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The source term η¯d is unaffected by the change of variables, but the source term c¯J becomes c¯J = c¯kα (τ )Jkα (τ ) = kα dτ1 d¯m (τ1 )t∗kα,m gkα (τ1 , τ ) Jkα (τ ) c¯kα (τ ) + kα m C The generating functional thus becomes Z[¯ η , J] = ¯ i ¯ d]eiSleads [¯c ,c ] eiSQD [d,d] e D[¯ c , c , d, C ¯ ∗ g)J ) dτ (η¯d+(¯ c +dt where the integration measure D[¯ c, c] = D[¯ c , c ] is invariant under the change of variables since it is only a shift. We now perform the functional differentiation Eq. 11). This generates two terms.

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. 61) which implies ∇k → ∇p but modifies the R derivative once more, ∇R → ∇R + 21 eB × ∇p so that the final expression reads H0 R ± 2i ∇p , p ∓ i 2 ∇R + eE ∂ + eB × ∇p ∂ω Hence the generalization of Eq. 62) and similarly the generalization of Eq. 42) is 1 ˆ −1 2 {G0 − U, GR }p,ω,R,T = ωGR − 21 H0 R + 2i ∇p , p − − 12 GR H0 R − 2i ∇p , p + i 2 i 2 ∇R + eE ∇R + eE ∂ + eB × ∇p ∂ω ∂ + eB × ∇p ∂ω All Green’s functions and self-energies are now functions of (p, ω, R, T ).

L. Hu, Phys. Rev. D 37, 2878 (1988). 7. P. Danielewicz, Ann. Phys. 152, 239 (1984). 8. L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics (Benjamin, New York, 1962). 9. S. Fujita, Introduction to Nonequilibrium Quantum Statistical Mechanics (Saunders, Philadelphia, 1966), and papers. 10. A. G. Hall, J. Phys. A 8, 214 (1975). 11. Yu. A. Kukharenko and S. G. Tikhodeev, Zh. Eksp. Teor. Fiz 83, 1444 (1982) [Sov. Phys. JETP 56, 831 (1982)]. 12. M. Wagner, Phys. Rev. B 44, 6104 (1991). 13. H. -P.

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