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Next: The Energy Density Functional Up: Density matrices in the Previous: Local densities


The $TC$ symmetry

When constructing the energy density functional (Sec. 4) from the local densities (38)-(50) one should ensure that it is invariant with respect to: 1$^\circ$ spatial rotations, 2$^\circ$ isospin rotations, 3$^\circ$ space inversion, and 4$^\circ$ time reversal. All the local densities of Sec. 3.2 have definite transformation properties with respect to the first three of those, 1$^\circ$-3$^\circ$, so one can easily construct the corresponding invariants by multiplying densities of the same type by one another. For example, a product of any pseudovector-isoscalar density with itself, or with any other pseudovector-isoscalar density, is an invariant.

The time-reversal symmetry cannot be immediately treated on the same footing, because the time-reversal and the isospin rotations do not commute. However, as noted in Ref. [174], for problems involving the isospin symmetry it is more convenient to use the $TC$ symmetry instead of the time-reversal. Indeed, since the charge-reversal $C$ is equivalent to a rotations by the angle $\pi$ about the iso-axis $k$=2, for conserved isospin the conservation of $TC$ is equivalent to conservation of $T$ alone. Therefore, in order to construct the energy density which is also time-reversal invariant, we should classify the local densities according to the $TC$ symmetry and then multiply by one another only densities with the same $TC$ transformation properties.

To this end, we split the p-h and p-p density matrices into parts that are symmetric and antisymmetric with respect to the $TC$ reversal, i.e., explicitly,

$\displaystyle \hat{ {\rho}}_\pm(x,x')$ $\textstyle =$ $\displaystyle {\textstyle{\frac{1}{2}}}\Big(\hat{ {\rho}}(x,x') \pm 16ss'tt'\hat{ {\rho}}^*(\bar{x},\bar{x}')\Big) ,$ (62)
$\displaystyle \hat{\breve{\rho}}_\pm(x,x')$ $\textstyle =$ $\displaystyle {\textstyle{\frac{1}{2}}}\Big(\hat{\breve{\rho}}(x,x') \pm 16ss'tt'\hat{\breve{\rho}}^*(\bar{x},\bar{x}')\Big) ,$ (63)

where we used a short-hand notation of $\bar{x}$$\equiv$ $\{\mbox{{\boldmath {$r$}}},$$-s,$$-t\}$. In conjunction with the $TC$ transformation properties of the Pauli matrices (32), one than immediately obtains that the corresponding non-local densities of Sec. 3.1 are either real or imaginary, i.e.,


$\displaystyle \rho_{0\pm} (\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')$ $\textstyle =$ $\displaystyle \pm\rho^{*}_{0\pm} (\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}'),$ (64)
$\displaystyle \vec{\rho}_\pm (\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')$ $\textstyle =$ $\displaystyle \mp\vec{\rho}^{\,*}_\pm (\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}'),$ (65)
$\displaystyle \mbox{{\boldmath {$s$}}}_{0\pm} (\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')$ $\textstyle =$ $\displaystyle \mp\mbox{{\boldmath {$s$}}}^{*}_{0\pm} (\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}'),$ (66)
$\displaystyle \vec{\mbox{{\boldmath {$s$}}}}_\pm (\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')$ $\textstyle =$ $\displaystyle \pm\vec{\mbox{{\boldmath {$s$}}}}^{\,*}_\pm (\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}'),$ (67)

and
$\displaystyle \breve{\rho}_{0\pm} (\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')$ $\textstyle =$ $\displaystyle \pm\breve{\rho}^{*}_{0\pm} (\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}'),$ (68)
$\displaystyle \vec{\breve{\rho}}_\pm (\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')$ $\textstyle =$ $\displaystyle \mp \vec{\breve{\rho}}^{*}_\pm (\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}'),$ (69)
$\displaystyle \breve{\mbox{{\boldmath {$s$}}}}_{0\pm} (\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')$ $\textstyle =$ $\displaystyle \mp \breve{\mbox{{\boldmath {$s$}}}}^{*}_{0\pm}(\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}'),$ (70)
$\displaystyle \vec{\breve{\mbox{{\boldmath {$s$}}}}}_\pm (\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}')$ $\textstyle =$ $\displaystyle \pm\vec{\breve{\mbox{{\boldmath {$s$}}}}}^{*}_\pm (\mbox{{\boldmath {$r$}}},\mbox{{\boldmath {$r$}}}') .$ (71)

This result shows that real and imaginary parts of non-local densities (20)-(26) have opposite $TC$ transformation properties. From Eqs. (64) one then obtains classification of local p-h densities, namely, the isoscalar densities $\rho_0(\mbox{{\boldmath {$r$}}})$, $\tau_0(\mbox{{\boldmath {$r$}}})$, and ${\mathsf J}_0(\mbox{{\boldmath {$r$}}})$ are $TC$ symmetric, and $\mbox{{\boldmath {$s$}}}_0(\mbox{{\boldmath {$r$}}})$, $\mbox{{\boldmath {$T$}}}_0(\mbox{{\boldmath {$r$}}})$, $\mbox{{\boldmath {$F$}}}_0(\mbox{{\boldmath {$r$}}})$, and $\mbox{{\boldmath {$j$}}}_0(\mbox{{\boldmath {$r$}}})$ are $TC$ antisymmetric, while the isovector densities $\vec{\rho}(\mbox{{\boldmath {$r$}}})$, $\vec{\tau}(\mbox{{\boldmath {$r$}}})$, and $\vec{{\mathsf J}}(\mbox{{\boldmath {$r$}}})$ are $TC$ antisymmetric, and $\vec{\mbox{{\boldmath {$s$}}}}(\mbox{{\boldmath {$r$}}})$, $\vec{\mbox{{\boldmath {$T$}}}}(\mbox{{\boldmath {$r$}}})$, $\vec{\mbox{{\boldmath {$F$}}}}(\mbox{{\boldmath {$r$}}})$, and $\vec{\mbox{{\boldmath {$j$}}}}(\mbox{{\boldmath {$r$}}})$ are $TC$ symmetric.

The rules of constructing the p-h energy density are thus identical to those valid in the case of no proton-neutron mixing [172]. On the other hand, from Eqs. (68) one obtains classification of local p-p densities, namely, real parts of all p-p densities are $TC$ antisymmetric and imaginary parts are $TC$ symmetric. The p-p energy density must therefore be built by multiplying real parts of different densities with one another, and separately imaginary parts also with one another. These rules are at the base of the energy density functional constructed in Sec. 4.

next up previous
Next: The Energy Density Functional Up: Density matrices in the Previous: Local densities
Jacek Dobaczewski 2004-01-03