Chapter

Clebsch–Gordan series for spinors

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/9780199662920.003.0007
Clebsch–Gordan series for spinors

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The derivation of the Clebsch–Gordan series starts from the observation that the object (C –1 γA)α β can be viewed as the transformation matrix between the space of antisymmetric tensors of rank A and the product of two 2m-dimensional spinors in so(2m+1). In so(2m) a similar statement holds after appropriate modifications due to the fact that the spinor is reducible into semispinors and the antisymmetric tensor of rank m is reducible into a self-dual and anti-self-dual part. These duality properties are particularly interesting for so(2) and so(4). In so(2) they provide for the reduction of the 2-dimensional defining representation into two 1-dimensional representations in accordance with the fact that so(2) is Abelian. In so(4) they provide for the isomorphism between so(4) and the direct sum of two so(3)s.

Keywords: Clebsch–Gordan series; antisymmetric tensors; duality

Chapter.  3946 words. 

Subjects: Mathematical and Statistical Physics

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