Angular Momentum and Isospin
Angular momentum and isospin are different symmetries and the transformations corresponding to them act in completely different spaces, but from the group theoretic standpoint they are very similar and can be treated using identical methods. This is because the group R(3) of rotations in our three-dimensional space is isomorphic to the group SU(2) describing the isospin symmetry [for details see Hamermesh (Ham 62)]. This is true only if isospin is described by the group U(2) instead of the direct-product subgroup U(N/2) × U(2) as in Section 7.2.1. It should be clearly understood that while we have the latter description for isospin there is no similar group structure for angular momentum. As we shall see a little later, this is the reason why we can easily carry out the isospin average—i.e., average over [U(N/2) × U(2)] states—but not the angular momentum average. Furthermore, it should be clear that we do not really need very much of formal group theory for R(3) or SU(2) since we can work out almost everything using the well-known angular momentum algebra.
KeywordsAngular Momentum Strength Distribution Spectroscopic Factor Reduce Matrix Element Analog Spin
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