Operator and Commutator Identities

The Levi-Civita symbol is denoted by iJk and is antisymmetric in all three indices. It is defined by 

Its primary use in our applications is for expressing various commutation relations in compact form and performing calculations involving them. The Levi-Civita symbol also arises naturally in vector analysis. Thus, if et, i = 1, 2, 3, are three mutually perpendicular unit vectors defining a right-handed coordinate system, then 

As mentioned in Section II, these unit vectors generate a Lie algebra if we define 

The following general commutator identities are quite useful for simplifying the commutation relations, which occur in the determination of realizations of a Lie algebra. The well-known relations 

In our discussion of scaling transformations in Section V we have to evaluate operator transformations of the form e BAeB [cf. Eq. (94)]. To evaluate such expressions define 

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