A generalization of the Baker–Hausdorff lemma

Abstract

Two operator identities involving a q-commutator, [A,B]≡AB+qBA, where A and B are two arbitrary (generally noncommuting) linear operators acting on the same linear space and q is a variable that commutes with these two operators, are formulated, proved and discussed. The first identity is a direct generalization of the Baker–Hausdorff lemma, whereas the second involves the time derivative of the exponential function of a time-dependant operator. It is indicated how these two identities can be used to good advantage to treat the Foldy–Wouthuysen transformation of the Dirac Hamiltonian for a particle in an external electromagnetic field.