Noninvariance Groups in the Second-Quantization Picture and Their Applications

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Abstract
We investigate the existence of noninvariance groups in the second‐quantization picture for fermions distributed in a finite number of states. The case of identical fermions in a single shell of angular momentumj is treated in detail. We show that the largest noninvariance group is a unitary group U(22j+1). The explicit form of its generators is given both in the m scheme and in the seniority—angular‐momentum basis. The full set of 0‐, 1‐, 2‐, ⋯, (2j + 1)‐particle states in the j shell is shown to generate a basis for the single irreducible representation [1] of U(22j+1). The notion of complementary subgroups within a given irreducible representation of a larger group is defined, and the complementary groups of all the groups commonly used in classifying the states in the j shell are derived within the irreducible representation [1] of U(22j+1). These concepts are applied to the treatment of many‐body forces, the state‐labeling problem, and the quasiparticle picture. Finally, the generalization to more complex configurations is briefly discussed.