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I'm trying to reproduce some of the results from this paper by B. Garbrecht but I'm having some difficulties getting his Eq. (110),

$$\mathrm{i} \delta S_{N i j}^{<,>}=\mathrm{i}\delta S_{N i}\left(\mathrm{i}\Sigma_{N i j}^{>}-\mathrm{i}\Sigma_{N i j}^{T}\right) \mathrm{i} S_{N j}^{T}-\mathrm{i} S_{N i}^{\bar{T}}\left(\mathrm{i}\Sigma_{N i j}^{>}-\mathrm{i}\Sigma_{N i j}^{\bar{T}}\right) \mathrm{i} \delta S_{N j}$$

from Eq. (109),

$$\mathrm{i} S_{N i j}^{\mathrm{wv}>}=\mathrm{i} S_{N i}^{>} \mathrm{i}\Sigma_{N i j}^{<} \mathrm{i} S_{N j}^{>}+\mathrm{i} S_{N i}^{\bar{T}} \mathrm{i}\Sigma_{N i j}^{>} \mathrm{i} S_{N j}^{T}-\mathrm{i} S_{N i}^{>} \mathrm{i}\Sigma^{T} \mathrm{i} S_{N j}^{T}-\mathrm{i} S_{N i}^{\bar{T}} \mathrm{i}\Sigma_{N i j}^{\bar{T}} \mathrm{i} S_{N j}^{>}\ .$$

The text gives the following suggestion:

In this expression, we have dropped terms that contain products of on-shell $\delta$-functions pertaining to $N_i$ and $N_j$ with $i\neq j$ that cannot be simultaneously satisfied.

Any comments would be very welcome, thanks.

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In case someone else has a similar question, here is the answer based on the discussion with the author of the paper.

The two things that are happening here are that one throws out terms where both of the $S_i$ are on-shell (since $i\neq j$, then the product of the $\delta$-functions yield zero) and also that the $T$ and $\bar T$ propagators have on-shell out-of-equilibrium contributions.

For example, in particular, in the second term of Eq. (109), one replaces one of the propagators by $\delta S$ and one remains an off-shell $T$ or $\bar T$ propagator, resulting in two terms in Eq. (110). On the other hand, the first term does not contribute at all.

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