Volume 31, Issue 1, January 1990
Index of content:

Quadratic alternative algebras
View Description Hide DescriptionQuadratic alternative algebras are completely classified, up to isomorphism, by means of the associated set of ‘‘vectors.’’ They include the quaternion and octonion algebras, but also many other nonassociative algebras used in physics.

Extended BRST quantization of gauge theories in the generalized canonical formalism
View Description Hide DescriptionThe rules of canonical quantization of gauge theories are formulated on the basis of the extended BRST symmetry principle. The existence of solutions of the generating equations of the gauge algebra is proved. Equivalence between the extended BRST quantization and the standard method of generalized canonical quantization is established. Ward identities corresponding to invariance of a theory under the extended BRST symmetry are obtained.

Superfield and matrix realizations of highest weight representations for osp(m/2n)
View Description Hide DescriptionA differential representation of the classical Lie superalgebra osp(m/2n) acting on superfield functions is given. This representation is used to construct matrix representations of the finite‐dimensional irreducible representations of the algebra. Inner products on the irreducible spaces are discussed and classes of star and grade‐star equivalent representations are identified.

On generalization of the Sinai theorem in the random site problem
View Description Hide DescriptionThe Sinai theorem is generalized to determine the transformations that preserve the critical concentration P _{ c } and the critical exponent β in the random site problem.

Wavelet analysis on the circle
View Description Hide DescriptionThe construction of a wavelet analysis over the circle is presented. The spaces of infinitely times differentiable functions, tempered distributions, and square integrable functions over the circle are analyzed by means of the wavelet transform.

Chern–Simons forms on principal superfiber bundles
View Description Hide DescriptionA graded Weil homomorphism is defined for principal superfiber bundles and the related transgression (or Chern–Simons) forms are introduced. As an example of the application of these concepts, a ‘‘superextension’’ of the Dirac monopole is discussed.

Space‐time geometry of relativistic particles
View Description Hide DescriptionA three‐dimensional space‐time geometry of relativistic particles is constructed within the framework of the little groups of the Poincaré group. Since the little group for a massive particle is the three‐dimensional rotation group, its relevant geometry is a sphere. For massless particles and massive particles in the infinite‐momentum limit, it is shown that the geometry is that of a cylinder and a two‐dimensional plane. The geometry of a massive particle continuously becomes that of a massless particle as the momentum/mass becomes large. The geometry of relativistic extended particles is also considered. It is shown that the cylindrical geometry leads to the concept of gauge transformations, while the two‐dimensional Euclidean geometry leads to a deeper understanding of the Lorentz condition.

Scattering of waves from a random spherical surface—Mie scattering
View Description Hide DescriptionThe stochastic theory developed by the authors for the scattering from a random planar surface is extended to the case of a random spherical surface, which is assumed to be a homogeneous random field on the sphere, homogeneous with respect to spherical rotations. Based on the group‐theoretical analogies between the two, the formulation of the theory is closely connected to the representation theory of the rotation group. The concept of the ‘‘stochastic’’ spherical harmonics associated with the rotation group and their several formulas are introduced and discussed at the beginning. For the plane wave incident on a random spherical surface, the scattered random wave field can be expanded systematically in terms of the stochastic spherical harmonics in much the same way as the nonrandom case, and several formulas are derived for the coherent scattering amplitude, the coherent and incoherent power flows, and the coherent and incoherent scattering cross sections. The power‐flow conservation law is cast into the stochastic version of the optical theorem stating that the total scattering cross section consisting of the coherent and incoherent power flow is equal to the imaginary part of the coherent forward‐scattering amplitude. Approximate solutions are obtained for the Mie scattering with a slightly random spherical surface where the single scattering approximation is valid due to the absence of a real resonance, as shown in the previous work on the two‐dimensional case. Some numerical calculations are made for the coherent and incoherent scattering cross sections.

The trivialization of constraints in quantum theory (working in a general gauge/parametrization)
View Description Hide DescriptionA way is found to trivialize and so deal with constraints in a gauge invariant manner.

Reduction, quantization, and nonunimodular groups
View Description Hide DescriptionIt is shown that even in relatively nice cases the naive approach to the quantization of constraints is not correct in general [i.e., the procedure that if f=0 is a classical constraint and τ(f) is the associated quantum operator, then the quantum constraint is τ(f)=0]. An explicit procedure for the quantization of constraints in the case of a configuration space with a symmetry group is provided and proven, where the reduced configuration space is the orbit space. It is not thought that the group acts freely, merely that all isotropy subgroups are conjugated to each other.

Unitary measurements of discrete quantities in quantum mechanics
View Description Hide DescriptionThe pure measurements of discrete physical quantities are characterized within quantum theory of measurement and their unitary representations are given. Probabilistic aspects of measurements related to the so‐called strong correlation conditions and a probabilistic characterization of the first kind measurements are examined. The problem of the objectification of the measurement result is analyzed in terms of a classical behavior of the measuring apparatus. As a by‐product a generalization of the Wigner–Araki–Yanase theorem is given.

Variational and perturbative schemes for a spiked harmonic oscillator
View Description Hide DescriptionA variational analysis of the spiked harmonic oscillator Hamiltonian operator −d ^{2}/ d x ^{2}+x ^{2}+l(l+1)/x ^{2}+λ‖x‖^{−α} , where α is a real positive parameter, is reported in this work. The formalism makes use of the functional space spanned by the solutions of the Schrödinger equation for the linear harmonic oscillator Hamiltonian supplemented by a Dirichlet boundary condition, and a standard procedure for diagonalizing symmetric matrices. The eigenvalues obtained by increasing the dimension of the basis set provide accurate approximations for the ground state energy of the model system, valid for positive and relatively large values of the coupling parameter λ. Additionally, a large coupling perturbative expansion is carried out and the contributions up to fourth‐order to the ground state energy are explicitly evaluated. Numerical results are compared for the special case α=5/2.

Some remarks on the Feynman–Kac formula
View Description Hide DescriptionA simple necessary condition for the existence of the representation of solutions of partial differential equations is found. This condition is applied to obtain the known results on the Schrödinger equation and the Dirac system in a unified way. Applications for further equations are also possible (Weyl’s equations are discussed).

Singular anharmonicities and the analytic continued fractions. II. The potentials V(r)=a r ^{2}+b r ^{−} ^{4}+c r ^{−} ^{6}
View Description Hide DescriptionThe c=0 results of Paper I [J. Math. Phys. 3 0, 23 (1989)] are extended. In spite of the presence of an additional coupling constant, the Laurent series solutions of the Schrödinger equation that are obtained remain similar to Mathieu functions. Indeed, the recurrences for coefficients preserve their three‐term character, their analytic continued fraction solutions still converge, etc. The formulas become even slightly simpler for c≠0 due to a certain symmetry of the equations to be solved. An acceleration of convergence is better understood and a few numerical illustrations of efficiency are also delivered.

Quantization of bi‐Hamiltonian systems
View Description Hide DescriptionOne of the distinguishing features of soliton equations is the fact that they can be written in Hamiltonian form in more than one way. Here we compare the different quantized versions of the soliton equations arising in the AKNS inverse scattering scheme. It is found that, when expressed in terms of the scattering data, both quantized versions are essentially identical.

The Yang–Baxter equations and differential identities
View Description Hide DescriptionThe solution of the Yang–Baxter equation for integrable systems is shown to be equivalent to the existence of a differential identity. Quantum integration formulas for the calculation of commutators of monodromy matrices are given. Based on the integration formulas and the systematic use of differential identities, the Yang–Baxter equations for the nonlinear Schrödinger model for the quantum case of both bosons and fermions are derived. The case for discrete models is also included. The parallelism between the classical and quantum case and the classical limiting process from the latter to the former are discussed.

Harmonic analysis of directing fields
View Description Hide DescriptionThe structure of a directing field is determined by the projective structure of space‐time and by various tensor (force) fields. Given a sufficient variety of such directing fields, which can be measured directly given only the ability to track material bodies with respect to an arbitrary coordinate system, it is shown how the projective and tensor fields involved can be determined (and hence measured). This method employs the technique of harmonic analysis on the forward unit hyperboloid. For the important and physically relevant case of an electromagnetic directing field, the projective structure and the electromagnetic field tensor can be determined using only o n e class of charged monopoles characterized by a given charge‐to‐mass ratio. The method also provides a new empirical criterion for determining whether or not a directing field is geodesic.

On Chandrasekhar’s perturbation analysis. I. The superfluity of the N.P. system
View Description Hide DescriptionChandrasekhar has developed a method of analyzing first‐order perturbations about some known metrics using the N.P. system of equations. In this paper it is shown that some of the intriguing aspects that have been noted in his method—the superfluity of the N.P. system, and the existence of very complicated integral identities—are not peculiar to this particular type of perturbation analysis; rather the underlying principles are fundamental properties associated with the differential structure of the N.P. system. Specifically the three different subsystems used in the three space‐times where Chandrasekhar’s method has been applied, are confirmed directly as sufficient subsystems for extracting all information from the complete N.P. system, for the respective situations in which they have been used.

Ricci and contracted Ricci collineations of the Robertson–Walker space‐time
View Description Hide DescriptionRicci collineations and contracted Ricci collineations of the Robertson–Walker metric, associated with a vector field of the form ξ=(ξ^{0}(t,r),ξ^{1}(t,r),0,0) are presented.

General exact solution for homogeneous time‐dependent self‐gravitating perfect fluids
View Description Hide DescriptionA procedure to obtain the general exact solution of Einstein equations for a self‐gravitating spherically symmetric static perfect fluid obeying an arbitrary equation of state is applied to time‐dependent Kantowski–Sachs line elements (with spherical, planar, and hyperbolic symmetry). As in the static case, the solution is generated by an arbitrary function of the independent variable and its first derivative. To illustrate the results, the whole family of (plane‐symmetric) solutions with a ‘‘gamma‐law’’ equation of state is explicitly obtained in terms of simple known functions. It is also shown that, while in the static plane‐symmetric line element, every metric is in one to one correspondence with a ‘‘partner metric’’ (both originated from the same generatrix function); in this case every generatrix function uniquely determines one metric.