spin chemical potential. The spin injection effect was briefly discussed based on our results. When the analysis is applied to the inhomogeneous magnetization case, we argued that the β torque in the current-induced magnetization dynamics can be anisotropic. Before finishing, we emphasize that the expression for the spin current and µ$_{s}$ are mean- ingless without specifying the physical observable to be measured. In the inverse spin Hall effect, which was originally proposed as [3, 23, 24] j$_{µ}$ ∝ ϵ$_{µνρ}$j ρ s $_{,ν}$, it has recently been demon- strated that the charge current is not directly proportional to the spin current [7, 25]. Solving for the spin current only does not therefore provide physical information. The authors thank E. Saitoh, J. Shibata, H. Kohno, S. Murakami for valuable discus- sions. This work was supported by a Grant-in-Aid for Scientific Research in Priority Areas, ”Creation and control of spin current” (1948027), the Kurata Memorial Hitachi Science and Technology Foundation and the Sumitomo Foundation. We calculate the commutators which appears in Eq. (13) with the total Hamiltonian H ≡ H$_{0}$ + H$_{so}$ 0 + H$_{so}$ A + H$_{em}$ + H$_{imp}$ . As for the free electron part, H$_{0}$ , the commutator reads The spin-orbit contributions read and the electromagnetic field contribution is 15 [ H$_{em}$, c ] = − i e m ∑ i A$_{i}$ ∇ $_{i}$c. (A3) [ H$_{so}$ $^{0}$, c ] = i ∑ ijk ϵ$_{ijk}$ ( ∇ $_{i}$v ( k ) so ) σ$_{k}$ ∇ $_{j}$c [ H$_{so}$ $^{A}$, c ] = e ijk ϵ$_{ijk}$ ( ∇ $_{i}$v ( k ) so ) A$_{j}$σ$_{k}$c, (A2) [ H$_{0}$, c ] = 1 2 m ∇ $^{2}$c. (A1) Appendix A: Derivation of the spin continuity equation Acknowledgment