# Physical interpretation of thermal 2-point function in QFT

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 $$y^0 < x^0$$) ?

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 $$\beta$$.