I study the jellium model: conduction electrons free except for their mutual repulsion. We have the following vertex: $$\hat{V}_{ee}~=~\sum_{k,k',q}V(q)a^{\dagger}_{k-q,\sigma}a^{\dagger}_{k'+q,\sigma'}a_{k',\sigma'}a_{k,\sigma}. $$ One sees that the electron spins z-projections are unchanged after the interaction $(\sigma,\sigma')\rightarrow(\sigma,\sigma')$ (consequence of the fact that the potential is scalar I guess) so or the mediating gauge boson (photon) isn't carrying any orbital momentum or helicity along the $z$-axis which is unphysical but maybe acceptable for a virtual particle, or there is on orbital momentum conservation at the vertex points. This point is a bit messy in my head, I'd appreciate if someone could share his interpretation or motivate the conservation of each electron's spin state.
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You're right the Coulomb interaction operator is a two-particle scalar operator not involving spin it must be diagonal in the spin indices of the particles. The matrix element can be intepreted as describing a transition from the initial states $(\mathbf{k},\sigma)$, $(\mathbf{k}',\sigma')$ to the final states $(\mathbf{k}-\mathbf{q},\sigma)$, $(\mathbf{k}'+\mathbf{q},\sigma')$ so that the particles exchange a momentun $\mathbf{q}$ [but total momentum is conserved since the interaction is invariant under global translations] and spin of the particles is conserved [no spin flip]. |
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