L. C. Biedenharn

On the Representations of the Semisimple Lie Groups. II

The explicit determination of the matrices of the generators of the unitary groups, SUn, is carried out and discussed in two alternative treatments: (a) by purely algebraic infinitesimal methods, and (b) by Young‐pattern techniques employing the Schwinger‐Bargmann boson operator methods. The implication of this result for a tableau calculus is discussed and a determination of the [λ] × [1] Wigner coefficient for all SUn is indicated.

On the Representations of the Semisimple Lie Groups. I. The Explicit Construction of Invariants for the Unimodular Unitary Group in N Dimensions

It is intended in the present series of papers to discuss explicit constructive determinations of the representations of the semisimple Lie groups SUn by an extension of the Racah‐Wigner techniques developed for the two‐dimensional unimodular unitary group (SU2). The present paper defines, and explicitly determines, a symmetric vector‐coupling coefficient for the group SUn. These coefficients are utilized to construct a series of canonical invariants for SUn, of which the first I2 is the familiar Casimir invariant, and it is proved (by construction) that these invariants form a complete system of independent invariants suitable for uniquely labeling the irreducible inequivalent representations of SUn.

Wigner Coefficients for the R4 Group and Some Applications

The local isomorphism of the R4 group to the group R3×R3 is utilized to obtain R4 Wigner coefficients for those representations in which the subgroup R3 is diagonal. The R4 Wigner coefficients so defined are then used to obtain recursion relations and differential equations for the representation coefficients, when the group is parametrized appropriately. The R4 spherical harmonics, and their properties, are explicitly obtained as specializations of the general formulas. Physical application to the problem of geometrizing the Coulomb field is briefly discussed.

Canonical Definition of Wigner Coefficients in Un

Two general results applicable to the problem of a canonical definition of the Wigner coefficient in Un are demonstrated: (1) the existence of a canonical imbedding of Un × Un into Un2 and (2) a general factorization lemma for operators defined in the boson calculus. Using these results, a resolution of the multiplicity problem for U3 is demonstrated, in which all degenerate operators are shown to split completely upon projection into U2.

On the Representations of the Semisimple Lie Groups. III. The Explicit Conjugation Operation for SUn

The analog of the SU2 (1 − j) symbol is defined and discussed in detail for SUn. An appropriate generalization to SUn of the Condon‐Shortley phase convention is explicitly given.

On the Representations of the Semisimple Lie Groups. IV. A Canonical Classification for Tensor Operators in SU3

It is shown that the multiplicity structure of the general SUn operators may be put in a one‐to‐one correspondence with the multiplicity structure of the corresponding states. This result allows a convenient labeling scheme to be devised for the general SUn Wigner operator and leads in a natural way to the concept of a reduced Wigner operator. The problem of multiplicity in tensor operators is shown to have a canonical resolution in the conjugation classification which is discussed in detail for the SU 3 case.

On the Representations of the Semisimple Lie Groups. V. Some Explicit Wigner Operators for SU3

A general method for the calculation of the [21 … 10] Wigner operators is presented and used to obtain specific results for the SU3 group. General expressions for the SU3 reduced Wigner operators are given for tensor operators transforming like the representations [100], [110], and [210].

Combinatorial Structure of State Vectors in Un. I. Hook Patterns for Maximal and Semimaximal States in Un

It is shown that, in the boson‐operator realization, the state vectors of the unitary groups Un—in the canonical chain Un⊃Un−1⊃⋯⊃U1—can be obtained ab initio by a combinatorial probabilistic method. From the Weyl branching law, a general state vector in Un is uniquely specified in the canonical chain; the algebraic determination of such a general state vector is in principle known (Cartan‐Main theorem) from the state vector of highest weight; the explicit procedure is a generalization of the SU(2) lowering‐operator technique. The present combinatorial method gives the normalization of these state vectors in terms of a new generalization of the combinatorial entity, the Nakayama hook, which generalization arises ab initio from a probabilistic argument in a natural way in the lowering procedure. It is the advantage of our general hook concept that it recasts those known algebraic results into a most economical algorithm which clarifies the structure of the boson‐operator realization of the Un representations.

SOME EXACT RADIAL INTEGRALS FOR DIRAC-COULOMB FUNCTIONS

The zero energy loss Dirac‐Coulomb integrals are evaluated using the technique of contour integration. The expressions obtained have a closed analytic form, showing that these integrals are formally similar to the corresponding classical and nonrelativistic quantum mechanical, zero energy loss integrals which also have exact elementary solutions.Application of the zero energy loss Dirac‐Coulomb integrals occurs in inelastic electron scattering and similar problems. The investigation of the finite energy loss Dirac‐Coulomb integrals requires a study of the zero energy loss integrals as a preliminary.

The pattern calculus for tensor operators in quantum groups

An explicit algebraic evaluation is given for all q‐tensor operators (with unit norm) belonging to the quantum group Uq(u(n)) and having extremal operator shift patterns acting on arbitrary Uq(u(n)) irreps. These rather complicated results are shown to be easily comprehensible in terms of a diagrammatic calculus of patterns. A more conceptual derivation of these results is discussed using fundamental properties of the q−6j coefficients.