respectively stand for $\delta x_i$, $\delta p_i$, $\overline{\delta x_i}$ and $\overline{\delta p_i}$. These equations define a dynamical system in the phase space $\Gamma = \mathbb C^6 \simeq \mathbb R^{12}$. It is clear that $\delta x_i = 0$ and $\delta p_i = 0$ ($\forall i$) is a fixed point of the dynamical system. Since the quantum fluctuations are small, the starting point of the dynamical system is in the neighbourhood of $0$. The Jacobian matrix of the dynamical system is $$\begin{aligned}
  {\partial F_{\hat i}}^j  =  \frac{\partial F_{\hat i}}{\partial \delta x_j} & = & -\|X-x\|^2\delta^i_j + (X_i-x_i)(X_j-x_j) \nonumber \\
  & & \quad - 2 \overline{\delta x^j} \delta x_i - 2 \overline{\delta x^k}\delta x_k \delta^i_j + 4 \overline{\delta x_i} \delta x^j \\
  {\partial F_{\hat i}}^{\bar j} = \frac{\partial F_{\hat i}}{\partial \overline{\delta x_j}} & = & -2\delta x_j\delta x_i + 2 \delta x^k \delta x_k \delta^i_j \\
  {\partial F_{i}}^{\hat j} = \frac{\partial F_{i}}{\partial \delta p_j} & = & \delta^i_j\end{aligned}$$ and ${\partial F_{i}}^{j} = {\partial F_{\hat i}}^{\hat j} = {\partial F_{i}}^{\tilde j} = {\partial F_{i}}^{\bar j} = {\partial F_{\hat i}}^{\tilde j} = 0$. It follows that the Jacobian matrix at $0$ (in the representation $(i,\hat i,\bar i, \tilde i)$) is $$\partial F_{|(\delta x,\delta p)=0} = 