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9) is the contour-ordered Green’s function of the scattering region, and we have −i Tc {ˆ ckα (τ )ˆ c†k α (τ )} = δαα δkk gkα (τ, τ ), because of translational invariance in the leads, where gkα (τ, τ ) ≡ −i Tc {ˆ ckα (τ )ˆ c†kα (τ )} is the contour-ordered Green’s function of the (isolated) leads. 3 Path Integral Derivation of the Mixed Green’s Function The path integral can also be used in the Keldysh technique. The only difference is that the action is obtained by integrating the Lagrangian over the Schwinger-Keldysh contour C instead of over the real axis.

B 6, 3189 (1972). 19. D. C. Langreth, in Linear and Nonlinear Electron Transport in Solids (Plenum Press, New York, 1976), vol. 17 of NATO Advanced Study Institute, Series B: Physics, edited by J. T. Devreese and V. E. van Doren. 20. J. M. Luttinger and J. C. Ward, Phys. Rev. 118, 1417 (1960). 21. G. Baym and L. P. Kadanoff, Phys. Rev. 124, 287 (1961). 22. G. Baym, Phys. Rev. 127, 1391 (1962). 23. M. Bonitz, Quantum Kinetic Theory (Teubner Stuttgart, Leipzig, 1998). 24. G. D. Mahan, Phys. Rep. 145, 253 (1987).

57) vanish since the quantities are all scalars. Furthermore, the term proportional to B in Eq. 64) vanishes because of the antisymmetry of the cross product. Hence we have simply {A, B}p,ω,R,T = 2AB, and Eq. 67) becomes ω− p + 1 8m ∇R + eE ∂ + eB × ∇p ∂ω 2 − Σ R GR = 1 The Kadanoff-Baym commutator in Eq. 3 Quantum Boltzmann Equation 31 ∂A ∂B ∂A ∂B − − ∇p A · ∇ R B + ∇ R A · ∇ p B ∂ω ∂T ∂T ∂ω ∂A ∂B +ieE · ∇p B − ∇p A + ieB · (∇p A × ∇p B) ∂ω ∂ω [A, B]p,ω,R,T = i so that the QBE Eq. 69) = Σ > G< − G> Σ < + i ∂pµ ∂Xµ ∂X µ ∂pµ i 1− where we use the four-vector notation pµ = (ω, p) and X µ = (T, R) with the Minkowski metric ηµν = (+ − −−).

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