Volume 15, Issue 6, June 1974
Index of content:

Theory of multibin tests: Definition and existence of extraneous tests
View Description Hide DescriptionDirac'ssets of commuting observables, in the guise of lists of mutually orthogonal projections which add to the unit matrix, are readily extended to a convex completion. But, furthermore, there exist lists of nonnegative Hermitian matrices which sum to the unit matrix which do not even belong to this convex completion. It is shown that these ``extraneous'' lists appear as tests in ordinary quantum‐mechanical experiments. This circumstance leads to simpler rules for injecting measurement theory into the social sciences than might otherwise be proposed. Various relationships between lists of orthogonal projections and more general tests are given. The problem of devising rules of inference by direct computation is very briefly engaged.

A physical system which can be forced to execute an arbitrary unitary transformation, and its use to perform arbitrary tests
View Description Hide DescriptionIt is shown that a system of N spin‐(1/2) magnets can be constructed so freely manipulable that an experimenter can impose any unitary transformation upon its state space, N any positive integer. This is then used with N ≥ b − 1 + 2 log_{2} n to construct arbitrary states over an n‐dimensional Hilbert space and to perform arbitrary b‐tests upon these states.

Eigenvalue problem for Lagrangian systems. VII
View Description Hide DescriptionMinimax principles for a subset of the real eigenvalues of the quadratic eigenvalue problem (ω^{2} A +ω B + C)ξ = 0 are presented, where A, B, and C are formally self‐adjoint operators mapping a dense subspace Δ of a complex Hilbert spaceE into E, and A >0. These results are applied to the problems of the small oscillations about equilibrium of a vertically stratified, viscous, heterogeneous incompressible fluid in a gravitational field and the oscillations of a rotating thin annular disk, and it is shown that the minimax principles characterize infinitely many eigenvalues of these systems.

Concerning conservation laws resulting from geometric invariance groups for field theories
View Description Hide DescriptionA geometric symmetry group is defined as a point transformation of a Riemannian manifold combined with a transformation law for the field as a geometrical object. A covariant definition of invariance of an action integral is given. It is shown that geometric invariance groups can be determined from knowledge of a tensor which can be computed from the density function L in the action integral. A general form for conservation laws due to geometric symmetry is given. Results are applied to electromagnetic fields and it is shown that the Bessel‐Hagen conservation laws represent all of the possible conservation laws for electromagnetic fields arising from geometric symmetry.

Eight‐vertex model on the honeycomb lattice
View Description Hide DescriptionThe most general vertex model defined on a honeycomb lattice is the eight‐vertex model. In this paper it is shown that the symmetric eight‐vertex model reduces to an Ising model with a nonzero real or pure imaginary magnetic fieldH. The equivalent Ising model is either ferromagnetic with e ^{2H/kT } real or antiferromagnetic with e ^{2H/kT } unimodular. The exact transition temperature and the order of phase transition in the former case are determined. As an application of the result we verify the absence of a phase transition in the monomer‐dimer system on the honeycomb lattice.

Heat conduction and sound transmission in isotopically disordered harmonic crystals
View Description Hide DescriptionWe investigate some kinetic properties of an isotopically disordered harmonic crystal. We prove rigorously that for almost all disordered chains the transmission coefficient of a plane wave with frequency ω, t_{N} (ω), decays exponentially in N, the length of the disordered chain, with the decay constant proportional to ω^{2} for small ω. The response of this system to an incident wave is related to the nature of the heat flux J(N) in a disordered chain of length N placed between heat reservoirs whose temperatures differ by ΔT>0. We clarify the relationship between the works of various authors in the heat conduction problem and establish that for all models J(N)→0 as N→∞ in a disordered system. The exact asymptotic dependence of J(N) on N eludes us, however. We also investigate the heat flow in a simple stochastic model for which Fourier's law is shown to hold. Similar results are proven for two‐dimensional systems disordered in one direction.

Lattice Green's function for the face centered cubic lattice
View Description Hide DescriptionWe have proved that Green's functionG(l,m,n) at an arbitrary lattice site (l,m,n) in face centered cubic lattice with nearest neighbor interactions is, in general, expressed in terms of linear combinations of products of complete elliptic integrals of the first and second kinds.

Geometrodynamics regained: A Lagrangian approach
View Description Hide DescriptionThe closing relation between two super‐Hamiltonians is cast into a condition on the super‐Lagrangian by a functional Legendre transformation. It is shown that the ADM super‐Lagrangian provides the unique representation of the ``group'' of deformations of a spacelike hypersurface embedded in a Riemannian space‐time when the intrinsic geometryg_{ij} of the hypersurface is allowed as the sole configuration variable. No such uniqueness exists for the super‐Lagrangians of source fields. As an illustration, the most general super‐Lagrangian for a scalar field with nonderivative gravitational coupling is recovered from the closing relation.

Closed‐form expressions of matrix elements and eigenfunctions from ladder‐operator considerations
View Description Hide DescriptionWithin the Schrödinger‐Infeld‐Hull factorization scheme, it is shown that, by suitable transformations, the ``accelerated'' or ``ν‐step'' ladder operator can always be brought to a simple canonical form, i.e., the νth derivative operation. Thus, one obtains a closed form expression of the eigenfunctions involving a Rodrigues' formula. The necessary and sufficient condition that this Rodrigues' formula generates classical orthogonal polynomials is found to be equivalent to the factorizability condition. Consequently, a closed form expression of any matrix element (diagonal or off‐diagonal) on the basis of the eigenfunctions of any factorizable equation is easily derived from the calculation of one unique particular integral. In most cases, this last integral is known analytically. The Kepler problem is reinvestigated as an example. As a concluding remark, further applications of the method are considered.

A fluid sphere in general relativity
View Description Hide DescriptionWe solve the Einstein field equations for the interior of a static fluid sphere in closed analytic form. The model sphere obtained has a physically reasonable equation of state, and a maximum mass of 2/5 the fluid radius (in geometric units). As the maximum mass is approached the central density and pressure become infinite, while for masses greater than about 0.35 times the fluid radius the velocity of sound in portions of the fluid exceeds the velocity of light, indicating that the fluid is noncausal in this mass range. In the low mass limit the solution becomes identical to the Schwarzschild interior solution.

A semi‐Euclidean approach to boson‐fermion model theories
View Description Hide DescriptionA formulation is presented for the study of semiboundedness of coupled boson‐fermion modelfield theories. Euclidean‐boson fields and ordinary fermion fields are employed. Expansion steps used to derive estimates are presented.

A pair of coupled quantum anharmonic oscillators
View Description Hide DescriptionThe energy levels of a pair of coupled anharmonic oscillators are studied. The technique employed is to find two approximately canonical coordinate momentum pairs. Particular emphasis is placed on the qualitative dependence of the coordinate and momentum operators on the quantum numbers.

Differential inequalities and stochastic functional differential equations
View Description Hide DescriptionConsider the system of stochastic functional differential equations,where σ is a n×m matrix, column vectors of σ, f are continuous, and z(t) is a normalized m‐vector Wiener process with.By developing a comparison principle, sufficient conditions are given for stability and boundedness in the mean of solutions of (S). The main technique here is the theory of functional differential inequalities and Lyapunov‐like functions.

The evaluation of ``Kondo'' and other integrals of arbitrary range
View Description Hide DescriptionIt is pointed out that integrals over arbitrary ranges and indefinite integrals may often be obtained very simply by the methods of contour integration.

The unitarity equation for scattering in the absence of spherical symmetry
View Description Hide DescriptionWe consider the problem of determining the scattering amplitude from the differential cross section at a fixed energy by using the unitarity equation when the scattering potential does not have spherical symmetry. We indicate some of the problems peculiar to this case. We prove two existence and uniqueness theorems. We give an example of nonuniqueness.

The power spectrum of the Mellin transformation with applications to scaling of physical quantities
View Description Hide DescriptionThe Mellin transform is used to diagonalize the dilation operator in a manner analogous to the use of the Fourier transform to diagonalize the translation operator. A power spectrum is also introduced for the Mellin transform which is analogous to that used for the Fourier transform. Unlike the case for the power spectrum of the Fourier transform where sharp peaks correspond to periodicities in translation, the peaks in the power spectrum of the Mellin transform correspond to periodicities in magnification. A theorem of Wiener‐Khinchine type is introduced for the Mellin transform power spectrum. It is expected that the new power spectrum will play an important role extracting meaningful information from noisy data and will thus be a useful complement to the use of the ordinary Fourier power spectrum.

Generating functions of the hypergeometric functions
View Description Hide DescriptionThe Lie algebra, which was introduced in a previous paper to treat the hypergeometric functions by Lie theory techniques, is used to derive generatingfunctions of the hypergeometric functions. Several generatingfunctions are obtained from the theory of multiplier representations. Weisner's method is also applied, giving another generatingfunction.

Optical equivalence theorem for unbounded observables
View Description Hide DescriptionThe optical equivalence theorem relating c‐number and q‐number formulations of quantum optics is rigorously extended to cover various unbounded operators, and in particular those operators that directly yield counting rates.

Duality of observables and generators in classical and quantum mechnics
View Description Hide DescriptionIn classical and in quantum mechanics physical variables play a dual role as observables and as generators of infinitesimal transformations in the invariance groups. We show that if the Lie algebra of generators is central simple, the observable‐generator duality restricts the structure of the algebra of observables to two cases: a commutative, associative algebra as in classical mechanics, or a central simple special Jordan algebra as in quantum mechanics.

Cancellation of the Green's function in the generation of continuum bound states by nonlocal potentials
View Description Hide DescriptionContinuum bound states (CBS) are known to appear in the spectra of some nonlocal scattering equations. We give a simple derivation of the presence of these states consistent with the requirement that such states occur for zeros of the Fredholm determinant. Examination of the form of the nonlocal potential necessary to the generation of a CBS shows that CBS solutions appear only when the potential has the effect of cancelling the Green's function in the kernel of the integral equation. Several examples from the literature are cited to demonstrate this characteristic feature of CBS.