Pauli gauge

The first choice is called the Cartan gauge the second one is the Pauli gauge (Altmann (1986), p. 108 see also Section 11.8). 

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. 

Electron motion is more generally formulated in a form of the Schrodinger equation, including the spin in the presence of external fields known as the Pauli equation. This equation is gauge invariant in the sense that a transformation as in (5) also changes the quantum wavefunction as... 

In Eq. (3.47)

field operator of two-component structure, a are the Pauli matrices and the electron-electron interaction reduces to the Coulomb interaction, denoted by H e- As usual, the gauge term proportional to... 

More complex schemes with several Higgs doublets have been considered in the literature. The 2x2 matrices representing the generators of 17(1) and SU(2) are just the unit matrix I and the Pauli matrices divided by two Tj, so

transform under the local gauge transformations as follows ... 

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. 

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. 

The most significant difference of Dirac s results from those of the non-relativistic Pauli equation is that the orbital angular momentum and spin of an electron in a central field are no longer separate constants of the motion. Only the components of J = L - - S and J, which commute with the Hamiltonian, emerge as conserved quantities [1]. Dirac s equation, extended to general relativity by the method of projective relativity [2], automatically ensures invariance with respect to gauge, coordinate and spinor transformations, but has never been solved in this form. 

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