y)\psi_2(x-\eta_1 y).\end{aligned}$$ It is also useful to introduce the Fourier transformed fields $$\begin{aligned}
\nonumber
\psi_i(x) &=& \int_p \psi_i(p)\, e^{-ipx}, \quad i=1,2,\quad p x=p_0 x_0-\vec p\vec x,\end{aligned}$$ with $$\int_p = \int_{p_0}\int_{\vec p}, \quad \int_{p_0}=\int_{-\infty}^\infty \frac{d p_0}{2\pi}, \quad \int_{\vec p}=\int \frac{d^3p}{(2\pi)^3}.$$ We also introduce an abbreviation for a composition of fields $(\psi_1\psi_2)$. Its position and momentum space representation reads $$\begin{aligned}
\nonumber
(\psi_1\psi_2)(x,\vec y) &=& \psi_1(x_0,\vec x+\eta_2 \vec y) \psi_2(x_0, \vec x-\eta_1 \vec y)\\
&=& \int_{p,q} e^{-ipx} e^{i\vec q\vec y} (\psi_1\psi_2)(p,\vec q),\end{aligned}$$ with $(\psi_1\psi_2)(p,\vec q)=\int_{q_0}\psi_1(\eta_1 p+q)\psi_2(\eta_2p-q)$. The conjugate field is $$(\psi_1\psi_2)^*(p,\vec q) = \int_{q_0} \psi_2^*(\eta_2 p-q) \psi^*(\eta_1 p+q),
\label{eq:}$$ where the reversed order of $\psi_1$ and $\psi_2$ is convenient when these are fermionic fields. Obviously, the first variable ($x=(x_0,\vec x)$ or $p=(p_0,\vec p)$) describes the center of mass motion, while the second variable ($\vec y$ or $\vec q$) describes the relative motion of the two particles. The fact that no relative time or energy appears is due to the instantaneous approximation for the interaction. For relativistic systems this point may be different and the generalization of the formalism to that case will be discussed in section \[sec:generalformalismandapproxschemes\].

Using the momentum representation, the interaction term in the microscopic action becomes $$\begin{aligned}
\nonumber
&&-\int_{x,\vec y} 