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We have obtained in configuration space the state functions of the q-isotropic oscillator by laddering the state functions of the q-linear oscillator. Starting from the q-Hermite functions of the linear oscillator one obtains the q-associated Laguerre functions of the isotropic oscillator. The energy spectrum is obtained from the curvature of the wave function at a point where the potential energy vanishes.

There exist operator solutions of the q‐Coulomb problem in both configuration and momentum space. Since the arguments of the corresponding amplitudes are noncommuting, however, there are problems of physical interpretation. Here we answer the question of physical interpretation by associating with the operator amplitude in momentum space a numerically valued amplitude lying in a Hilbert space defined by the SUq(2) algebra. This new amplitude, now depending on commuting arguments, may be interpreted by the usual rules of quantum mechanics and may also be Fourier transformed to yield the amplitude in configuration space.

A deformation of quantum mechanics obtained by replacing commutators by q‐commutators (where q is close enough to unity to be compatible with experiment) is studied. This idea is explored by studying the q‐harmonic oscillator in both configuration space and momentum space. The complete set of states in both representations as well as the transformation between x and p space is obtained. The states as well as the matrix elements lie in the SUq(2) algebra. To obtain expectation values and transition probabilities one must average over the SUq(2) algebra.

The orthogonality relations for SUq(2) are here formulated with the aid of the Woronowicz integral. The first part of this paper is devoted to a summary of the relevant parts of the Woronowicz theory and the second part to its implementation in terms of completely explicit representations of the irreducible unitary representations. The purpose of this presentation is to provide background for a separate study of the q deformation of the Coulomb problem.

The hydrogenic Schrodinger equation may be transformed into an integral equation on the group space of SO(3). The solutions of this integral equation are the Wigner functions Djmn, the matrix elements of the irreducible representations of SO(3). In this article we examine a q‐deformation of the Coulomb problem for which the new wave functions are the matrix elements of the irreducible representations of the quantum group SUq(2). The new integral equation is formulated in terms of the Woronowicz integral. The eigenvalues are given by a modified Balmer formula which lifts the Coulomb degeneracy and replaces n by [n].

The Grassmannian analogues of the Legendre polonomials are obtained from the corresponding Grassmannian Hermite multinomials. The generating function, recurrence relations, and differential equation are given. In contrast to the Hermitian case there is no Berezin weight function that orthogonalizes these functions. Generalizations to the Tchebyscheff and Gegenbauer cases are also possible.

By generalizing from systems characterized by finite Lie algebras the propagator of an extended particle that appears in a gauge theory based on an affine algebra is discussed.

The gravitational and electromagnetic fields of a charged, rotating source are obtained by an elementary algebraic method.

We describe an algorithm to solve the classical equations of supergravity for the coupled fields of helicity 2 and 3/2. The algorithm depends on an expansion in the associated Grassman algebra and leads to a sequence of coupled equations that may be solved in a step by step manner. The procedure begins with solutions of the usual empty space Einstein equations but the subsequent equations are all linear differential equations. To complete the method, it is necessary to generalize the supplementary conditions on the vector spinor field from flat to curved space. The algorithm also permits a classification of the complete solutions in terms of the associated gravitational fields. It is shown that vanishing curvature implies Minkowski space and does not permit other conceivable possibilities such as a Clifford space. Generalizations of the vanishing curvature space are suggested.