A projector formulation of the Van Vleck transformation. Part I. Degenerate case

What is it about?

(Uploaded: 2018-04-18). Co-author Thorvald Pedersen has sadly died in 2013. The following is therefore written by me, Flemming Jørgensen, alone. To understand Van Vleck's original approach to an effective Hamiltonian one can think of the matrix for the perturbed Hamiltonian H as the sum of an even part consisting of two diagonal blocks and the remaining odd part with one block on each side of the diagonal. He then showed how to construct a hermitian matrix S so that the unitary transformation exp(iS) gives a matrix whose odd part is diminished by one order of magnitude. The words even and odd used here are borrowed from Foldy and Wouthoysen who much later independently "re-invented" Van Vleck's method in the special context of Dirac's relativistic Hamiltonian for an electron. In principle the odd part of the transformed H can be further diminished by subsequent transformations - but with the original matrix notation this is overwhelmingly complicated in the general case. We have simplified the algebra very substantially by I: A formulation in terms of projection operators as in Bloch's much later approach. II: Finding the unitary transform by use of the (Baker–Campbell–) Hausdorff multicommutator expansion. III: Extensive use of the properties even and odd.

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Why is it important?

To our knowledge, the effective Hamiltonian as resulting from successive Van Vleck transformation, had only been formulated through the order three prior to the present paper. We now formulate it through the order eight in terms of a condensed multicommutator notation. In certain connections such commutators are most useful - actually necessary - for a meaningful interpretation of the results. This is for instance the case with Foldy and Wouthoysen effective Hamiltonian which they worked out through the order four. We find it as a simple special case - and work it out through the order six.

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