Dirac Operators and Wave Functions - Transformations

Leonardo Angelini

В. П. Незнамов

The paper describes conditions for transformation from the Dirac representation to the Foldy-Wouthuysen representation. The necessary condition is the block-diagonal transformation of Hamiltonian relative to the upper and lower components of the wave function. The sufficient condition is the wave function transformation law described by relation (6). It has been demonstrated that the unitary transformations offered by the authors of the papers [14], [15], [16], [17] do not satisfy the sufficiency condition (6) and, hence, they are not the Foldy-Wouthuysen transformations. In applications, the matrix elements of any operator in the FW representation can be calculated, according to (6), using the normalized two-component wave functions in the Dirac representation known for the given problem.

Heinz Otto Cordes

V. P. Neznamov, A. J. Silenko

The block diagonalization of the Hamiltonian is not sufficient for the transformation to the Foldy–Wouthuysen (FW) representation. The conditions enabling the transition from the Dirac representation to the FW one are formulated and proven. The connection between wave functions in the two representations is derived. The results obtained allow calculating expectation values of operators corresponding to main classical quantities.

Hong-Yi Fan, Ze Wu

The Foldy–Wouthuysen transformation was originally developed to elucidate the nonrelativistic limit of the Dirac equation. In this paper, we introduce a novel unitary transformation for the Hamiltonian of the Dirac equation. The unitary transformation operator is given by [Formula: see text] which results in the diagonal Hamiltonians [Formula: see text] and [Formula: see text]. We also apply this new unitary transformation to wave function calculations, and elaborate on its significance.