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 spatial
rotations, 2
isospin rotations, 3
space inversion,
and 4
time reversal. All the local densities of Sec. 3.2 have definite transformation properties with respect to
the first three of those, 1
-3
, 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
symmetry instead of the time-reversal. Indeed, since the
charge-reversal
is equivalent to a rotations by the angle
about the iso-axis
=2, for conserved isospin the conservation of
is equivalent to conservation of
alone. Therefore, in order
to construct the energy density which is also time-reversal
invariant, we should classify the local densities according to the
symmetry and then multiply by one another only densities with
the same
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
reversal, i.e., explicitly,
This result shows that real and imaginary parts of non-local
densities (20)-(26) have opposite
transformation properties. From Eqs. (64) one then obtains
classification of local p-h densities, namely, the isoscalar
densities
,
, and
are
symmetric, and
,
,
, and
are
antisymmetric, while the isovector
densities
,
, and
are
antisymmetric, and
,
,
, and
are
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 antisymmetric and imaginary
parts are
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.