AN EXPLICIT DESCRIPTION OF SL(2, C) IN TERMS OF SO+(3, 1) AND VICE VERSA

Authors : Frank KLINKER

Pages : 94-104

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Publication Date : 2015-04-30

Article Type : Research Paper

Abstract :In this note we present explicit and elementary formulas for the correspondence between the group of special Lorentz transformation SO+insert ignore into journalissuearticles values(3, 1);, on the one hand, and its spin group SLinsert ignore into journalissuearticles values(2, C);, on the other hand. Although we will not mention Clifford algebra terminology explicitly, it is hidden in our calculations by using complex 2 × 2-matrices. Nevertheless, our calculations are strongly motivated by the Clifford algebra glinsert ignore into journalissuearticles values(4, C); of fourdimensional space-time. It is well known that for a pseudo-euclidean vector space insert ignore into journalissuearticles values(V, g); the universal  cover of the special orthogonal group SOinsert ignore into journalissuearticles values(V, g); is given by the so called spin group  Spininsert ignore into journalissuearticles values(V, g);. For the case V = Rp+q and g = diaginsert ignore into journalissuearticles values(1q, −1p); we write SOinsert ignore into journalissuearticles values(p, q);  and Spininsert ignore into journalissuearticles values(p, q);. The covering map is 2:1 for dim V > 2. The theoretic setting in  which spin groups and related structures are best described is the Clifford algebra  C`insert ignore into journalissuearticles values(V, g);, see [2, 3, 8] for example. Although spin groups in general refrain from being described by classical matrix groups for dimensional reason, there are accidental  isomorphisms to such in dimension three to six, see Table 1. The isomorphisms are  a consequence of the classification of Lie algebras and can for example be seen by  recalling the connection to Dynkin diagrams. We use the notation from [4] and recommend this book for details on the definition of the classical matrix groups. Due  to the fact that the complexifications of the orthogonal groups are independent of  the signature of the pseudo-Riemannian metric the groups in each column of Table  1 are real forms of the same complex group for fixed dimension.
Keywords : Spin group, represenation, covering map