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On rotations as special cases of unitary transformations with some applications to the theory of spin
Author(s) -
Löwdin PerOlov
Publication year - 1992
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560410117
Subject(s) - mathematics , invariant (physics) , spinor , simple (philosophy) , rotation group so , pure mathematics , rotation (mathematics) , unitary state , geometry , mathematical physics , philosophy , epistemology , political science , law
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( i ω m ), where the “generator” m in the former case is an operator satisfying the relation m 2 = 1, and in the latter case, m 3 = 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.