Chapter

Representations of <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/9780199662920.003.0004
Representations of SU (2)

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Irreducible representations of SU(2) are defined. Eigenstates of J 3 to the eigenvalue m are introduced. Finite-dimensional representations have a so-called highest-weight state for which this eigenvalue is called j and the chapter shows that 2j must be a non-negative integer. It follows that these representations are (2j+1)-dimensional and that the quadratic Casimir operator has the eigenvalue j(j+1). Next addition of angular momentum is discussed, the Clebsch-Gordan series is described, and explicit values for Clebsch–Gordan coefficients are given when one of the angular momenta in the addition is 1/2 or 1. Spherical tensors are defined, the Wigner–Eckart theorem is demonstrated and reduced matrix elements are introduced. Inner and outer multiplicity is defined and SU(2) is shown to be multiplicity-free. Biographical notes on Casimir, Weyl, Clebsch, Gordan and Wigner are given.

Keywords: irreducible representations; highest-weight state; quadratic Casimir; angular momentum addition; Clebsch–Gordan series; Wigner–Eckart theorem; multiplicity

Chapter.  6811 words. 

Subjects: Mathematical and Statistical Physics

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