitational Fields with Geodesic Rays
Authors: Unti T. W. J. , Torrence R. J.
Published: January 2005

If one of the Ruse vectors of a field is assumed to be a geodesic having nonvanishing divergence θ, curl ω, and complex shear σ, the only vacuum metrics that exist are found to be of the cylindrical type, where the geodesic rays obey θ2 + ω2 = σσ̄.
A Perturbation Method for the Classical Time-Dependent Pair Correlation Function
Authors: Aronson Raphael
Published: January 2005

A perturbation technique useful for computing many-time thermal averages of classical quantities is developed. The canonical distribution function for the system is shown to evolve isothermally from that for a free-particle system as the interaction is switched on slowly. This permits convenient use of an interaction picture in which to perform thermal averaging. The technique is applied to the calculation of the time-dependent pair correlation function in position for a uniform gas. The correlation function is shown to be a sum of two components, one the solution of a kinetic equation and essentially a generalization of the autocorrelation in equilibrium, the other a generalization of the mutual correlation function in equilibrium. The equations arise as sums over diagrams. The equations resulting from the random phase approximation, valid for the short-time behavior, are solved exactly. It is shown directly that the generalized dielectric function given in terms of the correlation function is identical with that found by solution of the kinetic equation in the random phase approximation for all frequencies.
Symmetry Properties of the 3j-Symbols for an Arbitrary Group
Authors: Derome Jean-Robert
Published: January 2005

The symmetry properties of the 3j-symbols are studied for an arbitrary compact group. It is shown that when the three j's are all inequivalent it is possible to choose 3j-symbols which are invariant under any permutation of the j's and of the corresponding m's (generalized magnetic quantum numbers). When two of the three j's are equivalent, the 3j-symbols can be chosen in such a way that at most a minus sign appears when the j's and m's are permuted. It is also shown that when the three j's are equivalent it is in general not possible to choose the 3j-symbol such that its absolute value is invariant under every permutation of the m's.
A Continuous Representation of an Indefinite Metric Space
Authors: Gundzik M. G.
Published: January 2005

An overcomplete family of states (OFS) is constructed for a countably infinite linear vector space with an indefinite metric for the case that the metric is diagonal with eigenvalues (−1)n, where n is an integer. A continuous representation is indicated and the properties of a semiclassical description of a quantum mechanical system (the pseudo oscillator whose creation and destruction operators ā and ā+ satisfy [ā, ā+] = −1) defined in this vector space are studied. It is found that a consistent OFS |z> can be constructed if the operator G(z) which generates the state |z> from the vacuum is unitary. Furthermore, with the statistical state of this system specified by a bounded pseudo-Hermitian density matrix ρ̄, the related semiclassical complex function ρA(z) for antinormal ordering of operators in the indefinite metric space is found to be bounded, with ρA(z) and [ρA(z)]2 integrable, continuous, and a boundary value of an entire analytic function of two complex variables. The semiclassical function ρN(z) for normal ordering is associated with a sequence of functions ρN(ν)(z) whose square is integrable and related to a sequence of tempered distributions ρN(ν) such that the corresponding sequence of density matrices ρ̄(ν) converges to ρ̄ in the norm.
Effective Dielectric Constant, Permeability, and Conductivity of a Random Medium and the Velocity and Attenuation Coefficient of Coherent Waves
Authors: Keller Joseph B. , Karal Frank C.
Published: January 2005

Random media are considered in which the dielectric constant, permeability, and conductivity are random functions of position. For them, the average electric field, electric current, dielectric displacement, magnetic field, and magnetic induction are determined, assuming that these average quantities are time-harmonic plane waves. The proportionality factors between appropriate pairs of these quantities are found and defined to be the effective dielectric constant, permeability, and conductivity of the random medium. These effective parameters depend upon the frequency and propagation constant of the field in addition to the two-point auto- and cross-correlation functions of the random dielectric constant, permeability, and conductivity. For transverse fields thay are all scalars. In addition the dispersion equation for the propagation constant of the average or coherent field, derived previously, is analyzed and solved for high- and low-frequency fields. From the propagation constant, the phase velocity and attenuation coefficient can be found.