Chapter

Rotations: <i>SOi</i>(3) and <i>SU</i>(2)

Adam M. Bincer

in Lie Groups and Lie Algebras

Published in print October 2012 | ISBN: 9780199662920
Published online January 2013 | e-ISBN: 9780191745492 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199662920.003.0003
Rotations: SOi(3) and SU(2)

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SO(3) — the group of rotations in three dimensions, whose elements are 3x3 orthogonal unimodular real matrices — is introduced and SU(2) is introduced as the universal covering group of SO(3). This chapter starts with a discussion of rotations in two dimensions, which should be familiar to everybody. Next, it is shown that the determinant of real orthogonal nxn matrices equals +1 or –1 and therefore the O(n) group consists of two disconnected pieces: rotations, for which the determinant equals +1 and reflections, for which the determinant equals –1. Since rotations are generated by angular momentum I identify the infinitesimal generators of SO(n) with the Cartesian components of angular momentum in appropriate units. Explicit evaluation of the structure constants shows the so(3) and su(2) algebras are isomorphic. The corresponding groups are shown to be homomorphic, SO(3) being doubly connected, while SU(2) is the simply connected covering group.

Keywords: orthogonal matrices; rotations; reflections; connectivity; universal covering group

Chapter.  3003 words.  Illustrated.

Subjects: Mathematical and Statistical Physics

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