Mathematical Aspects of SU (2) and SO(3,R) Derived from Two-Mode Realization in Coordinate-Invariant Form  

作　　者：Alfred Wünsche Alfred Wünsche(Institut für Physik, Humboldt-Universitä,t, Newtonstr. 15, Berlin, Germany)

机构地区：[1]Institut für Physik, Humboldt-Universitä,t, Newtonstr. 15, Berlin, Germany

出　　处：《Journal of Modern Physics》2023年第3期361-413,共53页现代物理（英文）

摘　　要：Some mathematical aspects of the Lie groups SU (2) and in realization by two pairs of boson annihilation and creation operators and in the parametrization by the vector parameter  instead of the Euler angles and the vector parameter c of Fyodorov are developed. The one-dimensional root scheme of SU (2) is embedded in two-dimensional root schemes of some higher Lie groups, in particular, in inhomogeneous Lie groups and is represented in text and figures. The two-dimensional fundamental representation of SU (2) is calculated and from it the composition law for the product of two transformations and the most important decompositions of general transformations in special ones are derived. Then the transition from representation of SU (2) to of is made where in addition to the parametrization by vector  the convenient parametrization by vector c is considered and the connections are established. The measures for invariant integration are derived for and for SU (2) . The relations between 3D-rotations of a unit sphere to fractional linear transformations of a plane by stereographic projection are discussed. All derivations and representations are tried to make in coordinate-invariant way.Some mathematical aspects of the Lie groups SU (2) and in realization by two pairs of boson annihilation and creation operators and in the parametrization by the vector parameter  instead of the Euler angles and the vector parameter c of Fyodorov are developed. The one-dimensional root scheme of SU (2) is embedded in two-dimensional root schemes of some higher Lie groups, in particular, in inhomogeneous Lie groups and is represented in text and figures. The two-dimensional fundamental representation of SU (2) is calculated and from it the composition law for the product of two transformations and the most important decompositions of general transformations in special ones are derived. Then the transition from representation of SU (2) to of is made where in addition to the parametrization by vector  the convenient parametrization by vector c is considered and the connections are established. The measures for invariant integration are derived for and for SU (2) . The relations between 3D-rotations of a unit sphere to fractional linear transformations of a plane by stereographic projection are discussed. All derivations and representations are tried to make in coordinate-invariant way.

关 键 词：Boson Operators Lie Algebra Root Diagram Invariant Integration Hamilton-Cayley Identity Cayley-Gibbs-Fyodorov Parametrization Composition Law Quaternion Stereographic Projection Fractional Linear Transformation 

分 类 号：O17[理学—数学]