bluuebunny/crossref_metadata_2025 huggingface dataset:
Hermitian Analyticity and S-Matrix Singularity Structure
Authors: Polkinghorne J. C.
Published: January 2005

It is shown that the requirement of Hermitian analyticity can be used to determine the Riemann sheet structure of S-matrix singularities. The examples of the square-diagram Landau curve and of a single anomalous threshold are discussed for the (2 → 2)-particle amplitude.
Structure of Space and the Formalism of Relativistic Quantum Theory. IV
Authors: Ahmavaara Yrjö
Published: January 2005

The decay functions determining the lifetimes and the stability of particles in finite models of relativistic quantum theory are considered.
Transfer-Matrix Method for Gamma-Ray and Neutron Penetration
Authors: Aronson Raphael , Yarmush David Leon
Published: January 2005

The problem of radiation transport is formulated in terms of a transfer matrix H which is a 2 × 2 matrix of operators. H is simply related to the more intuitive transmission and reflection operators T and R. An explicit expression for H is derived in slab geometry for radiation distributions that depend on the angle with the slab normal and on energy. H for a multilayer slab is the matrix product of the transfer matrices for the individual layers. A formal expression for H for a homogeneous slab of finite thickness is found in terms of the T and R appropriate to an infinitestimally thin slab. These in turn are related to the single-scattering distribution and therefore can be computed from the microscopic cross section. For purposes of computation, finite matrix representations for the operators must be introduced corresponding to the finite vectors which approximate the distributions. Expansions of the distributions in the cosine of the angle and group representations in energy were chosen in the present work. Some numerical results are presented for gamma rays on aluminum. The extension to problems with internal sources and to nonplanar geometries is outlined.
Axiomatic Foundations of Quantum Theories
Authors: Guenin Marcel
Published: January 2005

A formalism is developed on an axiomatic basis and is shown to contain as special cases classical mechanics, the usual quantum mechanics, as well as the quantum theory of systems with continuous superselection rules. The structure of the symmetries of a general physical theory is studied and a classification of observables is exhibited.
On the Reduction of Direct Products of the Irreducible Representations of SU(n)
Authors: Harari H.
Published: January 2005

Explicit simple formulas are given for the reduction of some direct products of the irreducible representations of SU(n). All the interesting products from the point of view of the various symmetry models in particle physics are included as special cases of our results. In particular, all products of the totally symmetric and the regular representations of SU(n) are evaluated.
On the Space-Time Behavior of Schrödinger Wavefunctions
Authors: Hunziker W.
Published: January 2005

For the Schrödinger equation in Rl, with a potential V(x1 … xl) of the type considered by Kato, the following problem is solved: Given a monomial M(x1 … xl) of degree n in the coordinates, find sufficient conditions on the initial state u such that Me−iH tu is continuous in t and increasing in norm not faster than |t|n as |t| → ∞. In the special case where V(x1 … xl) is a bounded C∞-function with bounded derivatives, the result implies that (u, t) → e−iH tu is a continuous mapping of 𝒮(Rl) × R onto 𝒮(Rl), 𝒮(Rl) being the Schwartz space of rapidly decreasing functions in the usual topology.
Metrical Lattice and the Problem of Electricity
Authors: Lanczos C.
Published: January 2005

A reinterpretation of Einstein's tetrad geometry leads to new results if we abandon the over-simplified picture of the vacuum as an almost empty Minkowskian manifold. The basic tetrad is identified with the four principal axes of the matter tensor which belongs to a strongly curved fourfold periodic Riemannian world. The macroscopic perturbation of the metrical lattice is investigated, corresponding to a mere rotation of the principal axes and assuming the quadratic action principle of general relativity. The perturbation Lagrangian yields the scalar E2 − H2, (and thus the Maxwellian equations), although the basic manifold is strictly Riemannian, with a positive-definite line element.
Orbits in a Magnetic Universe
Authors: Melvin M. A. , Wallingford J. S.
Published: January 2005

A cylindrically symmetric parallel bundle of magnetic lines of force, in equilibrium under their mutual gravitational attraction (``magnetic universe''), has recently received attention. While a Newtonian analysis suggests that the equilibrium is unstable, the complete general relativity analysis shows that the equilibrium is stable. This discrepancy may have to do with the unusually slow falloff of the gravitational field at large distances in this geometry. In order to understand the gravitational field of the static magnetic universe somewhat better, we have studied its timelike and lightlike geodesics (i.e., the orbits in it of electromagnetically neutral test particles with unit or zero rest mass). Since the density of magnetic flux-and energy and stress and, therefore, ``gravitating mass''-is approximately uniform in the vicinity of the axis, the motion of test particles there is like that in a Newtonian simple harmonic oscillator field. ``Vicinity'' here means within a small fraction ρ of the range radius ā = (6.96/B0) × 1024 cm (B0 is the magnetic field on the axis measured in gauss). As is to be expected from the universality of the angular frequency ω0 in the harmonic oscillator field and the relation: orbital velocity ≅ ω0ρ, no motion can get too far from the axis. Otherwise the physical orbital velocity would exceed the speed of light. It is in this way that the strength of the attractive field, though it does not remain strictly of the harmonic oscillator type as one proceeds outward, implies that there is a critical straddling radius ρ = 1/√3. Circular or circular helical light tracks occur only at the critical radius, and with B0 = 105 G, the time required for light to circumnavigate the critical circle is about 200 years. The cylinder marked out by this radius plays a unique limiting role: All particles, whether of zero or nonzero mass, and no matter what their initial positions and velocities (except in the one singular subcase of light tracks parallel to the cylindrical axis), must have their orbits lying wholly or partially within the cylindrical region ρ < 1/√3; hence the use of the adjective ``straddling.'' Constants of motion which correspond closely to ζ-component linear momentum, angular momentum, and energy in Newtonian mechanics are defined. Bounds are placed on these dynamical constants and on the apsidal radii by the requirement that the range of motion be real. Finally, the magnetic universe is complete in the sense that ``no news can enter or leave''-all orbits are of infinite duration.
Correlation Functions for Eigenvalues of Real Quaternian Matrices
Authors: Mehta M. L. , Srivastava P. K.
Published: January 2005

The eigenvalue density, the two-, and the three-point correlation functions for the ensemble of real quaternion matrices are calculated. The forms suggest a generalization for the n-point correlation function. The probability that no eigenvalues lie inside a circle of radius r around the origin is also calculated for the ensemble of real quaternion matrices as well as for that of complex matrices. Upper and lower bounding functions for the last probability density are given.
VΘ-Bound State and Uniqueness in the Three-Particle Sector of the Lee Model
Authors: Pagnamenta Antonio
Published: January 2005

If the coupling constant is large enough, but still smaller than its critical value, there exists a Vθ-bound state in the Lee model. We find the bound-state wavefunction via the solution of the homogeneous Källen-Pauli equation. The condition for the existence of a Vθ-bound state is seen to give a statement about the uniqueness of the scattering solutions of both the Vθ- and N2θ-integral equations. There exists no N2θ-bound state if the V particle is unstable.
Internal Symmetry and Mass Splitting
Authors: Roskies Ralph
Published: January 2005

We prove two theorems concerning the possibility of embedding the Lie algebra S of an internal symmetry group 𝒮 and the Lie algebra P of the Poincare group 𝒫 into a larger algebra G to obtain intrinsic mass splitting among the multiplets belonging to the irreducible representations of 𝒮. We assume that G is a semi-direct sum of the invariant subalgebra S and of P. First, if 𝒮 is a compact Lie group, then G is in fact a direct sum of S and P and there is no mass splitting. Second, if 𝒮 is a semi-simple Lie group, with an appropriate definition of the mass operator, there is still no mass splitting. The relation of these results to previous papers is discussed.
Error Bounds for Approximations to Expectation Values of Unbounded Operators
Authors: Bazley Norman W. , Fox David W.
Published: January 2005

A bound for |(BΨ, Ψ) — (Bφ, φ)|, the error between the expectation value of a self-adjoint operator B on an eigenfunction Ψ of a self-adjoint Hamiltonian H and the value given by an approximating vector φ, is obtained when H is a radial Schödinger operator and B is multiplicative and unbounded. The analysis makes use of point and asymptotic estimates for Ψ.
Special Functions of Mathematical Physics from the Viewpoint of Lie Algebra
Authors: Kaufman Bruria
Published: January 2005

Families of special functions, known from mathematical physics, are defined here by their recursion relations. The operators which raise and lower indices in these functions are considered as generators of a Lie algebra. The general element of the corresponding Lie group thus operates on the function in two ways: on the one hand it shifts the argument of the function; on the other hand it produces an infinite sum of functions (at the unchanged argument) with shifted indices. Equating the two results of the operation gives us ``addition theorems,'' hitherto derived by analytical methods. The present paper restricts itself to the study of 2- and 3-parameter Lie groups.
Approximate Solution of Hill's Equation
Authors: Payne Herbert
Published: January 2005

A modified WKB approximation, amenable to successive corrections, to the solution of the linear differential equation of second order having a periodic coefficient in normal form is presented. Considered as an application of the related equation method, it uses the free-particle wave equation rather than, as in an alternative approach, Mathieu's equation. Particular attention is given the instances, where two simple turning points appear in the period and where there are no turning points. With respect to the one-dimensional crystal, it is shown how the energy band structure can be gleaned directly from the given periodic potential.
Euclidean Quantum Field Theory. I. Equations for a Scalar Model
Authors: Symanzik K.
Published: January 2005

The analytic continuations to imaginary time of the Green's functions of local quantum field theory define Euclidean Green's functions. Use of the proper-time method allows to represent these functions as multiple Wiener integrals of functionals that obey infinite systems of coupled integral equations which are similar to, and for the particular model of a complex scalar field in quadrilinear self-interaction considered here a limiting case of, systems studied in quantum statistical mechanics by Ginibre. As a consequence, the Euclidean Green's functions can for this model be obtained by a limiting process, with temperature and density going to infinity, from the reduced density matrices of a nonrelativistic Bose gas. Reduced functionals are defined and their equations determined as a prepartory step to renormalization in the super-renormalizable cases of two and three dimensions.
On Variational Principles for Electromagnetic Theory
Authors: Tao L. N.
Published: January 2005

New variational principles for electromagnetic theory are established. A functional consisting of the field vectors is defined through the use of a convolution, and it is shown that the variation of this functional subject to appropriate constraints is completely equivalent to Maxwell's equations, Ohm's law, and the constitutive equations, together with appropriate boundary and initial conditions. The present formulation does not have the defects of the classical variational principle for electromagnetic theory since it does not require the introduction of scalar and vector potentials and a priori knowledge of the field vectors at the final stage. Two variational formulations for the electric and magnetic field vectors alone are also presented.
Phase Transition of a Two-Dimensional Continuum—Ising Model
Authors: Thompson Colin J.
Published: January 2005

We decorate a plane two-dimensional Ising model by placing on each bond of the lattice a continuum spin which is allowed to interact only with the Ising spins at the ends of the bonds. The continuum spins are chosen to be either Gaussian (i.e., normally distributed with zero mean and unit variance) or spherical (i.e., constrained to lie on the surface of a sphere) after Berlin and Kac, and by integrating out the continuum spins, the partition function of the decorated lattice is expressed in terms of the Onsager partition function of the plane two-dimensional Ising model. The critical behavior of the model is as follows: For the Gaussian case, as for the plane Ising model, the specific heat has a logarithmic singularity at the critical point TcG given by 2 tanh2 [2(J/kTcG)2]=1and as t=TcG−T→0+, the spontaneous magnetization goes to zero like t1/8. For the spherical case, the specific heat is continuous and has a cusp at the critical point Tcs given by J/kTcs=[zc+(2+212)zc2]12, 2 tanh2 (2zc)=1, with slope going to ± ∞ like t−1 [ln|t|]−2 as t=Tcs−T→0±, and as t → 0+, the spontaneous magnetization goes to zero like [t/ln t]1/8.
Solution of the Bloch—Nordsieck Model
Authors: Tarski Jan
Published: January 2005

An operator solution is constructed for the Bloch—Nordsieck model, in the formulation of Bogolubov and Shirkov. The following are some of the features of the solution. (a) The method of solution depends on the following assumption: The renormalized electron current is identical with the free-field current. The resulting solution is consistent with this assumption. (b) The infraparticle structure of the electron is analogous to the infraparticle structure in Schroer's model. (c) The ordering of fields, i.e., the definition of their products, is considered in detail. A modification of Wick ordering (called F-ordering in the text) appears particularly satisfactory. (d) The solution allows a heuristic discussion of the infrared representations of the electromagnetic field. These depend, apparently, on the chosen velocity of the electron.
Statistical Mechanics of High-Temperature Quantum Plasmas Beyond the Ring Approximation
Authors: DeWitt Hugh E.
Published: January 2005

A quantum mechanical perturbation expansion of the partition function is used to evaluate the free energy of the electron gas and multicomponent plasmas to logarithmic accuracy in the particle number density, thus including the next important contribution beyond the ring approximation. The quantum generalization of Abe's work on the classical electron gas is given for the ladder interactions with the dynamic screened Coulomb potential, and each ladder is shown to be separately finite because of the finite size of the wave packets describing point electrons [of the order of the thermal de Broglie wavelength ƛ = h̄(β/2m)1/2]. The results show that quantum effects due to the uncertainty principle persist at high temperature, and that when kT > Ryd plasmas are quantum systems, rather than classical, because ƛ is greater than the average distance of closest approach, βe2. Results are also obtained for the Wigner-Kirkwood wave mechanical diffraction corrections to the classical electron-gas free energy which are valid for low temperature (kT < Ryd). The connection between the high- and low-temperature formula is discussed, and it is shown how the logarithmic divergence in the free energy that is cut off at βe2 in the low-temperature electron gas in the Abe theory is cut off at ƛ in the high-temperature case. Also it is shown that the quantum diffraction effects contained in the Montroll-Ward ring sum formula are valid only for kT > Ryd, even though the quantum ring sum formula contains the classical Debye-Hückel result.
Limiting Forms of the Screened Coulomb T Matrix
Authors: Ford William F.
Published: January 2005

In the complex energy plane, the pure Coulomb T matrix possesses branch points which would not appear if the force were properly defined. This is demonstrated by a study of the screened Coulomb T matrix in the limit as the screening radius R tends to infinity. No branch points develop if the proper order of limiting processes is observed and the results agree with previous calculations; however, the T matrix is discontinuous in the limit. A formula for the screened Coulomb T matrix is given which is valid to order 1/R for all energies.
Multipole Theory in the Time Domain
Authors: Granzow Kenneth D.
Published: January 2005

Spherical outgoing waves of arbitrary time dependence are first written in the usual way as a Fourier integral of a sinusoidally time-varying multipole expansion. It is then shown that the integrals over ω of the r- and t-dependent part of the multipole terms can be replaced by differential operators operating on arbitrary functions of retarded time. Thus a form of the multipole expansion is obtained that does not explicitly contain the frequency spectrum of the multipoles. Given the value of Er (for electric multipoles, or Br for magnetic multipoles) as a function of time on the surface of a sphere, expressions for the multipole expansions of all the spherical field components are derived. The method employs a convolution integral and is useful in problems involving a very broad frequency spectrum.
Poles of the Proper Vertex Function in the Bethe-Salpeter Formalism
Authors: Nakanishi Noboru
Published: January 2005

A formal general proof of the statement of Goebel and Sakita is presented on the basis of the Bethe-Salpeter formalism; namely, it is shown that the poles of a proper vertex function cannot appear in the corresponding scattering amplitude. Some related conjectures are also verified. An exactly solvable example is presented and discussed in this connection.
Permutation-Algebraic Formulation of Spin-Free Transition Density Matrices
Authors: Poshusta R. D. , Matsen F. A.
Published: January 2005

Spin-free transition density matrices are derived from spin-free kets which are symmetry-adapted to the symmetric group and its algebra. The Dirac identity establishes that these spin-free density matrices are identical to those obtained by integrating the spin from the full-spin density martices. Derivations are first given for arbitrary primitive kets which may be geminals of higher polymals, after which we consider products of orbitals, either orthonormal or nonorthonormal.

Correlation in the spin-free space is discussed and we show the influence of permutational symmetry on the probability of coincidence of pairs. A special case of this correlation is the well-known Fermi hole.
Generalizations of the Virial and Wall Theorems in Classical Statistical Mechanics
Authors: Siegert Arnold J. F. , Meeron Emmanuel
Published: January 2005

A generalized virial theorem which expresses inverse compressibility in terms of integrals of virials and canonical distribution functions through the four-particle distribution is transformed to the grand canonical ensemble and becomes an expression for compressibility in terms of the same integrals formed with grand canonical distribution functions. The integrals are of a mixed (virial and fluctuation) type.

While the thermodynamic functions expressed by the same integrals with canonical and grand canonical distribution functions are quite different, the two formulas agree in the thermodynamic limit because of the different asymptotic behavior of canonical and grand canonical distribution functions. We also derived an alternative form of the second virial theorem which expresses compressibility in terms of integrals over virials and grand canonical distribution functions through the three-particle distribution function only. It is shown that this form and its generalizations to higher derivatives of the density, as well as the hierarchy of fluctuation theorems and the fugacity expansions of distribution and correlation functions can all be very simply derived from a set of integro-differential equations satisfied by the grand canonical distribution functions. A generalization of the wall theorem (P/kT = ρwall) is derived and shown to be equivalent to the generalized virial theorem (canonical form).
Stationary Dust-Filled Cosmological Solution with Λ = 0 and without Closed Timelike Lines
Authors: Maitra S. C.
Published: January 2005

An analytic and complete solution of Einstein's field equations without the Λ term is presented for a dust-filled universe (p = 0). The solution is stationary and inhomogeneous and does not contain any closed time-like lines. Also some of the properties of the solution are studied.
A Short Simple Evaluation of Expressions of the Debye-Waller Form
Authors: Mermin N. David
Published: January 2005

Averages like those encountered in the theory of the Debye-Waller factor are evaluated in one sentence.
Derivation of the Generalized Boltzmann Equation in Quantum Statistical Mechanics
Authors: Morita Tohru
Published: January 2005

The hierarchy of the equations of motion for the reduced density matrices in quantum statistical mechanics is solved and the (cumulant) reduced density matrices at a time t are expressed in terms of those at an earlier time t0. Diagrams are introduced to express the results. With the aid of the technique of partial summations, the general term in the kinetic equation for the one-particle reduced density matrix or the generalized Boltzmann equation in quantum statistical mechanics is obtained. The equation is non-Markovian. A method of reducing the equation to Markovian is sketched.
Theorem on the Clebsch-Gordan Series in SU(n)
Authors: Nussinov S.
Published: January 2005

A theorem on the Clebsch-Gordan series conjectured by B. Vitale is proved using simple Young diagram techniques.
Ising Model Spin Correlations on the Triangular Lattice. II. Fourth-Order Correlations
Authors: Stephenson John
Published: January 2005

A class of fourth-order spin correlations whose sum is closely related to the specific heat is calculated exactly by Pfaffian perturbation theory. In the particular case when the four spins lie on a lattice axis, it is shown that the reduced fourth-order correlation function has the asymptotic form 〈σ0, 0σ1, 1σk, kσk+1, k+1〉−〈σ0, 0σ1, 1〉 〈σk, kσk+1, k+1〉≏A′(e−2βk/k2),where β−1 is the (positive) mean range of order. The decay of correlations with spin separation k is symmetric about the critical point β = 0. The amplitude A′ is 1/π2 at the critical point and is 1/(2π) at all other temperatures. Correlations on ferro- and antiferromagnetic triangular and quadratic lattices are discussed in some detail.
Topology of Some Spheroidal Metrics
Authors: Zipoy David M.
Published: January 2005

The solutions of Einstein's vacuum field equations, Rμν = 0, are found when quasi-oblate and prolate spheroidal coordinates are used. The solutions for the ``Newtonian'' potential can be written as a linear combination of Legendre polynomides of integral order l. For oblate coordinates the solutions for each l have a ring singularity and have a double sheeted topology; one can get from one sheet to the other by going through the ring. When the l = 0 and l = 1 solutions are combined an infinite-sheeted topology results from the nonlinear character of the field equations. In general the geometry is asymptotically flat on only one sheet; on the others it is highly distorted. In some cases the region near the ``ring singularity'' opens out into a multisheeted infinite space. For the prolate coordinates the solutions contain a line singularity of finite length. In general, the prolate coordinate solutions are much less rich in varied topologies than are the oblate solutions.
Unified Unitary Representation of the Poincaré Group for Particles of Zero and Positive Rest Mass
Authors: Chakrabarti A.
Published: January 2005

As is well known, Wigner's construction of the unitary representations in terms of little groups gives quite dissimilar forms for the cases m = 0 and m > 0, while the spinor representation which does give a unified description of the above two cases is not unitary. We point out that it is quite possible to give a unified unitary representation for both cases (Sec. 1). This is achieved simply by noting that, while the factorization of U(Λ) corresponding the choice of the little group of (1, 0) is not valid for m = 0, the explicit final expression for the Wigner rotation Rw and the corresponding infinitesimal generators remain perfectly well defined for m = 0, and continue to furnish a unitary (discrete spin) representation for this case, which is compatible with the restriction of helicity to a particular fixed value. Moreover, the representation thus obtained has a very simple and direct geometrical significance. The relation of our formulation with that of Wigner is studied (in Sec. 2) and the comparison with the spinor representations is given (Sec. 3). We rederive our representation, starting from a particular simple condition (4.1) (in Sec. 4) which holds for both cases (m = 0, m > 0). We then consider (Sec. 5) the application of our unified formulation to the reduction of direct products, involving particles of positive and zero rest mass, comparing the result with that of helicity coupling. In the Appendix we make certain remarks concerning the Hermiticity of the generators and the possibility of defining a position operator for m = 0. Comparison with Foldy's representation is given (in the Appendix and Sec. 3), explaining why his representation cannot be considered to be strictly unitary, though the relation with the unitary case is quite a simple one.
Nonlinear Theory of Elastic Directed Surfaces
Authors: Cohen H. , DeSilva C. N.
Published: January 2005

The present paper develops a nonlinear theory for the deformation of an elastic directed surface by assuming the existence of a strain energy function and postulating a principle of virtual work which governs its mechanical behavior. The equations of equilibrium and the boundary conditions are shown to involve both the classical stress as well as the double stress. Constitutive equations are derived which give the stress and double stress as functions of a complete set of strain measures which describe the deformation of directed surfaces.
Upper and Lower Bounds for Canonical Ensemble Averages
Authors: Falk Harold
Published: January 2005

Upper and lower bounds are obtained for canonical ensemble averages. The bounds are expressed entirely in terms of averages with respect to an arbitrary unperturbed canonical ensemble density operator. A weak form of the derived bounds is used to show that, for magnetic fields exceeding a given critical value, the magnetization of the antiferromagnetic linear chain approaches ferromagnetic saturation as the temperature approaches absolute zero.
Finite-Dimensional Representations of Some Non-Semisimple Lie Algebras
Authors: George Claude , Lévy-Nahas Monique
Published: January 2005

We study the finite-dimensional representations of non-semisimple Lie algebras. We give some general properties, and apply them to the case of the motion group and of the inhomogeneous Lorentz group.
Higher-Order Poles in the S-Matrix
Authors: Hagen C. R.
Published: January 2005

The question of the possible existence of multiple poles in the S-matrix is discussed. It is shown by means of a simple model that such poles are not generally incompatible with Lagrangian field theory and can only be excluded in axiomatic formulations by the assumption of a unique correspondence between stable particles and the poles of the S-matrix on the physical sheet.
Dynamics of Certain Spherical Charge Distributions
Authors: Liboff Richard L.
Published: January 2005

Two initial-value problems are solved in the Lagrange representation. The initial configuration of the first is an isolated, negative, finite spherical gas. In the second a fixed, infinitely thin, positive, permeable, spherical shell is placed concentric and exterior to the negative gas of the first configuration. The charge of the total system is zero. The first system is totally soluble. Application to the special case of uniform initial density gives that the charge density at a point far removed from the initial sphere decays as the inverse cube of time, and is independent of radius. In the second configuration if a is the radius of the positive shell and b the initial radius of the negative gas, then for uniform initial distribution, all charge in the shell b(b/a)½ ≤ r ≤ b escapes. The remaining charge oscillates and is randomized by phase mixing.
Majorization of Feynman Graphs
Authors: Chow Yutze
Published: January 2005

The different methods of majorization of Feynman graphs are reviewed and their respective domains of application and relative merits are discussed. The initial steps of Wu's majorization method are discussed in a variational formalism. Effort is made to present the rather complex arguments involved in Nakanishi's ``path method'' in a systematic manner.
Non-Markovian Model for the Approach to Equilibrium
Authors: Emch Gérard G.
Published: January 2005

This paper is intended to provide the axiomatic study of nonequilibrium quantum statistical mechanics with some simple and rigorously solvable models. The models considered here are obtained as generalizations of the Ising model. They illustrate and allow a rational discussion of the following concepts relevant to the theory of irreversible phenomena: coarse-graining and time-smoothing, ergodicity, recurrences, impossibility of a Markovian description of the approach to equilibrium for some physical systems, justification of the various random phase assumptions, properties of the interaction responsible for the approach to equilibrium, master equations, etc.
Exact Invariants for a Class of Nonlinear Wave Equations
Authors: Kruskal Martin D. , Zabusky Norman J.
Published: January 2005

A method is given for obtaining explicitly an infinite number of exact invariants for a physical system described by the coupled set of first-order hyperbolic partial differential equations∂ui/∂t=Aij(u0,u1,…un) ∂uj/∂x    (i,j=0,1,…,n). Temporal and spatial invariants are constructed as integrals of temporally and spatially invariant densities T and X, over appropriate spatial and temporal intervals, respectively. For physical systems the energy and momentum densities are temporal invariant densities. These invariant densities are solutions of the hodograph transformed equations corresponding to (A1). For the case n = 1 every invariant density T satisfies an equation in conservation form: (Tu0)t − (Tu1)x = 0. The methods are applied to the equationytt−(1+εyx) αyxx=0,and a denumerable infinity of invariant densities, each expressible as a polynomial, are calculated in two equivalent cases: the first when (A2) (with α = 1) is expressed in zero diagonal form ut = vx, vt = (1 + εu)ux, where u = yx and v = yt; and the second when (A2) is expressed in diagonal form rt−Φrx=0,st+Φsx=0, where Φ=(12 ε(2+α)(r+s))α/(2+α) and r and s, the Riemann invariants of (A2), are related to u and v. A theorem of Noether is used to construct from the invariant densities continuous transformation groups that leave the action functional invariant. Using the methods of Kruskal we derive the adiabatic invariant for the continuous system (A2) (α = 1) which has nearly periodic solutions. To order ε the adiabatic invariant is identical with one of the exact invariants and gives no new information about the system.
Global Covariant Conservation Laws in Riemannian Spaces. I
Authors: Shepelavey Bohdan
Published: January 2005

It is demonstrated that in the theory of general relativity integral conservation laws can be obtained from a vanishing covariant divergence of a tensor in a completely covariant form. This is achieved by introducing into tensor calculus a new operation, namely, tensor integration which is defined as an inverse of tensor differentiation. In particular, a one-dimensional absolute integration of a tensor along a curve is defined as an inverse operation to the same type of differentiation. A representation of absolute integration is developed by a perturbation method as an infinite series where each term consists of ordinary integrations only. As an example of absolute integration, a vector field is obtained whose components can be employed as a coordinate frame having a very close resemblance to the Riemannian coordinates. Covariant integration is then introduced as an inverse of covariant differentiation; however, its usefulness is severely limited by the conditions of integrability which have to be satisfied to make covariant integration possible. A set of unspecified covariantly constant base vectors is used to explain the idea of a covariantly constant tensor and to express symbolically absolute integration in terms of the ordinary integration. The one-dimensional absolute integration is then extended to higher dimensions in such a way that it is independent of the order of integrations. Finally, Gauss' theorem is proved for the absolute integration which enables one to convert a volume integral of a covariant divergence of a tensor into a corresponding surface integral of the same tensor.
Cylindrical Gravitational News
Authors: Stachel John J.
Published: January 2005

The concepts of news function and mass aspect are generalized to a class of cylindrically symmetric metrics containing both degrees of freedom of the gravitational field. It is proved that the mass/unit length always decreases if there is any cylindrical news. The asymptotic behavior of the Riemann tensor in the cylindrical case is analyzed and a peeling theorem proved for this case. An example is given to show that asymptotic conditions on the metric or the Riemann tensor which are analogous to the conditions used in the asymptotically spherical case do not exclude certain infinite incoming radiation trains in the cylindrical case. Pure incoming and outgoing solutions are defined for the cylindrical case, and their generalization to the asymptotically spherical case is suggested. An exactly conserved quantity is shown to exist which may be the cylindrical analog of the ten exactly conserved quantities recently discovered by Newman and Penrose.
A Closed Form for Elsasser Integrals
Authors: Wolfson Seymour J.
Published: January 2005

An Elsasser integral is defined as the definite integral of a particular product of three spherical harmonics, namely, ∫ 02π ∫ 0πYn1*m1[∂Yn2m2∂θ∂Yn3m3∂φ−∂Yn2m2∂φ∂Yn3m3∂θ]dθ dφ.

This integral occurs in connection with the ``dynamo theory'' of the magnetic field of the earth and thus it is desirable to have a value of the integral involving only the indices of the spherical harmonics. Such a closed form is developed and some properties of the integral are discussed.
A Class of Representations of the Generalized Bondi—Metzner Group
Authors: Cantoni Vittorio
Published: January 2005

A procedure is given for the construction of a faithful linear representation of the generalized Bondi—Metzner group from each faithful linear representation of the inhomogeneous orthochronous Lorentz group. Unitary representations can be obtained in this way.
Wave Mechanics in Classical Phase Space, Brownian Motion, and Quantum Theory
Authors: Riccia Giacomo Della , Wiener Norbert
Published: January 2005

A wave dynamics of fields φ(p, q; t) ∈ L2(Γ) over the phase space Γ(p, q) of a classical system 𝒮 is derived from the Liouville theorem. We define the energy contained in a given field φ(p, q; t). We show that for a special class of fields, selected on physical grounds, the energy spectrum is given by a time-independent Schrödinger equation. This allows us to associate with 𝒮 an ordinary quantum system Q such that the values of the quantized energy coincide for the fields in the phase space of 𝒮 and for Q. Then we make use of Wiener's stochastic integral based on the theory of Brownian motion to derive probabilities which are the same as those one would obtain through Born's statistical postulate of quantum theory. From this it follows that we can regard normalized fields φ(p, q; t) as ``probability amplitudes'' leading to a probability density function ρ(p, q; t) = φφ* in the sense of Gibbs' statistical mechanics. Our work therefore appears as a bridge between a statistical theory (in the sense of Gibbs) of a mechanical system 𝒮 and the usual quantum theory of the related quantum system Q.
The Definition of States in Quantum Statistical Mechanics
Authors: Emch Gérard G.
Published: January 2005

The controversy relative to the use of normal states in quantum physics is discussed in the light of ergodic theory. The nature of the spectrum of the Hamiltonian is shown to play a central role in the decision to enlarge the ordinary frame provided by the traditional density-matrix formalism. The connection of these considerations with the infinite-time, infinite-volume limits in nonequilibrium statistical mechanics is pointed out.
Wave-Packet Derivation of Field-Theoretic Scattering Amplitude
Authors: Frantz Lee M.
Published: January 2005

A formal expression is derived for the field-theoretic scattering amplitude in a Brillouin—Wigner perturbation expansion. Wave packets are used to introduce the initial conditions, thereby avoiding the necessity of adiabatically switching the coupling constant. The field of incident particles is second-quantized, and the target is first-quantized. The principal improvement on previous derivations is that the number of incident quanta, although finite, is otherwise unrestricted. The result is thus applicable, for example, to the nonrelativistic description of the scattering of a photon beam of arbitrary intensity by an atom or a charged particle.
Generalized Ward Identity and Unified Treatment of Conservation Laws
Authors: Lurié D. , Takahashi Y. , Umezawa H.
Published: January 2005

A technique for deriving conservation laws directly from field equations without recourse to the Lagrangians or Noether's theorem is reviewed and extended. The method allows a simple treatment of the so-called ``generalized'' conservation laws including Lipkin's ``zilch.'' An interesting feature which results from our approach is the existence of conserved currents for discrete as well as continuous symmetries. It is also pointed out that conservation laws do not always follow from the invariance of equation of motion if it is not derivable from a Lagrangian. Finally, we show how our method can be applied to the normalization of wavefunctions of composite particles such as Bethe-Salpeter wavefunctions.
Lattice Statistics of Hydrogen Bonded Crystals. I. The Residual Entropy of Ice
Authors: Nagle J. F.
Published: January 2005

The lattice statistical problem of calculating the residual entropy of ice has been considered in some detail for the hexagonal and cubic ice lattices as well as for a two-dimensional icelike lattice. Even for the two-dimensional lattice, this problem appears to be intractable using exact methods, so an approximation method is in order. The series method of DiMarzio and Stillinger has been developed so that the series is completely characterized by the numbers of various kinds of cycles on the lattice. The first five terms of the series have been evaluated and used to extrapolate values of the residual entropy S(0) within rather narrow limits for all practical purposes. The result for hexagonal ice and cubic ice is S(0) = .8145 ± .0002 cal/deg/mole which agrees with experiment even better than Pauling's original approximation. Some other methods are also discussed, and their results tend to confirm the series results.
Nonrelativistic Quantum Theory of an Electron in an Arbitrarily Intense Laser Field
Authors: Nickle H. H.
Published: January 2005

An essentially exact treatment of the time-dependent Schrödinger equation for a Bloch electron (or a free electron) in the presence of an arbitrarily intense laser field is described. Expressions for the wavefunction, current density, and energy of the electron state are presented in closed form for the case when the effective mass approximation is valid. The limitations of an ``almost exact'' solution of very simple form are investigated, the corrections to the almost exact solution being determined by the WKB approximation method. The exact solution for the wavefunction turns out to be quite different from that given by perturbation theory. However, the changes in the values of the current density and energy due to the presence of the laser field turn out to be, within the limitations imposed by the nonrelativistic nature of the Schrödinger theory, linear and quadratic in the field amplitude, and therefore agree with the results of perturbation theory.
A New Cluster Scheme in Statistical Physics
Authors: Ramanathan G. V.
Published: January 2005

A new cluster scheme is proposed, and some of its advantages over the Mayer expansion method are demonstrated. An explicit equation is derived for the s-particle correlation in a large equilibrium system of particles interacting through a two-body potential. This equation is solved to the leading order in the plasma limit. It is found that, in several systems, the equilibrium s-particle correlation is, to the leading order, a functional of the two-particle correlation, independent of the detailed form of the potential function.
Separation of the Interaction Potential into Two Parts in Statistical Mechanics. II. Graph Theory for Lattice Gases and Spin Systems with Application to Systems with Long-Range Potentials
Authors: Stell G. , Lebowitz J. L. , Baer S. , Theumann W.
Published: January 2005

The methods developed in a previous paper for treating systems with a pair potential of the form v(r) = q(r) + w(r) are here applied to lattice gases (isomorphic to Ising systems). We chose q(r) to be the ``hard-core'' potential preventing the multiple occupancy of a lattice site and w(r) the interaction between two particles (or parallel spins) separated by r. The resulting graphical formalism is similar to that obtained by other authors exclusively for spin systems. We are thus able to connect their work with the general Mayer theory as it was originally applied to fluids and also to find new interpretations for some of the quantities appearing in the spin-system expansion. The formalism is then used in the case where w(r) is a ``Kac potential'' of the form w(r, γ) ∼ γv φ(γ r), where v is the dimensionality of the space considered and γ−1 is the range or w, assumed very large. We then obtain systematic expansions in γ for the correlation functions and thermodynamic properties of the system. These expansions are, however, invalid inside the two-phase region and near the critical point of the ``van der Waals'' system; i.e., a system with γ → 0. To remedy this we introduce a new self-consistent type of approximation which is suggested by our graphical analysis of the γ expansion but is applicable also to systems with general interactions w(r), not necessarily parametrized by γ. The spatially asymptotic behavior of the two-body correlation function at the critical point is then discussed using these graphical methods. From the expansion procedures it seems possible to find specific subsets of graphs which will give any desired asymptotic behavior of the two-body correlation function including known exact ones. However, we could find no a priori reason for the retention of these subsets of graphs to the exclusion of all others.
A Theorem on the Clebsch-Gordan Series in A(l), B(l), C(l), and D(l)
Authors: Zaccaria F.
Published: January 2005

By a general method, conditions are derived for A(l), B(l), C(l), and D(l), in order that terms and multiplicities of the Clebsch-Gordan series for the decomposition of the direct product of two irreducible representations be dependent on only one of them.
The Enumeration of Homeomorphically Irreducible Star Graphs
Authors: Heap B. R.
Published: January 2005

In expressing properties of interacting systems in the form of series, in many cases only summation over star graphs is involved. The identification and classification of such graphs is simplified by reducing them to homeomorphically irreducible stars. These graphs can be regarded as being all the possible different topological types of star. A method is described which has been used to produce all the different homeomorphically irreducible stars which have cyclomatic numbers ≤ 5. In particular, it has been established that there are 118 such graphs with cyclomatic number 5, 111 of which are planar. Diagrams of these graphs are appended, and a table of their k weights, which are needed for obtaining series for percolation processes, is also given. An extension of the method has been used to count the numbers of homeomorphically irreducible stars containing up to 8 vertices and 12 edges, a table of which is given.
On Ordering and Identifying Undirected Linear Graphs
Authors: Nagle John F.
Published: January 2005

A general linear ordering relation is presented which completely orders any subset of the undirected linear graphs with the same number of vertices. The discussion is then specialized to the ordering and the identification of the unlabeled stars with no vertices of degree two, which faithfully represent the basic topologies of all stars.
Normal Thresholds in Subenergy Variables
Authors: Landshoff P. V. , Olive D. I. , Polkinghorne J. C.
Published: January 2005

The discontinuity across the normal threshold in a subenergy variable is evaluated by S-matrix methods. The effect on the discontinuity of continuing past higher Landau singularities is discussed.
The Icecream-Cone Singularity in S-Matrix Theory
Authors: Landshoff P. V. , Olive D. I. , Polkinghorne J. C.
Published: January 2005

The icecream-cone singularity is analysed by S-matrix theory methods. It is found to appear only in that part of the physical region predicted by finite-order perturbation theory. The corresponding discontinuity is derived, and the nature of the singularity is discussed.
Existence of Solutions of N/D Equations
Authors: Atkinson D.
Published: January 2005

The manifold of solutions of certain classes of N/D equations is considered, under the restriction that the scattering amplitude have a uniform bound in all complex directions. Two cases are treated: (i) The N integral equation is Fredholm, but the associated homogeneous equation may have solutions; (ii) the equation has the marginally singular behavior characteristic of an asymptotically constant left-hand cut discontinuity. The existence theorem in the latter case proceeds by the construction of the appropriate resolvent.
Homogeneous Dust-Filled Cosmological Solutions
Authors: Farnsworth David L. , Kerr Roy P.
Published: January 2005

The Einstein field equations with incoherent matter are discussed for the case of homogeneous space-time, i.e., for metrics allowing a four-parametric, simply transitive group of motions. It is proved that the only universes satisfying the above are those of Einstein, Gödel, and Ozsvath.