bluuebunny/crossref_metadata_2025 huggingface dataset:
Clebsch-Gordan Coefficients and Special Function Identities. I. The Harmonic Oscillator Group
Authors: Miller Willard
Published: February 2004

It is shown that by constructing explicit realizations of the Clebsch-Gordan decomposition for tensor products of irreducible representations of a group G, one can derive a wide variety of special function identities with physical interest. In this paper, the representation theory of the harmonic oscillator group is used to give elegant derivations of identities involving Hermite, Laguerre, Bessel, and hypergeometric functions.
Ray Theory of Diffraction by Open-Ended Waveguides. II. Applications
Authors: Lee S. W.
Published: February 2004

This series of papers presents an extension of Keller's diffraction ray method to problems involving two or more parallel plates by introducing a modified diffraction coefficient which takes care of coupling along a shadow boundary automatically. In Part I, a canonical problem was solved and the expression for the modified diffraction coefficient derived. In this part, we give a recipe for how to use this set of rays and illustrate it through several examples including (i) open-end parallel-plate waveguide, (ii) bifurcated waveguide, and (iii) an infinite array of parallel plates. The above three examples represent the only three types of problems in the edge diffraction theory that can be solved exactly by analytical techniques based on complex variables. In this paper it is demonstrated that all three exact solutions can be recovered by the present ray method. Moreover, in some problems where the analytical techniques cannot be conveniently applied, the ray method can often provide a useful approximate solution.
On the Stability of Periodic Orbits for Nonlinear Oscillator Systems in Regions Exhibiting Stochastic Behavior
Authors: Lunsford Gary H. , Ford Joseph
Published: February 2004

A computer has been used to determine the stability character of periodic orbits for the Hamiltonian oscillator system H=12(p12+p22+q12+q22)+q12q2−13q23. Using procedures developed by Greene [J. Math. Phys. 9, 760 (1968)], empirical evidence has been obtained indicating that this system has a dense or near dense set of unstable periodic orbits throughout its stochastic (unstable) regions of phase space. The extent to which such stochastic regions exhibit C-system behavior, i.e., ergodicity and mixing, is discussed. Finally, the above Hamiltonian system is shown to be intimately related to the Fermi-Pasta-Ulam system as well as to the Toda lattice.
About the Solutions of Dirac's Equations in the Presence of New Magnetic Fields
Authors: Hautot André
Published: February 2004

Dirac's equations are completely solved for four entirely new configurations of the external magnetic field. The use of the quaternionic formalism simplifies the problem of separation of variables. For three of those new magnetic fields the radial equations just obtained are not classical. Section II studies them and gives their solutions in term of usual transcendental functions. When energy is quantized, energy levels are given.
Canonical Dynamics of Spinning Particles in Gravitational and Electromagnetic Fields
Authors: Künzle H. P.
Published: February 2004

In terms of the canonical form and the connection form on the bundle of Lorentz frames P over a space-time manifold V, a presymplectic form ω is defined on P, which induces a Poisson bracket on the set of real valued functions on the phase space of the system representing a spinning particle in an exterior gravitational and electromagnetic field. This structure coincides with the unique Poincaré invariant one for the free particle. Moreover, the projections into V of the integral manifolds of the kernel of ω on P yield precisely the world lines of a spinning particle as obtained for the dipole approximation of Dixon's equations of motion for extended test bodies in general relativity.
Gårding Domains and Analytic Vectors for Quantum Fields
Authors: Hegerfeldt Gerhard C.
Published: February 2004

If one studies the canonical commutation relations (CCR's) of quantum field theory in the unitary Weyl form, one does not know if one can find a common dense domain for the field operators since their domain of definition depends on the test function. We consider here a general class of test function spaces including the spaces S and D of Schwartz and the space U0≃R(∞) of all finite linear combinations of a countable basis. It is shown that there exists an invariant Gårding domain D on which all fields are defined and strongly continuous. D consists of analytic vectors for the fields. It turns out that the test function space can be enlarged by continuity. For irreducible or factor representations it becomes even a Hilbert space. The basic idea of the proof is the same as in the Schrödinger representation for one degree of freedom and very transparent. We simply use rapidly decreasing functions in ``Q-space'' and ``P-space'' as smoothing factors. That this can be done in the infinite case also is due to a new and interesting measure theoretic result derived here. As an application of our results, we mention that the renormalized fields (after removing the cutoff) of the Φ22n model of Glimm and Jaffe possess a Gårding domain for test functions in S or D for each time.
Clebsch-Gordan Coefficients and Special Function Identities. II. The Rotation and Lorentz Groups in 3-Space
Authors: Miller Willard
Published: February 2004

It is shown that the construction of concrete models of Clebsch-Gordan decompositions for tensor products of irreducible group representations leads to a wide variety of special function identities. In this paper the representation theory of the rotation and Lorentz groups in 3-space is used to give elegant derivations of identities involving Languerre, Gegenbauer, hypergeometric, and generalized hypergeometric functions. Some of these identities may be new in this general form.
Spectrum Generating Algebras and Symmetries in Mechanics. I
Authors: Hermann Robert
Published: February 2004

Certain differential-geometric and Lie group theoretic facts that are useful in the systematic study and search for spectrum generating algebras are presented.
Local Existence of Solutions to the Equations Γjk;li=0,gij;k=0 When Rjkli≠0
Authors: Muraskin M.
Published: February 2004

We prove that solutions to the equations Γjk;li=0 and gij;k=0 exist locally when Rjkli≠0. Previously this was only shown for the special case of Rjkli=0.
The Weyl Basis of the Unitary Group U(k)
Authors: Seligman T. H.
Published: February 2004

Weyl's method for the construction of irreducible tensors of the unitary group is used to construct a basis for any irreducible representation of U(k) or GL(k) in terms of Bose creation operators. A simple way is indicated to select a complete but not over complete basis from the functions obtained. The basis obtained can be useful in nuclear or molecular calculations, as well as in some mathematical problems.
Decision Procedures in Quantum Mechanics
Authors: Benioff Paul A.
Published: February 2004

The results of an earlier paper on finite and infinite sequences of measurements are here extended to include decision procedures. It is shown that with each decision procedure Q there is uniquely associated a probability operator measure OQ, which gives the statistical properties of Q. None, some, or all of the paths of Q can be infinitely long. A result of this association is that there are two methods of measuring the probability that carrying out Q on a system in state ρ gives an outcome sequence in some set F. A remarkable aspect of this equivalence is that the purely physical operation of one method is equivalent to, or can replace, the physical operation and mathematical decision procedure of the other method.
Harmonic Analysis of Analytic Functions on Hyperspheres
Authors: Beers Brian Lee
Published: February 2004

The real analytic functions on the hypersphere Sn are shown to be in one-to-one correspondence with the family of series of hyperspherical harmonics with exponentially falling coefficients. These functions may be continued onto a larger complex manifold on which they represent holomorphic functions. The convergence of the harmonic expansions for the real analytic functions on Sn is governed by the singularity structure of the continued function on this complex manifold.
Studies in the C*-Algebraic Theory of Nonequilibrium Statistical Mechanics: Dynamics of Open and of Mechanically Driven Systems
Authors: Presutti E. , Scacciatelli E. , Sewell G. L. , Wanderlingh F.
Published: February 2004

We construct a C*-algebraic formulation of the dynamics of a pair of mutually interacting quantum-mechanical systems S̃ and Ŝ, the former being finite and the latter infinite. Our basic assumptions are that: (i) Ŝ, when isolated, satisfies the Dubin-Sewell dynamical axioms; (ii) the coupling between S̃ and Ŝ is energetically bounded and spatially localized; and (iii) the initial states of S̃ and Ŝ are mutually uncorrelated, with S̃ in an arbitrary normal state and Ŝ in a Gibbs state. Our formulation leads to a rigorous theory of (a) the dynamics of a finite open system, i.e., of a finite system (S̃) coupled to an ``infinite reservoir'' (Ŝ), and of (b) the dynamics of an infinite system (Ŝ), driven from equilibrium by a ``signal generator'' (S̃). As regards (a), we show that the state of S̃ always remains normal, and we derive a generalized master equation (in an appropriate Banach space) governing its temporal evolution. As regards (b), we show that the state of Ŝ always corresponds to a (time-dependent) density matrix in the representation space of the algebra of observables for Ŝ, induced by the initial Gibbs state. By formulating the linear part (appropriately defined) of the response of Ŝ to S̃, we generalize the fluctuation-dissipation theorem to infinite systems. Further, we show that the total effect of S̃ on Ŝ reduces to that of a ``classical'' time-dependent external force in cases where the initial state of S̃ possesses certain coherence properties similar to those of the Glauber type.
An O(2, 1) × O(3) Solution to a Generalized Quantum Mechanical Kepler Problem
Authors: Cunningham M. J.
Published: February 2004

A differential equation directly related to the generalized Kepler equation of Infeld and Hull is solved in an O(2, 1) × O(3) group scheme. This equation contains as special cases the Schrödinger, Klein-Gordon, and Dirac (two forms) hydrogen atoms. A generalized Pasternack and Sternheimer selection rule exists and some matrix elements can be evaluated group theoretically.
Amplitude Dispersion and Stability of Nonlinear Weakly Dissipative Waves
Authors: Cap Ferdinand F.
Published: February 2004

A method to solve nonlinear dissipative wave equations using ideas of Luke, Krylov, and Bogolyubov is presented. The method is compared to Whitham's theory. Dispersion relations for nonlinear dissipative waves, including amplitude dispersion, are discussed. Furthermore, stability problems of such waves are investigated.
Local and Covariant Quantization of Linearized Einstein's Equations
Authors: Bracci L. , Strocchi F.
Published: February 2004

An analysis is given of all the possible quantizations of the linearized Einstein equations in terms of a weakly local and/or covariant potential hμν(x). The discussion is done without making the apparently arbitrary choices which characterize the standard formulations, and special attention is paid in proving those general features which follow from basic principles and are therefore common to all local and/or covariant formulations. It is shown that the requirement of locality and/or covariance alone implies that the Einstein equations cannot hold as mean values on a dense set of states, and therefore unphysical states must be introduced in an essential way. Moreover, the requirement that the Einstein equations hold as mean values on the physical states forces the existence of states of negative norm in order to define hμν as a local and/or covariant operator. Thus the characteristic features of Gupta's formulation are shown to be shared by any local and/or covariant theory. The arbitrary choices which occur in the representation of the field operator hμν, in the definition of the metric operator and in the choice of the subsidiary condition which identifies the physical states, are shown to lead to only a one-parameter family of theories. They can be classified according to the subsidiary condition (∂μhμν+q∂νhμμ)+Ψ=0,each q ≠ −¼ identifying a possible theory. This arbitrariness, which makes the literature on the subject rather confusing, is resolved by proving that all the theories are (isometrically) equivalent. Such formulations are discussed in the framework of axiomatic quantum field theory, with particular emphasis on their group theoretical contents. Finally, an extensive treatment of Gupta's formulation is given along the lines discussed by Wightman and Gärding for quantum electrodynamics.
Stability of Finite-Sized Radiating Plasmas
Authors: Manley O. P. , Treve Y. M.
Published: February 2004

It is shown that the onset of instabilities in a finite-sized, optically thin, radiating plasma coincides with the loss of uniqueness of solutions to the governing nonlinear differential equations. Furthermore, conditions on the luminosity function are derived for the existence of metastable states of such plasmas with dimensions smaller than the critical maximum size derived from conventional normal mode stability analysis. The results are applied to a simple configuration modelling free-free emission from a confined radiating, high-temperature plasma.
The Reproducing Kernel Method. II
Authors: Devooght J.
Published: February 2004

The explicit solution of the Cauchy problem ∂N/∂t = HN by means of reproducing kernels is obtained under various forms: conformal mapping expansions, Sheffer polynomial expansion, polynomials orthogonal on a family of curves; the convergence is studied for both Szegö and Bergman kernels.
Dynamics of Harmonically Bound Semi-Infinite and Infinite Chains with Friction and Applied Forces
Authors: Lee K. H.
Published: February 2004

The dynamics of semi-infinite and infinite linear chains of identical masses and ideal springs is studied. In addition to the harmonic coupling between nearest neighbors, each particle is harmonically bound to ts equilibrium position and is subject to friction and time-dependent applied forces. The Laplace transform method is used to express the motion of all the particles. The exact solutions are found and discussed for four different cases: (a) an infinite chain, (b) a semi-infinite chain, (c) a semi-infinite chain with the position of the end particle specified as a function of time, and (d) an infinite chain with the position of one particle specified as a function of time. By specializing some results of the present work, those of previous calculations on simpler systems by other authors are recovered.
Physical Interpretation of Higher-Derivative Field Theories
Authors: Riewe F. , Green A. E. S.
Published: February 2004

Polynomial-type field equations are shown to have a realistic physical interpretation in terms of particle form factors, both for classical fields and for ``dipole-regularized'' quantized fields. Form factors arising from such field equations are found to give a reasonable description of the electromagnetic structure of the proton.
Maxwell's equations and complex Minkowski space
Authors: Newman Ezra T.
Published: February 2004

It is shown how, from the invariance of Maxwell's equations under the complex Poincaré group, new solutions can be obtained from already known ones. The technique is illustrated with the Coulomb solution. If the same ideas are applied to the linearized Einstein-Maxwell equations a simple classical ``derivation'' of the Dirac value of the gyromagnetic ratio is obtained.
Physical applications of multiplicative stochastic processes
Authors: Fox Ronald Forrest
Published: February 2004

The theory of multiplicative stochastic processes has been shown to lead to a density matrix description of nonequilibrium quantum mechanical phenomena. In the present paper a detailed treatment of the approach to the uniform, microcanonical, and canonical equilibrium density matrices is presented. The canonical equilibrium density matrix is approached by the density matrix which represents a subsystem in contact with a constant temperature heat reservoir.
Application of Kraichnan's direct interaction approximation to kinematic dynamo theory. I. Incompressible isotropic turbulence and a singular integral equation
Authors: Lerche I.
Published: February 2004

Using Kraichnan's direct interaction approximation in the induction equations we set up the kinematic equations governing behavior of an ensemble average magnetic field under homogeneous, incompressible, mirror symmetric isotropic velocity turbulence. We demonstrate that the normal modes of the field depend on the solutions to a nonlinear single integral equation. For a simple form of the velocity turbulence we have investigated some of the properties of the integral equation. In particular, we have been able to construct a particular solution. We also point out what remains to be done if we are to obtain all the modes of the ensemble average magnetic field. We have done this calculation so that the behaviour and normal modes of the field can be investigated at arbitrary magnetic Reynolds' numbers. This is in contrast to customary approximations (like first-order smoothing theory) which normally are valid only for very small Reynolds numbers, if at all, and which therefore omit large regimes of considerable physical interest.
The bra and ket formalism in extended Hilbert space
Authors: Prugovečki Eduard
Published: February 2004

The concept of bra-vector is endowed with a precise mathematical meaning by taking a Hilbert-space H (whose elements are the ket-vectors) and embedding it into a larger bra-space L. The space L is constructed by fitting together all the different spaces with ``negative'' norm corresponding to equipping operators D in H. It is shown that the formal manipulations of Dirac's formalism become theorems on the resulting extended Hilbert space structure H⊂L. On L we define topologies and bra-adjoints of operators in H and investigate their properties. We show how these results can be used in deriving rigorous versions of the typical relations involving distorted waves in time-independent scattering theory. By applying this formalism to Fock space we obtain an extended Fock space framework suitable for the rigorous formulation of the concept of a field at a point in configuration or momentum space.
A symmetry of Petrov-type N spaces
Authors: Sigal R.
Published: February 2004

It is shown that for empty Petrov-type N spaces the Lie derivative in the principal null direction of the Weyl conformal tensor and its dual is equal to a conformal transformation and a duality rotation of the original tensor. This type of symmetry is studied for general relativity and electromagnetism.
Radial Jost functions in scattering theory
Authors: Nesbet R. K.
Published: February 2004

Several methods have recently been proposed for representing oscillatory wavefunctions by relatively slowly varying modulations of known oscillatory functions. Direct computation of the modulating function leads to efficient numerical or variational procedures and to accurate interpolation over energy or other parameters of a scattering problem. A new method, based on the phase integral (WKB) formalism, is proposed here in a form applicable to multichannel scattering. The method generalizes the phase integral method to make use of arbitrary oscillatory comparison functions, rather than just plane waves as in the usual formalism. It makes use of the modulating factor of the radial Jost function. Several examples of the proposed method are given, including an application to a two-channel model problem.
Invariants of the equations of wave mechanics: Rigid rotator and symmetric top
Authors: Anderson Robert L. , Kumei Sukeyuki , Wulfman Carl E.
Published: February 2004

Applying the systematic method discussed in previous papers, we derive the invariants and the groups of the time-dependent Schrödinger equations for the rigid rotator and the symmetric top. The groups for these systems are found to be SO(3,2) (rigid rotator) and SU(2,2) (symmetric top). For the case of the symmetric top, it is found that under the symmetry breaking I1 = I2 = I3 → I1 = I2 ≠ I3, where I1, I2, and I3 are the moments of inertia of the top, two of the time-independent constants of the motion become time-dependent constants of the motion.
Positivity for some generalized Yukawa models in one space dimension
Authors: Federbush Paul
Published: February 2004

For the model (gψ̄ψφN+φ2M)1+1 in a box the energy is bounded below if M > N.
Scattering from a random rough surface: Diagram methods for elastic media
Authors: DeSanto John A.
Published: February 2004

In a previous paper [J. Math. Phys. 13, 1903 (1972)], Feynman diagram methods were used to construct the Green's function and its first two moments for scalar wave scattering from a random rough surface with a Neumann boundary condition. This paper extends these formal diagram methods to the calculation of the Green's function and its first two moments for an elastic half-space bounded by a random rough surface considered as a free boundary. The surface height is a single valued bounded function with Gaussian statistics. The Dyson and Bethe-Salpeter equations, for the mean and second moment respectively, of the Green's function, are derived. Some simplifications of these integral equations and some examples are presented.
Kinetic theory of a weakly coupled and weakly inhomogeneous plasma in a magnetic field
Authors: Tjo/tta Jacqueline Naze , O/ien Alf H.
Published: February 2004

A study of the evolution toward continuum of a weakly coupled, weakly inhomogeneous two-component electron-ion plasma in an external electromagnetic field using the multiple-time-scale method is presented. An electron-ion mass ratio parameter and a weak inhomogeneity parameter are introduced. Kinetic and macroscopic equations for a certain ordering between these parameters are obtained. The equations are solved when using simplified collision terms.
Asymptotic behavior of spacing distributions for the eigenvalues of random matrices
Authors: des Cloizeaux J. , Mehta M. L.
Published: February 2004

It is known that the probability Eβ(0, S) that an arbitrary interval of length S contains none of the eigenvalues of a random matrix chosen from the orthogonal (β = 1), unitary (β = 2) or symplectic (β = 4) ensemble can be expressed in terms if infinite products ∏n=0∞[1−λ2n(S)] and ∏n=0∞[1=λ2n+1(S)], where λn(S) is an eigenvalue of a certain integral equation. Using values of λn(S), valid for S large, obtained in connection with a recent study of spheroidal functions, we derive asymptotic expressions (S ≫ 1) for Eβ(0, S).
Optimal analytic extrapolation for the scattering amplitude from cuts to interior points
Authors: Ciulli S. , Nenciu G.
Published: February 2004

Given a data function together with an error corridor for the scattering amplitude along some finite part of the cuts, one can construct effectively the whole set of analytic functions (``admissible amplitudes''), compatible with these conditions and bounded by a certain number M on the remaining part of the cuts. Depending on the actual value of an important constant ε0 computed from the data function and the bound M, this set may be void. If not, in every point of the cut plane the set of values of the admissible amplitudes fills densely a circle; explicit formulas are given for its radius η̂(z) and center f̂(z), the latter being the best possible estimate for the whole set. In contrast to the linear extrapolation obtained by Poisson weighted dispersion relations, here nonlinear functional methods were used. This paper contains an appendix written by Professor C. Foias, on some functional analytical methods used in connection with the computation of the numerical value of the constant ε0.
Nonlinear Lee model. II
Authors: Calogero F.
Published: February 2004

The nonlinear Lee model is characterized by the Hamiltonian H = H0 + f(HI), where f(x) is a largely arbitrary (real) function and H0 resp. HI are the free resp. interaction parts of the usual Lee model. The exact solutions of this field-theoretical model, in some sectors of its Hilbert space, are explicitly displayed; they are similar to, but somewhat richer than, the corresponding solutions of the usual Lee model. The V - Nθ and V N - N Nθ sectors are treated in detail.
Simplex transformations of the spin-1/2 Ising model. I
Authors: Wood D. W. , Griffiths H. P.
Published: February 2004

The problems of developing power series expansions for the thermodynamic functions of the spin-1/2 Ising model are discussed. The basis of the discussion is a generalized Ising-type model which incorporates collective potentials between special groups of k particles on the lattice sites. The addition of these k-body potential functions throws a new light onto the problems of developing series expansions for the conventional spin-1/2 Ising model. A system of transformation equations connecting the high and low temperature series developments of the model is obtained which contains the well-known high-low transformation of the Ising model. It is likely that this generalized transformation will provide a useful additional technique in the theory of the conventional Ising model, as well as providing a basis for the discussion of other lattice models which exhibit critical phenomena.
Power statistics for wave propagation in one dimension and comparison with radiative transport theory
Authors: Kohler W. , Papanicolaou G. C.
Published: February 2004

We consider a one-dimensional medium with random index of refraction or a transmission line with random capacitance per unit length, allowing for impedance mismatch at the load and generator. We compute the expected value of the incident and reflected powers at any point between the generator and load in the limit of weak fluctuations and a long line. The results are compared with those of radiative transport theory and discrepancies show the limitations of that theory. Finally, we consider the spreading of pulses due to random fluctuations.
Diffusion, Einstein formula and mechanics
Authors: Emch Gérard G.
Published: February 2004

A mathematically rigorous discussion of the diffusion equation and of its connection with the Einstein relation linking the diffusion coefficient and the velocity autocorrelation function is presented. Diffusion is then propounded as a typical case for which the logical consistency of a purely mechanistic theory of dissipative phenomena can be established.
Parastatistics and the quark model
Authors: Bracken A. J. , Green H. S.
Published: February 2004

An extension of ordinary parastatistics is considered which makes use of all the representations of the parastatistics algebra obtained from the usual ansatz. Govorkov's demonstration that such an extension, for parastatistics of order 2, implies a U(2) symmetry, is generalized for parastatistics of order p. The parastatistics algebra, restricted to N dynamical states, is characterized by the irreducible representations of U(N), S O(2N), and S O(2N + 1) which it contains. It is shown that these representations have multiplicities equal to the dimensions of associated representations of U(p), O(p) and C(p), respectively, where C(p) is a subalgebra of the enveloping algebra of O(p), but is not a Lie algebra. The symmetric group S(p) also appears, as a subalgebra of the enveloping algebra of C(p). It is shown how a nondegenerate vacuum state may be defined for the generalized parastatistics algebra of order p, and how to construct state vectors corresponding to arbitrary numbers of quarklike particles and antiparticles. Such states belong to irreducible representations of U(N), and can be obtained by the application of one kind of creation and annihilation operators to certain basic states, here called reservoir states, which correspond to the different irreducible representations of S O(2N + 1). The specialization to parastatistics of order 3 is discussed in detail with the application to a quark model of the hadrons in view. It is shown how to define isospin and hypercharge in a significant way in this model, which, however, differs in some respects from Gell-Mann's well-known 3-fermion model, and also from Greenberg's 3-parafermion model. Some of the physical implications are examined.
A restricted Bäcklund transformation
Authors: McLaughlin David W. , Scott Alwyn C.
Published: February 2004

The Bäcklund transformation provides a mathematical tool which displays the interaction of solitons. Here a simple, systematic Bäcklund formalism is introduced which permits the explicit construction of these transformations for a restricted class of nonlinear wave equations. Traditionally a Bäcklund transformation has been viewed as a transformation of a solution surface of a partial differential equation into another surface which may not satisfy the same equation. In the present paper the term ``restricted Bäcklund transformation'' (hereafter abbreviated R-B) is used to refer to the case in which the transformed surface does satisfy the same equation. This formalism clarifies the nature of those transformations which have already been used to study nonlinear interactions in many physical problems. The formalism is introduced through a form of the linear Klein-Gordon equation. For this linear example a complete set of Fourier components is generated by a sequence of R-B transformations. This concrete example also indicates the type of results one can expect in the nonlinear case. For the nonlinear equation φx y = F(φ), a theorem is established which states that R-B transformations exist if and only if the nonlinearity F(·) satisfies F″ = κF, where κ is a constant. For such nonlinearities, the R-B transformations are explicitly constructed and are used to display exact nonlinear interactions. A relationship between the condition F″ = κF, the existence of an infinite number of conservation laws, and the transformation theory is briefly discussed.
Quantization of a general dynamical system
Authors: Ben-Abraham S. I. , Lonke A.
Published: February 2004

Invariance implies that the momentum operator should be represented by the covariant derivative. For a general dynamical system, both covariant canonical and path integral quantization yield in the Schrödinger equation a correction term proportional to the curvature. Earlier erroneous calculations are hereby corrected and reconciled.
Motion in radial magnetic fields
Authors: Hautot André
Published: February 2004

We study the problem of the motion of a charged particle in radial magnetic induction fields of the type B = h(θ)r/r3. A Coulomb electric field is also added. The mechanics of Newton, Einstein, Schrödinger, and Dirac are successively considered in view of exact solutions. The analogies between the four treatments are emphasized.
Off-energy-shelltmatrix for local potentials with nonlocal square well core interaction
Authors: Zohni O.
Published: February 2004

An analytic expression is obtained for the s-wave t matrix of a nonlocal square well core interaction combined with an outside local square well. The result is compared with the t matrix of the hard-core square well. In particular, the t matrix obtained is found to remain finite for large values of the energy parameter in contrast to the hard-core square well t matrix.
Quantum mechanics of the anharmonic oscillator
Authors: Halpern Francis R.
Published: February 2004

The dependence of the energy levels and the value of the matrix elements of the momentum and position operators of a quantum anharmonic oscillator are studied as functions of the quantum number and the strength of the anharmonic term. The principal technique employed is the construction of a canonical transformation.
Spaces admitting gravitational fields
Authors: Marathe Kishore B.
Published: February 2004

We start with a modern version of Einstein's definition of a gravitational field. Tensors of curvature type and the curvature product of symmetric tensors are defined. The interaction tensor is defined as the curvature product of the fundamental tensor and the energy-momentum tensor. The tensor W obtained by coupling the Riemann tensor and the interaction tensor is used to obtain a characterization of gravitational fields. The linear transformation of the space of second-order differential forms, induced by W, is used to give a new definition of a gravitational field. The field equations are expressed in terms of the gravitational sectional curvature function f. Thorpe's theorem characterizing Einstein spaces is obtained as a corollary. New formulations of the field equations are used to solve the problem of classification of gravitational fields. The mathematical foundations of the theory of classification are examined and a geometric interpretation of classification is obtained by using the critical point theory on a suitable manifold.
Coset symmetrization operators
Authors: Gilmore Robert
Published: February 2004

Coset symmetrization operators are presented and their properties discussed. Applications are indicated.
Boson realization of the unitary symplectic group
Authors: Quesne C.
Published: February 2004

A realization of the Lie algebra of the unitary symplectic group is proposed in terms of boson operators. The existence of a noncompact orthogonal group, complementary to the symplectic one, is shown. The highest weight polynomial of an irreducible representation of the latter group is constructed with the help of the former.
The evaluation of lattice sums. I. Analytic procedures
Authors: Glasser M. L.
Published: February 2004

Procedures are given for summing series of the form Σaijk …(i2 + j2 + k2 + …)−S and used to evaluate a number of planar lattice sums in terms of the zeta and related functions.
On the relaxation to quantum-statistical equilibrium of the Wigner-Weisskopf atom in a one-dimensional radiation field. IV. Exact solution for finite systems
Authors: Davidson Russell , Kozak John J.
Published: February 2004

A full study, both analytic and numerical, is made of an exact solution, given in a previous paper of the series, for the problem of the spontaneous emission of a Wigner-Weisskopf atom in a one-dimensional radiation field, when the system is considered to be finite in extent. The solution is obtained directly from the Schrödinger equation of the problem. The numerical solution and results are compared extensively with two separate weak-coupling approximations, treated earlier in the series, which were derived respectively from the Prigogine-Résibois master equation and from the solution of the Schrödinger equation. It is found that the latter corresponds much better, except for exceedingly small systems, with the exact results, and that it accordingly takes better account of the effects due to the nonanalyticity of the solution when the coupling tends to zero. Some proposals are made for the general exploitation in nonequilibrium statistical mechanics of this feature, and also for a possible application to the study of radiationless transitions in molecules.
On the relaxation to quantum-statistical equilibrium of the Wigner-Weisskopf atom in a one-dimensional radiation field. V. Exact solution for infinite systems
Authors: Davidson Russell , Kozak John J.
Published: February 2004

In pursuit of the study begun in the preceding paper of the spontaneous emission of a Wigner-Weisskopf atom in a one-dimensional radiation field, the exact solution of this problem in the thermodynamic limit of a system of infinite extent is examined. The analysis reveals the existence in general of three contributions to the probability of the atom's being excited at any moment: a constant contribution and two sorts of time-dependent ones, exponential and nonexponential. For a specific choice of coupling function, this probability is written down in closed form. The link between the constant contribution and the appearance of ghost states and ergodicity is investigated, and criteria given to determine when this contribution does not arise. A numerical study is made of the expressions obtained, and the nonexponential part of the solution compared with approximations for it which have been previously obtained by various means. It is concluded that the predictions of the weak-coupling Prigogine-Résibois master equation are inadequate except when the model is ergodic in a certain sense, and some suggestions are made concerning the relevance of these studies to nonequilibrium statistical mechanics.
On the Wigner coefficients of Sp(4) and SO(5)
Authors: Holman Wayne J.
Published: February 2004

The matrix elements of totally symmetric tensor operators in Sp(4), the spinor covering group of SO(5), are obtained by direct factorization of the corresponding Sp(4) * Sp(4) matrix element. The general Sp(4) * Sp(4) matrix element is factorized, and coefficients are obtained which possess a degree of symmetry higher than that of the canonical Wigner coefficients of Sp(4), but which do not form an orthonormal set in the multiplicity space. The canonical Wigner coefficients can, in principle, be derived from those obtained in different ways by a Schmidt orthogonalization process.
Global analysis of the Kerr-Taub-NUT metric
Authors: Miller J. G.
Published: February 2004

The Kerr-Taub-NUT metric is a local analytic solution of the vacuum Einstein-Maxwell equations. When the metric is expressed in Schwarzschild-like coordinates, two types of coordinate singularity are present. One occurs at certain values of the ``radial'' coordinate where grr becomes infinite and corresponds to bifurcate Killing horizons; the other occurs at θ=0,π, where the determinant of the components of the metric vanishes. It is shown that for nonzero NUT parameter the fixed points of the bifurcate Killing horizons and the degeneracies at θ=0,π cannot all be covered in one manifold. A maximal analytic manifold is constructed which covers the degeneracies at θ=0,π. It is non-Hausdorff but contains maximal Hausdorff subspaces, topologically S3×R, which reduce to Taub-NUT space for vanishing Kerr parameter. Kerr-Taub space can be interpreted as a closed, inhomogeneous electromagnetic-gravitational wave undergoing gravitational collapse. Another maximal analytic manifold is constructed which covers the fixed points of the bifurcate Killing horizons and the degeneracy at θ=0. It is suggested that this manifold represents the superposition of the Kerr geometry and a massless source of angular momentum at θ=π characterized by the NUT parameter.
Canonical transformations and accidental degeneracy. I. The anisotropic oscillator
Authors: Louck J. D. , Moshinsky M. , Wolf K. B.
Published: February 2004

The problem of accidental degeneracy in quantum mechanical systems has fascinated physicists for many decades. The usual approach to it is through the determination of the generators of the Lie algebra responsible for the degeneracy. In these papers we want to focus from the beginning on the symmetry Lie group of canonical transformations in the classical picture. We shall then derive its representation in quantum mechanics. In the present paper we limit our discussion to the anisotropic oscillator in two dimensions, though we indicate possible extensions of the reasoning to other problems in which we have accidental degeneracy.
The one-dimensional X-Y model in inhomogeneous magnetic fields
Authors: Case K. M. , Lau C. W.
Published: February 2004

The one-dimensional X - Y model in various kinds of inhomogeneous magnetic fields is analyzed. In each case we find the spectrum and eigenfunctions of the system, and study the equilibrium and relaxation behavior of the magnetization at any spin site.
Oscillator phase states, thermal equilibrium and group representations
Authors: Aharonov Y. , Lerner E. C. , Huang H. W. , Knight J. M.
Published: February 2004

Eigenstates of the annihilation type operator U = C + iS, where C and S are the ``cosine'' and ``sine'' operators for harmonic oscillator phase, are shown to be closely related to thermal equilibrium states of the oscillator and to provide a new interpretation of the thermal equilibrium density operator. The problem of creating such states is considered and a general theorem is established leading to the construction of interaction Hamiltonians which transform the eigenstates of U among themselves and, in particular, create them from the oscillator ground state. These Hamiltonians lead to representations of the Lie algebras of O(2,1) and O(3). It is suggested that the mathematical technique used, in which generalized U-type operators provide the link between a group and its representations, has its own intrinsic interest for the study of Lie groups.
Stochastic particle trajectories in turbulent flow
Authors: Tsokos C. P. , Telionis D. P.
Published: February 2004

The random position of a particle in turbulent flow r(t;ω) is a vector random function of time t ≥ 0, ω ∈ Ω, where Ω is the supporting set of the underlying probability space (Ω,A,μ). If the statistics of a Lagrangian velocity field is known, the random position vector r(t;ω) is the solution of a nonlinear stochastic integral equation. Depending on the particular application, this equation may be homogeneous or inhomogeneous and with deterministic or random limits. A theorem is presented on the existence and uniqueness of the solution of this integral equation. The proof is based on Banach's fixed point theorem.
Central decomposition of invariant states: Applications to the Galilei group
Authors: Loupias G. , Mebkhout M.
Published: February 2004

We follow the investigation initiated in an earlier paper. Here we treat the case of Galilei group G. The complete description of all closed subgroups H of G is given. Then among them we describe those such that the corresponding homogeneous spaces G/H bear a bounded invariant measure.
On Killing tensors and constants of motion
Authors: Sommers Paul
Published: February 2004

Some general properties belonging to constants of motion for geodesics and charged particle orbits are derived. A constant of motion for geodesics is seen to be a function on the cotangent bundle which has vanishing Poisson bracket with the ``energy function'' determined by the metric tensor. The resulting algebraic structure on the set of constants of motion is closely related to the Lie algebra of Killing tensors. Each constant of motion is shown to provide a family of mappings of geodesics into geodesics. Constants of motion for charged particles also possess a Lie algebra structure. The relationship of Killing tensors to charged particle constants of motion is derived. The linear and quadratic constants of motion for charged particle orbits in the charged Kerr metric illustrate the results. Examples of valence 2 Killing tensors are given in an appendix.
Exact envelope-soliton solutions of a nonlinear wave equation
Authors: Hirota Ryogo
Published: February 2004

Exact N-envelope-soliton solutions have been obtained for the following nonlinear wave equation, i∂ψ/∂t + i3α|ψ|2 ∂ψ/∂x + β∂2ψ/∂x2 + iγ∂3ψ/∂x3 + δ|ψ|2ψ = 0, where α, β, γ and δ are real positive constants with the relation αβ = γδ. In one limit of α = γ = 0, the equation reduces to the nonlinear Schrödinger equation which describes a plane self-focusing and one-dimensional self-modulation of waves in nonlinear dispersive media. In another limit, β = δ = 0, the equation for real Ψ, reduces to the modified Korteweg-de Vries equation. Hence, the solutions reveal the close relation between classical solitons and envelope-solitons.
Exact N-soliton solutions of the wave equation of long waves in shallow-water and in nonlinear lattices
Authors: Hirota Ryogo
Published: February 2004

Exact N-soliton solutions have been obtained for the nonlinear wave equation ∂2w/∂t2 − ∂2w/∂x2 − 6(∂2w2/x2) − ∂4w/∂x4 = 0 which describes motions of long waves in one-dimensional nonlinear lattices and in shallow-water under gravity. The solutions have the same functional form as N-solition solutions of the Korteweg-de Vries equation.
Correlation inequalities and phase transition in the generalized X-Y model
Authors: Suzuki Masuo
Published: February 2004

Correlation inequalities <σAz>≥0, <σAzσBz>≥<σAz><σBz>, ∂<σAz>/∂JBz≥0, and ∂<σAz>/∂JBx≤0 are proved for the generalized X-Y model with the Hamiltonian of the form H=−Σ(JAzσAz+JAxσAx), where σAz=ΠjεAσjz, σAx=ΠjεAσjx, JAz≥0,JAx≥0, A denotes an arbitrary subset of the N lattice points, and σjx, σjz are the Pauli matrices. This yields a simple extension of the Griffiths-Kelly-Sherman inequalities to the above quantal system. Applications to phase transitions are also discussed briefly.
Weyl systems for the (φ4)3 model
Authors: Fabrey James D.
Published: February 2004

Two Weyl systems, developed independently by Hepp and the author in connection with the (φ4)3 model, are shown to be equal. A cyclic vector is exhibited.
Unidirectional energy transfer in nonlinear wave-wave interactions
Authors: Wang P. K. C.
Published: February 2004

In this paper, the wave-energy transfer in a system of three nonlinearly interacting waves is studied. Sufficient conditions for the unidirectional transfer of energy into any wave from the remaining waves and bounds for the wave amplitudes are established. Also, a relation governing the rates of energy transfer between the waves is derived. The results are applied to a magnetized plasma with three nonlinearly interacting waves.
Analytic structure of the Laplace transforms of distributions with restricted support
Authors: Screaton G. R. , Truman A.
Published: February 2004

A relationship between the support property of a distribution and the analytic structure of its Laplace transform is obtained. This relationship is more general than the usual tube theorems.
Product-vector basis and occupation-number basis in Fock space for bosons and fermions
Authors: Leaf Boris
Published: February 2004

A basis of products of one-particle vectors (symmetric or antisymmetric for bosons or fermions) is defined, and related to the occupation-number basis in an abstract Fock space, a generalization of the wave-mechanical configuration representation of the space of second quantization. Creation and destruction operators are simply defined in the product-vector basis and shown to have their usual properties in occupation-number representation. Algebraic properties of product vectors are developed and illustrated. The representation of operators in the product-vector basis is described, together with a brief discussion of the effects on these operators of transformations from one set of one-particle vectors to another. A simple treatment of density and reduced density operators for systems of bosons or fermions is given in the product-vector basis. The relationship between degeneracy in the Fock-space spectrum of an additive operator NA1 and degeneracy in the spectrum of one-particle operator A1 is obtained.
Note on Green's functions for open lattices
Authors: Montet G. L.
Published: February 2004

The method of Horiguchi is modified to discuss the case in which functions become infinite. The modified method is then used to derive Green's functions for the diamond lattice from those for the face-centered-cubic lattice.