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### Continuity equation for the scalar-isoscalar density

The CE now results from specifying to the local gauge transformation [13,Dobaczewski and Dudek(1995)] that is defined as

 (21)

Then, Eq. (13) gives:
 (22)

Matrix elements of the local-gauge angle are given by local integrals,
 (23)

therefore, from Eq. (13) again, the average value of the gauge angle, , depends on the scalar-isoscalar local density , that is,
 (24)

Now, the assumed local-gauge invariance of the potential energy implies the equation of motion for the average value , which from Eq. (20) reads

 (25)

where the standard scalar-isoscalar current is defined as [16] .

We note here [13,Dobaczewski and Dudek(1995)], that the gauge invariance that corresponds to a specific dependence of the gauge angle on position, , represents the Galilean invariance of the potential energy for the system boosted to momentum . Then, equation of motion (25) simply represents the classical equation for the center-of-mass velocity,

 (26)

In the general case, that is, when the potential energy is gauge-invariant and the gauge angle is an arbitrary function of , Eq. (25) gives the CE that reads

 (27)

Thus for a gauge-invariant potential energy density, the TDHF or TDDFT equation of motion implies the CE, that is, the gauge invariance is a sufficient condition for the validity of the CE. By proceeding in the opposite direction, we can prove that it is also a necessary condition. Indeed, the CE of Eq. (27) implies the first-order condition (19), and then the full gauge invariance stems from the fact that the gauge transformations form local U(1) groups.

Next: Continuity equation for densities Up: Time-dependent density functional theory Previous: Time-dependent density functional theory
Jacek Dobaczewski 2011-11-11