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I was reading an article on wiki about the Breit equation:

http://en.wikipedia.org/wiki/Breit_equation

And I'm having a hard time understanding a few thing about this equation. The first thing is that, from what I can gather, the Breit Hamiltonian is basically a Dirac Hamiltonian, plus a the coulomb interaction operator (in the coulomb gauge), plus a correction that accounts for the fact that electron-electron interaction is felt in a retarded fashion. Why is it that this equation is/was useful? Is it not possible to write the exact relativistic Hamiltonian for a multi-electron system?

I'm reading a book where it is stated that to generalize the Dirac equation to a two electron system you basically "do a tensor product" of the two Hamiltonians with the 4x4 Id matrix then add them up. One issue that arises is that you have different time for each Hamiltonian. A way to simplify this is to assume that both times are the same. So far so good. The second issue that arises is the choosing of (A,[tex]\phi[/tex]) for each one of the electrons. As I understand it the Breit operator is a way of doing this (although exactly how this happens still eludes my understanding). Isn't there a way to write out these potentials exactly?

My second doubt concerns the wikipedia article. The section entitled "Breit Hamiltonians" seems misleading to me. The Hamiltonians described in that section are the so-called two component Hamiltonians, which come from the reduction of the dirac-coulomb-breit hamiltonian into "quasi-relativistic" forms. But this can be done (and I have done it by hand) with the simple Dirac Hamiltonian, and also leads to Darwin, Mass-velocity, SO terms. Am I wrong about this?

I have third doubt. This is related to the statement that the Breit hamiltonian accounts for the fact that interactions between electrons are retarded. This statement is made time and time again in many sources, however I've come across one book that states this the last term in hamiltonian (called B in the wiki article) is basically the quantization of the classical expression for the interaction energy between two charged particles in which retardation is explicitly neglected!

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# Breit equation/relativistic QM and simplifications

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