Volume 28, Issue 12, December 1987
Index of content:

Representations and invariant equations of E(3)
View Description Hide DescriptionUsing methods analogous to those introduced by Gel’fand e t a l. [R e p r e s e n t a t i o n s of the rotation and Lorentz Groups and Their Applications (Pergamon, New York, 1963)] for the Lorentz group the matrix elements for the representations of the Lie algebra of the Euclidean group in three dimensions E(3) are explicitly derived. These results are then used to construct invariant equations with respect to this group and to show, in particular, that the nonrelativistic analog to the Dirac equation is not unique.

Weight‐2 zeros of 3j coefficients and the Pell equation
View Description Hide DescriptionAll weight‐2 zeros of the Wigner 3j coefficients may be obtained from the quadratic Diophantine equation known as Pell’s equation. These zeros may then be classified by the orbits of a discrete, infinite‐order subgroup of the Lorentz group SO(1,1). This is carried out by transforming the ‘‘polynomial part’’ of a weight‐2 3j coefficient to Pellian form and obtaining the fundamental zeros numerically. The relation of this polynomial to a family of binary quadratic forms is also given, together with a discussion of the invariance group.

Construction of space‐time by gauge translations
View Description Hide DescriptionThe following procedure is described: Starting with a connection in a principal fiber bundle P(M,G), where G is either the Poincaré group or one of the de Sitter groups, a connection in the bundle of linear frames of a submanifold N of M is constructed by using the translational components of the original connection for frame identification. The dimension of N is gauge dependent, and the flat four‐dimensional Minkowski space‐time may appear of dimension less than 4 when certain gauges are used. It is shown that in the case of a de Sitter group, the minimum dimension to which the flat four‐dimensional space‐time can be reduced is 1, while the number is 0 for the Poincaré group. The gauge transformation that achieves the maximum dimension reduction in the de Sitter case is constant and leads to infinite strings as a result. Variable continuous gauge transformations that can reduce the dimension over a finite region of the base manifold are also considered.

Quantization: Towards a comparison between methods
View Description Hide DescriptionIn this paper it is shown that the procedure of geometric quantiztion applied to Kähler manifolds gives the following result: the Hilbert spaceH consists, roughly speaking, of holomorphic functions on the phase space M and to each classical observable f (i.e., a real function on M) is associated an operator f on H as follows: first multiply by f+ 1/4 ℏΔ_{dR} f (Δ_{dR} being the Laplace–de Rham operator on the Kähler manifoldM) and then take the holomorphic part [see G. M. Tuynman, J. Math. Phys. 2 7, 573 (1987)]. This result is correct on compact Kähler manifolds and correct modulo a boundary term ∫_{ M } dα on noncompact Kähler manifolds. In this way these results can be compared with the quantization procedure of Berezin [Math. USSR Izv. 8, 1109 (1974); 9, 341 (1975); Commun. Math. Phys. 4 0, 153 (1975)], which is strongly related to quantization by *‐products [e.g., see C. Moreno and P. Ortega‐Navarro; Amn. Inst. H. Poincaré Sec. A: 3 8, 215 (1983); Lett. Math. Phys. 7, 181 (1983); C. Moreno, Lett. Math. Phys. 1 1, 361 (1986); 1 2, 217 (1986)]. It is shown that on irreducible Hermitian spaces [see S. Helgason, Differential Geometry,Lie Groups and Symmetric Spaces (Academic, Orlando, FL, 1978] the contravariant symbols (in the sense of Berezin) of the operators f as above are given by the functionsf+ 1/4 ℏΔ_{dR} f. The difference with the quantization result of Berezin is discussed and a change in the geometric quantization scheme is proposed.

A class of integrable potentials
View Description Hide DescriptionA class of time independent two‐dimensional integrable potentials, all possessing an invariant of the same general form, is constructed. One of these potentials is superintegrable, its invariants realize the symmetry algebra sO(3) for negative energies, e(2) for zero energy, and sO(2,1) for positive energies. A transformation of coupling constants reveals that in parabolic coordinates this potential is the harmonic oscillator acted on by constant forces. This and another potential in the class may be considered as successive extensions of the Kepler potential. The analytic properties of these integrable systems in the complex time plane are also discussed.

Second‐order equation fields and the inverse problem of Lagrangian dynamics
View Description Hide DescriptionThe transformation properties of determined, autonomous systems of second‐order ordinary differential equations, identified as vector fields on the tangent bundle of the space of dependent variables, are derived and studied. The inverse problem of Lagrangian dynamics is studied from this transformation viewpoint as well as the problem of alternative Lagrangians. In particular, regular Lagrangians which are analytic as functions of the first derivatives are considered. Finally, the inverse problem for second‐order systems corresponding to the geodesic flow of a symmetric linear connection is investigated.

The inverse scattering problem for the soft ellipsoid
View Description Hide DescriptionA soft triaxial ellipsoid, of unknown semiaxes and orientation, is excited into secondary radiation by a plane acoustic wave of a fixed low frequency. It is proved that one measurement of the leading low‐frequency coefficient and exactly six measurements of the second low‐frequency coefficient of the real part of the forward or the backward scattering amplitude are enough to specify completely both the semiaxes, as well as the orientation of the ellipsoid. Therefore, only the first two low‐frequency coefficients of the real part of the scattering amplitude are needed in order to solve the inverse scattering problem for the soft ellipsoid. For the case of spheroids, the number of measurements is restricted to one for the first and three for the second coefficient. Finally, the sphere is specified by a single measurement of the leading coefficient. The special cases where the orientation or the semiaxes are known are also discussed.

Integrability of restricted multiple three‐wave interactions
View Description Hide DescriptionUsing a Hamiltonian framework with complex canonical variables allows for the determination of irreducible forms, which serve as building blocks for polynomial invariants. All the independent invariants in involution are thus obtained for the restricted multiple three‐wave interactions, where all triads are coupled through a common pump (or daughter) wave, in the case of equal coupling strengths in all triads. The mixed, common pump/daughter wave case is not integrable.

Quantum observables: Compatibility versus commutativity and maximal information
View Description Hide DescriptionTwo different approaches to a characterization of the degree of (in)compatibility of quantum observables are investigated. First, recent examples of the (partial) commutativity of spectral measures of incompatible observables are proved to be generic. The analysis is extended to the case of compatible or incompatible unsharp, or stochastic observables, leading to a general criterion for commutativity of position and momentum effects. Further, a recently proposed information theoretic quantification of the (in)compatibility of noncommuting observables is generalized, and the relation between ‘‘maximal information,’’ ‘‘minimal uncertainty,’’ partial commutativity, and strict correlation is further clarified. Both approaches are illustrated in a number of examples.

The problem of moments in the phase‐space formulation of quantum mechanics
View Description Hide DescriptionLong ago, Moyal [Proc. Cambridge Philos. Soc. 4 5, 99 (1949)] formulated a moment problem in the context of the Wigner–Weyl phase‐space formulation of quantum mechanics. The problem amounts to giving necessary and sufficient conditions for a sequence of numbers to be moments of a Wigner function. In this paper, that problem is solved, and so is a truncated version of it.

Composite systems in quaternionic quantum mechanics
View Description Hide DescriptionThe natural composition of systems in quaternionic quantum mechanics is examined via their lattices of propositions and it is shown that the criticisms that have been made of such a composition are unconvincing.

Component states of a composite quaternionic system
View Description Hide DescriptionThe problem of finding component states given a composite state is examined for quaternionic quantum mechanics. It is shown that under very loose conditions the component state is forced to be complex.

Absolutely continuous spectra of quasiperiodic Schrödinger operators
View Description Hide DescriptionSeveral aspects of the general and constructive spectral theory of quasiperiodic Schrödinger operators in one dimension are discussed. An explicit formula for the absolutely continuous (a.c.) spectral densities that yields an immediate proof of the fact that the Kolmogorov–Arnold–Moser (KAM) spectrum constructed by Dinaburg, Sinai, and Rüssmann [Funkt. Anal. Prilozen. 9, 8 (1975); Ann. Acad. Sci. 3 5 7, 90 (1980)] is a subset of the a.c. spectrum is provided. Some quasiperiodicity properties of the Deift–Simon a.c. eigenfunctions are proved, namely, that the normalized phase of such eigenfunctions is a quasiperiodic distribution. In the constructive part the Dinaburg–Sinai–Rüssmann theory is extended to quasiperiodic perturbations of periodic Schrödinger operators using a KAM Hamiltonian formalism based on a new treatment of perturbations of harmonic oscillators. Particular attention is devoted to the dependence upon the eigenvalue parameter and a complete control of KAM objects is achieved using the notion of Whitney smoothness.

On the construction of perfect Morse functions on compact manifolds of coherent states
View Description Hide DescriptionPerfect Morse functions on the manifold of coherent states are effectively constructed. The case of a compact, connected, simply connected Lie group of symmetry, having the same rank as the stationary group of the manifold of coherent states, such that the manifold of coherent states is a Kählerian C‐space, is considered. It is proved that the set of perfect Morse functions is dense in the set of energy functions for linear Hamiltonians in the elements of the Cartan algebra of the Lie algebra of the representation of the group considered. It is proved that the maximum number of orthogonal vectors on a coherent vector manifold is equal to the Euler–Poincaré characteristic of the manifold.

Exponential time‐evolution operator for the time‐dependent harmonic oscillator
View Description Hide DescriptionThe time‐evolution operator for the time‐dependent harmonic oscillatorH= (1)/(2) {α(t)p ^{2} +β(t)q ^{2}} is exactly obtained as the exponential of an anti‐Hermitian operator. The method is based on the equations of motion for the coordinate and momentum operators in the Heisenberg representation. The problem is reduced to solving the classical equations of motion.

Quadratic zeros of Racah 6j coefficients: A geometrical approach
View Description Hide DescriptionIt is shown that the projective symmetries of the polynomial Φ of quadratic 6j coefficients form the symmetrical group S _{6}. Nonlinear rational symmetries of Φ are found. Partial parametrizations of the zeros of Φ are presented.

On discrete Schrödinger equations and their two‐component wave equation equivalents
View Description Hide DescriptionAn approach to inverse scattering problems for discrete Schrödinger equations, which are discrete three‐term recursions, is presented by systematically transforming them into discrete two‐component wave‐propagation equations. The wave‐propagation equations permit the immediate application of certain computationally efficient and physically insightful ‘‘layer‐peeling’’ algorithms for inverse scattering. The mapping of three‐term recursions to two‐component evolution equations is one to many, because the relation between the ‘‘potential’’ sequence parametrizing Schrödinger equations and the ‘‘reflection coefficient’’ sequence determining local wave interaction is a nonlinear difference equation. This mapping is examined in some detail and it is used to study both direct and inverse scattering problems associated with discrete Schrödinger equations.

Time‐dependent canonical formalism of thermally dissipative fields and renormalization scheme
View Description Hide DescriptionThe canonical formalism of thermally dissipative semifree fields in the time‐dependent situation is presented. The use of thermal covariant derivatives simplifies the formulation considerably. With this formalism one can unambiguously obtain the interaction Hamiltonian under any thermal situation which together with the free propagator enables perturbative calculations to be performed. The ‘‘on‐shell’’ renormalization condition in the time‐dependent case is also discussed. The model of a system with a thermal reservoir illustrates how the present formalism works in time‐dependent situations.

There is no isolated p‐p wave
View Description Hide DescriptionA theorem is presented that basically states that there are no nontrivial well‐behaved, spatially asymptotically flat space‐times which are p‐p waves at infinity.

Some solutions of Einstein’s equations with shock waves
View Description Hide DescriptionThe Einstein field equations in the presence of a polytropic fluid performing self‐similar motion are reduced to a dynamical system. Qualitative properties of the dynamical system are investigated in the case when the fluid motion is with shock waves.