  We propose a quantum approach to "electron-hole exchange", better named
electron-hole pair exchange, that makes use of the second quantization
formalism to describe the problem in terms of Bloch-state electron operators.
This approach renders transparent the fact that such singular effect comes from
interband Coulomb processes. We first show that, due to the sign change when
turning from valence-electron destruction operator to hole creation operator,
the interband Coulomb interaction only acts on spin-singlet electron-hole
pairs, just like the interband electron-photon interaction, thereby making
these spin-singlet pairs optically bright. We then show that when written in
terms of reciprocal lattice vectors ${\bf G}_m$, the singularity of the
interband Coulomb scattering in the small wave-vector transfer limit entirely
comes from the ${\bf G}_m = 0$ term, which renders its singular behavior easy
to calculate. Comparison with the usual real-space formulation in which the
singularity appears through a sum of "long-range processes" over all ${\bf
R}\not= 0$ lattice vectors once more proves that periodic systems are easier to
handle in terms of reciprocal vectors ${\bf G}_m$ than in terms of lattice
vectors $\bf R$. Well-accepted consequences of the electron-hole exchange on
excitons and polaritons are reconsidered and refuted for different major
reasons.
