Transcription of Chapter 9 Unitary Groups and SU(N)
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Chapter 9 Unitary Groups and SU(N)
Chapter 9 Unitary Groups and SU(N) The irreducible representations of SO(3) are appropriate for describingthe degeneracies of states of quantum mechanical systems which haverotational symmetry in three dimensions. But there are many systemsfor which operations on classical coordinates must be supplementedby operations on \internal" degrees of freedom which have no classicalanalogue. For example, the Stern{Gerlach experiment showed thatelectrons are endowed with an internal degree of freedom called \spin"which has the properties of an angular momentum. The two spin statesare therefore inconsistent with the dimensionalities of the irreduciblerepresentations of SO(3), so another group|SU(2)|must be used todescribe these states.}
148 Unitary Groups and SU(N) ties and the basis functions of irreducible representations derived from direct products. 9.1 SU(2) As with orthogonal matrices, the unitary groups can be deflned in terms of quantities which are left invariant. Consider a general complex trans-formation in two dimensions, x0= Axwhich, in matrix form, reads: ˆ x0 ...
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