The quantum numbers of a single-particle state in coordinate space are defined by the variables $$ x=(\boldsymbol{r},\sigma), $$ where $$ \boldsymbol{r}\in {\mathbb{R}}^{d}, $$ with \( d=1,2,3 \) represents the spatial coordinates and \( \sigma \) is the eigenspin of the particle. For fermions with eigenspin \( 1/2 \) this means that $$ x\in {\mathbb{R}}^{d}\oplus (\frac{1}{2}), $$ and the integral $$ \int dx = \sum_{\sigma}\int d^dr = \sum_{\sigma}\int d\boldsymbol{r}. $$ Since we are dealing with protons and neutrons we need to add isospin as a new degree of freedom.