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

Year 2015, Volume: 8 Issue: 1, 94 - 104, 30.04.2015

Frank KLINKER

Abstract

In this note we present explicit and elementary formulas for the correspondence between the group of special Lorentz transformation SO+(3, 1), on the one hand, and its spin group SL(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 gl(4, C) of fourdimensional space-time.It is well known that for a pseudo-euclidean vector space (V, g) the universal cover of the special orthogonal group SO(V, g) is given by the so called spin group Spin(V, g). For the case V = Rp+q and g = diag(1q, −1p) we write SO(p, q) and Spin(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`(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

References

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