Definition of an essentially RPA field

A pure RPA fields can be defined as one in which all the phases and amplitudes of the Fourier modes make a set of $2N$ statistically independent variables and in which all phase factors $\psi$ are uniformly distributed on their respective unit circles. In such pure form RPA never survives except for in the un-interesting state of complete thermodynamic equilibrium. However, WT closure only requires an approximate RPA which holds up to certain order in small $\epsilon$ and $1/N$ and only in a coarse-grained sense, i.e. for the reduced $M$-particle objects with $M \ll N$. Below we give a relaxed definition of an (essentially) RPA property which, on one hand, is sufficient for the WT closure and, on the other hand, is preserved over the nonlinear time.

Definition: We will say that the field $a$ is of an essentially RPA type if:

1. The phase factors are statistically independent and uniformly distributed variables up to $O({\epsilon}^2)$ corrections, i.e.
   

   $${\cal P}^{(N)} \{s, \xi \} = {1 \over (2 \pi)^{N} } {\cal P}^{(N,a)} \{s \} \; [1 +O({\epsilon}^2)],$$ (6)

   
   
   where
   

   $${\cal P}^{(N,a)} \{s \} = \left( \prod_{ l {\cal 2 B}_N... ... } \vert d \xi_l\vert \; \right) {\cal P}^{(N)} \{s, \xi \},$$ (7)

   
   
   is the $N$-particle amplitude PDF. In terms of the generating functional
   

   $$Z^{(N)} \{\lambda, \mu \} = Z^{(N,a)} \{\lambda \} \, \prod_{l \in {\cal B}_N } \delta(\mu_l) \; [1 +O({\epsilon}^2)],$$ (8)

   
   
   where
   

   $$Z^{(N,a)} \{\lambda \} =\langle \prod_{l \in {\cal B}_N... ...da_l A_l^2} \rangle = Z^{(N)} \{\lambda, \mu\}\vert _{\mu=0}$$ (9)

   
   
   is an $N$-particle generating function for the amplitude statistics.

2. The amplitude variables are independent in a coarse-grained sense, i.e. for each $M \ll N$ modes the $M$-particle amplitude PDF is equal to the product of the one-particle PDF's up to $O(M/N)$ and $o({\epsilon}^2)$ corrections,
   

   $${\cal P}^{(M,a)}_{j_1, j_2, \dots , j_M} = P^{(a)}_{j_1}... ..._2} \dots P^{(a)}_{j_M} \; [1 + O(M/N) + O({\epsilon}^2)].$$ (10)

   
   

As a first step in validating the RPA property we will have to prove that the generating functional remains of the form (8) over the nonlinear time provided it has this form at $t=0$.