Including isospin \( \tau \) we have $$ x=(\boldsymbol{r},\sigma,\tau), $$ where $$ \boldsymbol{r}\in {\mathbb{R}}^{3}, $$ For nucleons, which are fermions with eigenspin \( 1/2 \) and isospin \( 1/2 \) this means that $$ x\in {\mathbb{R}}^{d}\oplus (\frac{1}{2})\oplus (\frac{1}{2}), $$ and the integral $$ \int dx = \sum_{\sigma\tau}\int d\boldsymbol{r}, $$ and $$ \int d^Ax= \int dx_1\int dx_2\dots\int dx_A. $$ We will use the standard nuclear physics definition of isospin, resulting in \( \tau_z=-1/2 \) for protons and \( \tau_z=1/2 \) for neutrons.