Browsing by Author "Sundius, Tom"

H. Weyl describes in his book “Gruppentheorie und Quantenmechanik” how the multidimensional representations of the unimodular unitary group in two dimensions can be constructed using polynomials in two complex variables. This “spinor method” was further developed by B. L. van der Waerden 1932 and was later frequently used by H.A. Kramers. In 1962 V. Bargmann showed that these polynomials generate a finite-dimensional subspace of a Hilbert space, which is constructed from analytical functions in two variables. However, using the spinor algebra of P. Kustaanheimo one can restrict the treatment to finite dimensional spinor spaces, and the existence of a Hilbert space need not be presumed. In the previously used spinor methods the spinor components were utilized, while the main point in this description lies on the spinor basis. In chapter I it is shown, how one in this way can construct symmetrical spinor spaces and the corresponding unitary representations. In addition, spinor polynomials with the same transformation properties as the basis spinors are constructed. They are analogous with the spin generating operators that formed the basis for J. Schwinger’s representation of the rotation group in 1952. In chapter II the Clebsch-Gordan series, which describes the complete reduction of two symmetric spinor spaces in irreducible subspaces, will be derived, and in the final chapter irreducible spinor operators are constructed using the spin generation operators.