On the use of Young tableaux and Dynkin diagrams in the analysis of symmetry structure in mathematical physics. (Volumes I and II)

James Seal, Purdue University

Abstract

The purpose of this investigation is to develop a unified approach to the classical Lie algebras and apply these techniques to various physical situations. In the first chapter, we review the traditional approach of Lie, Cartan, Dynkin, and Weyl to the theory of representations and invariant structure. The author extends these ideas and presents a new algorithm for generating weights and determining their multiplicities. We apply it to SU(3) and SO(5) $\sim$ SP(4) to obtain a general method for generating weight spaces for these groups. This algorithm also provides a two dimensional method for displaying the hyperdimensional weight spaces. The second chapter differs from the first in that we consider the classical Lie algebras as a unified whole rather than dwelling upon their differences. In this approach, the classical Lie groups are "rotations" on vector spaces over some appropriate field of numbers. This approach enables us to derive simple algorithms for dimensionality, branching rules, direct products, etc., for the classical Lie algebras, based on the method of Young, Littlewood, and Robinson, for SU(N) and the symmetric group. This differs from other authors, such as Fischler, in that it does list a long series of rules which apply to individual algebras. We apply these techniques in an attempt to find a theory that explains the generation problem and is also consistent with grand unification and superstring models. We shall also consider some of the predictions of that theory at the TeV level. Finally the methods of Chapter II are applied to the invariant structure of the classical Lie algebras. We investigate the Casimir invariants of all orders and show that there exists a unified method of enumerating all such invariants. The symmetries related to the diagramatic method is also pointed out and used to simplify the calculations. We collect information about the various Lie algebras in the appendices. In unpublished work, the author collects the basic results for algebras of rank 2 through 8.

Degree

Ph.D.

Advisors

Kuo, Purdue University.

Subject Area

Particle physics

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