Lawrence Berkeley National Laboratory

We present the calculation of the generating functions and the $r$th-order correlations for densities of the form $\rho(\xv)=g(s(\xv))$ where $g(s)$ is a non-negative function of the quadratic "action" $s(\xv)=\sum_i,j H_ij x_ix_j$, where $\xv=(x_1,x_2,\cdots,x_n)$ is a real $n$-dimensional vector and $H$ is a real, symmetric $n\times n$ matrix whose eigenvalues are strictly positive. In particular, we find the connection between the $(r+2)$th-order and $r$th-order correlations, which constitutes a generalization of the Gaussian moment theorem \cite gardiner, which corresponds to the particular choice $g(s)=e^-s/2$. We present several examples for specific choices for $g(s)$, including the explicit expression for the generating function for each case and the subspace projection of $\rho(\xv)$ in a few cases. We also provide the straightforward generalizations to: 1) the case where $g=g(s(\xv)+\av\dotprod\xv)$, where $\av=(a_1,a_2,\cdots,a_n)$ is an arbitrary real $n$-! dimensional vector, and 2) the complex case, in which the action is of the form $s(\zv)=\sum_i,j H_ij z^*_i z_j$ where $\zv=(z_1,z_2,\cdots,z_n)$ is an $n$-dimensional complex vector and $H$ is a Hermitian $n\times n$ matrix whose eigenvalues are strictly positive.