the end of the section that this formalism is suited much better for the generalization away from the nonrelativistic and instantaneous approximation. In section \[sec:approximationschemes\] we suggest approximation schemes and apply our formalism to a nonrelativistic Yukawa potential. Finally, section \[sec:generalformalismandapproxschemes\] gives a discussion of a somewhat generalized formalism that can be applied for example in relativistic quantum field theory or at nonzero density and we draw some conclusions in section \[sec:conclusions\].

Microscopic model for nonrelativistic particles {#sec:nonrelativisticmodel}
===============================================

Let us consider the problem of two nonrelativistic particles interacting via a potential $V(\vec x-\vec y)$. We will use a quantum field theoretic description using the functional integral. The microscopic action we employ is $$\begin{aligned}
\nonumber
S_\psi&=&\int_x \psi_1^*(x)\left(i\partial_t+\frac{1}{2M_1}\vec \nabla^2\right) \psi_1(x) \\
\nonumber
&&+ \int_x \psi_2^*(x)\left(i\partial_t+\frac{1}{2M_2}\vec \nabla^2\right) \psi_2(x) \\
\nonumber
&&- \int_{x_1,x_2} \psi_2^*(x_2)\psi_1^*(x_1) \; V(\vec x_1-\vec x_2) \\
&&\times\; \delta((x_1)_0-(x_2)_0)\; \psi_1(x_1)\psi_2(x_2).
\label{eq:microscopicaction2}\end{aligned}$$ We use here $\int_x=\int_{x_0}\int_{\vec x}=\int dx_0 \int d^3x$. For the Coulomb problem the interaction is of the form $$V(\vec x_1-\vec x_2) = \frac{-e^2}{4\pi |\vec x_1-\vec x_2|}.
\label{eq:Coulombpot}$$ It is useful to change variables of integration according to $x=\eta_1 x_1+\eta_2 x_2$, $y=x_1-x_2$ with $\eta_1=M_1/(M_1+M_2)$, $\eta_2=M_2/(M_1+M_2)$. The interaction term becomes then $$\begin{aligned}
\nonumber
&&-\int_{x,y} \psi_2^*(x-\eta_1 y)\psi_1^*(x+\eta_2 y) V(\vec y)\\
&& \delta((x_1)_0-(x_2)_0) \psi_1(x+\eta_2 