Gauge term Breit interaction

The two parts of this formula are derived from the same QED Feynman diagram for interaction of two electrons in the Coulomb gauge. The first term is the Coulomb potential and the second part, the Breit interaction, represents the mutual energy of the electron currents on the assumption that the virtual photon responsible for the interaction has a wavelength long compared with system dimensions. The DCB hamiltonian reduces to the complete standard Breit-Pauli Hamiltonian [9, 21.1], including all the relativistic and spin-dependent correction terms, when the electrons move nonrelativistically. 

The last term in is the gauge term. The addition to the Coulomb interaction in the Feynman gauge is called the Gaunt interaction, and in the Coulomb gauge it is the Breit interaction. 

The modified operators for the Gaunt and Breit interactions can be derived in an analogous manner. The derivation is somewhat more involved than for the Coulomb interaction due to the presence of the a matrices. Here we derive the Gaunt terms, but the gauge term for the Breit interaction is considerably more complicated (and the derivation may be found in appendix F). 

The gauge term, which is the difference between the Gaunt interaction and the Breit interaction, produces a spin-free operator that can be interpreted as an orbit-orbit interaction. Thus, both the Gaunt interaction and the gauge term of the Breit interaction give rise to spin-free contributions to the modified Dirac operator. We will use the developments of this section in chapter 17 to derive the Breit-Pauli Hamiltonian. 

The contributions to the two-electron operator from the gauge term of the Breit interaction may be developed in the same manner, using the representation for the modified Dirac operator. The derivation is more complicated, and details may be found in appendix I. The only term that contributes to 0(c ) is a spin-free term ... 

The gauge term, which comprises the difference between the Gaunt and the Breit interactions (see (5.48) and (5.49)), is more complicated than the Gaunt term due to the scalar quadruple product involving the alpha mafiices ... 

Gauge Term Contributions from the Breit Interaction to the Breit-Pauli Hamiltonian... 

The reduction of the remaining contributions from the gauge term is extremely tedious, and in fact gives a zero contribution to the operator to 0(c ). Thus, there is no spin-dependent contribution from the gauge term for the relativistic correction to the electron-electron interaction to the Breit-Pauli Hamiltonian. 

Obviously it is the Coulomb gauge which leads to the familiar Breit interaction, obtained by adding the second term in (20) to the expression obtained in 2 )l... 

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