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Dec 1975

Volume 16, Issue 12, pp. 2341-2526

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Representations of a local current algebra: Their dynamical determination

Ralph Menikoff and David H. Sharp

J. Math. Phys. 16, 2341 (1975); http://dx.doi.org/10.1063/1.522495 (12 pages) | Cited 7 times

Online Publication Date: 3 September 2008

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Local currents are used to describe nonrelativistic many‐body quantum mechanics in the thermodynamic limit. The problem of determining a representation of the local currents corresponding to a given Hamiltonian is studied. We formulate the dynamics in such a way that one solves simultaneously for the ground state and the representation of the local currents. This leads to two coupled functional equations relating the generating functional to a functional which describes the ground state. Together these functionals determine a representation of the local currents in which the Hamiltonian is a well‐defined operator. The functional equations are equivalent to a set of integro–differential equations for expansion coefficients of the two functionals.
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11.40.-q Currents and their properties

Approximate representations of a local current algebra

Ralph Menikoff and David H. Sharp

J. Math. Phys. 16, 2353 (1975); http://dx.doi.org/10.1063/1.522496 (8 pages) | Cited 1 time

Online Publication Date: 3 September 2008

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An approximate method for dealing with nonrelativistic many‐body quantum systems having short range interactions is developed using local currents. The scheme is based on determining approximate representations of subalgebras of the local currents. This mathematical framework is used to discuss several approximation schemes.
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11.40.-q Currents and their properties

The Weyl tensor and shear‐free perfect fluids

E. N. Glass

J. Math. Phys. 16, 2361 (1975); http://dx.doi.org/10.1063/1.522497 (3 pages) | Cited 14 times

Online Publication Date: 3 September 2008

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It is proved that a necessary and sufficient condition for a shear‐free perfect fluid to be irrotational is that the Weyl tensor be pure electric type. For shear‐free isentropic flow with unit tangent uα, we find the conservation law ∇α(n1/3iω uα) =0, where i is the relativistic specific enthalpy, n is the conserved particle number density, and ω is the vorticity scalar.
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04.40.-b Self-gravitating systems; continuous media and classical fields in curved spacetime

Singularities in nonsimply connected space–times

Dennis Gannon

J. Math. Phys. 16, 2364 (1975); http://dx.doi.org/10.1063/1.522498 (4 pages) | Cited 32 times

Online Publication Date: 3 September 2008

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Space–times with asymptotically flat nonsimply connected spacelike slices are shown to possess enough intrinsic geometric structure to guarantee the existence of singularities under conditions usually considered insufficient. In particular, it is shown that if the normal geodesics to the spacelike slice are converging on a suitable compact set, and the space–time satisfies a standard energy condition, then it is timelike geodesically incomplete. A similar result holds if the space–time satisfies the chronology and generic conditions.
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04.20.Cv Fundamental problems and general formalism

Ergodicity of quantum mechanical systems

Koichiro Matsuno

J. Math. Phys. 16, 2368 (1975); http://dx.doi.org/10.1063/1.522499 (4 pages) | Cited 1 time

Online Publication Date: 3 September 2008

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The ergodicity of pure quantum states is maintained in the space of orthonormal quantum states which diagonalizes an observable. The ergodicity of mixed quantum states, which is met in quantum statistical mechanics admitting an ensemble of many similar systems, is identical to the principle of equal a priori probabilities.
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05.30.-d Quantum statistical mechanics

Note on the computation formula of the boost matrices of SO(n−1,1) and continuation to the d matrices of SO(n)

Takayoshi Maekawa

J. Math. Phys. 16, 2372 (1975); http://dx.doi.org/10.1063/1.522500 (3 pages) | Cited 7 times

Online Publication Date: 3 September 2008

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The general formula of computing the boost matrices of SO(n−1,1), which is valid for the single‐ and double‐valued representations and is similar to that of Vilenkin and Wolf, is given. It is noted that a phase factor of a unit magnitude in the boost matrices must be taken into account in analytic continuation to the d matrices of SO(n), and then the formula of computing the d matrices of SO(n) is given. It is remarked that the d matrices of SO(n) are expressed in terms of those of SO(n−1).
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02.20.Qs General properties, structure, and representation of Lie groups

Retarded multipole fields and the inhomogeneous wave equation

W. E. Couch and R. J. Torrence

J. Math. Phys. 16, 2375 (1975); http://dx.doi.org/10.1063/1.522501 (3 pages)

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The inhomogeneous wave equation for a class of driving terms that arise in certain physical problems is analyzed by introducing a kind of ’’inner product’’ g with respect to which the 2l‐pole solutions, ψl, of the homogeneous wave equation are an orthogonal basis. This allows the condition that δ, the Lth multipole part of the driving term, will give rise to a nonspreading solution to be expressed as g (ψL,r2δ) =0. The complete solution is found in terms of its spreading and nonspreading parts, and the backscattered radiation is calculated from the spreading part.
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41.20.Jb Electromagnetic wave propagation; radiowave propagation

On the computation of the prolate spheroidal radial functions of the second kind

B. P. Sinha and R. H. MacPhie

J. Math. Phys. 16, 2378 (1975); http://dx.doi.org/10.1063/1.522502 (4 pages) | Cited 9 times

Online Publication Date: 3 September 2008

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The series expansion for the prolate spheroidal radial function of the second kind (or its derivative) is found to be slowly convergent when the eccentricity of the spheroid is large (a thin spheroid). To overcome this difficulty, a method is presented in which a small finite number of terms are summed in the conventional manner, and then the infinite remainder series is approximated by an integral of a continuous function. The validity of the method is confirmed by comparing the computed Wronskian with the theoretical. Satisfactory agreement (three to five significant figures) and a very substantial reduction in computation time are achieved.
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02.30.Gp Special functions

A Bäcklund transformation in two dimensions

Hsing‐Hen Chen

J. Math. Phys. 16, 2382 (1975); http://dx.doi.org/10.1063/1.522503 (3 pages) | Cited 21 times

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Bäcklund transformation method is applied to find solutions of a nonlinear evolution equation. This equation describes weakly dispersive nonlinear shallow water wave in two space dimensions.
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47.10.-g General theory in fluid dynamics

Lorentz covariant treatment of the Kerr–Schild geometry

Metin Gürses and Feza Gürsey

J. Math. Phys. 16, 2385 (1975); http://dx.doi.org/10.1063/1.522480 (6 pages) | Cited 60 times

Online Publication Date: 3 September 2008

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It is shown that a Lorentz covariant coordinate system can be chosen in the case of the Kerr–Schild geometry which leads to the vanishing of the pseudo energy–momentum tensor and hence to the linearity of the Einstein equations. The retarded time and the retarded distance are introduced and the Liénard–Wiechert potentials are generalized to gravitation in the case of world‐line singularities to derive solutions of the type of Bonnor and Vaidya. An accelerated version of the de Sitter metric is also obtained. Because of the linearity, complex translations can be performed on these solutions, resulting in a special relativistic version of the Trautman–Newman technique and Lorentz covariant solutions for spinning systems can be derived, including a new anisotropic interior metric that matches to the Kerr metric on an oblate spheroid.
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04.20.-q Classical general relativity
02.40.Ky Riemannian geometries

Direct use of Young tableau algebra to generate the Clebsch–Gordan coefficients of SU (2)

Jack Nachamkin

J. Math. Phys. 16, 2391 (1975); http://dx.doi.org/10.1063/1.522504 (4 pages) | Cited 1 time

Online Publication Date: 3 September 2008

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Although it is well known that the irreducible representations of the SU (N) groups may be generated by using Young tableau algebra, this technique seems to have found little use for the derivation of closed algebraic expressions for the Clebsch–Gordan coefficients of these groups. A frontal attack on the derivation of these coefficients using tableau symmetrizers is described. As an example, the SU (2) group illustrates the fundamental ideas behind the process.
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02.20.Qs General properties, structure, and representation of Lie groups

Some solutions of complex Einstein equations

J. F. Plebañski

J. Math. Phys. 16, 2395 (1975); http://dx.doi.org/10.1063/1.522505 (8 pages) | Cited 166 times

Online Publication Date: 3 September 2008

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Complex V4’s are investigated where math =0 and therefore a fortiori equations Gab=0 are fulfilled. A general theory of spaces of this type is outlined and examples of nontrivial solutions of all degenerate algebraic types are provided.
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04.20.Jb Exact solutions

Null geodesic surfaces and Goldberg–Sachs theorem in complex Riemannian spaces

J. F. Plebañski and S. Hacyan

J. Math. Phys. 16, 2403 (1975); http://dx.doi.org/10.1063/1.522506 (5 pages) | Cited 45 times

Online Publication Date: 3 September 2008

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An extension of the Goldberg–Sachs theorem for the case of a complex V4 is given with a simple proof. The interpretation of the theorem, however, no longer applies the concept of the geodesic and shearless congruence of null directions; instead, the existence of a geodesic 2‐surface (complex), the tangent vectorial space to which (i) contains only null vectors, (ii) is parallelly propagated along the surface, is now essential.
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04.20.Cv Fundamental problems and general formalism
02.40.Ky Riemannian geometries

Structural properties of the canonical U(3) Racah functions and the U(3) : U(2) projective functions

J. D. Louck, M. A. Lohe, and L. C. Biedenharn

J. Math. Phys. 16, 2408 (1975); http://dx.doi.org/10.1063/1.522481 (19 pages) | Cited 14 times

Online Publication Date: 3 September 2008

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The class of U(3) Racah functions which are identically zero are determined from the canonical splitting of the multiplicity. These results imply the form of a special class of (projective) tensor operators. The function Gq associated with the ’’stretched’’ (maximal null space) Wigner operator is generalized and shown to be applicable in determining the denominator for the minimal null space operator.
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02.20.Qs General properties, structure, and representation of Lie groups
03.65.Fd Algebraic methods

Group theory and propagation of operator averages

C. Quesne

J. Math. Phys. 16, 2427 (1975); http://dx.doi.org/10.1063/1.522482 (5 pages) | Cited 9 times

Online Publication Date: 3 September 2008

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The propagation of operator averages, which is the basis of French’s spectral distribution method, is reformulated in the framework of group theory. The concept of complementary groups is extensively used. It is shown that the possibility of propagating averages is intimately connected with the absence of state labeling problem. The construction of the propagation operators is examined, and for those cases where it is not trivial, a new way of approach is suggested by establishing a link with recent group theoretical advance in the construction of subgroup invariants in the universal covering algebra of a group. Finally the discussion is illustrated by some examples taken, or not, from current literature.
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03.65.Fd Algebraic methods
31.15.-p Calculations and mathematical techniques in atomic and molecular physics

The electromagnetic field on a simplicial net

Rafael Sorkin

J. Math. Phys. 16, 2432 (1975); http://dx.doi.org/10.1063/1.522483 (9 pages) | Cited 39 times

Online Publication Date: 3 September 2008

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See Also: Erratum

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The ’’Regge calculus’’ approach is extended to the electromagnetic case. To this end an ’’affine’’ tensor formalism and associated exterior calculus are developed. The simplicial approach to linear field equations is illustrated by the two‐dimensional scalar wave equation, on which also a discussion of the treacherous character of the continuum limit is based.
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03.50.-z Classical field theories

Dynamic stability and thermodynamics in kinetic theory and fluid mechanics

Miroslav Grmela

J. Math. Phys. 16, 2441 (1975); http://dx.doi.org/10.1063/1.522484 (9 pages) | Cited 5 times

Online Publication Date: 3 September 2008

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The study of the relationship between thermodynamic and local dynamic stability that has been developed in the previous papers is further extended. The dynamical systems considered include the multicomponent fluid dynamics and the two component Enskog–Vlasov dynamics.
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47.10.-g General theory in fluid dynamics
51.10.+y Kinetic and transport theory of gases

Long time behavior of solutions to the linearized two component Enskog–Vlasov kinetic equations

Miroslav Grmela

J. Math. Phys. 16, 2450 (1975); http://dx.doi.org/10.1063/1.522485 (8 pages) | Cited 4 times

Online Publication Date: 3 September 2008

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Fluid dynamics is obtained from the study of the long time behavior of solutions to the two component Enskog–Vlasov kinetic equations. Both thermodynamic and dynamic phenomenological coefficients of fluid mechanics are expressed in terms of the phenomenological quantities entering the Enskog–Vlasov‐type kinetic equations.
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47.10.-g General theory in fluid dynamics
51.10.+y Kinetic and transport theory of gases

Solutions for the general cylindrically symmetric stationary dust model

John Charles Zimmerman

J. Math. Phys. 16, 2458 (1975); http://dx.doi.org/10.1063/1.522486 (3 pages) | Cited 2 times

Online Publication Date: 3 September 2008

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For a dust‐filled space–time possessing cylindrical symmetry, the field equations form an underdetermined set. As demonstrated by King [Commun. Math. Phys. 38, 157 (1974)], by carefully selecting a function it is possible to generate solutions which are either well‐behaved or are characterized by one of a number of different types of singularity. The three particular choices for the functions we take produce two nonsingular solutions and one with a Weyl singularity.
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04.20.Jb Exact solutions
98.80.-k Cosmology

Invariance transformations, invariance group transformations, and invariance groups of the sine‐Gordon equations

Sukeyuki Kumei

J. Math. Phys. 16, 2461 (1975); http://dx.doi.org/10.1063/1.522487 (8 pages) | Cited 33 times

Online Publication Date: 3 September 2008

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We investigate a structure of continuous invariance transformations connected to the identity transformation. The transformations considered do not necessarily form a group. We clarify the relationship between the infinitesimal invariance transformation and the finite invariance transformation by showing explicitly how the infinitesimal transformations are woven into the finite one. The analysis leads to a new method of finding generators of the invariance group transformation. The results are useful in the study of symmetry properties, or group theoretic structure, of differential equations. We use the results in studying the group properties of the sine‐Gordon equation uxt=sinu, and indicate that the equation is invariant under an infinite number of one‐parameter groups; the groups obtained are of a more general type than that dealt with by Lie. These findings are used to prove the group theoretic origin of the well‐known conservation laws associated with the sine‐Gordon equation.
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02.30.Hq Ordinary differential equations

General techniques for single and coupled quantum anharmonic oscillators

Francis R. Halpern

J. Math. Phys. 16, 2469 (1975); http://dx.doi.org/10.1063/1.522488 (7 pages) | Cited 3 times

Online Publication Date: 3 September 2008

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Matrix mechanic methods are used to find approximate equations and solutions for quantum anharmonic oscillator problems. A series of hypotheses are introduced that truncate and partially decouple the infinite set of coupled equations that specify the problem in the matrix mechanics formulation. The dependent variables or unknowns in these equations are the matrix elements of the coordinate and momentum operators. The independent variables are the matrix indices and coupling strengths. The equations themselves specify that the off diagonal matrix elements of the Hamiltonian and the commutators expressed in terms of the unknowns vanish and that the diagonal commutator matrix elements vanish except for canonical pairs in which case they are equal to −ih. The truncation and decoupling hypotheses offer an orderly procedure for dealing approximately with the vast array of equations of the exact problems. Only the leading behavior of the coordinate and momentum operator matrix elements is found in terms of the matrix indices and coupling parameters. Although general techniques are presented to find the equations, the solutions discussed and the applications are brief extensions of problems that have already been treated.
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03.65.Ge Solutions of wave equations: bound states

Separation of variables in the Hamilton–Jacobi, Schrödinger, and related equations. II. Partial separation

Peter Havas

J. Math. Phys. 16, 2476 (1975); http://dx.doi.org/10.1063/1.522489 (14 pages) | Cited 10 times

Online Publication Date: 3 September 2008

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Sufficient conditions are given for coordinate systems in which the Hamilton–Jacobi equation and the Schrödinger and related equations are partially separable in n dimensions. For the first equation, the solution is assumed to be a sum of ν+τ functions of a single variable, and of μ[0⩽μ⩽ (n−ν−τ)/2] groups of other variables; for the other equations, products of such functions are assumed. These assumptions lead to ν+τ completely separated differential equations, of which ν are linear in the separation constants, and μ partially separated equations depending on the remaining n−ν−τ variables. The general forms of the various metric tensors gkl of the Riemannian spaces Vn as well as of the allowed potentials V corresponding to the different possible types of such equations are determined; they are identical for the Hamilton–Jacobi equation and for the other equations studied, except that for the latter some of the metrics are further restricted by a condition on their determinants. The results are established by methods similar to those used in Paper I of this series for complete separation, and include the results obtained there as special cases. In the course of determining the allowed forms of the gkl it is also established that there exist ν+τ+μ independent first integrals linear and quadratic in the momenta for the dynamical systems descrived by the Hamilton–Jacobi or the Schrödinger equation. The ν linear ones are homogeneous, and the τ+μ quadratic ones correspond to homogeneous quadratic integrals of the geodesics of the Vn. These results imply the existence of ν Killing vectors and of τ+μ Killing tensors of rank two for the Vn. Further polynomial integrals can be constructed; those integrals of degree r which are independent of the original ν+τ+μ integrals each correspond to an independent Killing tensor of rank r.
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45.05.+x General theory of classical mechanics of discrete systems
02.30.Jr Partial differential equations

Similarity solution for viscous and thermal gas flow in a cone

N. Liron and H. E. Wilhelm

J. Math. Phys. 16, 2490 (1975); http://dx.doi.org/10.1063/1.522490 (4 pages) | Cited 1 time

Online Publication Date: 3 September 2008

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The nonlinear, partial differential equations describing the compressible flow of a viscous and heat conducting gas in a cone are reduced to two coupled, ordinary, nonlinear differential equations by means of a self‐similar transformation. These are solved numerically for the velocity and temperature distributions in the cone. It is shown that for given flow numbers R, P, and M, laminar flows exist only up to a critical cone angle ϑ0.
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47.10.-g General theory in fluid dynamics
47.20.Gv Viscous and viscoelastic instabilities

SU(4) Clebsch–Gordan coefficients

Veronika Rabl, George Campbell, and Kameshwar C. Wali

J. Math. Phys. 16, 2494 (1975); http://dx.doi.org/10.1063/1.522491 (13 pages) | Cited 25 times

Online Publication Date: 3 September 2008

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We give tables of Clebsch–Gordan coefficients for the products of SU(4) representations 15⊗15 and 20⊗15, decomposed with respect to SU(3).
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02.20.Qs General properties, structure, and representation of Lie groups

Lie theory and separation of variables. 8. Semisubgroup coordinates for Ψtt − Δ2Ψ = 0

E. G. Kalnins and Willard Miller

J. Math. Phys. 16, 2507 (1975); http://dx.doi.org/10.1063/1.522492 (10 pages) | Cited 7 times

Online Publication Date: 3 September 2008

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We classify and study all coordinate systems which permit R‐separation of variables for the wave equation in three space–time variables and such that at least one of the variables corresponds to a one‐parameter symmetry group of the wave equation. We discuss 33 such systems and relate them to orbits of commuting operators in the enveloping algebra of the conformal group SO (3,2).
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02.20.Sv Lie algebras of Lie groups
02.30.Jr Partial differential equations
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