Electron correlation - difference between correlation and dependence

Question

When we talk about electron correlation in condensed matter physics or chemical physics, we usually refer to the fact that the pair-density $$ P(r,r') = N(N-1) \int |\psi(r,r',r_3,...,r_N)|^2 \; \textrm d^3 r_3 ... \textrm d^3 r_N $$ cannot be written as the product of the electron density. $$ P(r,r') \neq \rho(r_1)\rho(r_2) \;, $$ where $$ \rho(r) = N \int |\psi(r,r_2,r_3,...,r_N)|^2 \; \textrm d^3 r_2 ... \textrm d^3 r_N \;. $$ However, a mathematician would say, that the statement "the pair-density cannot be written as a product of the electron density" implies electron dependence and not necessarily electron correlation.

Are physicist simply not making a distinction between dependence and correlation?

For example in the Wikipedia article correlation and dependence one can read:

In informal parlance, correlation is synonymous with dependence.

If, however, I want to clearly distinguish between correlation and dependence, then the common approach as "the pair-density cannot be written as a product of the electron density" only defines electron dependence and does simply not define electron correlation.

So what is electron correlation?

Answer 1

Mathematicians have a formal technical term for the word "correlation"? Great! so does quantum physics.

The usage in physics is related to that in mathematics (or, at least, it shares the same basis in the informal term), but mathematics does not have a monopoly on how technical terms should be understood. The formal understanding of the term 'correlation' in statistics is nice to know, but it isn't in any way determinative about how the word is used in physics.

Electron correlation is a hard-enough topic to begin with, and it is generally a reasonably nebulous concept that's rather hard to define precisely. (So: don't weigh yourself down further by insisting on useless usages from maths.) I've most often seen it used in opposition, i.e., as a bunch of phenomena that Hartree-Fock theory cannot describe effectively. If you want one key discriminant as to whether a given set of electrons is said to be 'correlated', though, your best bet is to ask whether the system's multi-electron wavefunction can be described as a single configuration (i.e. a single Slater determinant) or whether it can only ever be described as a superposition of multiple configurations.

As for "electron dependence", though - that's not a term that anyone uses or that anyone would understand.