Local Densities and Currents

The complete density matrix $\rho(\vec{r}\sigma t, \vec{r}'\sigma' t')$ in spin-isospin space as defined in (1) can be decomposed into the sum of scalar $\rho_{t t_3} (\vec{r}, \vec{r}')$ and vector densities $\vec{s}_{t t_3} (\vec{r}, \vec{r}')$, where the subscripts denote the isospin quantum numbers:

$${\rho(\vec{r}\sigma \tau, \vec{r}'\sigma' \tau') }$$

$${\textstyle\frac{{1}}{{4}}} \Big[ \rho_{00} (\vec{r}, \vec{r}') \... ...}') \cdot \mbox{{\boldmath {$\sigma$}}}_{\sigma \sigma'} \, \delta_{\tau \tau'}$$ =

$$+ \delta_{\sigma \sigma'} \! \sum_{t_3 = -1}^{+1} \rho_{1 t_3} (\... ...dot \mbox{{\boldmath {$\sigma$}}}_{\sigma \sigma'} \tau^{t_3}_{\tau\tau'} \Big]$$

The quantities $\mbox{{\boldmath {$\sigma$}}}_{\sigma \sigma'}$ and $\tau^{t_3}_{\tau\tau'}$ are matrix elements of the Pauli matrices in spin and isospin space. In terms of these, the local density $\rho$, spin density $\vec{s}$, kinetic density $\tau$, kinetic spin density $\vec{T}$, current $\vec{j}$, and spin-orbit tensor $\stackrel{\leftrightarrow}{J}$ are

$$\rho_{t t_3} (\vec{r}) \rho_{t t_3} (\vec{r}, \vec{r}) \phantom{\Big\vert _{\vec{r}=\vec{r}'}}$$ =

$$\vec{s}_{t t_3} (\vec{r}) \vec{s}_{t t_3} (\vec{r}, \vec{r}) \phantom{\Big\vert _{\vec{r}=\vec{r}'}}$$ =

$$\tau_{t t_3} (\vec{r}) \nabla \cdot \nabla' \rho_{t t_3} (\vec{r},\vec{r}') \Big\vert _{\vec{r}=\vec{r}'}$$ =

$$\vec{T}_{t t_3} (\vec{r}) \nabla \cdot \nabla' \vec{s}_{t t_3} (\vec{r},\vec{r}') \Big\vert _{\vec{r}=\vec{r}'}$$ =

$$\vec{j}_{t t_3} (\vec{r}) - {\textstyle\frac{{\mbox{\rm\scriptsize {i}}}}{{2}}} ( \nabla - \nabla' ) \rho_{t t_3} (\vec{r},\vec{r}') \Big\vert _{\vec{r}=\vec{r}'}$$ =

$$J_{t t_3, ij} (\vec{r}) - {\textstyle\frac{{\mbox{\rm\scriptsize {i}}}}{{2}}} ( \nabla - ... ...la' )_i \; s_{t t_3,j} (\vec{r},\vec{r}') \Big\vert _{\vec{r}=\vec{r}'} \quad .$$ = (31)

The densities $\rho$, $\tau$, and $\stackrel{\leftrightarrow}{J}$ are time-even, while $\vec{s}$, $\vec{T}$, and $\vec{j}$ are time-odd. See [20] for a more detailed discussion.