Volume 16, Issue 12, December 1975
Index of content:

Representations of a local current algebra: Their dynamical determination
View Description Hide DescriptionLocal 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.

Approximate representations of a local current algebra
View Description Hide DescriptionAn 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.

The Weyl tensor and shear‐free perfect fluids
View Description Hide DescriptionIt 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 u ^{α}, we find the conservation law ∇_{α}(n ^{1/3} iω u ^{α}) =0, where i is the relativistic specific enthalpy,n is the conserved particle number density, and ω is the vorticity scalar.

Singularities in nonsimply connected space–times
View Description Hide DescriptionSpace–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.

Ergodicity of quantum mechanical systems
View Description Hide DescriptionThe 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 p r i o r i probabilities.

Note on the computation formula of the boost matrices of SO(n−1,1) and continuation to the d matrices of SO(n)
View Description Hide DescriptionThe 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).

Retarded multipole fields and the inhomogeneous wave equation
View Description Hide DescriptionThe 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 2^{ l }‐pole solutions, ψ_{ l }, 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 (ψ_{ L },r ^{2}δ) =0. The complete solution is found in terms of its spreading and nonspreading parts, and the backscattered radiation is calculated from the spreading part.

On the computation of the prolate spheroidal radial functions of the second kind
View Description Hide DescriptionThe 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.

A Bäcklund transformation in two dimensions
View Description Hide DescriptionBäcklund 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.

Lorentz covariant treatment of the Kerr–Schild geometry
View Description Hide DescriptionIt 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.

Direct use of Young tableau algebra to generate the Clebsch–Gordan coefficients of S U (2)
View Description Hide DescriptionAlthough it is well known that the irreducible representations of the S U (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 S U (2) group illustrates the fundamental ideas behind the process.

Some solutions of complex Einstein equations
View Description Hide DescriptionComplex V _{4}’s are investigated where =0 and therefore a f o r t i o r i equations G _{ a b }=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.

Null geodesic surfaces and Goldberg–Sachs theorem in complex Riemannian spaces
View Description Hide DescriptionAn extension of the Goldberg–Sachs theorem for the case of a complex V _{4} 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.

Structural properties of the canonical U(3) Racah functions and the U(3) : U(2) projective functions
View Description Hide DescriptionThe 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 G _{ q } 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.

Group theory and propagation of operator averages
View Description Hide DescriptionThe 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.

The electromagnetic field on a simplicial net
View Description Hide DescriptionThe ’’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.

Dynamic stability and thermodynamics in kinetic theory and fluid mechanics
View Description Hide DescriptionThe 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.

Long time behavior of solutions to the linearized two component Enskog–Vlasov kinetic equations
View Description Hide DescriptionFluid 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.

Solutions for the general cylindrically symmetric stationary dust model
View Description Hide DescriptionFor 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 generatesolutions 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.

Invariance transformations, invariance group transformations, and invariance groups of the sine‐Gordon equations
View Description Hide DescriptionWe 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 u _{ x } _{ t }=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.