Raj Wilson

Quantum theory of infinite component fields

The quantum theory of the infinite component SO(4,2) fields is formulated as a model for relativistic composite objects. We discuss three classes (timelike, lightlike, and spacelike) of physical solutions to a general class of infinite component wave equations. These solutions provide a definite physical interpretation to infinite component wave equations and are obtained by reducing SO(4,2) with respect to its orthogonal, pseudo‐orthogonal, and Euclidean subgroups. The analytic continuations among these solutions are established. In the nonrelativistic limit the timelike physical states exactly reduce to the Schrodinger solution for the hydrogen atom—the simplest composite object. The wave equations for the three classes are studied in two different realizations. In one case the equations describe a three‐dimensional internal Kepler motion with a discrete and a continuous energy spectrum and in the other case the equations describe a four‐dimensional internal oscillatory motion with attractive as well as...

Algebraic treatment of second Poschl-Teller, Morse-Rosen and Eckart equations

The method of an earlier paper (see ibid., vol.20, p.4075 (1987)) is applied to the non-compact case to solve a family of second Poschl-Teller Morse-Rosen and Eckart equations with quantised coupling constants. Both discrete and continuous spectra, bound state and scattering wavefunctions (transmission coefficients) are found from the matrix elements of group representations,.

A new realisation of dynamical groups and factorisation method

A new method of algebraisation of quantum mechanical eigenvalue equations is presented. In this method the dynamical algebra is represented on the space of group matrix elements. The ladder operators of the dynamical algebra are obtained from Infeld-Hull-Miller factorisations. The method is used to study the first Poschl-Teller equation even in the non-symmetric case. The energy spectrum and the exact normalised solutions are obtained in agreement with the results of non-algebraic methods.

Path-integral treatment of the morse oscillator

Abstract The path integral for the Green function of the Morse oscillator is evaluated by changing local variables and local time parameters. The exact energy spectrum for the Morse oscillator is obtained.

Some new identities of the Clebsch–Gordan coefficients and representation functions of SO(2,1) and SO(4)

A systematic derivation of various relations and identities among the Clebsch–Gordan coefficients and for the representation functions of SO(4) and SO(2,1), is given. These relations are essential in work involving the matrix elements of arbitrary group elements in higher noncompact groups such as O(4,2).

The generalized Morse oscillator in the SO(4,2) dynamical group scheme

A family of the Morse oscillators with certain quantized coupling constants are described as composite objects in the framework of the SO(4,2) dynamical group scheme. Although a single Morse oscillator can be solved by the subgroup SO(2,1) of SO(4,2) this SO(2,1) is not the spectrum generating group], the set of all energy levels is given by the representation of another particular one‐parameter subgroup of SO(4,2), which is the dynamical group of a single Morse oscillator. The continuous spectra of this oscillator and other variations of the Morse potential are also discussed by making an analytic continuation from the Morse potential well to the Morse barrier.

Path integral realization of a dynamical group

A way to realize a dynamical group in terms of a path integral is illustrated by using the Poschl-Teller oscillator.

A unified description of the representations of the graded Lie algebra Gsl(2)

Indecomposable representations of the graded Lie algebra Gsl(2) are analyzed in detail. It is further shown that the study of the irreducible representations (finite‐ and infinite‐dimensional) is intimately related to the study of these indecomposable representations.

Ladder operators of group matrix elements

All ladder operators and some recurrence relations of the matrix elements of certain group elements of SO(3), SO(2,1), E(2), SO(4), SO(3,1), and E(3) have been explicitly determined and the underlying factorizations of the second‐ and the fourth‐order linear ordinary differential equations in terms of first‐ and second‐order ladder operators have been transparently demonstrated as an extension to the Schrodinger–Infeld–Miller factorization. These ladder operators are very useful in physical applications where the corresponding matrix elements represent certain physical transitions.

A generalized Dirac current in O(4,2) theory

Abstract A relativistic Lagrangian field theory of spin-half particles using a four-dimensional Dirac-representation of O(4,2) is considered. In this theory a generalized Dirac current with a “convective” term is used. From the field equation a point-quantum mechanical Hamiltonian is derived and certain integrated quantities are investigated. A general divergence term that can be added to a free Lagrangian density without changing the corresponding field equation is suggested and illustrated. The scale invariance of the theory is examined. Also the theory is second-quantized and is studied in the infinite momentum limit.