Rotational symmetry SO(3)

If we assume that the density matrices $\hat{\rho}$ and $\hat{\breve{\rho}}$, Eqs. (1) and (2), are invariant under the three-dimensional rotations forming the SO(3) group, it immediately follows from Eqs. (5) and (6) that the densities of type $\rho$ are the SO(3) scalars while the densities of type $\bm{s}$ are the SO(3) vectors (note that the spin Pauli matrices are the SO(3) vectors). Therefore, the nonlocal density of type $\rho$ takes a form of Eq. (20):

$$\rho (\bm{r},\bm{r}')=\varrho_0(r^2,\bm{r}\cdot\bm{r}',r^{\prime 2}),$$ (26)

while the nonlocal density of type $\bm{s}$ has a form of Eq. (21):

$$\bm{s}(\bm{r},\bm{r}')= \varrho_{11}(r^2,\bm{r}\cdot\bm{r}',r^{\p... ...)\bm{r}' +\varrho_{13}(r^2,\bm{r}\cdot\bm{r}',r^{\prime 2})\bm{r}\times\bm{r}',$$ (27)

where $\varrho_0$, $\varrho_{11}$, $\varrho_{12}$, and $\varrho_{13}$ are arbitrary scalar functions. All local differential densities (11)-(18) can be calculated by differentiating Eqs. (26) and (27), like it was done in I. But to establish general forms of all local densities, it is sufficient to realize that all of them are local isotropic fields with definite transformation rules under SO(3) rotations. These rules can be deduced from the definitions (9)-(18). We see that the local densities $\rho (\bm{r})$, $\tau (\bm{r})$, and $J(\bm{r})$ are scalar fields and all take the form of Eq. (23). Similarly, the densities $\bm{s}(\bm{r})$, $\bm{j}(\bm{r})$, $\bm{J}(\bm{r})$, $\bm{T}(\bm{r})$, and $\bm{F}(\bm{r})$ are the SO(3) vectors; hence, are given by the form of Eq. (24). Finally, the spin-current density $\underline{\mathsf{J}}(\bm{r})$ is the traceless symmetric tensor of the form (25).