R. Jagannathan

Quantum theory of magnetic electron lenses based on the Dirac equation

A quantum theory of magnetic electron lenses based on a convenient formulation of the Dirac theory is outlined. It is shown that the passage from the conventional scalar theory to the spinor theory can be accomplished through a simple algebraic rule in analogy with the passage from scalar to vector light optics.

Realizations of su(1,1) and Uq(su(1,1)) and generating functions for orthogonal polynomials

Positive discrete series representations of the Lie algebra su(1,1) and the quantum algebra Uq(su(1,1)) are considered. The diagonalization of a self-adjoint operator (the Hamiltonian) in these representations and in tensor products of such representations is determined, and the generalized eigenvectors are constructed in terms of orthogonal polynomials. Using simple realizations of su(1,1), Uq(su(1,1)), and their representations, these generalized eigenvectors are shown to coincide with generating functions for orthogonal polynomials. The relations valid in the tensor product representations then give rise to new generating functions for orthogonal polynomials, or to Poisson kernels. In particular, a group theoretical derivation of the Poisson kernel for Meixner–Pollaczek and Al-Salam–Chihara polynomials is obtained.

TAMM-DANCOFF DEFORMATION OF BOSONIC OSCILLATOR ALGEBRAS

The Tamm-Dancoff (TD) deformation of the boson oscillator incorporates a high energy cutoff in its spectrum. It is found that one can obtain a similar deformation of any generalized bosonic oscillator algebra. The Hopf (or ‘quantum’) algebraic aspects of the TD-deformation are discussed. Examples are given.

Lie algebraic treatment of dioptric power and optical aberrations

The dioptric power of an optical system can be expressed as a four-component dioptric power matrix. We generalize and reformulate the standard matrix approach by utilizing the methods of Lie algebra. This generalization helps one deal with nonlinear problems (such as aberrations) and further extends the standard matrix formulation. Explicit formulas giving the relationship between the incident and the emergent rays are presented. Examples include the general case of thick and thin lenses. The treatment of a graded-index medium is outlined.

Generalized q-fermion oscillators and q-coherent states

The algebra of q-fermion operators, developed earlier by two of the present authors is re-examined. It is shown that these operators represent particles that are distinct from usual spacetime fermions except in the limit q=1. It is shown that it is possible to introduce generalized q-oscillators defined for - infinity (q<or=1. In the range - infinity (q(0, these coincide with the q-boson operators and for 0(q<or=1 they coincide with q-fermions. The ordinary bosons and fermions may be identified with the limits q=-1 and +1 respectively. Generalized q-fermion coherent states are constructed by utilizing a nonlinear shift automorphism of the algebra of q-fermion operators. These are compared with the coherent states defined as eigenstates of annihilation operator. Matrix elements of the shift operator in the Fock space basis are evaluated.

Group theoretical basis for the terminating 3F2(1) series

It is shown that a recursive use of the transformation for a terminating 3F2(1) series used by Weber and Erdelyi (1952), which belongs, as shown by Whipple (1925), to a set of equivalent 3F2(1) functions obtained by Thomae (1979), results in a 72-element group associated with 18 terminating series. The generators, conjugacy classes, invariant subgroups, characters and dimensions of irreducible representations for this group are presented.

Dynamical maps and nonnegative phase-space distribution functions in quantum mechanics

Using the formalism of dynamical maps it is shown that if a quantum measurement process is to be described in terms of a non-negative phase-space distribution obtained by smoothing the Wigner distribution of the quantum state, then the smoothing kernel characterizing the measuring apparatus cannot be an arbitrary Wigner distribution.

Finite-dimensional quantum mechanics of a particle

This paper analyzes the possible implications of interpreting the finitedimensional representations of canonically conjugate quantum mechanical position, and momentum operators of a particle consistent with Weyls form of Heisenbergs commutation relation as the actual position, and momentum operators of the particle when it is confined to move within a finite spatial domain, and regarding the application of current quantum mechanical formalism based on Heisenbergs relation to such a situation as an asymptotic approximation. In the resulting quantum mechanical formalism the discrete and finite position and momentum spectra of a particle depend on its rest mass and the spatial domain of confinement. Such a “finite-dimensional quantum mechanics” may be very suitable for describing the physics of particles confined to move within very small regions of space.

On Up,q(gl(2)) and a (p,q)-Virasoro algebra

The quantum algebra Up,q(gl(2)), with two independent deformation parameters (p, q), is studied and, in particular, its universal R-matrix is constructed using Reshetikhins method. A contraction procedure then leads to the (p, q)-deformed Heisenberg algebra Up, q(h(1)) and its universal R-matrix. Using a Sugawara construction employing an infinite number of copies of these Heisenberg modes, a (p, q)-deformed Virasoro algebra is obtained. The closure property of the (p, q)-Virasoro algebra necessitates two parameters ( alpha , beta ) for the generators (Lm( alpha , beta )). While the parameter alpha may be taken as an integer, the parameter beta is continuous on a complex path and imparts an integral equation structure to the (p, q)-deformed Virasoro algebra.

On the number operators of multimode systems of deformed oscillators covariant under quantum groups

For multimode systems of deformed oscillators covariant under the actions of the quantum groups SUq(n), SUq(n mod m), GLp,q(n) and GLp,q(n mod m) the number operators are constructed explicitly in terms of the creation and annihilation operators. The relation between the various kinds of deformed oscillator systems, representations of these oscillator algebras in terms of coordinates and deformed derivatives, realizations of classical groups in non-commutative spaces, and some aspects of the physical behaviour of quantum group covariant oscillator systems are also discussed.