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Let $\phi$ be a scalar field, $\rvert \psi_i \rangle$ a set of multiparticle states living in the Fock space of the theory indexed over the naturals, with definite 4-momentum. Let $$\rho_i = \frac{e^{-\beta E_i}}{Z}$$ be the eigenvalue of the density matrix $\rho$ when acting on one of the above states, $Z$ the partition function which normalizes, $\beta$ the inverse of the temperature of the ensemble of states $\rvert \psi_i \rangle$. Consider the $2$-point thermal correlation function $$\langle{G(x,y)}\rangle_\beta = \sum_i \rho_i \langle \psi_i\rvert \phi(x)\phi(y)\rvert \psi_i \rangle = \mathrm{Tr} \left[\rho \phi(x)\phi(y)\right]$$ What is the physical interpretation of this quantity? Is it similar to the physical interpretation we give to the non-thermal $$G(x,y)= \langle 0\rvert \phi(x)\phi(y)\rvert 0 \rangle$$ which is the probability amplitude for a particle of the field $\phi$ to travel from $y$ to $x$ (if $y^0 < x^0$) ?

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At every position, $\phi(x)$ is a random variable. So the two-point correlation function $\mathrm{Tr}[\rho \phi(x) \phi(y)]$ (where the probability density $\rho$ doesn't have to be thermal) is a covariance matrix. I find the physical content of this much easier to think about than a time ordered Green's function.

Nevertheless, you are right that they are related since $e^{-\beta H}$ is the Boltzmann factor while $e^{-it H}$ is the time evolution operator. You can go from one to the other with the Wick rotation $t \mapsto i\tau$ and compactifying $\tau$ on a circle of radius $\beta$.

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