A.1 Many-Body Operators

An operator $\hat{U}$ is a one-body operator if the action of $\hat{U}$ on a state $\vert\alpha_{1}...\alpha_{N})$ of $N$ particles^A.1 is the sum of the action of $\hat{U}$ on each particle:

$$\hat{U}\vert\alpha_{1}...\alpha_{N})=\sum_{i=1}^{N}\hat{U}_{i}\vert\alpha_{1}...\alpha_{N}),$$ (A.1)

where the operator $\hat{U}_{i}$ operates only on the $i$-th particle.

An operator $\hat{U}$ is a two-body operator if the action of $\hat{U}$ on a state $\vert\alpha_{1}...\alpha_{N})$ of $N$ particles is the sum of the action of $\hat{U}$ on all distinct pairs of particles:

$$\hat{U}\vert\alpha_{1}...\alpha_{N})=\frac{1}{2}\sum_{1\leqslant i\neq j\leqslant N}\hat{U}_{ij}\vert\alpha_{1}...\alpha_{N}),$$ (A.2)

where $\hat{U}_{ij}$ operates only on particles $i$ and $j$.

In general an $n$-body operator $\hat{U}$ is defined as an operator which acts on a state $\vert\alpha_{1},...,\alpha_{N})$ in the following way:

$$\hat{U}\vert\alpha_{1}...\alpha_{N})= \frac{1}{n!}\sum_{1\leqslan... ...neq i_{n}\leqslant N}\hat{U}_{i_{1}i_{2}...i_{n}}\vert\alpha_{1}...\alpha_{N}),$$ (A.3)

where $\hat{U}_{i_{1}i_{2}...i_{n}}$ acts on the subset of $n$-particles $i_{1},i_{2},...,i_{n}$. S. Smirnov: