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Continuity equation for densities in spinisospin channels
We can now repeat derivations presented in
Eqs. (21)(25) by considering the spinisospin
localgauge groups, and derive CEs in other spinisospin channels.
To this end, we first express the nuclear onebody density
matrix (13)(14)
as a linear combination of nonlocal
spinisospin densities
[Carlsson and Dobaczewski(2010)],







(28) 
where the sums run over the spin () and isospin ()
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
was included so as to cancel the
corresponding values of the ClebschGordan coefficients, and to
maintain the standard normalization of the spinisospin densities.
The spinisospin densities can be conversely expressed as the following
traces of the density matrix,
The CEs for densities in the
scalarisoscalar (, ), scalarisovector (, ),
vectorisoscalar (, ), and vectorisovector (, ) channels,

(30) 
where
and
, are now equivalent to the
local gauge invariances, respectively, with respect to the four local
spinisospin groups:
Of course, the standard CE derived in Sec. 2.2.1
corresponds to
. Note that the
four gauge groups are different: gives the standard
abelian gauge group U(1), and form
the nonabelian gauge groups SU(2), whereas
corresponds to the nonabelian gauge group SU(2)SU(2).
Next: The NLO quasilocal functional
Up: Timedependent density functional theory
Previous: Continuity equation for the
Jacek Dobaczewski
20111111