From the rotation group to the Poincare group
Larry Biedenham is, perhaps, best known for his penetrating insights into the relevance of abstract mathematics for the structure of physical theory and his ability to implement this mathematics in a style comprehensible to physicists. This is well-illustrated by his many recent contributions to q-algebras. He has also written numerous papers concerned with the pedagogy of physical theory -- papers with the purpose of clarifying the content and meaning of a subject. It is in the spirit of these latter contributions that the present paper is presented. The subject is the quantum rotation group, SU(2), the group of 2 {times} 2 unitary unimodular matrices, and its fundamental role in physical theory. The fact that the irreducible representations (irreps) of the group SU(2) can be realized by unitary transformation of state vectors whose domain of definition is the usual Euclidean 3-space, R{sup 3}, is usually obscured by the fact that one is accustomed to thinking of the underlying group of transformations on R{sup 3} the group SO(3,R) of proper orthogonal matrices. We show in the first part of this paper how one can formulate the usual ``orbital angular momentum theory`` from the viewpoint of SU(2) alone. The motivation for this approach is to eliminate from the language of quantum mechanics the ambiguities arising from the highly unsatisfactory notion of ``double-valued functions.`` In the second part of the paper, we show how the irreps of SU(2), extended to GL(2,C), enter into the description of the unitary irreps of the Poincare group. We emphasize again the uniform role of SU(2), in contrast to SO(3,R), and point out the several ways that the standard irreps of SU(2) occur. In the third part of the paper, we discuss the parametrization of SL(2,C) in terms of biquaternions and mention the problem of determining the discrete subgroups of SL(2,C).
- Research Organization:
- Los Alamos National Lab., NM (United States)
- Sponsoring Organization:
- USDOE, Washington, DC (United States)
- DOE Contract Number:
- W-7405-ENG-36
- OSTI ID:
- 10187356
- Report Number(s):
- LA-UR-92-3200; CONF-9208164-1; ON: DE93000864
- Resource Relation:
- Conference: 6. symmetries in science,Bregenz (Austria),2-7 Aug 1992; Other Information: PBD: [1992]
- Country of Publication:
- United States
- Language:
- English
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