Myth of scattering in finite volume

In this notes, we illustrate why the infinite volume scattering amplitude is in fact dispensable when it comes to formulating few-body quantization condition in finite volume. Only subprocess interactions or interactions associated subprocess amplitudes are essential and fundamental ingredients of quantization conditions. After these ingredients are determined, infinite volume scattering amplitude can be computed separately. The underlying reasons are rooted in facts that (1) the final physical process is generated by all subprocess or interactions among particles; (2) the ultimate goal of quantization condition in finite volume is to find stationary solutions of few-body system. That is to say, in the end, it all comes down to the solving of eigenvalue problem, $\hat{H} | n \rangle = E_n | n \rangle $.