I am currently studying lecture notes by Akhmedov on interacting scalar field theory in de Sitter space. In these notes, he considers a scalar field theory of the form $$\mathcal{L}=\partial_\mu\phi_1\partial^\mu\phi_1+m^2\phi_1^2+\frac{\lambda}{3!}\phi^3_1-\partial_\mu\phi_2\partial^\mu\phi_2-m^2\phi_2^2-\frac{\lambda}{3!}\phi_2^3$$ This Lagrangian arises from the fact that we are in a non-equilibrium situation so that we have to 'close the time contour' and consider fields on the forward as well as the backward time 'branches' (this is where the minuses come from). There is a matrix of four propagators, which can be labeled by a combination of $1$'s and $2$'s:

$$\hat{G}=\begin{pmatrix} G^{11} & G^{12}\\ G^{21} & G^{22}\\ \end{pmatrix} $$ They satisfy the identity $G^{11}+G^{22}=G^{12}+G^{21}$.

In order to exploit this identity, he now performs the change of coordinates known as the Keldysh rotation: $$\phi_a=\frac{1}{2}(\phi_1+\phi_2),\ \phi_b=\phi_1-\phi_2 $$ In terms of these new variables, the Lagrangian is $$\mathcal{L}=\partial_\mu\phi_a\partial^\mu\phi_b+m^2\phi_a\phi_b+\frac{\lambda}{3!}\biggl(3\phi_a^2\phi_b+\frac{1}{4}\phi_b^3\biggr) $$ And the matrix of propagators becomes $$\hat{D}=\begin{pmatrix} D^{K}&D^{R}\\ D^{A}&0\\ \end{pmatrix} $$ We now only have three propagators left. They are related to the 'old' propagators by $$D^K=G^{12}+G^{21},\; D^R=G^{11}-G^{12},\; D^A=G^{11}-G^{21} $$

My question now is: How do I derive the Feynman rules for this theory in terms of the new variables, which doesn't look much like anything I've encountered before (e.g. a 'mixed' kinetic term?!)? Akhmedov himself just provides an image which is supposed to summarize them, which I have reproduced below. Note that my $\phi_a$ and $\phi_b$ correspond to his $\phi_\text{cl}$ and $\phi_\text{q}$ respectively. However, I cannot make much sense out of it (I also don't understand where he got his vertex factors, they seem to be wrong to me. The ratio should at least be 12, right?)

enter image description here

  • $\begingroup$ I don't think that there is anything wrong with those vertex factors. For the $\phi_b^3$ term, recall from $\phi^4$ theory that there is a permutation factor 4! comes when you do wick contraction. Here you have 3! term which cancels to give $\frac{\lambda}{4}$ term as desired. After it you have 2! times 1! coming from the $\phi_a^2\phi_b$ term, which gives you the desired vertex factor for other diagram. $\endgroup$ – user44895 May 17 '14 at 18:50
  • $\begingroup$ @user44895 Ah, so there's no contradiction between the Lagrangian and the vertex factors! Thanks a lot for pointing that out! Would you know more about how to derive the Feynman rules from scratch for this theory? If you'd be able to expand your comment into a more comprehensive answer I'd be glad to accept it. $\endgroup$ – Danu May 17 '14 at 19:32

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