Volume 18, Issue 2, February 1977
Index of content:

Jauch–Piron states
View Description Hide DescriptionWe argue that a finite proposition system in the sense of Jauch and Piron that admits a unital set of states is necessarily purely classical. Based on this result, we investigate the extensibility of σ‐additive states on projection lattices to all projections of a separable Hilbert space.

Nonuniqueness in the inverse source problem in acoustics and electromagnetics
View Description Hide DescriptionA recently developed formulation of the inverse source problem as a Fredholm integral equation of the first kind provides motivation for the development of analytical characterizations of the nonuniqueness in the inverse source problem. Nonradiating sources, i. e., sources for which the field is identically zero outside a finite region, are introduced. It is then shown that the null space of the Fredholm integral equation is exactly the class of nonradiating sources.

Subalgebras of the similitude algebra and their invariants
View Description Hide DescriptionThe subalgebras of the similitude algebra have previously been classified into conjugacy classes; in this article these classes are classified into isomorphism classes. For each conjugacy class of subalgebras, the invariants are also calculated. All the results are summarized in tables.

Exact occupation statistics for two‐dimensional lattices of single particles
View Description Hide DescriptionA general expression is developed which describes exactly the ensemble average number of one‐ or two‐dimensional structures per arrangement, created when indistinguishable single particles are arranged on a two‐dimensional lattice. The expression obtained is applied to the calculation of some physically important structures which appear on a rectangular and a closest‐packed hexagonal lattice. The problem of nearest‐neighbor pairs is then solved as a special case from the general expression.

Systems of imprimitivity and representations of quantum mechanics on fuzzy phase spaces
View Description Hide DescriptionThe problem of expressing quantum mechanical expectation values as averages with respect to nonnegative density functions on phase space, by analogy with classical mechanics, is reexamined in the light of some earlier work on fuzzy phase spaces. It is shown that such phase space representations are possible if ordinary phase space is replaced by a so‐called fuzzy phase space, on which the usual marginal distribution functions are redefined to conform to the fact that arbitrarily precise simultaneous measurements on position and momentum are not compatible with quantum mechanics. In the process a generalization of Wigner’s theorem on the nonexistence of phase space representations of quantum mechanics, which also satisfy the standard (classical) marginality conditions in position and momentum, is obtained. It is shown that a (continuous) representation of quantum mechanics exists on a given fuzzy phase space if an only if the corresponding confidence functions for position and momentum measurements satisfy the Heisenberg uncertainty relations.

On the definition of scattering subspaces in nonrelativistic quantum mechanics
View Description Hide DescriptionA physically motivated definition of scattering subspaces is given for problems in nonrelativistic quantum mechanics. The definition is more stringent than earlier similar definitions. It is applicable to potential scattering and to n‐body problems.

A new family of solutions of the Einstein field equations
View Description Hide DescriptionAn ordinary differential equation is presented, from solutions of which may be constructed solutions of the Einstein field equations. The study of these solutions may shed light upon the still obscure systematics of the Tomimatsu–Sato spinning mass fields.

Percolation theory on directed graphs
View Description Hide DescriptionThe pair connectivity P _{ u v } of a directed graph G between vertices u and v is the probability that there is a path from u to v when each edge and vertex has a given probability of being deleted, deletions being made independently. We consider the coefficient in the expansion P _{ u v }(G) = Σ_{ A′⊇A } _{ u v }(G′) Π_{ aεA′} p _{ a } Π_{ wεV′} p _{ w }, where A and V are respectively the arc and vertex sets of G, and p _{ a }(p _{ w }) is the probability that the arc a (vertex w) is not deleted. G′ is the arc set A′ together with its set of incident vertices V′. It is shown that _{ u v }(G′) is nonzero if and only if G′ is coverable by some set of (directed) paths from u to v and has no circuit. When these conditions are satisfied, _{ u v }(G′) = (−1)^{ t } ^{ u v } ^{+1} where the number of independent paths from u to v is t _{ u v }. Moreover, t _{ u v } is shown to have the value of ν (G)+1, ν (G) being the cyclomatic number of the graph G.

The Killing form for graded Lie algebras
View Description Hide DescriptionA Killing form with nice symmetry and invariance properties is constructed for an arbitrary graded Lie algebra. When this form is nondegenerate, Casimir operators can be constructed, and the graded Lie algebra possesses properties analogous to those possessed by semisimple ordinary Lie algebras.

Plane wave solutions in scalar tensor theories and solutions of source‐free Einstein–Maxwell theory
View Description Hide DescriptionVacuum solutions of the field equations that satisfy the original notion of the plane wave, namely g _{μν}=g _{μν}(Z), Z=Σ_{μ} aμx ^{μ}, a _{μ}’s being constants and g ^{μν} Z _{,μ} Z _{,ν}=0, are sought for both Brans–Dicke scalar–tensor theory and the more recent scalar–tensor theory due to Sen and Dunn. (Here Greek letters range from 1 to 4 and Latin letters from 1 to 3.) A complete set of solutions are obtained for both cases. Although not required at the outset, it turns out that, in both cases, the scalar field is also function of Z alone. As a by‐product, one gets the complete set of solutions of the Einstein–Maxwell equations for the null electromagnetic field for the cases when g _{μν} satisfies the above‐mentioned requirement.

Closure in anisotropic cosmological models
View Description Hide DescriptionThe closure result of Friedmann cosmology is briefly reviewed. A new closure result is presented for nonrotating, dust filled, but otherwise anisotropic, inhomogeneous cosmological models. The isotropy assumptions are replaced by much milder physical assumptions while the crucial Friedmann condition, 8πκρ−3h ^{2}≳0, is replaced by the only slightly stronger 4πκρ−3h ^{2}≳0.

Higher spin states in the stochastic mechanics of the Bopp–Haag spin model
View Description Hide DescriptionThe author has previously shown that the dynamically natural random variable representing angular momentum in the stochastic mechanics of the Bopp–Haag spin model has the expectation values predicted by quantum mechanics for spin=1/2. The result is generalized to all higher values of the spin.

Group theoretic aspects of conservation laws of nonlinear dispersive waves: KdV type equations and nonlinear Schrödinger equations
View Description Hide DescriptionGroup theoretic properties of nonlinear time evolution equations have been studied from the standpoint of a generalized Lie transformation. It has been found that with each constant of motion of the KdV type equationf _{ x x x }+a (f) f _{ x }+f _{ t }=0 and of the coupled nonlinear Schrödinger equationf _{ x x } +a (f,g)+i f _{ t }=0, g _{ x x }+a (g,f) −i g _{ t }=0 one invariance group of the equations is always associated. The well‐known series of constants of motion of the KdV equation and the cubic Schrödinger equation will be recovered from the invariance groups of the equations. The doublet solution of the KdV equation will be characterized as the invariant solution of one of the groups. In a more general context, it will be shown that the well‐known equation of quantum mechanics (d/d t) 〈U〉=〈[i H,U] +∂U/∂t〉 can be generalized to a class of nonlinear time evolution equations and that if U is a generator of an invariance group of the equation then (d/d t) 〈U〉=0. The class includes equations such as the KdV, the cubic Schrödinger, and the Hirota equations.

On plane symmetric Einstein–Maxwell fields
View Description Hide DescriptionIt is shown that the results given by Letelier and Tabensky [J. Math. Phys. 15, 594 (1974)] regarding the solution of Einstein–Maxwell equations for plane symmetry will be modified after suitable correction of an error in their paper. Further appropriate exterior solutions, which satisfy the conditions of fit at the boundaries of plane symmetric charged dust distributions, are obtained. One such exterior nonstatic solution can be transformed to a static one, which is a member of the general class.

Causal solutions of nonlinear wave and spinor equations obtained by Gel’fand–Shilov regularization
View Description Hide DescriptionBy extending the Gel’fand–Shilov regularization method to products of locally integrable functions it can be shown that the nonlinear Klein–Gordon equations ∂^{2} u/∂t ^{2}−∂^{2} u/∂x ^{2} _{1}− ⋅⋅⋅ −∂^{2}/∂x ^{2} _{ n } +k u ^{2p+1}=0, where k=const≳0, n⩾3, p=integer⩾1, have causal solutions which have no δ singularities. It is further shown that this also can be expected for the nonlinear Dirac equation γ_{λ}∂ψ/∂x _{λ}+l ^{2}ψ (⩾ψ) =0.

Lie theory and the wave equation in space–time. 3. Semisubgroup coordinates
View Description Hide DescriptionWe classify and study those coordinate systems which permit R separation of variables for the wave equation in four‐dimensional space–time and such that at least one of the variables corresponds to a one‐parameter symmetry group of the wave equation. We discuss over 100 such systems and relate them to orbits of triplets of commuting operators in the enveloping algebra of the conformal group SO(4,2).

Conservative neutron transport theory
View Description Hide DescriptionA functional analytic development of the Case full‐range and half‐range expansions for the neutron transport equation for a conservative medium is presented. A technique suggested by Larsen is used to overcome the difficulties presented by the noninvertibility of the transport operator K ^{−1} on its range. The method applied has considerable advantages over other approaches and is applicable to a class of abstract integro–differential equations.

Prolongation structures and nonlinear evolution equations in two spatial dimensions. II. A generalized nonlinear Schrödinger equation
View Description Hide DescriptionThe prolongation structure approach of Wahlquist and Estabrook is used to determine an inverse scattering formulation for a generalization of the nonlinear Schrödinger equation to two spatial dimensions.

Axisymmetric stationary Brans–Dicke vacuum fields
View Description Hide DescriptionIt is shown that the axisymmetric stationary Brans–Dicke vacuum solutions can be obtained from the solutions of the axisymmetric stationary Einstein vacuum fields and also the axisymmetric static Brans–Dicke vacuum fields.

The n‐bubble series in the theory of the classical one‐component plasma
View Description Hide DescriptionWith the aid of the Mellin transform, an exact expression for the three‐dimensional Fourier transformG (k) of the renormalized sum exp[−Λexp(−r)/r]−1+Λexp(−r)/r for n‐bubble graphs is given in two equivalent forms, where the plasma parameter Λ is not necessarily smaller than unity. Its several properties, such as small and large k limits, are discussed in detail.