Appropriate theory for low-energy processes:

• degrees of freedom: neutrons and protons ("What do we measure?")
• symmetry: Galilei $\approx$ low-energy Lorentz (translational and rotationally invariant potentials)
• particle statistics and identity: quantum mechanics

with   $\hat{H}=-\frac{\hbar^2}{2}\sum\limits_i\frac{\vec{\nabla}^2}{m_i}+\sum\limits_{i,j}v_{ij}+\sum\limits_{i,j,k}v_{ijk}$

$$\begin{eqnarray} v_{ij}&=&\sum\limits_na_n\frac{e^{-n\mu r_{ij}}}{\mu r_{ij}}+\sum\limits_nc_n\frac{e^{-n\mu r_{ij}}}{\mu r_{ij}}\vec{L}\cdot\vec{S}\\ &&+\left\lbrace\frac{b_1}{\mu r_{ij}}\left[\left(\frac{1}{3}+\frac{1}{\mu r_{ij}}+\frac{1}{(\mu r_{ij})^2}\right)e^{-\mu r_{ij}}-\left(\frac{b_0}{\mu r_{ij}}+\frac{1}{(\mu r_{ij})^2}\right)e^{-b_0\mu r_{ij}}\right]+\sum\limits_nb_n\frac{e^{-n\mu r_{ij}}}{\mu r_{ij}}\right\rbrace \left[3\left(\vec{\sigma}_1\cdot\hat{\vec{r}}\right)\left(\vec{\sigma}_2\cdot\hat{\vec{r}}\right)-\vec{\sigma}_1\cdot\vec{\sigma}_2\right] \end{eqnarray}$$

But how can we derive this from a fundamental theory?

relativistic quantum field theory $\to$ non-relativistic dynamics (Schrödinger, Lippmann-Schwinger) $\oplus$ few-body potentials

Understand the dynamical equation as a wave equation resulting from a Lagrangean/Hamiltonian density:

$\mathcal{L}_0=\frac{i}{2}\Psi^*\partial_t\Psi-\frac{i}{2}\Psi\partial_t\Psi^*-\frac{\left(\nabla\Psi^*\right)\left(\nabla\Psi\right)}{2m}$

where the field $\Psi(\vec{x},t)$ replaces the position $\vec{x}(t)$ as the fundamental degree of freedom and

$\left[\Psi(\vec{x},t),\Psi^*(\vec{x}',t)\right]=\delta(\vec{x}-\vec{x}')$

Renormalized form of an $A$-body contact theory.

Challenges: