On rotations as special cases of unitary transformations with some applications to the theory of spin


  • Per-Olov Löwdin

    1. Quantum Theory Project, 362 Williamson Hall, University of Florida, Gainesville, Florida 32611
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    • Professor Emeritus at Uppsala University, Uppsala, Sweden


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.