Origin of the Pairing Interaction in the Theory of Superconductivity

A strongly coupled many-body system (a system of fermions confined in a shell surrounding the Fermi surface for which the variation in the single-particle kinetic energy is neglected) is defined. Some general properties of this system are obtained, in particular a criterion for permissible approximations. It is shown that a set of diagrams in which an electron of momentum $-\overrightarrow{\mathrm{k}}$ and of spin ↓ interacts with many electrons of momentum $\overrightarrow{\mathrm{k}}+\overrightarrow{\mathrm{q}}$ and spin ↑, where $\overrightarrow{\mathrm{q}}$ does not grow too large, can be consistently extracted from the totality of graphs generated by this strongly coupled Hamiltonian. When analyzed, these lead to a ground-state and single-particle spectrum displaying the important qualitative features of a BCS superconductor.