bluuebunny/crossref_metadata_2025 huggingface dataset:
Renormalization group for a system of continuous spins on a lattice
Authors: Subbarao K.
Published: March 2003

The renormalization group for a field theory on a lattice, or equivalently, in the language of statistical mechanics, a system of continuous spins on a lattice, described by Wilson, is considered in detail. The conditions under which a class of transformations are renormalization group transformations are studied.
On the duality condition for quantum fields
Authors: Bisognano Joseph J. , Wichmann Eyvind H.
Published: March 2003

A general quantum field theory is considered in which the fields are assumed to be operator-valued tempered distributions. The system of fields may include any number of boson fields and fermion fields. A theorem which relates certain complex Lorentz transformations to the TCP transformation is stated and proved. With reference to this theorem, duality conditions are considered, and it is shown that such conditions hold under various physically reasonable assumptions about the fields. Extensions of the algebras of field operators are discussed with reference to the duality conditions. Local internal symmetries are discussed, and it is shown that these commute with the Poincaré group and with the TCP transformation.
Lie theory and separation of variables. 10. Nonorthogonal R-separable solutions of the wave equation ∂t
tψ=Δ2ψ
Authors: Kalnins E. G. , Miller W.
Published: March 2003

We classify and discuss the possible nonorthogonal coordinate systems which lead to R-separable solutions of the wave equation. Each system is associated with a pair of commuting operators in the symmetry algebra so(3,2) of this equation, one operator first order and the other second order. Several systems appear here for the first time.
On a phase interchange relationship for composite materials
Authors: Schulgasser Kalman
Published: March 2003

A theorem exists relating the transverse conductivity of a fiber reinforced material in a determinate manner to the conductivity of the composite with the phase properties interchanged. It is shown that no such theorem can exist in the three-dimensional case, e.g., for a statistically isotropic composite material. However, an inequality is established relating the two effective conductivities.
Dynamical quantization of the Kepler manifold
Authors: Onofri Enrico
Published: March 2003

The dynamical quantization of the ’’Kepler manifold’’ in any number of degrees of freedom is constructed. The Kepler manifold is the phase space of the regularized Kepler motion and is shown to be an SO(n,2) -homogeneous symplectic manifold, corresponding to an extremely singular orbit in the co-adjoint representation; the quantization is obtained by ’’approximating’’ this orbit by more regular ones, which are equivalent to homogeneous bounded domains. The most relevant result is that the usual quantum-mechanical ’’hydrogen atom’’ model is recovered in the particular representation introduced by Fock in 1935 [SO(n) -homogeneous integral equation in momentum space].
Conformal algebra in superspace and supergauge theory
Authors: Freund Peter G. O.
Published: March 2003

The conformal algebra in a superspace with orthosymplectic metric is found to be an orthosymplectic algebra with two extra Bose dimensions. It is argued that all supergauge fields are also Nambu–Goldstone fields. In supergauge theory the nature of the gauged internal symmetry is found to be severely restricted: It must be O(2N), where N is the number of fundamental Dirac fermions or alternatively one-eightth of the number of Fermi dimensions of superspace.
Matrix elements of multiple electron exchange between spin functions of maximum multiplicity
Authors: Girardeau M. D.
Published: March 2003

A general formula for such matrix elements is obtained, in terms of a sum of terms each proportional to the square root of a product of eight binomial coefficients.
Multipole expansion of the density of states about a crystal cell
Authors: Strinati G. , Fano U.
Published: March 2003

We construct the expansion of a Bloch wave with energy E into a complete set of multipole waves around a ’’center’’ of a crystal as an analog of the expansion in spherical waves in free space. Crystal point group symmetry is used to classify the set. The density of states in a cell is then analyzed into multipole components whose magnitude depends on the cell’s distance from the ’’center.’’
Stationary scattering for N-body systems involving Coulomb potentials
Authors: Zorbas J.
Published: March 2003

A stationary Hilbert space scattering theory is derived for N-body systems involving Coubomb-like potentials. The derivation is based on stationary representations of the α-channel renormalized wave operators, Ω(α)±, having the form Ω(α)±= s-limε→+∞ W(α)±εP(α) F(α)±ε, where W(α)±εP(α) are the stationary operators which form the basis of short-range stationary scattering theory and F(α)±ε* are appropriate stationary ’’renormalization’’ terms.
A class of field theories with unique well-defined functional integration
Authors: Ringwood G. A.
Published: March 2003

A class of field theories is considered where the codomain of the field is taken to be a compact Hausdorff topological group G. The product space GRn is then a compact Hausdorff topological group GRn and as such there exists a unique measure on this space.
Critical-exponent inequalities in the Ornstein–Zernike theory
Authors: Alexanian Moorad
Published: March 2003

A recent generalization of the classical Ornstein–Zernike theory of critical scattering is used to derive inequalities for the critical-point exponents which characterize a first-order phase transition. These inequalities for fluids are found to differ from the corresponding Buckingham–Gunton inequalities for Ising ferromagnets and the Josephson inequality.
L
p
-space techniques in potential scattering
Authors: Blaauw H. J. , Tip A.
Published: March 2003

We consider potential scattering of structureless particles for potentials V (x) contained in certain Lp-spaces. In particular we study the compactness properties of K (t) =V1/2 exp(−iH0t) V1/2, H0 being the free Hamiltonian. By interpreting K̃ (z) =∫∞0dt exp(izt) K (t), Imz⩾0, as a Bochner integral, we find the following property for the total cross section σtot(ω) (ω is the energy variable): If V∈L1(↛3) ∩Lp(↛3), 3/2⩾p⩾2, then ∫∞ω0dω[ω1/2σtot(ω)]q is finite for some suitable ω0⩾0 and 2p/(2p−3) <q<∞.
Determination of unitary, analytic representations of the conformal group in 2+1 dimensions using the operator formalism of the Gel’fand–Naimark Z basis
Authors: Güler Y.
Published: March 2003

Operator formalism of the Gel’fand–Naimark Z basis is used to determine the unitary, analytic representations of the conformal group in two space dimensions and one time dimension [SO(3,2)]. For this purpose an operator Z, which is an operator valued function of the generators, is formed. Common eigenstates, labeled by d, s, and Z of three commuting operators Z, I1, I2, are taken as the basis states of the representation space. Two series of the representations d=σ and d=−1/2+id1 are obtained for S=0. An invariant scalar product is obtained in the space of homogenous functions f (Z).
A multipoint interpolation method based on variational principles for functionals of the solution to linear equations
Authors: Cheng Edward T. , Conn Robert W.
Published: March 2003

A method is derived for using variational expressions to interpolate among known values of a functional of the solution to linear equations. For linear functionals of the solution to an inhomogeneous equation, the interpolation expression is exact at N distinct points when N distinct functions are used, each of which is the solution of the underlying Euler equation. Two point variational interpolation is derived to interpolate on the value of an eigenvalue using the Rayleigh quotient. Illustrative examples are given based on neutron transport studies of fusion reactor blanket systems and applications to sensitivity and optimization studies in reactor theory are discussed.
The relationship between the normalization coefficient and dispersion function for the multigroup transport equation
Authors: Feigenbaum Mitchell J.
Published: March 2003

An explicit formula for the discrete Case normalization coefficient is presented in terms of functions related to the dispersion function. These functions are easily determined and provide the normalization coefficient without need of prior evaluation of the eigenvectors.
Algebraically special ℋ-spaces
Authors: Fette C. W. , Janis Allen I. , Newman Ezra T.
Published: March 2003

All diverging algebraically special solutions of the complex vacuum Einstein equations which are left (or right) conformally flat (ℋ-spaces) are found explicitly. These metrics contain four arbitrary functions of two variables.
Bethe–Salpeter spinor equation at Pμ=0 and SO(5) spinor spherical harmonics
Authors: Floratos E. G.
Published: March 2003

We study the B−S equaiton, in the ladder approximation, for the zero energy bound states of a spinor and a scalar particle interacting via the exchange of a massless scalar particle. Constructing and using a complete set of SO(5) spinor spherical harmonics, we find the SO(5) degenerate spectrum of the coupling constant and the bound state amplitudes up to a normalization constant. It turns out that the SO(5) symmetry is broken by these amplitudes in a peculiar way.
Monotonicity of correlation functions
Authors: Pearce Paul A.
Published: March 2003

Counterexamples to pair correlation monotonicity inequalities, analogous to Griffiths’ second inequality for the Ising model, are presented for the finite spin Heisenberg model, the spin−(1/2) X–Y model, the anisotropic planar classical Heinsberg model, and the spherical model.
Kinematics of tensor fields in hyperspace. II
Authors: Kucha? Karel
Published: March 2003

Various kinematical relations, holding between hypersurface projections of spacetime tensor fields in an arbitrary Riemannian spacetime, are studied in terms of differential geometry in hyperspace. A criterion is given that a collection of hypertensor fields is generated by the projections of a single spacetime tensor field intersected by the embeddings. From here, it is shown that the super-Hamiltonian of an arbitrary tensor field splits into two parts, Hφ↑ and Hφ−, Hφ↑ being local in the field momenta and Hφ− containing their first derivatives. The form of Hφ− for an arbitrary tensor field is determined from the field behavior under hypersurface tilts. The kinematical equations for the intrinsic metric and the extrinsic curvature are written in a quasicanocical form, and their connection with the closing relations for the gravitational super-Hamiltonian is exhibited. The conservation laws of charge, energy and momentum, and the contracted Bianchi identities, are written as hypertensor equations.
Dynamics of tensor fields in hyperspace. III
Authors: Kucha? Karel
Published: March 2003

The dynamics of tensor fields with derivative gravitational coupling on a given Riemannian background is formulated as Hamiltonian dynamics of hypersurface projections of these fields propagating in hyperspace. The first-order spacetime action is transformed into an equivalent hypersurface form. The supermomentum and different parts of the super-Hamiltonian are identified with projected pieces of the (symmetrical, canonical, and spin) energy–momentum tensors, and their kinematical and dynamical roles are analyzed. Hypersurface variables are included among the canonical variables, and the resulting first-order generalized Hamiltonian dynamics of hypertensor fields is discussed. The closing relations for the constraint functions in the generalized Hamiltonian dynamics are derived from the foliation independence of the hypersurface action. The elimination of the λ-multipliers, which are characteristic to the first-order theory, is accomplished. The general formalism is specialized to the n-form fields with nonderivative gravitational coupling.
Extension of the statistical mechanics of equilibrium to noncommutative constraints
Authors: Lubkin Elihu
Published: March 2003

A formula similar to the Gibbs canonical and grand canonical ensembles is proven for the ensemble of maximal entropy among those ensembles of common mean values of possibly noncommuting operators. This is done over a Hilbert space of finite dimension Δ. A partition matrix Π becomes important; the partition function Z=TrΠ displaces Π in the thermodynamics only in the commutative case. Generalization to Δ infinite is discussed informally.
Irreducible tensor method for a chain SU2⊃⋅⋅⋅⊃ G″⊃G′⊃G and molecular physics
Authors: Kibler Maurice R.
Published: March 2003

Three things are discussed in this note around two results in the theory of angular momentum. First, certain Clebsch–Gordan coefficients of a compact topological group, which has a representation generalizing the antisymmetric representation [1n] of Sn, are shown to possess an interesting symmetry property. Second, for physical purpose this property is applied to the octahedral group O≈S4 and its double (i.e., covering) group O* (i.e., O*/S2≈O) considered as subgroups of SO3 and SU2, respectively. We take this opportunity for briefly connecting some papers on the representation theory of SU2 recently published in this journal with some previous works by the author on the Wigner–Racah algebra of a noncanonical chain SU2⊃G. Third, the material is used for rationalizing the well-known (j=1) - and less known (j=1/2) -isomorphisms of molecular physics in terms of isoscalar factors for the chains SO3⊂O and SU2⊂O*, respectively.
Massive and massless supersymmetry: Multiplet structure and unitary irreducible representations
Authors: Jarvis P. D.
Published: March 2003

UIR’s of the supersymmetry algebra for the massive and massless cases are analyzed covariantly (without the use of induced representations) in terms of their component spins. For the massive case normalized basis vectors ‖p2≳0, j0; σ; pjλ〉 are constructed, where j0 is the ’’superspin’’ and σ is an additional quantum number serving to distinguish the different ‖pjλ〉, the constituent p2≳0, spin-j UIR’s of the Poincaré group. For the massless case, normalized basis vectors ‖p2=0, λ0; pλ〉 are similarly constructed, where λ0 is the ’’superhelicity.’’ Matrix elements of the supersymmetry generators, in these bases, are explicitly given. The ’’σ basis’’ is used to define weight diagrams for the massive UIR’s of supersymmetry, and their properties are briefly described. Eigenfunctions ωσ(ϑ) are also defined, and their connection with the reduction of higher spin massive superfields ΦJ(x,ϑ) is discussed. Finally, it is shown how gauge dependence necessarily arises with certain massless superfields. The massless scalar superfield, both gauge-dependent and gauge-independent, is discussed as an example.
Quasiclassical scattering above barriers in one dimension
Authors: Meyer R. E.
Published: March 2003

The Schrödinger equation is studied at energy levels above the highest values of the potential for wavelengths smaller than the scale of the potential variations. The first approximation to the scattering matrix is then expressed in terms of a single parameter a+ representable as, effectively, a Fourier transform. This is used to prove the transcendental property of reflection for smooth potentials and to obtain the precise, first approximation to the scattering matrix for a very large class of analytic potentials.
Optimal factor group for nonsymmorphic space groups
Authors: Car R. , Ciucci G. , Quartapelle L.
Published: March 2003

The construction of the irreducible representations of single and double nonsymmorphic space groups is discussed. The proof is given that for any symmetry element where the nonsymmorphism plays a role there is a finite group of lowest order such that its irreducible representations engender all the allowable representations of the little group. For most high symmetry elements the order of this optimal factor group is only twice the order of the corresponding point group of the wave vector. The computational advantages of using this group instead of other known factor groups are stressed.
Classic theory of direct intermembrane interaction
Authors: Letelier Patricio S.
Published: March 2003

The theory of action-at-a-distance interaction between objects of two dimensions is studied. The theory developed has some similarities to the corresponding theory of one-dimensional objects (strings). The fundamental invariants of the theory are found. A particular interaction is studied, and it is found that the interaction between closed membranes takes place via a pseudoscalar field.
The Galilei group and its connected subgroups
Authors: Sorba Paul
Published: March 2003

All the connected subgroups of the Galilei group G—and of its central extension G̃—are determined and classified up to a conjugation of G—respectively G̃—and also up to an isomorphism. In order to construct these subgroups, some general properties on subalgebras of a given Lie algebra have been proved. It is interesting to note that the subgroups of G̃ are derived from the subgroups of G.
On the extrapolation of optical image data
Authors: Viano G. A.
Published: March 2003

In this paper we show that the extrapolation of an image’s piece as well as the object-reconstruction problem are improperly posed in the sense that the solutions do not depend continuously on the data. We try to restore the stability for these problems introducing suitable additional constraints. In the present work we treat in detail only the extrapolation of the image data. At this purpose we use and illustrate two numerical methods, which are based on the doubly-orthogonality of the linear-prolate-spheroidal-functions. Finally a probabilistic approach to these questions is outlined.
Integration formula for Wigner 3-j coefficients
Authors: Winch D. E.
Published: March 2003

An integration formula is given for Wigner 3-j coefficients having 72 different forms associated directly with the symmetry properties of the Regge square. The integration formula permits a direct derivation of the integral of a product of three rotation matrix elements, and is shown to include an integration formula for spin projection coefficients as a special case.
Eigenvalues of invariants of U(n) and SU(n)
Authors: Aguilera-Navarro M. C. K. , Aguilera-Navarro V. C.
Published: March 2003

In this paper, we present four distinct ways of obtaining the eigenvalues of invariants of unitary groups, in any irreducible representation. The invariants are defined according to a different contraction convention. Their eigenvalues can be given in terms of special partial hooks different from those found by other authors.
Diffraction corrections to the equilibrium properties of the classical electron gas. Pair correlation function
Authors: Deutsch C. , Gombert M. M.
Published: March 2003

We perform a systematic study of the diffraction corrections (h/≠0) in the high-temperature range (kBT≳1 Ry) for the pair correlation function of the one-component classical electron gas with a neutralizing background, up to the third-order in the plasma parameter Λ=e2/kBTλD. This program is achieved through the effective interaction Vee(r) = (e2/r)(1−e−cr) with c ∼ (thermal De Broglie wavelength)−1, allowing for a straightforward and tractable generalization of the one-component classical plasma model. The nodal expansion of the potential of average force is performed order-by-order with finite Mayer-Salpeter diagrams. The classical results of De Witt (second-order) and Cohen–Murphy (third-order) are recovered in the h/→0 limit. The resummation to all orders of the bubble diagrams gives access to the short-range behavior of the pair correlation function, which is found similar with the Monte-Carlo results.
Cluster expansions for fermion fields by the time dependent Hamiltonian approach
Authors: Brydges David C.
Published: March 2003

A cluster expansion is given for a fermion field moving in an external field according to the interaction ψ̄ψφ in one space dimension.
On canonical quantum field theories
Authors: Herbst Ira W.
Published: March 2003

We consider time reversal invariant canonical quantum field theories in which π=φ. We show that if the theory has a mass gap, the vacuum is an analytic vector for the time zero field φ (f). With the additional assumptions of Poincaré covariance, cyclicity of the vacuum for the time zero fields, and a domain condition on the Hamiltonian, we show that the Schwinger functions of the theory determine a Euclidean covariant Markov field theory. We also consider the implications of a bound of the form ±φ (f)j⩽H+γ (f) for the behavior of the ground state at large field strength. We show that such a bound implies that the vacuum is an analytic vector for ‖φ (f) ‖j/2+1.
Theory of irradiance distribution function in turbulent media—cluster approximation
Authors: Furutsu K.
Published: March 2003

The equations of the correlation functions of irradiance in turbulent media are formally solved for arbitrary order by use of an operator method. A cluster approximation is then applied to the fourth and higher order moments of irradiance to express those in terms of the lower order moments. The values of moments predicted by this approximation show a very good agreement with the recent experimental values observed up to the fourth order moment of irradiance. The irradiance distribution function is then analytically derived from the obtained expression of the moments of irradiance and is found to have the following features: (1) It is close to the Gaussian distribution with respect to the logarithm of irradiance, (2) it has a threshold value for irradiances giving nonvanishing probability, and (3) it has a very small but sharp distribution of the form of the δ function at the threshold value. The spectrum of medium as well as the conditions of the initial wave, e.g., of whether it is a plane wave or is a beam wave do not directly enter in the expression but appear only through the first three moments of irradiance. The condition of applicability of the cluster approximation is also discussed in some details based on the Kolmogorov spectrum of turbulence.
N-level system in contact with a singular reservoir
Authors: Gorini Vittorio , Kossakowski Andrzej
Published: March 2003

We study a model of an N-level atom coupled linearly to an infinite free boson bath whose time correlation functions are Gaussian. We prove that, in the limit when the decay time of the correlations of the bath goes to zero, the reduced dynamics of the atom is given by a completely positive dynamical semigroup. By varying the interaction parameters, any such semigroup can be obtained in the limit. We also discuss the formally analogous situation of an N-level atom whose Hamiltonian contains an external fluctuating Gaussian stationary contribution.
Quantum theory of anharmonic oscillators. II. Energy levels of oscillators with x2α anharmonicity
Authors: Hioe F. T. , MacMillen Don , Montroll E. W.
Published: March 2003

This is an investigation of the energy levels of anharmonic oscillators characterized by the potentials (1/2) x2+λx2α with α=2,3,⋅⋅⋅ and λ≳0. Two regimes of (λ,n) space are distinguishable: In one, the energy levels differ only slightly from the harmonic ones En=n+1/2 and in the other they differ only slightly from the purely anharmonic oscillators with En?Cαλ1/(1+α) (n+1/2)2α/(1+α), C being a constant which depends on α. The magnitude of the combination η≡λ (n+1/2)α−1 determines the regime. If this combination is ≪1, one is in the harmonic regime, and if it is ≫1, one is in the anharmonic regime. As n→∞ the ’’boundary layer’’ between the two regimes narrows. The small parameter of a perturbation theory should thus be η rather than λ. Several rapidly convergent algorithms have been developed for the calculation of the energy levels of our anharmonic oscillators, and energy level tables and graphs are presented.
Symmetry breaking interactions for the time dependent Schrödinger equation
Authors: Boyer C. P. , Sharp R. T. , Winternitz P.
Published: March 2003

A systematic study of the symmetry porperties of the Schrödinger equation uxx + iut = F (x,t,u,u*) is performed. The free particle equation (for F=0) is known to be invariant under the six-dimensional Schrödinger group 𝒮1. In this paper we find all continuous subgroups of 𝒮1 and for each subgroup we construct the most general interaction term F (x,t,u,u*), reducing the symmetry group of the equation from 𝒮1 to the considered subgroup. Since we allow for an arbitrary dependence of F on the wavefunction u (and its complex conjugate u*) the considered Schrödinger equation is in general a nonlinear one [the ordinary Schrödinger equation with a time dependent potential is recovered if F (x,t,u,u*) =uG (x,t)]. For each symmetry breaking interaction F the remaining symmetry group is used to obtain special solutions of the equations or at least to separate variables in the equation and to obtain some properties of the solutions.
Ruelle’s cluster property for nonstrictly localizable fields derived from symmetry of their Wightman functions
Authors: Lücke W.
Published: March 2003

Hermitian fields are considered which fulfill all the Wightman axioms, except local commutativity, formulated for the Gel’fand space S1(R4) instead of the Schwartz space 𝒮 (R4). For fields of this class, containing the Jaffe class and even the larger class of essentially local fields on S1(R4), it is shown that symmetry of the Wightman functions and mass gap imply Ruelle’s cluster property. Hence the Haag–Ruelle–Hepp scattering formalism applies to these fields, which need not be local.
On the structure of the multiplicity-free Wigner coefficients of U(n)
Authors: Wong M. K. F.
Published: March 2003

It is shown that the isoscalar factor (or reduced Wigner coefficient) in U(n) is essentially a doubly stretched 9-j symbol in U(n−1). The connection between the isoscalar factorand the 9-j symbol of U(n) and U(n−1) is also noted. This result immediately implies that the Weyl coefficients of U(n) are basically 6-j symbols of U(n−1), a result first noted by Holman. The finite transformation matrix D[m]n(m′)n−1(m)n−1 either in terms of generalized Euler angles or double bosons can thus be written down in a simple way. The stretched 6-j symbols of U(n) are obtained in a simple form, involving no summations. The generalized beta functions of Gel’fand and Graev for U(n) are found to be connected with the stretched 6-j symbols of U(n−1) and an isoscalar factor of U(n−1). The 144 Regge symmetries of the 6-j symbol of U(2) can be interpreted as the symmetries of the Weyl coefficients of the double boson state of U(3) -U(3). In the Appendix we give the phase relations between the Wigner coefficients and 3-j symbols of U(n), a result which is by no means trivial, and is of some practical importance.
Branching rules and Clebsch–Gordan series for E8
Authors: McKay W. , Patera J. , Sharp R. T.
Published: March 2003

Branching rules for the second lowest dimensional representation of the exceptional simple Lie algebra of type E8 are given with repsect to all its 14 maximal semisimple subalgebras. This representation of E8 is of dimension 3875, the maximal subalgebras are of types A8, D8, A1-A7, A1-E7, A2-E6, A3-D5, A4-A4, A1-A2-A5, G2-F4, A1-A2, C2, and three nonconjugate subalgebras all of type A1. The Clebsch–Gordan series, necessary for decomposition of the direct product of three representations of dimension 248, are given.
A new analytic continuation of Appell’s hypergeometric series F2
Authors: Sud Krishan , Wright L. E.
Published: March 2003

The doubly infinite series for Appell’s function F2(α,a1,a2,b1,b2;x,y) is written in terms of four of Appell’s F3 functions. Analytic continuations are given for the F3 series, thereby allowing one to obtain a new analytic continuation for the F2 series. The new doubly infinite series are all absolutely convergent if their variables satisfy ‖x‖<1 and ‖y‖<1, whereas the F2 series is absolutely convergent only in the domain ‖x‖+‖y‖<1. The analytic continuations given here are very useful for evaluating the Appell F2 series when one of the variables is near unity. In particular, our results are useful for calculating radial matrix elements over products of Dirac–Coulomb functions and the electromagnetic interaction Green’s function.
On the solution of nonlinear matrix integral equations in transport theory
Authors: Bowden R. L. , Zweifel P. F. , Menikoff R.
Published: March 2003

The coupled nonlinear matrix integral equations for the matrices X (z) and Y (z) which factor the dispersion matrix Λ (z) of multigroup transport theory are studied in a Banach space X. By utilizing fixed-point theorems we are able to show that iterative solutions converge uniquely to the ’’physical solution’’ in a certain sphere of X. Both isotropic and anisotropic scattering are considered.
Soluble classical spin model with competing interactions
Authors: Stephenson John
Published: March 2003

The partition function and spin pair correlation functions have been calculated exactly for a classical linear chain model with alternate next-nearest-neighbor (nnn) interactions, in which the interaction energies between pairs of nearest (nn) and next-nearest (nnn) neighbor spins are arbitrary functions of the angles between the relevant spins. Of special interest is the cosine interaction model described by the Hamiltonian ℋ=−ΣNi=1[J1(cosϑ2i−1,2i +cosϑ2i,2i+1)+J2cosϑ2i−1,2i+1]. When the nnn interaction is antiferromagnetic (J2<0) it competes with the nn interaction J1, and there can be disorder point(s) at which nnn correlations change from monotonic to oscillatory. The ground state is ferromagnetic when the interaction ratio r≡J2/‖J1‖ ≳−1/2=rc, but is disordered for more negative values. The disorder point locus has been determined. It terminates at zero temperature at rD=−1/21/2, at which point the ground state energy is a maximum. The result that rD differs from rc is thought to be peculiar to one-dimensional models. Over a limited range of values of r there can be two disorder points. The low temperature asymptotic behavior of the partition and correlation functions is analyzed in detail. Also a novel summation formula for spherical Bessel functions is obtained.
Gauge theories and nonrelativistic cosmological symmetries
Authors: Roman P. , Haavisto J.
Published: March 2003

It is shown that the application of the locality principle in a uniformly curved space leads to the emergence of a dynamical quantum mechanical group which is precisely the Hooke group. The interaction structure is also studied.
Binary mixture with nearest and next nearest neighbor interaction on a one-dimensional lattice
Authors: Yan C. C.
Published: March 2003

A mixture of two kinds of molecules on a one-dimensional lattice with free ends is considered. The energy term is assumed to consist of interactions of nearest neighbors and next nearest neighbors and interaction with the uniform external field. The partition function is evaluated by determining the degeneracies.
On Wentzel’s proof of the quantization condition for a single-well potential
Authors: Fröman Nanny , Fröman Per Olof
Published: March 2003

Wentzel’s elegant method for deriving the quantal generalization of the Sommerfeld quantization condition, which has been criticized by other authors, is justified and put on an irrefutable basis. Clarifying comments on some questions related to the discussion in the present paper are also made.
Geometric formulation of electrodynamics and general relativity in discrete space–time
Authors: Weingarten Don
Published: March 2003

Classical electrodynamics is reconstructed in discrete space–time, then combined with Regge’s discrete version of general relativity. Quantization using path integrals is considered briefly.
Maxwell and Wien processes as special cases of the generalized Feller diffusion process
Authors: Lehnigk Siegfried H.
Published: March 2003

It is shown that Maxwell and Wien type processes are special cases of the generalized Feller diffusion process. In particular, both are obtained for specific parameter values from the delta function initial condition solution of the generalized Feller equation. For specific values of the independent time variable, one obtains the well-known distribution laws of Maxwell and Wien of statistical physics.
Hamiltonians with x−2-like singularity
Authors: Pick Štěpán
Published: March 2003

A class of singular Hamiltonians with deficiency indices (1,1) is studied. By using asymptotic behavior of functions from their domain, all self-adjoint extensions are described. The Friedrichs extension is shown to be (in a certain sense) a limit of nonsingular Hamiltonians.
Convergent low-frequency expansions for penetrable scatterers
Authors: Dassios George
Published: March 2003

We develop convergent series solutions in powers of the wavenumber (k=2π/λ) for the field (ψ) and the normalized (dimensionless) scattering amplitude (g) for scattering by lossless penetrable obstacles whose physical properties are specified by two real parameters. The first two terms of ψ are solutions of Laplace’s equation and the term of order kn, n⩾2, satisfies a two-parameter Poisson equation whose inhomogeneous term is proportional to the kn−2 term. The leading term of g is of order k3 (as obtained originally, by Rayleigh); the k4 term is zero for shapes that have inversion symmetry, and vanishes in the forward direction for all shapes; the kn terms, n⩾3, are expressed as volume integrals of functions involving the terms of ψ up to order kn−2. Equivalent expressions in terms of surface integrals are included. For a plane wave of arbitrary direction of incidence and a triaxial ellipsoid, we obtain explicitly the first four nonvanishing terms of ψ (to order k3) and the first two nonvanishing terms of Img (to order k5) and Reg (to order k8). Corresponding results for spheroids, needle, disc, sphere, and for the one-parameter problems are obtained as special cases. The necessary transformation of the ellipsoidal harmonics are also provided.
Solitons, pseudopotentials, and certain Lie algebras
Authors: Corones James
Published: March 2003

It is shown that there is a common algebraic structure in the pseudopotentials of equations solvable by the generalized Zakharov–Shabat eigenvalue problem. It follows that an arbitrarily large number of prolongation variables can be associated with these equations and that a recently developed geometric interpretation of solitons can be given for each of these equations.
The Kähler structure of asymptotic twistor space
Authors: Ko M. , Newman E. T. , Penrose R.
Published: March 2003

Asymptotic twistor space 𝒯 is a 4-complex-dimensional Kähler manifold (of signature ++−−) which can be constructed from an asymptotically flat space–time containing gravitational radiation. The properties of this Kähler structure are investigated, the Kähler metric being of a particular type, arising from a scalar Σ with special homogeneity properties. The components of the Kähler curvature Kαβγδ are found explicitly in terms of the asymptotic Weyl curvature of the space–time. When gravitational radiation is present, Kαβγδ ≠0, whereas for a stationary field Kαβγδ=0. The ’’Ricci-flat’’ condition Kαβαγ=0 is found always to hold.
The quantum mechanical representations of the anisotropic harmonic oscillator group
Authors: Major M. E.
Published: March 2003

A group G associated with the n-dimensional anisotropic harmonic oscillator is constructed : G is essentially a group generated by the position and momentum observables, the identity operator, and the Hamiltonian of the system. All the quantum mechanical irreducible representations of G are evaluated, using Mackey’s theory of induced representations.
On the accidental degeneracy of the anisotropic harmonic oscillator. I
Authors: Major M. E.
Published: March 2003

A group G associated with the n-dimensional anisotropic harmonic oscillator is shown to be embedded in a semidirect product L of the Weyl group N and the symplectic group Sp(2n,R). A particular induced representation of the group L, when restricted to G, is proved to be unitarily equivalent to ⊕sdω,sUGv,−(sgnv)s) where dω,s is the degeneracy of the energy level Eω,s of the n-dimensional anisotropic harmonic oscillator with frequencies (ω1, ω2,...,ωn) =ω, UGv,−(sgnv)s is an irreducible representation of G, and s may be regarded as indexing all distinct energy levels of the system.
A finite quantum electrodynamical base for many photon collective phenomena
Authors: Stelzer M. , Mühlstein J. , Ernst V.
Published: March 2003

We analyze the interaction of many ’’nonoverlapping’’ atoms with the common radiation field of transverse photons over a continuum of modes. Typical effects of quantum optics arising from this interaction require extensions to very high powers in the coupling constant e and diverge therefore within a systematic perturbation approach. Since renormalization to such high orders is not feasible in practice, we formulate for the purpose of quantum optics ’’a new peace treaty between QED and its infinities.’’ This is a compromise between the necessity of including effects of very high powers in e and the trivial demands for finite results. We show that a unitary time evolution operator exists if, in essence, only any finite number of levels of each atom is treated as ’’existent.’’ We convince ourselves that the limits of the concept of ’’nonoverlapping atoms’’ are reached long before any infinities can enter. The resulting theories meet still reasonable requirements with respect to the dualism of light, Dicke principle, and causality. They are formulated so that some practical demands of quantum optics (homogeneous and inhomogeneous line broadening, Doppler effects) can be met easily if desired.
Evolution of a stable profile for a class of nonlinear diffusion equations with fixed boundaries
Authors: Berryman James G.
Published: March 2003

A class of quasilinear parabolic equations with fixed boundaries arising in studies of cross-field diffusion in toroidal multipole plasmas is presented. It is well known that these equations have separable solutions which decay in time. Surprisingly, both octupole and numerical experiments show, in particular cases, that the separable solution evolves from an arbitrary initial distribution of particles. The evolution and stability properties of these solutions are demonstrated in this paper. When the coefficients of the equations are independent of the spatial variable, infinitesimal perturbations decay as the fourth power (or higher) of the separable solution time dependence; the separable solution is therefore stable. When the initial particle distribution has no nulls except at the boundaries, an approximate analysis shows that large perturbations decay exponentially causing the rapid evolution of the separable solution. The analysis allows the asymptotic behavior of the system to be predicted approximately from knowledge of the initial particle distribution.
A necessary condition for the validity of Huygens’ principle on a curved space–time
Authors: Goldoni Riccardo
Published: March 2003

It is proved that a necessary condition for the validity of Huygens’ principle on a curved space–time V4 is that V4 be an Einstein space. In connection with this result some remarks about the strong and weak formulations of Mach’s principle are also pointed out.
Spaces of positive and negative frequency solutions of field equations in curved space–times. I. The Klein–Gordon equation in stationary space–times
Authors: Moreno Carlos
Published: March 2003

In stationary space–times Vn×R with compact space-section manifold without boundary Vn, the Klein–Gordon equation is solved by the one-parameter group of unitary operators generated by the energy operator i−1T−1 in the Sobolev spaces Hl(Vn) ×Hl−1(Vn). The canonical symplectic and complex structures of the associated dynamical system are calculated. The existence and the uniqueness of the Lichnerowicz kernel are established. The Hilbert spaces of positive and negative frequency-part solutions defined by means of this kernel are constructed.
Analytic evaluation of an important integral in collision theory
Authors: Detrich John , Conn Robert W.
Published: March 2003

A commonly occurring integral in collision theory when the radial part of the potential has the form rne−αr is In;ll′(α;k,k′)  = ℱ∞0dr rn+1 e−αrJl+1/2(kr) Jl′+1/2(k′r). Analytic results for this integral have been found for the most important cases, including those with negative values of n. This permits efficient evaluation of Born matrix elements required in many scattering theory applications based on perturbation methods.
Perturbation theory for Green’s functions as an effective mass formalism
Authors: Krause F.
Published: March 2003

A formal solution for Green’s functions of the type [∂2+m2+gV (x)]G (x,y) =δ (x−y) is presented which has the structure of an effective mass formalism. One first solves the free case [gV (x) =0] for given boundary conditions and then replaces the parameter m in the solution by a quantity depending on V (x). The rules for this replacement are given, a connection with the Baker–Campbell–Hausdorff formula is established, and it is shown how the formalism unites different perturbation and approximation schemes.
A variational principle for the Gel’fand–Levitan equation and the Korteweg–de Vries equation
Authors: Kanal M. , Moses H. E.
Published: March 2003

A functional is constructed from the spectral density used in the general Gel’fand–Levitan equation and an arbitrary function N of two sets of variables. This functional is shown to be an absolute maximum when N satisfies the Gel’fand–Levitan equation. In the case of the Gel’fand–Levitan equation for the one-dimensional and radial Schrödinger equations and certain generalizations, this result can be translated into a theorem about the area under a curve to a given point (x or r), considered as a functional of N. This curve is given by the scattering potential to the given point when the functional takes on its maximum value. The functional may thus be considered a method of obtaining the scattering potential from the spectral data through a variational technique. In the case that the Gel’fand–Levitan equation is that for the one-dimensional Schrödinger equation the results can be interpreted as a theorem about the area to a given point x under the curve given by the solution of the Korteweg–de Vries equation. That is, at a given time t the area under a curve to a given point x, considered as a known functional of N, takes on its maximum value for all x and t when the curve represents the solution of the Korteweg–de Vries equation with appropriate initial conditions.
Continuous subgroups of the fundamental groups of physics. III. The de Sitter groups
Authors: Patera J. , Sharp R. T. , Winternitz P. , Zassenhaus H.
Published: March 2003

An algorithm for classifying the closed connected subgroups S of a given Lie group G into conjugacy classes, presented in earlier papers, is further refined so as to provide us with ’’normalized’’ lists of representatives of subalgebra classes. The normalized lists contain the subgroup normalizer NorGS (NorGS is the largest subgroup of G for which S is an invariant subgroup) for each subgroup representative. The advantage of having normalized lists is that the problem of merging several different sublists (e.g., the lists of all subgroups of each maximal subgroup of G) into a single overall list becomes greatly simplified. The method is then applied to find all closed connected subgroups of the two de Sitter groups O(3,2) and O(4,1). The classification group in each case is the group of inner automorphisms.
Lie theory and the wave equation in space–time. 3. Semisubgroup coordinates
Authors: Kalnins E. G. , Miller W.
Published: March 2003

We classify and study those coordinate systems which permit R separation of variables for the wave equation in four-dimensional space–time and such that at least one of the variables corresponds to a one-parameter symmetry group of the wave equation. We discuss over 100 such systems and relate them to orbits of triplets of commuting operators in the enveloping algebra of the conformal group SO(4,2).
The n-bubble series in the theory of the classical one-component plasma
Authors: Furutani Y. , Deutsch C.
Published: March 2003

With the aid of the Mellin transform, an exact expression for the three-dimensional Fourier transform G (k) of the renormalized sum exp[−Λexp(−r)/r]−1+Λexp(−r)/r for n-bubble graphs is given in two equivalent forms, where the plasma parameter Λ is not necessarily smaller than unity. Its several properties, such as small and large k limits, are discussed in detail.
A classical Markov process in nonequilibrium quantum statistical mechanics
Authors: Torres P. L.
Published: March 2003

We obtain, from the exact microscopic, reversible, dynamics of an oscillator coupled to a heat bath, a classical, Markov stochastic process, for the energy of the oscillator. Also, the corresponding macroscopic phenomenological equation, describing an irreversible approach to equilibrium, is obtained.
Exact occupation statistics for two-dimensional lattices of single particles
Authors: Miyazaki Eizo , Yasumori Iwao
Published: March 2003

A general expression is developed which describes exactly the ensemble average number of one- or two-dimensional structures per arrangement, created when indistinguishable single particles are arranged on a two-dimensional lattice. The expression obtained is applied to the calculation of some physically important structures which appear on a rectangular and a closest-packed hexagonal lattice. The problem of nearest-neighbor pairs is then solved as a special case from the general expression.
Percolation theory on directed graphs
Authors: Arrowsmith D. K. , Essam J. W.
Published: March 2003

The pair connectivity Puv of a directed graph G between vertices u and v is the probability that there is a path from u to v when each edge and vertex has a given probability of being deleted, deletions being made independently. We consider the coefficient d⃗ in the expansion Puv(G)  = ΣA′⊇A d⃗uv(G′) ΠaεA′pa ΠwεV′pw, where A and V are respectively the arc and vertex sets of G, and pa(pw) is the probability that the arc a (vertex w) is not deleted. G′ is the arc set A′ together with its set of incident vertices V′. It is shown that d⃗uv(G′) is nonzero if and only if G′ is coverable by some set of (directed) paths from u to v and has no circuit. When these conditions are satisfied, d⃗uv(G′)  =  (−1)tuv+1 where the number of independent paths from u to v is tuv. Moreover, tuv is shown to have the value of ν (G)+1, ν (G) being the cyclomatic number of the graph G.
Remarks on the second order error in the variational calculation of expectation values
Authors: Epstein Saul T.
Published: March 2003

A formula due to Gerjuoy [J. Math. Phys. 16, 761 (1975)] is derived in a straightforward way by use of second order perturbation theory.
Half-space analysis basic to the time-dependent BGK model in the kinetic theory of gases
Authors: Siewert C. E. , Burniston E. E.
Published: March 2003

The elementary solutions of the linearized time-dependent BGK equation are shown to have, for the case of no discrete eigenvalues, the half-range expansion property necessary for half-space analysis. Also the partial indices corresponding to the basic matrix Riemann problem encountered are shown, for the general case, to be nonnegative, as required for the half-space analysis.