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I'm having some troubles following the derivation of the scalar field thermal propagator. I'm following the article "Finite Temperature Quantum Field Theory in Minkwoski space" by Niemi and Semenoff (equations 2.24-2.28).

I want to find $D(x)$ such that $(\square_c -m^2)D(x)=\delta_c (x)$ where the delta function and the square operator are defined on the Schwinger Keldysh contour (but I don't think this matters now).

I have the ansatz $D(x)=\theta_c(t_x)D^>(x)+\theta_c(-t_x)D^<(x)$ and the KMS conditions $D^>(t-i \beta,x)=D^<(t,x)$

Now, the article just says that the solution for $D$ is (after Fourier transforming the spatial variables)

$D(t,\omega_k)=\frac{-i}{2\omega_k}\frac{1}{1-e^{-\beta \omega_k}} \{ [e^{- i \omega_k t}+e^{-\beta \omega_k+ i \omega_k t }] \theta_c(t) +[e^{i \omega_k t}+e^{-\beta \omega_k- i \omega_k t }] \theta_c(-t) \}$

where $\omega_k=\sqrt{k^2+m^2}$

Now, I'm used to see this propagator with $T=0$ (which you get by Fourier transforming $D(x)$ and then closing the contour in the complex plane), and right now I don't understand how to get this. To be more specific, how do I put the KMS condition when I'm solving the equation for $D(x)$?

Thanks

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