The question is about an unusual looking version of the Hartree or mean field approximation. The context is several papers I've been reading recently about the out of equilibrium dynamics of phase transitions in the early universe [1-3]. The procedure is to shift a (scalar) field $\Psi \to \psi + \langle \Psi \rangle$, where $\psi$ are fluctuations and $\langle \Psi \rangle$ is a background field. This is perfectly fine. Then you simplify the dynamics of the fluctuations by replacing cubic and quartic terms by quadratic terms like so:

$$ \psi^4 \to 6 \langle \psi^2 \rangle \psi^2 - 8 \langle \psi \rangle^3 \psi + 6 \langle \psi \rangle^4 - 3 \langle \psi^2 \rangle^2, $$

$$ \psi^3 \to 3 \langle \psi^2 \rangle \psi - 2 \langle \psi \rangle^3. $$

See footnote 11 of [3] for these full formulas; the other refs simplify these by using $\langle \psi \rangle = 0$. Also note this is supposed to work for *bosonic* fields.

In the papers these substitutions just come out of the blue.

I get that the idea is to make the problem solvable by reducing the order of the interaction terms (and then eventually determining a self-consistent $\langle\Psi\rangle$), but I'm very confused about the numerical coefficients. Where do they come from? Is there a systematic way to derive them, or consistency condition they must satisfy? Why the alternating signs? What is the relation of this "Hartree" approximation to *the* Hartree approximation which has to do with minimizing the energy for separable wavefunctions or summing up a particular class of Feynman diagrams?

I've tried the seemingly likely stories:

* It looks at first like maybe something to do with Wick contractions, but the signs are a problem even if all the numbers came out the right size - which they don't.
* Repeatedly applying the mean field theory rule $AB \to A\langle B\rangle + \langle A\rangle B - \langle A\rangle\langle B\rangle$ doesn't seem to work out either.
* Neither does writing $\psi^4 = (\psi - \langle\psi\rangle)^4 + \cdots$ and crossing off the $(\psi - \langle\psi\rangle)^4$ term, or same thing for $\psi^4 = (\psi^2 - \langle\psi^2\rangle)^2 + \cdots$
* It looks similar to the [cumulant expansion](http://www.scholarpedia.org/article/Cumulants) but I can't get it out of that either.
* The [online encyclopedia of integer sequences](http://oeis.org/) was completely useless!

A literature pointer would be acceptable if the derivation is long, but please give me an article that *explains* the result! So far I've only found things which *quote* it and act as if it's the most obvious thing in the world. Sorry if it is... I feel like I'm missing something very basic. :)

1. Boyanovsky, D., Cormier, D., de Vega, H., & Holman, R. (1997). Out of equilibrium dynamics of an inflationary phase transition. Physical Review D, 55(6), 3373–3388. doi:10.1103/PhysRevD.55.3373
2. Boyanovsky, D., & Holman, R. (1994). Nonequilibrium evolution of scalar fields in FRW cosmologies. Physical Review D, 49(6), 2769–2785. doi:10.1103/PhysRevD.49.2769
3. Chang, S.-J. (1975). Quantum fluctuations in a φ^{4} field theory. I. Stability of the vacuum. Physical Review D, 12(4), 1071–1088. doi:10.1103/PhysRevD.12.1071