Debabrata Basu

The unitary irreducible representations of SL(2, R) in all subgroup reductions

We use the canonical transform realization of SL(2, R) in order to find all matrix elements and integral kernels for the unitary irreducible representations of this group. Explicit results are given for all mixed bases and subgroup reductions. These provide the full multiparameter set of integral transforms and series expansions associated to SL(2, R).

A Molecular Insight into the Early Events of Chickpea (Cicer arietinum) and Fusarium oxysporum f. sp. ciceri (Race 1) Interaction Through cDNA-AFLP Analysis

Wilt of chickpea caused by Fusarium oxysporum f. sp. ciceris is one of the most severe diseases of chickpea throughout the world. Variability of pathotypes of F. oxysporum f. sp. ciceris and breakdown of natural resistance are the main hindrances to developing resistant plants by applying resistant breeding strategies. Additionally, lack of information of potential resistant genes limits gene-transfer technology. A thorough understanding of Fusarium spp.-chickpea interaction at a cellular and molecular level is essential for isolation of potential genes involved in counteracting disease progression. Experiments were designed to trigger the pathogen-challenged disease responses in both susceptible and resistant plants and monitor the expression of stress induced genes or gene fragments at the transcript level. cDNA amplified fragment length polymorphism followed by homology search helped in differentiating and analyzing the up- and downregulated gene fragments. Several detected DNA fragments appeared to have relevance with pathogen-mediated defense. Some of the important transcript-derived fragments were homologous to genes for sucrose synthase, isoflavonoid biosynthesis, drought stress response, serine threonine kinases, cystatins, arginase, and so on. Reverse-transcriptase polymerase chain reaction performed with samples collected at 48 and 96 h postinfection confirmed a similar type of differential expression pattern. Based on these results, interacting pathways of cellular processes were generated. This study has an implication toward functional identification of genes involved in wilt resistance.

High level expression of soybean trypsin inhibitor gene in transgenic tobacco plants failed to confer resistance against damage caused byHelicoverpa armigera

Helicoverpa armigera is a major pest of many tropical crop plants. Soybean trypsin inhibitor (SBTI) was highly effective against the proteolytic activity of gut extract of the insect. SBTI was also inhibitory to insect growth when present in artificial diet. The gene coding for SBTI was cloned from soybean (Glycine max, CVBirsa) and transferred to tobacco plants for constitutive expression. Young larvae ofH. armigera, fed on the leaves of the transgenic tobacco plants expressing high level of SBTI, however, maintained normal growth and development. The results suggest that in certain cases the trypsin inhibitor gene(s) may not be suitable candidates for developing insect resistant transgenic plants.

The Clebsch–Gordan coefficients of the three‐dimensional Lorentz algebra in the parabolic basis

Starting from the oscillator representation of the three‐dimensional Lorentz algebra so(2,1), we build a Lie algebra of second‐order differential operators which realizes all series of self‐adjoint irreducible representations. The choice of the common self‐adjoint extention over a two‐chart function space determines whether they lead to single‐ or multivalued representations over the corresponding Lie group. The diagonal operator defining the basis is the parabolic subgroup generator. The direct product of two such algebras allows for the calculation of all Clebsch–Gordan coefficients explicitly, as solutions of Schrodinger equations for Poschl–Teller potentials over one (D×D), two (D×C), or three (C×C) charts. All coefficients are given in terms of up to two 2F1 hypergeometric functions.

Representations of SL(2,R) in a Hilbert space of analytic functions and a class of associated integral transforms

It is shown that the boson operators of SL(2,R) realized as hyperdifferential operators in Bargmann’s Hilbert space of analytic functions yield, on exponentiation, a parametrized continuum of integral transforms. Each value of the group parameters yields an integral transform pair. For the metaplectic representation the resulting integral transform is essentially the mapping of the Moshinsky–Quesne transform in Bargmann’s Hilbert space B(C). The formula for the inversion of this transform is obtained simply by replacing the group element by its inverse. The corresponding Hilbert space for arbitrary representations of the discrete series is B(C2), where C2 is the two‐dimensional complex Euclidean space. To carry out the reduction of B(C2) into the eigenspaces Bk(C) (k= 1/2 ,1, (3)/(2) ,...) of irreducible representations of the positive discrete class, the complex polar coordinates (z1=z cos φ, z2=z sin φ) in C2 are introduced. The ‘‘reduced Bargmann space’’ Bk(C) has many interesting features. The element...

The Barut–Girardello coherent states

It is shown that the completeness problem of the SL(2, R) coherent states proposed by Barut and Girardello leads to a moment problem, not a Mellin transform. This moment problem, which also appears in the theory of para‐Bose oscillators, has been solved following the Sharma–Mehta–Mukunda–Sudarshan solution of the problem. The matrix element of finite transformation in the coherent state basis is shown to satisfy a ‘‘quasiorthogonality’’ condition analogous to the orthogonality condition of the matrix element in the canonical basis. Finally, the Barut–Girardello ‘‘Hilbert space of entire analytic functions of growth (1,1)’’ turns out to be only a subspace of Bargmann’s well‐known Hilbert space of analytic functions. This subspace, which has been called ‘‘the reduced Bargmann space’’ in a previous paper, is an invariant subspace of SL(2,R). With this identification the generators of the group in this realization turn out to be the well‐known boson operators of Holman and Biedenharn.

The Lorentz group in the oscillator realization. I. The group SO(2,1) and the transformation matrices connecting the SO(2) and SO(1,1) bases

The unitary transformation connecting the SO(2) and SO(1,1) bases for the principal and discrete series of representations of the three‐dimensional Lorentz group is determined by using the oscillator representation technique. The Hilbert space and the SO(1,1) basis, in this realization, have a simple appearance while the compact basis appears as the solution of an ordinary differential equation reducible to the confluent hypergeometric equation by a simple substitution. The Taylor expansion of this solution obtained by the use of certain functional identities yields the continuous spectrum of the SO(1,1) representations and the unitary transformation from the compact to the noncompact basis after the Sommerfeld–Watson transformation.

A unified treatment of the characters of SU(2) and SU(1,1)

The character problems of SU(2) and SU(1,1) are re-examined from the standpoint of a physicist by employing the Hilbert space method which is shown to yield a completely unified treatment for SU(2) and the discrete series of representations of SU(1,1). For both the groups the problem is reduced to the evaluation of an integral which is invariant under rotation for SU(2) and Lorentz transformation for SU(1,1). The integrals are accordingly evaluated by applying a rotation to a unit position vector in SU(2) and a Lorentz transformation to a unit SO(2,1) vector which is time-like for the elliptic elements and space-like for the hyperbolic elements in SU(1,1). The details of the procedure for the principal series of representations of SU(1,1) differ substantially from those of the discrete series.

The master analytic function and the Lorentz group. III. Coupling of continuous representations of O(2,1)

The Clebsch–Gordan problem for continuous representations belonging to the principal series of O(2,1) is treated by a method developed previously for the coupling of a discrete and a continuous representation. The values of the complex variable x occurring in the fundamental differential equation of the problem are restricted to lie on the unit circle, and the Clebsh–Gordan coefficients are identified with the Fourier coefficients of solutions of this equation. If j belongs to the discrete class there is only one acceptable solution of the second order equation. But, if j1,j2,j all belong to the continuous class any two independent solutions of the equation give a possible Clebsch–Gordan series. The problem of orthogonalizing the solutions in the latter case is solved and the normalization factor is determined using the Sturm–Liouville theory of differential equations. The Clebsch–Gordan coefficients generated by an orthogonal pair of solutions become automatically orthogonal. To determine the j values ap...

The master analytic function and the Lorentz group. I. Reduction of the representations of O(3,1) in O(2,1) basis

The reduction of the principal and supplementary series of representations of SL(2,C) in the SU(1,1) basis is carried out by using a basis function which formally resembles the coupled state of two angular momenta. The spectrum of the SU(1,1) representations contained in SL(2,C) and the transformation coefficients are obtained by expanding the SU(2) in terms of the SU(1,1) bases with the help of the Sommerfeld–Watson transformation. The orthogonality conditions for the principal and supplementary series are discussed. For the principal series this follows easily from the standard Sturm–Liouville theory of the second order differential equations. For the supplementary series the orthogonality condition is obtained from the fourth order differential equation satisfied by the Fourier transform of the basis function.