Norm-preserving transformations, effective operators

B. Effective Operators Based on Norm-Preserving Transformations... 

Section II and Table I show that state-independent effective operators can be obtained with norm-preserving mappings K, or with any of the three kinds of non-norm-preserving mappings K, L), K, L), and K, L). This section first proves that the commutation relations between two arbitrary operators cannot generally be conserved upon transformation to any of these state-independent effective operators. A determination is then made of operators whose commutation relations are preserved by at least some state-independent effective operator definitions, and a few applications are then presented. Particular interest is focused on operators which commute with H, including constants of the motion. 

Hence, the commutation relation between A and B is conserved iff the right hand sides of Eqs. (4.8) and (4.9) are equal to each other, thereby leading to Theorem VII as follows the commutation relation between two operators A and B is preserved upon transformation to state-independent effective operators obtained with norm-preserving mappings iff A and B satisfy... 

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