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Volume 28, Issue 7, July 1987

On subalgebras of the Lie algebra of the extended Poincaré group P̃(1,n)
View Description Hide DescriptionSome general results on the subalgebras of the Lie algebra AP̃(1,n) of the extended Poincaré group P̃(1,n) (n≥2) with respect to P̃(1,n) conjugation have been obtained. All subalgebras of AP̃(1,4) that are nonconjugate to the subalgebras of AP(1,4) are classified with respect to P̃(1,4) conjugation. The list of representatives of each conjugacy class is presented.

Time‐ordered operators and Feynman–Dyson algebras
View Description Hide DescriptionAn approach to time‐ordered operators based upon von Neumann’s infinite tensor product Hilbert spaces is used to define Feynman–Dyson algebras. This theory is used to show that a one‐to‐one correspondence exists between path integrals and semigroups, which are integral operators defined by a kernel, the reproducing property of the kernel being a consequence of the semigroup property. For path integrals constructed from two semigroups, the results are more general than those obtained by the use of the Trotter–Kato formula. Perturbation series for the Feynman–Dyson operator calculus for time evolution and scattering operators are discussed, and it is pointed out that they are ‘‘asymptotic in the sense of Poincaré’’ as defined in the theory of semigroups, thereby giving a precise formulation to a well‐known conjecture of Dyson stated many years ago in the context of quantum electrodynamics. Moreover, the series converge when these operators possess suitable holomorphy properties.

A uniqueness theorem for an inverse Sturm–Liouville problem
View Description Hide DescriptionA new uniqueness theorem is established for the inverse Sturm–Liouville problem. It is shown that the measurement of a particular eigenvalue for an infinite set of different boundary conditions is sufficient to determine the unknown potential.

Ovsiannikov’s method and the construction of partially invariant solutions
View Description Hide DescriptionA plausibility argument presented by the first two authors in an earlier paper [J. Math. Phys. 2 6, 3042 (1985)] concerning the existence of partially invariant solutions for some equations of the Fokker–Planck type is made precise by the explicit construction of one such solution. In the process a substantial simplification of Ovsiannikov’s method for finding partially invariant solutions is achieved. In addition, the class of partially invariant solutions obtained by Ovsiannikov for the equations of transonic flow of a gas is enlarged.

On the stability of the telegraph equation
View Description Hide DescriptionThe linear first‐order boundary conditions that will lead to a stable (well‐posed) problem for the telegraph equation in quarter space are established.

Multidimensional inverse scattering: An orthogonalization formulation
View Description Hide DescriptionThe three‐dimensional Schrödinger equationinverse scattering problem is solved using an orthogonalization approach. The plane waves propagating in free space are orthogonalized with respect to an inner product defined in terms of a Jost operator. The resulting integral equation is identical to the generalized Gel’fand–Levitan equation of Newton, although the present derivation is simpler and more physical than that of Newton. Newton’s generalized Marchenko equation is derived from the defining integral equation for the Jost operator. These integral equations are shown to be solved by fast algorithms derived directly from the properties of their solutions. This paper thus presents a simple interpretation of Newton’s two integral equations, two fast algorithms for solving these integral equations, and relations between the various approaches. This is a generalization of previously obtained results, which are also reviewed here, for the one‐dimensional inverse scattering problem.

Any physical, monopole equation of motion structure uniquely determines a projective inertial structure and an (n−1)‐force
View Description Hide DescriptionIt is proved that, in the context of a conformal causal structure, (a) any acceleration field decomposes uniquely into the sum of an affine structure that is compatible with the conformal structure and an n‐force, and (b) any directing field, such that the n‐force of the corresponding family of acceleration fields is due to tensor fields and is orthogonal to the n‐velocity, uniquely decomposes into a projective structure that is compatible with the conformal structure and an (n−1)‐force. Moreover, if there are no second clock effects and variable rest masses do not exist, there exists a unique pseudo‐Riemannian metric on space‐time that determines the unique standard of no acceleration for all massive monopoles. It follows from this that a non‐null result for the Eötvös experiment entails the existence of a f i f t h f o r c e rather than a violation of the universality of free fall.

Canonical structures for dispersive waves in shallow water
View Description Hide DescriptionThe canonical Hamiltonian structure of the equations of fluid dynamics obtained in the Boussinesq approximation are considered. New variational formulations of these equations are proposed and it is found that, as in the case of the KdV equation and the equations governing long waves in shallow water, they are degenerate Lagrangian systems. Therefore, in order to cast these equations into canonical form it is again necessary to use Dirac’s theory of constraints. It is found that there are primary and secondary constraints which are second class and it is possible to construct the Hamiltonian in terms of canonical variables. Among the examples of Boussinesq equations that are discussed are the equations of Whitham–Broer–Kaup which Kupershmidt has recently expressed in symmetric form and shown to admit tri‐Hamiltonian structure.

The Dirac wave equation in the presence of an external field
View Description Hide DescriptionThe method of characteristics is applied to the Diracwave equation in the presence of an external field. The retarded Green’s function for the minimal coupling to an external electromagnetic field is calculated explicitly, and a general coupling is discussed.

Extensions of Wigner’s distribution to particles with spin 1/2
View Description Hide DescriptionFor particles of spin 1/2 a class of distributions closely related to Wigner’s is introduced on dynamical grounds. It is found that they may be refined to give correct expectation values of higher powers of spin components, but depart somewhat from a criterion that has been used in characterizing the Wigner distribution. For a special choice amongst this class, a more subtle refinement is possible satisfying this criterion exactly. This requires, however, a dubious distinction at every point between positive spin about a direction and negative spin about the reverse direction.

Witten index, axial anomaly, and Krein’s spectral shift function in supersymmetric quantum mechanics
View Description Hide DescriptionA new method is presented to study supersymmetric quantum mechanics. Using relative scattering techniques, basic relations are derived between Krein’s spectral shift function, the Witten index, and the anomaly. The topological invariance of the spectral shift function is discussed. The power of this method is illustrated by treating various models and calculating explicitly the spectral shift function, the Witten index, and the anomaly. In particular, a complete treatment of the two‐dimensional magnetic field problem is given, without assuming that the magnetic flux is quantized.

An SU(8) model for the unification of superconductivity, charge, and spin density waves
View Description Hide DescriptionA model Hamiltonian for a many‐electron system which unifies superconductivity,charge density waves, and spin density waves is analyzed. It is shown that the spectrum generating algebra for this system is su(8), and all 63 generators of this Lie algebra are identified. The seven symmetry operators that are broken in transition to the condensed state are identified, together with 56 order operators, whose expectations give the order parameters of the various phases present in the model. The discrete symmetry properties of these operators are tabulated. A chain of subalgebras of submodels with corresponding decoupled phases is constructed. Finally, how the finite temperature Green’s functions may be obtained and used to solve the problem of self‐consistency of the order parameters in the model is indicated.

Separability of the Killing–Maxwell system underlying the generalized angular momentum constant in the Kerr–Newman black hole metrics
View Description Hide DescriptionThe concept of a Killing–Maxwell system may be defined by the relation Â_{[μ;v];ρ} =(4π/3)ĵ_{[μ} g _{ν]}. In such a system the one‐form Â_{μ} is interpretable as the four‐potential of an electromagnetic field F̂_{μv }, whose source currentĵ ^{μ} is an ordinary Killing vector. Such a system determines a canonically associated duality class of source‐free electromagnetic fields, its own dual being a Killing–Yano tensor, such as was found by Penrose [Ann. N.Y. Acad. Sci. 2 2 4, 125 (1973)] (with Floyd) to underlie the generalized angular momentumconservation law in the Kerr black hole metrics, the existence of the Killing–Yano tensor being also a sufficient condition for that of the Killing–Maxwell system. In the Kerr pure vacuum metric and more generally in the Kerr–Newman metrics for which a member of the associated family of source‐free fields is coupled in gravitationally, it is shown that the gauge of the Killing–Maxwell one‐form may be chosen so that it is expressible (in the standard Boyer–Lindquist coordinates) by 1/2 (a ^{2} cos ^{2} θ−r ^{2})d t+ 1/2 a(r ^{2}−a ^{2})sin^{2} θ dφ, the corresponding source current being just (4π/3)(∂/∂t). It is found that this one‐form (like that of the standard four‐potential for the associated source‐free field) satisfies the special requirement for separability of the corresponding coupled charged (scalar or Dirac spinor) wave equations.

A new self‐similar space‐time
View Description Hide DescriptionA new self‐similar solution of the Einstein field equations is presented. In the new space‐time, the density is zero at time zero and follows an inverse square law for large t. The new solution may have interesting astrophysical applications since it has the same reference lengths as that of the Friedmann universe.

On the solution of the Tolman–Oppenheimer–Volkov equation with the ultrarelativistic equation of state
View Description Hide DescriptionThe Tolman–Oppenheimer–Volkov equation is studied in the case of the ultrarelativistic equation of state. The original system of two first‐order differential equations is turned into one first‐order equation that is independent of the central density, plus an integral. It is shown how the physical solutions are related to the analytically known infinite central density solution. The results are further generalized into the arbitrary γ‐law equation of state,p(γ−1)ρ. Finally, the case of a nonzero bag constant is briefly discussed.

Spin fluids in stationary axis‐symmetric space‐times
View Description Hide DescriptionThe relations establishing the equivalence of an ordinary perfect fluid stress‐energy tensor and a spin fluid stress‐energy tensor are derived for stationary axis‐symmetric space‐times in general relativity. Spin fluid sources for the Gödel cosmology and the van Stockum metric are given.

Homogeneous space‐times of Gödel‐type in higher‐derivative gravity
View Description Hide DescriptionA general theorem concerning any Gödel‐type solution of higher‐derivative gravity field equations, which may be produced by any reasonable physical source with a constant energy‐momentum tensor, is analyzed. The resulting class of metrics depends on two parameters, one of which is related to the vorticity. A general class of solutions of Gödel‐type space‐time‐homogeneous universes in the context of the higher‐derivative theory is exhibited. This is the most general higher‐derivative solution of such type of metric and includes all known solutions of Einstein’s equations related to these geometries as a special case. A number of completely causal rotating models is also obtained. Some of them present the interesting feature of having no analogs in the framework of general relativity.

Symmetric tensor spherical harmonics on the N‐sphere and their application to the de Sitter group SO(N,1)
View Description Hide DescriptionThe symmetric tensor spherical harmonics (STSH’s) on the N‐sphere (S ^{ N }), which are defined as the totally symmetric, traceless, and divergence‐free tensor eigenfunctions of the Laplace–Beltrami (LB) operator on S ^{ N }, are studied. Specifically, their construction is shown recursively starting from the lower‐dimensional ones. The symmetric traceless tensors induced by STSH’s are introduced. These play a crucial role in the recursive construction of STSH’s. The normalization factors for STSH’s are determined by using their transformation properties under SO(N+1). Then the symmetric, traceless, and divergence‐free tensor eigenfunctions of the LB operator in the N‐dimensional de Sitter space‐time which are obtained by the analytic continuation of the STSH’s on S ^{ N } are studied. Specifically, the allowed eigenvalues of the LB operator under the restriction of unitarity are determined. Our analysis gives a group‐theoretical explanation of the forbidden mass range observed earlier for the spin‐2 field theory in de Sitter space‐time.

Exact solutions in 1+1 dimensions of the general two‐velocity discrete Illner model
View Description Hide DescriptionThe Illner model is the most general two‐velocity discrete model of a Boltzmann equation in one spatial dimension which satisfies an H‐theorem. It includes, as particular cases, both the Carleman and the McKean models. ‘‘Solitons’’ (one‐dimensional solutions) and ‘‘bisolitons’’ (two‐dimensional, space‐plus‐time, solutions), which are defined as rational fractions, and solutions with one or two exponential variables are determined. The model is treated as a nonintegrable nonlinear one, and from the solitons the possible class of bisolitons is guessed. Two classes of physically acceptable bisolitons are found. The first class is distributions positive only along one semiaxis and identically zero outside. These are interpreted physically by introducing elastic walls plus source or sink terms which become negligible at infinite time. The second class is periodic solutions which can be seen as damped sound waves. Essentially the same tools are used as in a companion paper for the six‐velocity Broadwell model, where the two bisoliton classes mentioned above also exist. This suggests that general methods for obtaining nontrivial exact solutions could exist for the hyperbolic semilinear discrete Boltzmann models.

The effect of a one‐dimensional potential of finite range on the statistical parameters of an incident ensemble of particles. I. Pure states
View Description Hide DescriptionIf an ensemble of particles moving parallel to the x axis in the positive direction impinges on a piecewise continuous potential confined to the interval [−a,a] (a>0) it will divide into a transmitted ensemble and a reflected ensemble. It is shown that the classical results for the means, variances, and covariance of position and velocity of the transmitted and reflected ensembles hold in quantum mechanics if the incident state is assumed to be pure and defined by a C _{∞}wave function whose Fourier transform has bounded, positive support, on which the modulus and argument of the transmission and reflection coefficients are C _{∞}.