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G. D. Mahan, Phys. Rep. 145, 253 (1987). 25. Y. Meir and N. S. Wingreen, Phys. Rev. Lett. 68, 2512 (1992). 58 References 26. Y. Meir, N. S. Wingreen, and P. A. Lee, Phys. Rev. Lett. 66, 3048 (1991). 27. Y. Meir, N. S. Wingreen, and P. A. Lee, Phys. Rev. Lett. 70, 2601 (1993).

In Wigner coordinates, we have H0 (x1 , −i∇1 − eA1 ) → H0 R + 12 r, −i( 12 ∇R + ∇r ) + 12 e(R + 12 r) × B H0 (x2 , i∇2 − eA2 ) → H0 R − 12 r, i( 12 ∇R − ∇r ) + 21 e(R − 12 r) × B After Fourier transformation, we obtain terms like H0 R ± 2i ∇k , k + 21 eR × B ∓ 2i (∇R + 12 eB × ∇k ) As before, we perform the Mahan-H¨ansch transformation Eq. 38) which changes the derivative with respect to R according to Eq. 40), and obtain H0 R ± 2i ∇k , k + 12 eR × B ∓ i 2 ∇R + eE ∂ + 1 eB × ∇k ∂ω 2 The Mahan-H¨ ansch transformation got rid of the ∝ E · R term.

Hence we have 50 3 Applications (vp + ∇p Re Σ)Γ + σ∇p Γ = vp Γ + (ω − p )∇p Γ + O(n2i ) → vp Γ where in the last step we have neglected the term proportional to ω − p since it is multiplied by A2 (p, ω) in Eq. 29) which is a strongly peaked function around ω = p in the dilute limit (Re Σ ∝ ni is small). The QBE thus becomes ∂nF i 2 2 A (p, ω) ∂ω eE · vp Γ (p, ω) = iA(p, ω)Γ (p, ω) ×eE · vp Λ(p, ω) − ni Γ (p, ω) ∂nF ∂ω d3 q |Vp−q |2 A(q, ω)vq Λ(q, ω) (2π)3 from which we extract an integral equation for the unknown function Λ(p, ω), vp Λ(p, ω) = 21 A(p, ω)vp + ni Γ (p, ω) d3 q |Vp−q |2 A(q, ω)vq Λ(q, ω) (2π)3 which is reminiscent of the integral equation satisfied by the vector vertex function in the ladder approximation of the diagrammatic Kubo analysis.

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