General Parametrization of Unitary Transformations

After the free-particle Foldy-Wouthuysen transformation the magnitude of the small component 

We have already seen that the free-particle Foldy-Wouthuysen transformation can be expressed in a couple of ways which seem very different at first sight. The only boundary for the explicit choice of a unitary transformation is that the off-diagonal blocks and therefore all odd operators vanish. If many different choices are possible — and we will see in the following that this is actually the case — the question arises how are they related and what this implies for the resulting Hamiltonians. 

In general, a unitary transformation U can be parametrized by an odd and antihermitean operator W. The antihermiticity of this operator W, 

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Within any decoupling scheme there are only a few restrictions on the choice of the transformations U. First, they have to be unitary and analytic (holomorphic) functions on a suitable domain of the one-electron Hilbert space V, since any parametrization has necessarily to be expanded in a Taylor series around W = 0 for the sake of comparability but also for later application in nested decoupling procedures (see chapter 12). Second, they have to permit a decomposition of in even terms of well-defined order in a given expansion parameter of the Hamiltonian (such as 1/c or V). It is thus possible to parametrize U without loss of generality by a power-series ansatz in terms of an antihermitean operator W, where unitarity of the resulting power series is the only constraint. In the next section this most general parametrization of U is discussed. 

By using the general power series expansion for U all the infinitely many parametrizations of a unitary transformation are treated on an equal footing. However, the question about the equivalence of these parametrizations for application in decoupling Dirac-like one-electron operators needs to be studied. It is furthermore not clear a priori whether the antihermitean matrix W can always be chosen in the appropriate way the mandatory properties of W, i.e., its oddness, antihermiticity and behavior as a certain power in the chosen expansion parameter, have to be checked for every single transformation U applied to the untransformed or any pre-transformed Hamiltonian. Since the even expansion coefficients follow from the odd coefficients, the radius of convergence Rc of the power series depends strongly on the choice of the odd coefficients. 

For simplicity it has been assumed that all unitary transformations Lf, feature the same parametrization with = 1, i.e., the cubic coefficient is the first coefficient to be chosen freely. The odd operators Wi = W (A = 0) are the familiar perturbation-independent operators parametrizing the standard unitary DKH transformations Lfj see chapter 12. As an important consequence, no terms containing the energy-damped property (X) occur within Xdkhoo-In order to emphasize this general structure and to avoid any misconceptions, we give the explicit expression of the first-order term of the transformed property [764],... 

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