Transformed Breit Contribution

Analogously to the transformed Coulomb interaction, the anticommutator with the Breit operator in Eq. (13.27), 

It must be emphasized that the momentum operators in Eq. (13.52) (as in most equations to follow) only operate on the wave function and not on any other position-dependent quantity in the operators. Thus, they must be read as, e.g., (Pi P2)/ 12 = I/H2 (Pi P2)- The second term in the Breit operator is more 

In the following, we omit the special case for which the two coordinates coincide, ri = T2. Accordingly, we find for the product of scalar products with exchanged indices, 

the second term of the Breit operator is more difficult to transform. We first consider the action of the momentum operator for which we need to evaluate 

---

Here Hd, is the Dirac Hamiltonian for a single particle, given by Eq. [30]. Recall from above that the Coulomb interaction shown is not strictly Lorentz invariant therefore, Eq. [59] is only approximate. The right-hand side of the equation gives the relativistic interactions between two electrons, and is called the Breit interaction. Here a, and a, denote Dirac matrices (Eq. [31]) for electrons i and /. Equation [59] can be cast into equations similar to Eq. [36] for the Foldy-Wouthuysen transformation. After a sequence of unitary transformations on the Hamiltonian (similar to Eqs. [37]-[58]) is applied to reduce the off-diagonal contributions, one obtains the Hamiltonian in terms of commutators, similar to Eq. [58]. When each term of the commutators are expanded explicitly, one arrives at the Breit-Pauli Hamiltonian, for a many-electron system " ... 

---