Non-local densities

The density matrices in the spin and isospin spaces can be expressed as linear combinations of the unity and Pauli matrices. To write the corresponding formulae the following notation is assumed. Vectors and vector operators in the physical three-dimensional space are denoted with boldface symbols, e.g., $\mbox{{\boldmath {$r$}}}$ or $\mbox{{\boldmath {$\nabla$}}}$, and the second rank tensors - with sans serif symbols, e.g., $\mathsf{J}$. Scalar products of three-dimensional space vectors are, as usual, denoted with the central dot: $\mbox{{\boldmath {$r$}}}$$\cdot$ $\mbox{{\boldmath {$\nabla$}}}$. The components of vectors and tensors are labelled with indices $a,b,c$ and the names of axes are $x$, $y$, and $z$, e.g., $\mbox{{\boldmath {$r$}}}$= $(\mbox{{\boldmath {$r$}}}_x,\mbox{{\boldmath {$r$}}}_y,\mbox{{\boldmath {$r$}}}_z)$. In order to make a clear distinction, vectors in isospace are denoted with arrows and scalar products of them -- with the circle: $\vec{v}\circ\vec{w}$. The components of isovectors are labelled with indices $i,k$, and the names of iso-axes are 1, 2, and 3, e.g., $\vec{v}$=$(v_1,v_2,v_3)$. Finally, isoscalars are marked with subscript ``0'', and we often combine formulae for isoscalars and isovectors by letting the indices run through all the four values, e.g., $k$=0,1,2,3.

With this convention the density matrices have the following form

$$\hat{\rho}(\mbox{{\boldmath {$r$}}}st,\mbox{{\boldmath {$r$}}}'s't') = {\textstyle{\frac{1}{4}}}\rho_0(\mbox{{\boldmath {$r$}}},\mbox{{\... ...}')\cdot \hat{\mbox{{\boldmath {$\sigma$}}}}_{ss'}\circ \hat{\vec{\tau}}_{tt'},$$ (18)

$$\hat{\breve{\rho}}(\mbox{{\boldmath {$r$}}}st,\mbox{{\boldmath {$r$}}}'s't') = {\textstyle{\frac{1}{4}}}\breve{\rho}_0(\mbox{{\boldmath {$r$}}},... ...}}')\cdot\hat{\mbox{{\boldmath {$\sigma$}}}}_{ss'}\circ \hat{\vec{\tau}}_{tt'},$$ (19)

where $\hat{\vec{\tau}}_{tt'}=(\hat{\tau}^1_{tt'},\hat{\tau}^2_{tt'},\hat{\tau}^3_{tt'})$ and $\hat{\mbox{{\boldmath {$\sigma$}}}}_{ss'} =(\hat{\mbox{{\boldmath {$\sigma$}}}}... ...x{{\boldmath {$\sigma$}}}}^y_{ss'},\hat{\mbox{{\boldmath {$\sigma$}}}}^z_{ss'})$ are the isospin and spin Pauli matrices, respectively, which are accompanied by the corresponding unity matrices, $\hat{\tau}^0_{tt'}=\delta _{tt'}$ and $\hat{\sigma}^u_{ss'}=\delta _{ss'}$. The density matrices defined in Eqs. (1) and (4) are now expressed by several functions of the pair of position vectors $\mbox{{\boldmath {$r$}}}$ and $\mbox{{\boldmath {$r$}}}'$. To avoid confusion, the functions appearing on the right-hand sides of Eqs. (18) will be called the (non-local) density functions or, simply, densities, unlike the density matrices of Eqs. (1) and (4) appearing on the left-hand sides.

The densities are traces in spin and isospin indices of the following combinations of the density and the Pauli matrices:

$\bullet$ scalar densities:

-

p-h isoscalar and isovector densities:

$$\rho_0(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}') = \sum_{st}\hat{\rho} (\mbox{{\boldmath {$r$}}}st,\mbox{{\boldmath {$r$}}}'st),$$ (20)

$$\vec{\rho}(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}') = \sum_{stt'}\hat{\rho} (\mbox{{\boldmath {$r$}}}st,%% \mbox{{\boldmath {$r$}}}'st')\hat{\vec{\tau}}_{t't},$$ (21)

-

p-p isoscalar and isovector densities:

$$\breve{\rho}_0(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}') = \sum_{st}\hat{\breve{\rho}}(\mbox{{\boldmath {$r$}}}st,\mbox{{\boldmath {$r$}}}'%% st),$$ (22)

$$\vec{\breve{\rho}}(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}') = \sum_{stt'}\hat{\breve{\rho}}(\mbox{{\boldmath {$r$}}}st,%% \mbox{{\boldmath {$r$}}}'st')\hat{\vec{\tau}}_{t't},$$ (23)

$\bullet$ vector densities:

-

p-h spin isoscalar and isovector densities:

$$\mbox{{\boldmath {$s$}}}_0(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}') = \sum_{ss't}\hat{\rho}(\mbox{{\boldmath {$r$}}}st,%% \mbox{{\boldmath {$r$}}}'s't)\hat{\mbox{{\boldmath {$\sigma$}}}}_{s's},$$ (24)

$$\vec{\mbox{{\boldmath {$s$}}}}(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}') = \sum_{ss'tt'}\hat{\rho}(\mbox{{\boldmath {$r$}}}st,%% \mbox{{\bol... ...h {$r$}}}'s't')\hat{\mbox{{\boldmath {$\sigma$}}}}_{s's}\hat{\vec{\tau}}_{t't},$$ (25)

-

p-p spin isoscalar and isovector densities:

$$\breve{\mbox{{\boldmath {$s$}}}}_0(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}') = \sum_{ss't}\hat{\breve{\rho}}(\mbox{{\boldmath {$r$}}}st,%% \mbox{{\boldmath {$r$}}}'s't)\hat{\mbox{{\boldmath {$\sigma$}}}}_{s's},$$ (26)

$$\vec{\breve{\mbox{{\boldmath {$s$}}}}}(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')%% = \sum_{ss'tt'}\hat{\breve{\rho}}(\mbox{{\boldmath {$r$}}}st,\mbox{... ...$r$}}}'s't')%% \hat{\mbox{{\boldmath {$\sigma$}}}}_{s's}\hat{\vec{\tau}}_{t't}.$$ (27)

Since the p-h density matrix and the Pauli matrices are both hermitian, all the p-h densities are hermitian too,

$$\rho_0(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}') = \rho^{*}_0(\mbox{{\boldmath {$r$}}}',\mbox{{\boldmath {$r$}}}),$$ (28)

$$\vec{\rho}(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}') = \vec{\rho}^{\,*}(\mbox{{\boldmath {$r$}}}',\mbox{{\boldmath {$r$}}}),$$ (29)

$$\mbox{{\boldmath {$s$}}}_0(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}') = \mbox{{\boldmath {$s$}}}^{*}_0(\mbox{{\boldmath {$r$}}}',\mbox{{\boldmath {$r$}}}),$$ (30)

$$\vec{\mbox{{\boldmath {$s$}}}}(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}') = \vec{\mbox{{\boldmath {$s$}}}}^{\,*}(\mbox{{\boldmath {$r$}}}',\mbox{{\boldmath {$r$}}}),$$ (31)

and hence, their real parts are symmetric, while the imaginary parts are antisymmetric, with respect to transposition of spatial arguments $\mbox{{\boldmath {$r$}}}$ and $\mbox{{\boldmath {$r$}}}'$.

On the other hand, the unity matrices $\hat{\sigma}^u_{ss'}=\delta _{ss'}$ and $\hat{\tau}^0_{tt'}=\delta _{tt'}$ (scalar and isoscalar) are $TC$-symmetric, while the vector and isovector Pauli matrices are $TC$-antisymmetric, i.e.,

$$\hat{\mbox{{\boldmath {$\sigma$}}}}_{ss'} = - 4ss'\hat{\mbox{{\boldmath {$\sigma$}}}}^*_{-s\,-s'},$$ (32)

$$\hat {\vec{\tau}}_{tt'} = - 4tt'\hat {\vec{\tau}}^*_{-t\,-t'}.$$ (33)

We should stress here again that operation $TC$ is antilinear, and therefore, complex conjugation appears in all right-hand-sides of Eqs. (32), although only the Pauli matrices $\sigma_y$ and $\tau_2$ are imaginary.

Since the p-p density matrix transforms under $TC$ as in Eq. (12), the p-p densities are either symmetric (scalar-isovector and vector-isoscalar) or antisymmetric (scalar-isoscalar and vector-isovector) under the transposition of their arguments, namely:

$$\breve{\rho}_0(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}') = - \breve{\rho}_0 (\mbox{{\boldmath {$r$}}}',\mbox{{\boldmath {$r$}}}),$$ (34)

$$\vec{\breve{\rho}}(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}') = \phantom{-} \vec{\breve{\rho}} (\mbox{{\boldmath {$r$}}}',\mbox{{\boldmath {$r$}}}),$$ (35)

$$\breve{\mbox{{\boldmath {$s$}}}}_0(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}') = \phantom{-} \breve{\mbox{{\boldmath {$s$}}}}_0 (\mbox{{\boldmath {$r$}}}',\mbox{{\boldmath {$r$}}}),$$ (36)

$$\vec{\breve{\mbox{{\boldmath {$s$}}}}}(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}') = - \vec{\breve{\mbox{{\boldmath {$s$}}}}} (\mbox{{\boldmath {$r$}}}',\mbox{{\boldmath {$r$}}}) .$$ (37)

Equations (28) and (34) are fulfilled independently of any other symmetries conserved by the system; they result from general properties (11) of density matrices $\hat{\rho}$ and $\hat{\breve{\rho}}$.