Matrix Elements Of Unitary Representations Research Articles

The multivariate Meixner polynomials as matrix elements of SO(d, 1) representations on oscillator states

The multivariate Meixner polynomials are shown to arise as matrix elements of unitary representations of the SO(d, 1) group on oscillator states. These polynomials depend on d discrete variables and are orthogonal with respect to the negative multinomial distribution. The emphasis is put on the bivariate case for which the SO(2, 1) connection is used to derive the main properties of the polynomials: orthogonality relation, raising/lowering relations, generating function, recurrence relations and difference equations as well as explicit expressions in terms of standard (univariate) Krawtchouk and Meixner polynomials. It is explained how these results generalize directly to d variables.

Unitary representations of the quantum group SU q (1, 1): II ? Matrix elements of unitary representations and the basic hypergoemetric functions

Some series of unitary representations of the quantum group SU q (1, 1) are introduced. Their matrix elements are expressed in terms of the basic hypergeometric functions. Operator realization of the coordinate elements of SU q (1, 1) and aq-analogue of some classical identities are discussed.

Horospheric basis of the SL(2, C) group

Representations of theSL(2,C) groups are reduced with respect to the horospheric subgroup\(\left( {\begin{array}{*{20}c} 1 \\ 0 \\ \end{array} \begin{array}{*{20}c} \beta \\ 1 \\ \end{array} } \right)\) isomorphic to the translation group in two dimensions. The matrix elements of unitary representations of the principal and supplementary series are derived in a basis defined by a two-dimensional momentum corresponding to the above subgroup. Matrix elements of unitary representations expressed in this basis assume a simpler form than in any other basis considered so far. The two-dimensional momentum introduced proves to be essentially identical with the impact parameter.

Deformation of the SO(2, C) subgroup of the Lorentz group

The two-dimensional complex sphere S12+S22+S32=S2 forms a homogeneous space under the SL(2,C) group. The little group of a point in this space is the SO(2,C) group or the horospheric group T(2), according to whether S ≠ 0 or S = 0. Deformation of the SO(2,C) group into T(2) is investigated and is demonstrated on unitary representations. This deformation is a counterpart to that of the little groups SO(3) or SO(2,1) into E(2). We conclude with a formula relating the matrix elements of unitary representations of the SL(2,C) group in SO(2) × SO(1,1) = SO(2,C) basis to those in horospheric basis.

Unitary representations of the SL (2, C) group in horospheric basis

The matrix elements of unitary representations of the S L (2, C) group are derived in a basis defined by two-dimensional momenta corresponding to the horospheric subgroup (1β01). A parametrization fitting the above basis well is chosen on the analogy of the rotation group by sandwiching a complex rotation about the y axis with two horospheric translations. In this way the two outer subgroups can be factored out immediately in terms of plane waves which are counterparts to the exponentials formed by the Euler angles φ and ψ occuring in the rotation group. Finally, matrix elements of complex rotations about y axis are found by solving the simultaneous eigenvalue problem of the two Casimir operators. Unitary representations obtained in this way are expressed in a rather simple form in terms of Bessel functions.

Quasi-potential approach and the expansion in relativistic spherical functions

The expansion in matrix elements of unitary representations of the Lorentz groups is applied in the relativistic quasi-potential scattering theory. A relativistic analogue of the three-dimensional configuration space is found. The formalism developed is very close to the nonrelativistic one. It allows us to hope that in the framework of this approach, an adequate relativistic generalization of Regge's ideas can be found.