Continuity equation for densities in spin-isospin channels

We can now repeat derivations presented in Eqs. (21)-(25) by considering the spin-isospin local-gauge groups, and derive CEs in other spin-isospin channels. To this end, we first express the nuclear one-body density matrix (13)-(14) as a linear combination of nonlocal spin-isospin densities $\rho_v^t(\bm{r},\bm{r}')$ [Carlsson and Dobaczewski(2010)],

$$\hspace*{-1cm}\rho(\bm{r}\sigma \tau, \bm{r}'\sigma' \tau') =$$

$$\frac{1}{4} \sum_{v=0,1, t=0,1}\left(\sqrt{3}\right)^{v+t} \left[... ...\sigma'} \left[\tau_{\tau\tau'}^t \rho_v^t(\bm{r},\bm{r}') \right]^0\right]_0 ,$$ (28)

where the sums run over the spin ($v=0,1$) and isospin ($t=0,1$) indices denoted by subscripts and superscripts, respectively, coupled to total scalar and isoscalar. Here and below we use the coupling of spherical tensors both for angular momentum and isospin tensors; therefore, in Eq. (28) the factor of $\left(\sqrt{3}\right)^{v+t}$ was included so as to cancel the corresponding values of the Clebsch-Gordan coefficients, and to maintain the standard normalization of the spin-isospin densities. The spin-isospin densities can be conversely expressed as the following traces of the density matrix,

$$\rho_v^t(\bm{r},\bm{r}') = \sum_{\sigma \tau,\sigma' \tau'} \sigma_v^{\sigma'\sigma}\tau_{\tau'\tau}^t \rho(\bm{r}\sigma \tau, \bm{r}'\sigma' \tau') .$$ (29)

The CEs for densities in the scalar-isoscalar ($v=0$, $t=0$), scalar-isovector ($v=0$, $t=1$), vector-isoscalar ($v=1$, $t=0$), and vector-isovector ($v=1$, $t=1$) channels,

$$\frac{{\rm d}}{{\rm d} t} \rho^t_v({\bm r}) = -\frac{\hbar}{m} \bm{\nabla}\cdot\bm{J}^t_v(\bm{r}) ,$$ (30)

where $\bm{J}^t_v(\bm{r})=\frac{1}{2i}\left(\bm{\nabla}-\bm{\nabla}'\right) \rho^t_v(\bm{r},\bm{r}')\vert _{\bm{r}'=\bm{r}}$ and $\rho^t_v(\bm{r})=\rho^t_v(\bm{r},\bm{r})$, are now equivalent to the local gauge invariances, respectively, with respect to the four local spin-isospin groups:

$$U^t_v(\bm{r}) = \exp\left(i\left[\left[\gamma^t_v(\bm{r})\sigma_v\right]_0\tau^t\right]^0\right) .$$ (31)

Of course, the standard CE derived in Sec. 2.2.1 corresponds to $\gamma(\bm{r})\equiv\gamma^0_0(\bm{r})$. Note that the four gauge groups are different: $U^0_0(\bm{r})$ gives the standard abelian gauge group U(1), $U^0_1(\bm{r})$ and $U^1_0(\bm{r})$ form the non-abelian gauge groups SU(2), whereas $U^1_1(\bm{r})$ corresponds to the non-abelian gauge group SU(2)$\times$SU(2).