Rotational and mirror symmetry O(3)

When the density matrices are also invariant under mirror reflections, it follows from Eqs. (5), (6), and (26) that $\rho (\bm{r},\bm{r}')$ should have positive parity. The spin Pauli matrices form an O(3) pseudovector and thus the nonlocal spin density should be a pseudovector as well. On the right-hand side of Eq. (27), only the last term is a pseudovector. Therefore, in the case of the O(3) symmetry, the spin nonlocal density takes the form:

$$\bm{s}(\bm{r},\bm{r}')= \varrho_1''(r^2,\bm{r}\cdot\bm{r}',r^{\prime 2}) (\bm{r}\times\bm{r}'),$$ (28)

meaning that the local spin density $\bm{s}(\bm{r})=0$. It is impossible to build a pseudovector from one vector $\bm{r}$; therefore, local densities $\bm{T}(\bm{r})$ and $\bm{F}(\bm{r})$, being pseudovectors, must vanish, as well as pseudoscalar $J(\bm{r})=0$ and pseudotensor $\underline{\mathsf{J}}(\bm{r})=0$. On the other hand, vectors $\bm{j}(\bm{r})$ and $\bm{J}(\bm{r})$ do not vanish and take the form (24).