Baker Campbell Hausdorf Theorem

This is usually referred to as the Zassenhaus expansion or the Baker Campbell Hausdorf theorem [11]. As an aside a symmetrized version of this expansion terminated at the Cl term results in the Feit and Fleck [12] approximate propagator. We have shown [13] that an infinite order subset of these commutators, may be resummed exactly as an interaction propagator ... 

It applies for both formulations above that the expansion in principle contains an infinite number of terms. The convergence to a few lowest order terms relies on the ability to orderly separate influences of the dominant rf irradiation terms (through a suitable interaction frame) from the less dominant internal terms of the Hamiltonian. In principle, this may be overcome using the spectral theorem (or the Caley-Hamilton theorem [57]) providing a closed (i.e., exact) solution to the Baker-Campbell-Hausdorf problem with all dependencies included in n terms where n designates the dimension of the Hilbert-space matrix representation (e.g., 2 for a single spin-1/2, 4 for a two-spin-1/2 system) [58, 59]. 

Now, using Baker-Hausdorf-Campbell theorem, we have... 

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See also in sourсe #XX -- [ ]

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