We consider a case where electric field E ( x , t )( ≡ − ˙ A ( x , t )) is position and time dependent. The electromagnetic interaction is written as where Ω is the frequency of the electric field. We will consider the limit of small Ω and small q . The scattering by the normal impurities is represented by where v$_{i}$ represents the strength of the impurity potential, R$_{i}$ represents the position of random impurities, N$_{imp}$ is the number of impurities, and N ≡ V/a 3 is number of sites. To estimate physical quantities, we take the random average over impurity positions in a standard manner [9]. To derive the spin continuity equation, Eq. (1), we derive the equation of motion for the spin density, s ( x , t ) ≡ 〈 c $^{†}$( x , t ) σ c ( x , t ) 〉 ( 〈 〉 represents the quantum average). The time development of the spin density reads with the spin current given as where and 6 j so ,α s ,i ≡ − ∑ j ϵ$_{ijα}$ ( ∇ $_{j}$v ( α ) so ) 〈 c $^{†}$c 〉 . (16) j (n) ,α s ,i ≡ − i 2 m 〈 c $^{†}$σ α ↔ ∇ i c 〉 − e m A$_{i}$ 〈 c $^{†}$σ$_{α}$ c 〉 ≡ j (0) ,α s ,i + j A,α s ,i , (15) j α s ≡ j (n) ,α s + j so ,α s , (14) ˙ s α = −∇ · j α s + T $^{α}$, where H is the total Hamiltonian of the system. The commutators are calculated as in Appendix A, and Eq. (13) turns out to be Eq. (1), namely ˙ s α = i 〈 [ H, c $^{†}$] σ $^{α}$c + c $^{†}$σ $^{α}$[ H, c ] 〉 , (13) H$_{imp}$ = N$_{imp}$ ∑ i =1 ∑ kk ′ v$_{i}$ N e i ( k − k $^{′}$) · $^{R$_{i}$}$c $^{†$_{k}$$_{′}$}$c$_{k}$ , (12) H$_{em}$ = − e m ∑ k , q ∑ i k$_{i}$A$_{i}$ ( q , Ω)( c †$_{k}$ − q $_{2}$c$_{k}$$_{+}$ q $_{2}$) , (11)