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Journal of Mathematical Physics

Representations of the three-dimensional rotation group by the direct method

A. J. Torruella

Published Aug 1, 1975• 10 citations
Publisher

Abstract

The irreducible representations of the three-dimensional rotation group are obtained directly from the irreducible representations of its infinitesimal generators (the spin matrices), parametrized in terms of the rotation angle and the direction of the rotation axis. Expressions are given for the rotation operator exp(iψn ⋅ S) in terms of two different bases of 2j+1 elements for spin j. The results are related to the spectral decomposition of the rotation operator and expressions obtained for spin projection operators along any spatial direction for arbitrary spin.

DOI
10.1063/1.522718

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The irreducible representations of the three-dimensional rotation group are obtained directly from the irreducible representations of its infinitesimal generators (the spin matrices), parametrized in terms of the rotation angle and the direction of the rotation axis. Expressions are given for the rotation operator exp(iψn ⋅ S) in terms of two different bases of 2j+1 elements for spin j. The results are related to the spectral decomposition of the rotation operator and expressions obtained for spin projection operators along any spatial direction for arbitrary spin.
Abstract A general expression of Clebsch‐Gordan coefficients is derived by means of de Broglie‐like méthode de fusion: the second angular momentum is added to the first in n steps of half each, the maximum symmetric composition of the n spin functions being taken into account explicitly. The procedure is operatorial. Group‐theoretically speaking the irreducible representations of the direct product of two irreducible representations of the rotation group are realized via symmetric group Sn with the aid of projection operators (equivalent to Young's symmetrizers). The decomposition of the representations of Sn therein is performed inductively by utilizing the chain of subgroups: Sn ⊃ Sn−1 … ⊃ S2 ⊃ S1.
Abstract Some problems in elementary geometry are approached from the point of view of linear algebra and generalized to the theory of linear spaces of finite or infinite dimensions having a positive definite binary product. The angle ω between two elements of the linear space is defined from the concept of length by means of the cosine‐theorem. A rotation is then defined as a special case of a unitary transformation moving all elements the same angle ω, except that under certain circumstances, some elements may stay invariant. In the former case, one speaks of a rotation around an “external axis,” and in the latter case, of a rotation around an “internal axis” defined by the invariant elements. It is shown that the finite rotations U of both types may be expressed in the simple exponential form U = exp(iωm), where the “generator” m in the former case is an operator satisfying the relation m2 = 1, and in the latter case, m3 = m. The structure of the group of finite rotations in the former case is clarified in some detail. As an illustration of the theory, some applications to the three‐ and two‐dimensional spaces as well as to the theory of spin are given. The coupling between the ordinary three‐dimensional rotations and the spinor transformations is considered in somewhat greater detail.