The heuristic understanding of polarization operator?

[tghg]

I find that some book e.g. Many-body theory of solids by John C. Inkson P145
says that :" By analogy with the classical polarization, we define a polarization
propagator through the relationship ... ε=1-vP" , where ε is the dielectronic response function, and v is the bare Coulomb potential.
I do not remember such relationship in electrodynamics. Can anybody remind me of it?

In fact, on the next page in the book, I find a new relationship i.e. P=δρ/δV， where V is the total potential instead of external potential and ρ is the electron density.
But is there any similarity between this polarization function P and the classical polarization?

 

[BigJDubs]

The relationship that the book mentions is actually a variation of the classical polarization formula. In classical electrodynamics, the polarization P is defined as the ratio of the induced electric dipole moment (δρ) to the applied electric field (δV). This is the same relationship that is mentioned in the book, but using the total potential instead of just the external potential. Therefore, while there is a similarity between the classical polarization formula and the one mentioned in the book, the two formulas are not exactly the same.