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Continuity equation for the scalarisoscalar 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 localgauge angle
are
given by local integrals,

(23) 
therefore, from Eq. (13) again, the average value of the
gauge angle,
, depends on the scalarisoscalar local density
,
that is,

(24) 
Now, the assumed localgauge invariance of the potential energy
implies the equation of motion for the average value
,
which from Eq. (20) reads

(25) 
where the standard scalarisoscalar 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 centerofmass velocity,

(26) 
In the general case, that is, when the potential energy is gaugeinvariant
and the gauge angle
is an arbitrary function of ,
Eq. (25) gives the CE that reads

(27) 
Thus for a gaugeinvariant 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 firstorder
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: Timedependent density functional theory
Previous: Timedependent density functional theory
Jacek Dobaczewski
20111111