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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 ).

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