bluuebunny/crossref_metadata_2025 huggingface dataset:
Continuous subgroups of the fundamental groups of physics. I. General method and the Poincaré group
Authors: Patera J. , Winternitz P. , Zassenhaus H.
Published: March 2003

We present a general method for reducing the problem of finding all continuous subgroups of a given Lie group G with a nontrivial invariant subgroup N, to that of classifying the subgroups of N and the subgroups of the factor group G/N. The method is applied to classify all continuous subgroups of the Poincaré group (PG) and of the Lorentz group extended by dilatations [the homogeneous similitude group (HSG)]. Lists of representatives of each conjugacy class of subalgebras of the Lie algebras of the groups PG and HSG are given in the form of tables.
Decay of correlations for infinite-range interactions
Authors: Duneau M. , Souillard B. , Iagolnitzer D.
Published: March 2003

Strong cluster properties are proved at low activity and in various other situations for classical systems with infinite-range interactions. The decay of the correlations is exponential, resp. like an inverse power of the distance, if the potential decreases itself exponentially, resp. like an inverse power of the distance. The results allow us to extend to the case of exponentially decreasing potentials the equivalence theorem between strong cluster properties and analyticity with respect to the activity, previously proved for finite-range interactions.
Theory of tachyons for arbitrary spin
Authors: Shay D.
Published: March 2003

It is shown that, even though the finite-dimensional representations of the little group for tachyons, particles with spacelike momentum, are not unitary in general, a fully covariant theory for spin-s tachyons can be obtained. This occurs because a helicity dependent factor (−1)σ, where σ denotes helicity, appears in the expressions for the invariant integral. This factor acts as a metric for the nonunitary representations and a covariant theory results. As has been found for spin- (1/2), the ideas of conserved particle number or total charge do not work for tachyons. The conserved quantities depend on helicity.
Ray-optical analysis of fields on shadow boundaries of two parallel plates
Authors: Lee S. W. , Boersma J.
Published: March 2003

The electromagnetic diffraction by two parallel plates of semi-infinite length is treated by ray methods. Two special problems are considered: (i) calculation of the fields in the forward and backward directions due to diffraction of a normally incident plane wave by two nonstaggered parallel plates; (ii) calculation of the field due to a line source in the presence of two staggered parallel plates when the source, the two edges, and the observation point are on a straight line. The crucial step in the ray-optical analysis is the calculation of the interaction between the plates. This calculation is performed by two methods, namely, the uniform asymptotic theory of edge diffraction and the method of modified diffraction coefficient. The relative merits of the two methods are discussed. The ray-optical solution of problem (i) agrees with the asymptotic expansion (plate separation large compared to wavelength) of the exact solution.
Random function theory revisited: Exact solutions versus the first order smoothing conjecture
Authors: Lerche I. , Parker E. N.
Published: March 2003

We remark again that the mathematical conjecture known as first order smoothing or the quasilinear approximation does not give the correct dependence on correlation length (time) in many cases, although it gives the correct limit as the correlation length (time) goes to zero. In this sense, then, the method is unreliable.
Equivalence of a class of Wigner coefficients of S
U (1,1) with those of S
U (2)
Authors: Chacón E. , Levi D. , Moshinsky M.
Published: March 2003

The purpose of this note is to establish the relation (λ1λ2μ1μ2‖ΛM)nc= (−1)μ2−λ2〈 (1/2) M+λ1−λ2−1), (1/2)(M+λ2−λ1−1), (1/2)(μ1−μ2+λ1+λ2−1), (1/2)(μ2−μ1+λ1+λ2−1) ‖Λ−1,λ1+λ2−1〉, where the left-hand side is a Wigner coefficient of the noncompact group SU (1,1) and in the right-hand side appears a standard Wigner coefficient of SU (2). The parameters λ1, λ2, Λ characterize unitary irreducible representations in the positive discrete series of SU (1,1), and thus they take positive integer or half-integer values. The other parameters are restricted by μ1=λ1,λ1+1,λ1+2,⋅⋅⋅, μ2=λ2,λ2+1,λ2+2,⋅⋅⋅, M=Λ,Λ+1,Λ+2,⋅⋅⋅. Besides we have μ1+μ2=M and Λ=λ1+λ2,λ1+λ2+1,⋅⋅⋅. A similar result is obtained [cf. our Eq. (3.28)] for Wigner coefficients involving unitary irreducible representations of the negative discrete series of SU (1,1).
Orthonormality properties of double coset matrix elements
Authors: Sullivan John J.
Published: March 2003

The orthogonality and completeness conditions of group representation theory are shown to provide complementary orthonormality relations for weighted double coset matrix elements. The double coset matrix elements are appropriate for symmetry adaption of the basis to two subgroup sequences using the double coset decomposition H-G/K. When applied to the symmetric and unitary groups, the double coset matrix elements assume the role of recoupling coefficients. The orthonormality properties give nontrivial relations between coefficients coupling tensors of different rank and/or different dimensions.
Jacobi’s principle for the case of time-varying Hamiltonian
Authors: Cawley Robert G.
Published: March 2003

We present a generalization of Jacobi’s principle to the case where the particle Hamiltonian varies with time, illustrating the formalism with an example from particle mechanics. It can be applied also to Stueckelberg mechanics to the case of variable mass, and we give an example.
Canonical transformations and path integrals
Authors: Fanelli Robert
Published: March 2003

A limited class of canonical transformations is introduced into the Lagrangian path integral method of quantization. Path integral quantization in different representations is discussed and a simple example is given.
Discrete space quantum mechanics
Authors: Vasholz David P.
Published: March 2003

The general problem of finding exactly soluble quantum systems is considered. It is argued that discrete space quantum mechanics emerges in a natural way as an avenue of approach. Discrete space quantum mechanics is formulated and applied to one-dimensional quantum systems with emphasis on single-channel models. It is found that a large variety of systems are exactly soluble in the sense that they only require the inversion of a finite-dimensional matrix. The interaction may in general be both nonlocal and non-time-reversal-invariant. The analytic structure of the resolvent is worked out in detail for a simple class of examples. It is shown that a slightly modified version of the usual continuous space Schrödinger equation may in principle be solved exactly for any finite range local potential by writing the solution in terms of corresponding discrete space solutions. It is also shown that from an algebraic viewpoint the models constructed are realizations of generalized versions of the Weyl relations.
Statistical theory of effective electrical, thermal, and magnetic properties of random heterogeneous materials. VI. Comment on the notion of a cell material
Authors: Hori Motoo
Published: March 2003

The concept of a cell material introduced by Miller is reinvestigated in connection with Brown’s assertion that an asymmetric cell material is not self-consistent [J. Math. Phys. 15, 1516 (1974)]. We construct a simple example of the asymmetric cell material which is in fact self-consistent. The misleading interpretation of the asymmetric cell material is due to Miller rather than to Brown.
Some examples of symmetrical perturbation problems in several complex variables
Authors: Gherman O. , Méder L.
Published: March 2003

We consider two examples of symmetrical perturbation problems in several complex variables and show that special singularities appear. We then discuss the spectral properties of the perturbed operators in connection with the involved symmetries.
A solution of the Korteweg–de Vries equation in a half-space bounded by a wall
Authors: Moses H. E.
Published: March 2003

We give a solution of the Korteweg–de Vries equation in the half-space 0<r<∞ with the boundary condition V (0) =0. The boundary condition may be interpreted as the requirement that the plane which bounds the half-space be a rigid wall. Aside from possible physical interest, this solution, which is obtained from one of the potentials for the radial Schrödinger equation which do not scatter, appears to indicate that the radial Schrödinger equation and the corresponding Gel’fand–Levitan equation play a role in the case of the half-space bounded by a wall similar to that of the one-dimensional Schrödinger equation (−∞<x<∞) and its corresponding Gel’fand–Levitan equation in the more usual full space treatment of the KdV equation. A possible interpretation of the solution presented in this paper is that it corresponds to the reflection of a wave by a wall, in which the incident wave is singular and the reflected wave is nonsingular but highly dispersive.
Multigroup neutron transport. I. Full range
Authors: Bowden R. L. , Sancaktar S. , Zweifel P. F.
Published: March 2003

A functional analytic approach to the N-group, isotropic scattering, particle transport problem is presented. A full-range eigenfunction expansion is found in a particularly compact way, and the stage is set for the determination of the half-range expansion, which is discussed in a companion paper. The method is an extension of the work of Larsen and Habetler for the one-group case.
On a convergent nonlinear perturbation theory without small denominators or secular terms
Authors: Eminhizer Charles R. , Helleman Robert H. G. , Montroll Elliott W.
Published: March 2003

We demonstrate a method for solving the dynamics of systems of extremely nonlinear coupled anharmonic oscillators with polynomial force laws. Several convergent perturbative and iterative techniques are described and applied, yielding the Fourier frequencies and coefficients of the time dependent solution. The usual stumbling block in a series analysis of coupled nonlinear systems is the appearance of terms with arbitrarily small denominators resulting in a rapidly divergent series. We avoid this classic ’’small denominators problem’’ by concentrating on the periodic solutions of a system. We employ the ’’Poincaré recurrence time’’ Tr of a system as the period of the solution, i.e., each Fourier frequency can be expressed as n (2π/Tr), with some integer n. These periodic solutions are dense among all possible solutions and suffice for our practical calculations (just as the rational numbers suffice for most practical computations). Hence we construct the solutions with rationally dependent Fourier frequencies whereas the Kolmogorov–Arnold–Moser theorems demonstrate the existence of certain solutions whose frequencies are rationally independent. As an example we treat a two oscillator system equivalent to the ’’Duffing equation,’’ a driven anharmonic oscillator appearing in models of the laser, the driven pendulum, etc. In our analysis no small denominators arise. We extend the region of bounded solutions beyond the (numerical) estimates in the literature and add to the list of peculiar phenomena exhibited by the Duffing equation.
Solution of initial-value problems for some infinite chains of harmonic oscillators
Authors: Christian W. G. , Law A. G. , Martens W. F. , Mullikin A. L. , Sledd M. B.
Published: March 2003

Procedures are given for solving the equations of motion of infinite and half-infinite chains of linear spring-mass oscillators with nearest-neighbor coupling. Arbitrary initial displacements and initial velocities may be prescribed for any finite number of the masses in the chains, and external forces may be applied to any finite number of them. The solutions appear as integrals with integrands involving orthogonal polynomials generated by three-term recurrence relations whose coefficients are determined from the equations of motion. Tables of the polynomicals needed for solving the equations of motion of a number of special chains are included.
On conserved quantities in kinematic dynamo theory
Authors: Lerche I.
Published: March 2003

Using a Lagrangian approach to the magnetic induction equation in an infinite medium, we demonstrate that there exist seven conserved quantities which, by analogy with classical mechanics, we label as ’’energy,’’ ’’momentum,’’ and ’’angular momentum.’’ For prescribed fluid motions we spell out the detailed conservation equations. For a fluid motion which is turbulent we also give the average conserved quantities. In a pragmatic sense it is expected that these conservation laws will be of use in attempts to obtain numerically accurate solutions to the turbulent kinematic dynamo equations. Since the magnetic induction equation is not self-adjoint, numerical attempts to date have to impose some extraneous ad hoc ’’criteria of goodness’’ at any given level of numerical truncation. The conserved quantities given here provide an internal check of the accuracy of any numerical calculation without the necessity for arbitrarily imposed external criteria of accuracy. As such they should be a powerful tool in rapidly increasing the accuracy of numerical solutions to the kinematic dynamo equations. We also point out that the conserved quantities can be used to indicate the possibility of kinematic dynamo activity ahead of any detailed calculations.
The generalized Wiener–Feynman path integrals
Authors: Maheshwari Amar
Published: March 2003

The generalized Wiener–Feynman path integrals are defined by the primitive mappings of the canonical Gaussian measure on a Hilbert space of real square integrable functions. The expressions of the covariance of the pro (pseudo) measures are found to be form covariant. The measures known in the literature by names as Wiener–Feynman and Uhlenbeck–Ornstein appear as special cases of no particular remark in our general definition. The connection of the primitive mapping with the general class of linear Cameron–Martin transformations is established.
A theorem on stress–energy tensors
Authors: Lord Eric A.
Published: March 2003

The equality of the symmetrized Noether stress–energy tensor (Belinfante’s tensor) and the canonical stress–energy tensor (functional derivative of the Lagrangian density with respect to the metric) is established by methods based on the formalism of tetrads and Ricci rotation coefficients. The result holds for any Lagrangian which contains no derivatives of the fields higher than first order.
Multigroup neutron transport. II. Half range
Authors: Bowden R. L. , Sancaktar S. , Zweifel P. F.
Published: March 2003

This paper accompanies a preceding one in which a functional analytic method was used to obtain the full-range expansion in multigoup neutron transport. In the present paper the analysis is extended to obtain the half-range expansion. The method used is an extension of the work of Larsen and Habetler for the one-group case. The results are given in terms of certain matrices which are solutions of coupled integral equations and which factor the dispersion matrix.
Spectrum generating group of the symmetric top molecule
Authors: Bohm Arno , Teese R. B.
Published: March 2003

The spectrum generating group for the symmetric top is found and applied to derive spectra, energy values and selection rules. All the well-known results are reproduced and some generalizations are obtained as a consequence of the group property, except for the Coriolis splitting for which an additional dependence upon the factor k/j is derived.
Prolongation structures and a generalized inverse scattering problem
Authors: Morris Hedley C.
Published: March 2003

The prolongation structure method of Wahlquist and Estabrook is used to determine a generalized inverse scattering problem for the equation utt=uxx+6(uux)x+uxxxx which describes the motion of shallow-water waves under gravity. The relevant Gel’fand–Levitan equation is solved for the single soliton solutions.
Prolongation structures and nonlinear evolution equations in two spatial dimensions
Authors: Morris H. C.
Published: March 2003

The prolongation structure approach of Wahlquist and Estabrook is used to determine nonlinear evolution equations in two spatial dimensions for which an inverse scattering formulation exists. The equations of nonlinear wave–envelope interactions and the Kadomtsev–Petviashvilli–Dryuma equation are considered in detail.
Octonionic Hilbert spaces, the Poincaré group and SU(3)
Authors: Günaydin Murat
Published: March 2003

A formalism based on real octonions is developed in order to construct an octonionic Hilbert space for the description of colored quark states. The various possible forms of scalar products and related scalars are discussed. The choice of a direction in the space of octonion units leads naturally to a representation of the Poincaré group in terms of complex scalar products and complex scalars. The remaining octonion directions span the color degrees of freedom for quarks and anti-quarks. In such a Hilbert space, product states associated with color singlets are shown to form a physical quantum mechanical Hilbert space for the description of hadrons. Color triplets, on the other hand, correspond to unobservable parafermion states of order three.
Euclidean field theory. II. Remarks on embedding the relativistic Hilbert space
Authors: Challifour John L.
Published: March 2003

Starting from extensions of the Schwinger functions to a positive, symmetric linear functional on the Borchers algebra we give a theorem on the embedding of the physical Hilbert space as a closed subspace of a Euclidean Hilbert space. The case of locally L1 strongly positive extensions is discussed as is the relation to Nelson’s Markov property for Euclidean fields.
Tensor spherical harmonics and tensor multipoles. I. Euclidean space
Authors: Daumens M. , Minnaert P.
Published: March 2003

Two bases in the Hilbert space of tensor fields on the unit sphere are discussed: the tensor spherical harmonics and the tensor multipoles. For vector fields these two bases are related by an orthonormal transformation whose coefficients are shown to be Clebsch–Gordan coefficients. This remark suggests a method of building multipole bases for higher order tensor fields. The second order tensor multipoles are studied in detail as well as their relations with the symmetric ones defined by several authors for application to the gravitational radiation.
Resistance inequalities for the isotropic Heisenberg ferromagnet
Authors: Powers Robert T.
Published: March 2003

Inequalities concerning the spin correlations of states of finite energy of the isotropic ferromagnetic Heisenberg model are proved. These inequalities estimate the spin correlation between two lattice sites in terms of the total energy and the electrical resistance between the lattice sites as calculated using the inverse of the coefficients occurring in the Heisenberg Hamiltonian. These resistance inequalities are combined with the resistance properties of a regular three-dimensional lattice to yield the result that all states of finite energy for the isotropic Heisenberg model in three dimensions have long range order.
Perfect fluids and symmetry mappings leading to conservation laws
Authors: Oliver D. R. , Davis W. R.
Published: March 2003

Some of the results recently obtained by Glass relating to shear-free perfect fluids are extended and further interpreted. In particular, it is pointed out that certain symmetry methods are fundamental to this type of investigation. This approach of using symmetry mappings (e.g., characterizing vanishing shear) at the level of the matter tensor is seen to naturally lead to considerations of an important family of symmetry properties which include Ricci collineations as a special case. These considerations are used to obtain new conservation expressions holding for perfect fluids.
Principal null directions without spinors
Authors: Stehney Ann K.
Published: March 2003

The method used by Petrov to obtain the first classification of vacuum space–times has since been overshadowed by methods which yield a finer classification based on principal null directions. It is shown that this finer classification can be expressed in Petrov’s terms and a simple algorithm is obtained for finding all repeated principal null directions by matrix methods.
The high conductivity limit in mean field electrodynamics
Authors: Krause F. , Roberts P. H.
Published: March 2003

An attempt is made to resolve conflicting views on the reliability of first order smoothing theory when applied to electromagnetic induction of magnetic field by a turbulently moving conductor, in the astrophysically most interesting case of large microscale magnetic Reynolds numbers.
On the analyticity of certain stationary nonvacuum Einstein space–times
Authors: Lindblom Lee
Published: March 2003

It is shown that certain nonvacuum solutions of Einstein’s general relativistic field equations are analytic space–times, i.e., an analytic atlas exists with respect to which the components of the metric tensor, and all material fields are analytic functions. The two specific cases discussed here are interesting from an astrophysical point of view. The first is the class of space–times containing a source free electromagnetic field: the exterior of a charged black hole, for example. The second is the class of space–times filled with a rigidly moving perfect fluid, often used to describe the interior of a rotating star.
Steady, one-dimensional multigroup neutron transport with anisotropic scattering
Authors: Larsen Edward W. , Zweifel Paul F.
Published: March 2003

The solution of steady, one-dimensional half-space multigroup transport problems with degenerate anisotropic scattering is obtained for L1 sources and incident distributions. The solution is expressed in terms of contour integrals of the resolvent operator (λI−K)−1, where K is the ’’separated’’ transport operator. The connection between this method and the ’’Case eigenfunction’’ method is briefly discussed, and the half-space albedo problem is treated in detail. This problem reduces to obtaining the Wiener–Hopf factorization of the dispersion matrix, hence to solving two coupled nonlinear, nonsingular matrix integral equations.
A simple proof of the angular momentum Helmholtz theorem and the relation of the theorem to the decomposition of solenoidal vectors into poloidal and toroidal components
Authors: Moses H. E.
Published: March 2003

Vector spherical harmonics are used in a simple proof of the angular-momentum Helmholtz theorem. The decomposition of vectors defined on a sphere into two components which this theorem gives is carried out explicitly. Furthermore, the potentials which occur in the theorem are given explicitly in terms of the original vector. The decomposition of solenoidal vectors into poloidal and toroidal components is also carried out explicitly. It is shown how these components are related to the components given by the angular-momentum Helmholtz theorem.
Gauge theories for space–time symmetries. I
Authors: Harnad J. P. , Pettitt R. B.
Published: March 2003

The general formulation of gauge invariant field theories based upon space–time symmetries is developed and given its geometrical interpretation. The consequences of gauge invariance, in the form of identities and conservation laws, are derived and the field equations are obtained from a class of gauge invariant Lagrangians. This is the first paper of a series, the subsequent work treating specific cases, in particular, conformal invariance.
Markovian subdynamics in quantum dynamical systems
Authors: Chen Etang
Published: March 2003

Subdynamical systems induced from a given quantum dynamical system are studied in the framework of operator algebras. Sufficient conditions are shown for induced subdynamics to be Markovian. It is also proved that the ergodicity of states in Markovian subsystems is preserved.
A soluble dispersion relation for a three-dimensional band structure
Authors: da Costa R. C. T. , Lobo R.
Published: March 2003

We show that an exact dispersion relation can be obtained for a cubic lattice made of spherically-symmetric attractive potentials. This result is obtained in a limit case where the potentials have zero range and infinite intensity.
On zeroes of the pion electromagnetic form factor
Authors: Raszillier I. , Schmidt W. , Stefanescu I.-S.
Published: March 2003

We develop a general procedure for the location of possible zeroes of the pion form factor, which relies on interpolation theory for analytic functions. The zeroes are confined (in the unit disk) to regions bounded by (real) roots of algebraic equations and by algebraic curves. These regions depend both on the interpolation data and the class of functions, which is suitable for the physical problem.
On intelligent spin states
Authors: Aragone C. , Chalbaud E. , Salamó S.
Published: March 2003

In this paper we give a more compact representation of the intelligent spin states defined by Aragone, Guerri, Salamó, and Tani. Using this new representation, we discuss the differences between minimum uncertainty states, coherent Bloch spin states and intelligent states. The evolution of these states under a particular time dependent Hamiltonian is studied, showing the relevance of the noncompact subgroup K of the Lorentz group. Finally we analyze the radiative properties connected with the intelligent states for a pointlike medium. The main results are: (I) they have a nonvanishing dipole moment (as the Bloch states) and (II) the proper intelligent states give a spontaneous emission intensity which is different from the one provided by the Bloch states.
On the irreducible representations of the Lie algebra chain G2⊇A2
Authors: Perroud M.
Published: March 2003

In the first part of this article we solve the ’’state labelling problem’’ for the irreducible finite dimensional representations of the G2⊆A2 chain, using a method applicable to other algebras-subalgebras chains. In the second part we define, for these representations, operators analogous to those introduced by Nagel and Moshinsky for the An⊆An−1 chain and explicitly construct the representations belonging to two equivalent classes.
Propagator techniques for equilibrium strongly coupled Fermi fluids
Authors: Reichl L. E.
Published: March 2003

This is Paper III of a series of three papers in which a self-consistent propagator resummation of self-energy effects in a strongly coupled Fermi fluid is performed. In the present paper, a Laplace transformation of the reaction matrix expansion of the grand potential and the magnetization is obtained, and a self-consistent propagator resummation of self-energy effects due both to the medium and to constant and spatially varying external magnetic fields is performed. The procedure for resumming polarization diagrams is discussed.
A calculation of SU(4) Clebsch–Gordan coefficients
Authors: Haacke E. M. , Moffat J. W. , Savaria P.
Published: March 2003

All the Clebsch–Gordan coefficients for SU(4) that would be required for particle physics, including those decomposed with respect to SU(3), are obtained by using the general formalism of Baird and Biedenharn.
Green’s functions for a face centered orthorhombic lattice
Authors: Hoshen J. , Kopelman R.
Published: March 2003

Diagonal and off-diagonal matrix elements of the Green’s functions for a face centered orthorhombic lattice are presented in terms of integrals of complete elliptic integrals of the first and third kind. These Green’s functions are also applicable to structures like that of the benzene crystal (space group D152h, interchange symmetry D2).
Automorphisms of the Lie algebra of polynomials under Poisson bracket
Authors: Wollenberg L. S.
Published: March 2003

By constructing the one-parameter group of automorphisms generated by a typical derivation and generalizing certain special cases arising, we find all the automorphisms of the Lie algebra of polynomials under Poisson bracket. We introduce the notion of quasi-Hamiltonian equations, and investigate the effect of transformations (q,p) → (α (q),α (p)) (α an arbitrary automorphism) on such equations. By considering linear quasi-Hamiltonian equations with constant coefficients we obtain a conserved quantity for an arbitrary (2 × 2) linear system with constant coefficients.
On the tensor representation for compact groups
Authors: Backhouse Nigel , Gard Patricia
Published: March 2003

A recent paper of Kasperkovitz and Dirl [J. Math. Phys. 15, 1203 (1974)] concerning the tensor representation for compact groups is examined critically. The flaw which is found in the main theorem fortunately does not affect the deductions which are made from that theorem.
N-level systems in contact with a singular reservoir. II
Authors: Frigerio Alberto , Gorini Vittorio
Published: March 2003

We study an N-level system coupled linearly to an infinite quasifree Fermi or Bose reservoir in the vacuum state or in a state corresponding to an arbitrary temperature. We show that the singular reservoir limit can be performed in the vacuum state and at infinite temperature, thus leading to a completely positive Markovian reduced time evolution for the system, which, in the infinite temperature case, preserves the central state. On the other hand, no such limit is possible for KMS states (finite temperature) and at zero temperature. Some extension to norm-continuous semigroups of an infinite-dimensional ℬ (ℋ) is possible.
A note on energy bounds for boson matter
Authors: Brydges David , Federbush Paul
Published: March 2003

Two new proofs are given of the Dyson and Lenard lower bound for the energy of matter with boson electrons. Another result is a new inequality for the two-point correlation function.
Shock waves in relativistic magnetohydrodynamics under general assumptions
Authors: Lichnerowicz André
Published: March 2003

We present a rigorous theory of magnetohydrodynamical shock waves in the framework of a given curved space–time, under general assumptions corresponding both to a plasma and to a condensed medium. The results can be of use in astrophysics. We prove the timelike character of the wavefronts, the main thermodynamic inequalities, the relative location of the speeds of the shock waves with respect to the magnetosonic and Alfvén speeds, and show some existence and uniqueness theorems. In particular, we show that there can exist initial states giving slow shocks, but no weak shocks.
Differential equations satisfied by Fredholm determinants and application to the inversion formalism for parameter dependent potentials
Authors: Cornille Henri
Published: March 2003

We consider the two first order differential operators Ax=μ (x) ∂/∂x+λ (x), Bx=−(∂/∂x) μ (x)+λ (x), associate two kernels f, g satisfying both well-defined boundary conditions and Axf=Byg, Ayf=Bxg, and construct the Fredholm determinants corresponding to these kernels. From these determinants we can build up solutions of second order differential equations. These solutions have an interpretation in the Schrödinger inversion formalism. For instance, for the inversion at fixed angular momentum l, these solutions for k=0 correspond to the classical Gel’fand–Levitan and Marchenko equations, whereas, for k≠0, they correspond to k dependent potentials. Similarly, for the inversion at fixed k, these solutions for l=0 correspond to the classical Regge–Newton equations, whereas, for l≠0, they correspond to l-dependent potentials. More generally we show, in a generalized inversion formalism, how parameter dependent potentials appear very naturally in the theory.
A note on certain multiple integrals
Authors: Mehta M. L.
Published: March 2003

An analytical expression is given for the multiple integral ∫⋅⋅⋅∫dμ (xn+1)  dμ (xn+2) ⋅⋅⋅dμ (xN) Π1⩽j<k⩽N‖xj −xk‖β, where 0⩽n⩽N, β=1,2, or 4 and the positive measure dμ (x) is such that all its moments exist, ∫dμ  (x) xj<∞, j=0,1,2,⋅⋅⋅. The case dμ (x) =dx, for −1⩽x⩽1, and dμ (x) =0, for ‖x‖≳1, is given as an example. In the limit N→∞ the correlation functions of this example, the so-called Legendre ensembles, coincide with those of the circular or the Gaussian ensembles of random matrices.
Some basic properties of Killing spinors
Authors: Hacyan Shahen , Plebański Jerzy
Published: March 2003

The concept of Killing spinor is analyzed in a general way by using the spinorial formalism. It is shown, among other things, that higher derivatives of Killing spinors can be expressed in terms of lower order derivatives. Conformal Killing vectors are studied in some detail in the light of spinorial analysis: Classical results are formulated in terms of spinors. A theorem on Lie derivatives of Debever–Penrose vectors is proved, and it is shown that conformal motion in vacuum with zero cosmological constant must be homothetic, unless the conformal tensor vanishes or is of type N. Our results are valid for either real or complex space–time manifolds.
The intrinsic spinorial structure of hyperheavens
Authors: Finley J. D. , Plebański J. F.
Published: March 2003

Following Plebański and Robinson, complex V4’s which admit a congruence of totally null surfaces are shown to have coordinates which, in pairs, have a spinor structure which generates the usual spinor structure of the 2-forms over the space. This structure allows Einstein’s vacuum equations to fracture into three triples and a singlet, which allow for easy reduction of the entire set to one nonlinear partial differential equation needed for consistency. An inhomogeneous GL̃ (2,C) group of coordinate transformations, constrained to leave the tetrad form invariant, is constructed and used to simplify the equations and clarify the geometrical meaning of the parameters introduced during the integration process.
Monopoles, vortices and the geometry of the Yang–Mills bundles
Authors: Ezawa Z. F. , Tze H. C.
Published: March 2003

A topological classification of monopoles and vortices is formulated in terms of fibre bundles. The distinction between Dirac and ’t Hooft monopoles is made in the light of the energy finiteness problem. Finite-length vortices with Dirac monopoles at the end points are also discussed.
Functional calculus for the symmetric multigroup transport operator
Authors: Greenberg William
Published: March 2003

A rigorous treatment of the symmetric multigroup transport equation is given by developing the functional calculus for the transport operator. Von Neumann spectral theory is applied to nonorthogonal cyclic subspaces, and the isometries onto C (N) are explicitly evaluated.
The Case eigenfunction expansion for a conservative medium
Authors: Greenberg William , Zweifel Paul F.
Published: March 2003

By using the resolvent integration technique introduced by Larsen and Habetler, the one-speed, isotropic scattering, neutron transport equation is treated in the infinite and semi-infinite media. It is seen that the results previously obtained by Case’s ’’singular eigenfunction’’ approach are in agreement with those obtained by resolvent integration.
The master analytic function and the Lorentz group. I. Reduction of the representations of O(3,1) in O(2,1) basis
Authors: Basu Debabrata , Majumdar S. Datta
Published: March 2003

The reduction of the principal and supplementary series of representations of SL(2,C) in the SU(1,1) basis is carried out by using a basis function which formally resembles the coupled state of two angular momenta. The spectrum of the SU(1,1) representations contained in SL(2,C) and the transformation coefficients are obtained by expanding the SU(2) in terms of the SU(1,1) bases with the help of the Sommerfeld–Watson transformation. The orthogonality conditions for the principal and supplementary series are discussed. For the principal series this follows easily from the standard Sturm–Liouville theory of the second order differential equations. For the supplementary series the orthogonality condition is obtained from the fourth order differential equation satisfied by the Fourier transform of the basis function.
The master analytic function and the Lorentz group. II. The Clebsch–Gordan problem for O(2,1)
Authors: Majumdar S. Datta
Published: March 2003

The Clebsch–Gordan coefficients of the noncompact group O(2,1) representing Lorentz transformations in three-dimensional space–time are calculated in the compact O(2) basis. Considerable simplification is achieved by introducing a variable x and replacing all algebraic equations by differential equations. The coupled state appears in the theory as a solution of an ordinary differential equation reducible to the hypergeometric equation by a simple substitution. The coefficients in the Taylor–Laurent expansion of this solution in powers of x are shown to be identical with the Clebsch–Gordan coefficients. The inverse expansion, obtained by the use of certain identities for the hypergeometric function and the Sommerfeld–Watson transformation, yields the normalization factor and the values of j appearing in the reduction.
Stability of nonlinear parametric-decay interactions in finite homogeneous plasma
Authors: Fuchs V. , Beaudry G.
Published: March 2003

We prove the existence and uniqueness of a stable steady state of stimulated backscattering from a bounded homogeneous lossless interaction region, with boundary conditions corresponding to a steady incident pump wave-mode Poynting flux and zero-flux input to the decay wave modes. In steady state, once the excitation of the interaction region exceeds a certain critical value, the boundary value problem is characterized by a finite number of eigenvalues, and associated nontrivial eigenfunctions equilibria of the system, corresponding to mutually distinct states of anomalous reflection of the pump wave. A stability analysis of these equilibria with respect to small phase and amplitude perturbations reveals that (i) in the vicinity of the nonfundamental equilibria the phase perturbations exhibit singularities, preventing phase locking from occurring, and (ii) in the vicinity of the fundamental equilibrium both the phase and amplitude perturbations asymptotically vanish. A WKBJ phase-integral stability condition is derived to show that growing normal modes of the amplitude-perturbation boundary-value problem cannot propagate in the potential formed by the field of the depleted, spatially inhomogeneous, pump of the fundamental equilibrium.
Gurtin-type properties associated with wave propagation in a superfluid
Authors: Murray J. C.
Published: March 2003

Variational and reciprocity principles of the Gurtin type are established for the initial-boundary value problem associated with wave propagation in a superfluid.
Approach of the statistical theory of light scattering to the phenomenological theory
Authors: Varatharajulu V. , Vezzetti D. J.
Published: March 2003

A fully statistical treatment of the spectrum of light scattered by a simple fluid is given. The results are shown to be in close accord with the phenomenological theory of the same process.
Asymptotic properties of generalized Chaba and Pathria lattice sums
Authors: Hall George L.
Published: March 2003

For an arbitrary Bravais τ lattice in an m-dimensional Euclidean space amd for 0<a<∞, we present an extension of the Chaba and Pathria method of evaluating the lattice sums Σ′ττ−2k exp(−aτ2) from integral k values to all positive k values. We use the extension to study the asymptotic properties of these sums as the parameter a approaches zero. The leading term is given by πm/2 [Γ (k)(m/2−k) am/2−k]−1 for 0<2k<m, by πm/2Γ (k)−1 ln(1/a) for 2k=m, and by Σ′ττ−2k for 2k≳m. Thus 2k=m gives a transition point from structure independence to structure dependence.
Scalar potentials in the Dirac equation
Authors: Critchfield Charles L.
Published: March 2003

The Dirac equation for motion in a central potential is generalized to include scalar potentials proportional to r and to r−1. It is solved by analytic methods. The linear dependence upon radius leads to spectra similar to that of the harmonic oscillator except that the approximately constant level distance applies to E2 instead of E. A large negative term in the rest mass displaces the equilibrium point of the oscillator to a large radius.
Lie theory and separation of variables. 9. Orthogonal R-separable coordinate systems for the wave equation ψt
t−Δ2ψ=0
Authors: Kalnins E. G. , Miller W.
Published: March 2003

A list of orthogonal coordinate systems which permit R-separation of the wave equation ψtt−Δ2ψ=0 is presented. All such coordinate systems whose coordinate curves are cyclides or their degenerate forms are given. In each case the coordinates and separation equations are computed. The two basis operators associated with each coordinate system are also presented as symmetric second order operators in the enveloping algebra of the conformal group O(3,2).
Lie theory and separation of variables. 11. The EPD equation
Authors: Kalnins E. G. , Miller Willard
Published: March 2003

We show that the Euler–Poisson–Darboux equation {∂tt−∂rr−[(2m+1)/r]∂r}Θ=0 separates in exactly nine coordinate systems corresponding to nine orbits of symmetric second-order operators in the enveloping algebra of SL(2,R), the symmetry group of this equation. We employ techniques developed in earlier papers from this series and use the representation theory of SL(2,R) to derive special function identities relating the separated solutions. We also show that the complex EPD equation separates in exactly five coordinate systems corresponding to five orbits of symmetric second-order operators in the enveloping algebra of SL(2,C).
Spectrum generating algebras and Lie groups in classical mechanics
Authors: Andrié M.
Published: March 2003

We give a general framework for a geometric foundation of time dependent classical mechanics. The theory is based on the concept of evolution space which is phase space extended by time. Lie algebras of constants of motion which may possess explicit time dependence are constructed, and general conditions for getting global Lie group actions from infinitesimal actions are derived. In a natural way these groups map solutions of the Hamiltonian equations of motion onto one another and act on the orbit space via symplectic transformations. The theory is applied to the nonrelativistic free particle, the harmonic and damped oscillator, nonstationary quadratic systems, and to the motion of a particle in constant electromagnetic fields.
Duality transformation in a many-component spin model
Authors: Wu F. Y. , Wang Y. K.
Published: March 2003

It is shown that the duality transformation relates a spin model to its dual whose Boltzmann factors are the eigenvalues of the matrix formed by the Boltzmann factors of the original spin model. The duality relation valid for finite lattices is obtained, and applications are given.
Solutions of the three magnon bound state equation. I
Authors: Majumdar Chanchal K.
Published: March 2003

Recently several unphysical solutions of the three magnon bound state equation for the isotropic Heisenberg Hamiltonian have been found, and one unphysical solution for the Hamiltonian with longitudinal anisotropy has been computed. Here we complete the work for such unphysical solutions for all anisotropy from the Ising to the isotropic Heisenberg limit by directly solving the integral equation. Two types of wavefunctions are constructed. The eigenvalue of the first type satisfies a cubic equation in general, but gives only a real root in the Ising limit and a pair of complex conjugate roots in the isotropic limit. The other type has a single eigenvalue; this one, previously known numerically, is shown to have a simple analytic form.
Solutions of the three magnon bound state equation. II
Authors: Mukhopadhyay S. K. , Majumdar Chanchal K.
Published: March 2003

A unified derivation of all the unphysical bound state solutions of the three magnon bound state equation found so far is given by making a simple algebraic transformation of the variables in the equation.
Application of coherent state representation to classical x6 and coupled anharmonic oscillators
Authors: Dutt R. , Lakshmanan M.
Published: March 2003

The problem of obtaining perturbative solutions to the nonlinear differential equations which describe the motion of the x6 and quartically coupled oscillators is treated by the use of the well-known coherent state representation. The results exhibit the basic qualitative features of nonlinearities and the characteristics of a coupled system in the weak coupling limit.
Canonical transformations and phase space path integrals
Authors: Fanelli Robert
Published: March 2003

A previous discussion of canonical transformations and path integrals is extended to the phase space path integral method. Within this approach a broader class of canonical transformations can be introduced than within the Lagrangian approach, including coordinate transformations and essentially all infinitesimal tranformations.
Comultiplicator of finite magnetic groups
Authors: Rudra P.
Published: March 2003

A matrix algebraic method for constructing the comultiplicator group for any finite magnetic group is given. Each element of this comultiplicator group corresponds to an inequivalent factor system of the magnetic group.
Removal of the nodal singularity of the C-metric
Authors: Ernst Frederick J.
Published: March 2003

The charged C-metric is transformed into another exact solution of the Einstein–Maxwell field equations corresponding to a massive charged particle accelerated by an electric field. When the appropriate equations of motion are satisfied, the nodal singularity associated with the C-metric disappears.
Wigner coefficients for the theory of five-dimensional quasispin
Authors: Hongoh M.
Published: March 2003

We present some Gaunt-type formulas and several classes of multiplicity-free Wigner coefficients for the noncanonical reduction O(5) ⊆SU(2) ×U(1).
On path integral solutions of the Schrödinger equation, without limiting procedure
Authors: Mizrahi Maurice M.
Published: March 2003

Techniques for integration in function spaces which are not necessarily vector spaces are presented in the light of DeWitt-Morette’s recent redefinition of path integrals, which does not involve the usual limiting process, and is therefore free from the ambiguities inherent in this approach. General translated Gaussian measures defined by their Fourier transforms are introduced on various path spaces and used to derive generalized moments formulas, and what is essentially the Feynman–Kac formula for the expression of solutions of the Schrödinger equation as functional integrals.
Geometrical optics in general relativity: A study of the higher order corrections
Authors: Anile A. M.
Published: March 2003

The higher order corrections to geometrical optics are studied in general relativity for an electromagnetic test wave. An explicit expression is found for the average energy–momentum tensor which takes into account the first-order corrections. Finally the first-order corrections to the well-known area-intensity law of geometrical optics are derived.
Models of Zermelo Frankel set theory as carriers for the mathematics of physics. I
Authors: Benioff Paul A.
Published: March 2003

This paper is a first attempt to explore the relationship between physics and mathematics ’’in the large.’’ In particular, the use of different Zermelo Frankel model universes of sets (ZFC models) as carriers for the mathematics of quantum mechanics is discussed. It is proved that given a standard transitive ZFC model M, if, inside M, B (ℋM) is the algebra of all bounded linear operators over a Hilbert space ℋM, there exists, outside M, a Hilbert space ℋ and an algebra B (ℋ), along with isometric monomorphisms UM and VM from ℋM into ℋ and from B (ℋM) into B (ℋ). UM and VM are used to relate quantum mechanics based on M to quantum mechanics based on the usual ZFC model. It is then shown that, contrary to what one would expect, all ZFC models may not be equivalent as carriers for the mathematics of physics. In particular, it is proved that if one requires that an outcome sequence, associated with an infinite repetition of measuring a question observable on a system prepared in some state, be random, and if a strong definition of randomness is used, then the minimal standard ZFC model cannot be a carrier for the mathematics of quantum mechanics.