Lie Groups and Their Algebras
So far in our discussion the group has been discrete and finite, but in nuclear physics one mainly considers continuous groups whose transformations depend upon a finite number of continuously varying parameters. If the parameters vary over a finite range in the parameter space and if at the same time a certain measure for the density of operators in the group is finite, the group is said to be compact. Also if the transformations are analytic (nonsingular) functions of the parameters (say r in number) then one gets an r-parameter Lie group. These are of primary interest to us. The transformations in quantum mechanics have to be unitary, and if we consider these to be acting on a finite number N of single-particle states, then they generate the unitary group U(N). The group U(N) is compact and hence so are all its subgroups.
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