Next: The NLO quasilocal functional Up: Time-dependent density functional theory Previous: Continuity equation for the

### 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 [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 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, (29)

The CEs for densities in the scalar-isoscalar ( , ), scalar-isovector ( , ), vector-isoscalar ( , ), and vector-isovector ( , ) channels, (30)

where and , are now equivalent to the local gauge invariances, respectively, with respect to the four local spin-isospin groups: (31)

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 non-abelian gauge groups SU(2), whereas corresponds to the non-abelian gauge group SU(2) SU(2). Next: The NLO quasilocal functional Up: Time-dependent density functional theory Previous: Continuity equation for the
Jacek Dobaczewski 2011-11-11
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