Volume 22, Issue 3, March 1981
Index of content:

Some examples of compact supermanifolds with non‐Abelian fundamental group
View Description Hide DescriptionA detailed construction is given of a supermanifold which is compact and has far from trivial topology. Higher dimensional examples are also described. The Grassmann algebra from which the supermanifolds are constructed may have finite or infinite dimension.

Conserved densities for nonlinear evolution equations. II. Odd order case
View Description Hide DescriptionThis paper extends to nonlinear evolution equations of odd order the analysis of existence and structure of the polynomial conserved densities. The results for low order densities are similar to the case of even order. The situation for densities with high order derivatives is now radically different. An asymptotic algorithm is presented for the search of such densities, which are shown to be quadratic in the highest derivatives. The very existence of just one high order conserved density is shown to severely restrict the evolution equation, and in the third order case it leads, with some minor additional hypothesis, to the KdV family.

On the nature of the Gardner transformation
View Description Hide DescriptionIt is shown that every higher Korteveg‐de Vries equation can be included in a one‐parameter family of integrable equations.

Analytical expansions for Fermi–Dirac functions
View Description Hide DescriptionWe obtain a fast convergent series expansion for the Fermi–Dirac function F _{s}(a) for −10<a⩽1. We give values of F _{s}(a) for s = n+ 1/2 (n = 0,1,⋅⋅⋅,6) with a in the same range.

On nearest neighbor degeneracies of indistinguishable particles
View Description Hide DescriptionArrangement degeneracies suggested by sufficient statistics associated with binary stationary mth order Markov chains are discussed, and are shown to correspond and generalize some degeneracies arising when indistinguishable particles are placed on a one‐dimensional lattice with n compartments. From these statistics it is possible to define an mth order unit. The arrangement degeneracy obtained from s 1’s and n−s 0’s so that lower order units are placed in higher order is difficult. For this case only the third order arrangement degeneracy is obtained, the first and second orders being relatively simple. These results are applied in determining the asymptotic distributions of rare events.

Moment problem approximants to the Chandrasekhar H‐equation
View Description Hide DescriptionApproximate solutions to the Chandrasekhar H‐equation are obtained by considering a truncated moment problem. Convergence to the physical solution is proved and a numerical example is outlined.

An exact invariant for a class of time‐dependent anharmonic oscillators with cubic anharmonicity
View Description Hide DescriptionAn exact invariant is constructed for a class of time‐dependent anharmonic oscillators using the method of the Lie theory of extended groups. The presence of the anharmonic term imposes a constraint on the nature of the time dependence. For a subclass it is possible to obtain an energy‐like integral and a condition under which the motion is bounded.

Scattering of a scalar wave from a slightly random surface
View Description Hide DescriptionScalar wavescattering from a slightly random surface is analyzed by a probabilistic method. We make use of the homogeneity of an infinite random surface, that is, the shift invariance property of the strictly homogeneous random field. By the group‐theoretic consideration of such a shift invariance property, the wave solution proves to be a homogeneous random field multiplied by an exponential function. Then such a homogeneous random field is approximately solved for a slightly random surface to yield a wave solution involving multiple scattering. Several statistical properties of the scattering are calculated and shown in the figures. The accuracy of the approximate solution is examined in terms of the error of the boundary‐value equation.

Covariant electrodynamics of a dyon in a medium
View Description Hide DescriptionA covariant formulation of electrodynamics of a dyon in a uniform, isotropic, and transparent medium is discussed. Using generalized Maxwell’sequations admitting magnetic charge, the 4‐potentials of a dyon of electric chargee and magnetic chargem and the corresponding field tensor are calculated. This field tensor is then used to calculate the various stress tensors of Minkowski, Abraham, and Marx. And it is found that the dyon behaves like an electrically charged particle of ’’effective’’ charge e*, where e* = e[1+(e/m)(m/e)^{2}]^{1/2}, where e and m are the electric and magnetic permeabilities of the medium.

A multipole formalism for material media
View Description Hide DescriptionIt is shown that it is possible to make a meaningful multipole expansion for the electromagnetic fields produced by localized charge and current distributions embedded in material media—provided the media possess certain reasonable properties. The result is similar to the well‐known multipole expansions for localized sources in vacuum but differs from it primarily because of induced charge and current density contributions to the various multipole coefficients.

Energy‐particle‐number inequality in nonlinear complex‐scalar field theory
View Description Hide DescriptionFor a self‐interaction energy density U(‖ψ‖^{2}) that is positive‐definite, monotone‐increasing with increasing ‖ψ‖^{2}, and concave‐saturating, it is shown that the total field energy E and particle number N satisfy the general inequalityE⩾ (function of ξ) ‖N‖^{(3−ξ)/2(2−ξ)} in which the positive parameter ξ≡min [dlnU( ρ)/dlnρ] is less than unity.

Measurement in stochastic mechanics
View Description Hide DescriptionStochastic mechanics is an explanation of nonrelativistic quantum phenomena in terms of stochastic differential equations. In this note a simple example of a measurement is constructed and the behavior of the sample paths of the corresponding stochastic differential equation is examined. The sample demonstrates that stochastic mechanics provides a natural explanation of the ’’reduction of the wave packet.’’

Quantum mechanical Hamiltonian models of discrete processes
View Description Hide DescriptionHere the results of other work on quantum mechanical Hamiltonian models of Turing machines are extended to include any discrete process T on a countably infinite set A. The models are constructed here by use of scattering phase shifts from successive scatterers to turn on successive step interactions. Also a locality requirement is imposed. The construction is done by first associating with each process T a model quantum system M with associated Hilbert spaceH_{ M } and step operator U _{ T }. Since U _{ T } is not unitary in general, M, H_{ M }, and U _{ T } are extended into a (continuous time) Hamiltonian model on a larger space which satisfies the locality requirement. The construction is compared with the minimal unitary dilation of U _{ T }. It is seen that the model constructed here is larger than the minimal one. However, the minimal one does not satisfy the locality requirement.

The Lipkin model and coherent states
View Description Hide DescriptionIt is shown that the Bloch or angular momentumcoherent states furnish a particularly efficacious basis for a discussion of various aspects of the Lipkin model of the ’’nucleus.’’ The Hartree–Fock description (as well as its projected version) is elegantly obtained in this framework. It is demonstrated that the ’’transition probability’’ between the first excited and ground states is proportional to the square of the number of ’’nucleons,’’ representing (in contrast to what obtains in the random phase approximation) a cooperativity of the ’’super‐radiant’’ type. The extension of the model through the introduction of bosons permits, with the use of Bloch and Glauber coherent states, a succinct description of the phenomenon of boson condensation.

Expansion around instantons in quantum mechanics
View Description Hide DescriptionWe calculate numerically a few terms of the corrections to the large‐order behavior of the ground state energy of the O(N) anharmonic oscillator by analyzing the perturbation series. We then generate 94 terms of the perturbative expansion of the difference between the energies of the two low lying states of the double‐well potential and analyze their large‐order behavior.

On decoupling of finite singularities in the scattering theory for the Schrödinger operator with a magnetic field
View Description Hide DescriptionIn this paper, the authors extend the theory of the effect of local singularities on scattering theory to include magnetic fields.

Self‐adjointness and spectrum of Hamiltonians in nonrelativistic quantum electrodynamics
View Description Hide DescriptionThe nonrelativistic quantum electrodynamics is formulated in a mathematically rigorous way. The self‐adjointness and the basic spectral property of the Hamiltonians are proved.

Contacts of space–times
View Description Hide DescriptionThe concept of contact between manifolds is applied to space–times of general relativity. For a given background space–time a contact approximation of second order is defined and interpreted both from the point of view of a metric pertubation and of a higher order tangent manifold. In the first case, an application to the high frequency gravitational wave hypothesis is suggested. In the second case, a constant curvature tangent bundle is constructed and suggested as a means to define a ten parameter local space–time symmetry.

Intrinsic isometry groups in general relativity
View Description Hide DescriptionWe derive necessary and sufficient tensor conditions for the existence of a four‐parameter isometry group G _{4} which acts multiply transitively on a Riemannian V _{3}. We then apply these results to determine which spatially homogeneous cosmological models have induced 3‐metrics which are invariant under such a four‐parameter group.

Fiber bundles, superspace, and the gravitational field
View Description Hide DescriptionWe develop a new theory of the pure gravitational field by treating superspace as a fiber bundle with space–time as the base space, and Fermi space, with anticommuting Majorana spinor coordinate q^{ i }, as the typical fiber. In the fiber bundle geometry, a spin‐3/2 field arises automatically. It comes in through the commutation relations of the basis vectors in the horizontal lift basis, or through the metric in the local direct‐product basis. The Lagrangian is taken to be the scalar curvature of the fiber bundle, in analogy with general relativity with no source terms. The theory is self sourced and the spin‐3/2 field and usual spin‐2 field appear as gauge fields. The theory is also completely basis invariant. The field equations correctly describe a spin‐3/2 field coupled to general relativity, with the correct ’’energy–momentum tensor’’ of the spin‐3/2 field appearing automatically. We thus end up with a very simple, geometrical theory which contains far fewer fields that the geometrical work of Arnowitt and Nath while keeping their elegance of formulation. The resulting field equations are similar to those of simple supergravity with only a spin‐3/2 field appearing in addition to the spin‐2 Einstein field, however, supersymmetry invariance seems to play little or no direct role in the present theory.