Matsubara Field Theory - what does imaginary time $\tau$ in $G(\tau,\mathbf{x})$ mean? Consider the free, real scalar field $\phi$ in Matsubara Finite-Temperature quantum field theory, where our system is kept in equilibrium with a heat bath at temperature $\frac{1}{\beta}$.
Then the field $\phi$ is an operator $\phi(\tau, \mathbf{x})$ where $\mathbf{x}$ is a position in $\mathbb{R}^3$ and $\tau$ is an imaginary time. Furthermore, the propagator for two points $(\tau_1, \mathbf{x}_1)$ and $(\tau_2, \mathbf{x}_2)$ is:
$$
G\left(\tau_1, \mathbf{x}_{1};\tau_2, \mathbf{x}_{2}\right) \ = \ < \phi(\tau_1, \mathbf{x}_1)\phi(\tau_2, \mathbf{x}_2) >_{\beta}
$$
Where the $\beta$ denotes a thermal average. Because of the KMS condition, we have the boundary condition $G\left(\tau_1, \mathbf{x}_{1};\tau_2+\beta, \mathbf{x}_{2}\right) = G\left(\tau_1, \mathbf{x}_{1};\tau_2, \mathbf{x}_{2}\right)$. Following the usual derivation, we then get a propagator that looks like:
$$
G\left(\tau_1, \mathbf{x}_{1};\tau_2, \mathbf{x}_{2}\right) \propto \sum_{n=-\infty}^{\infty} \int d^{3}\mathbf{p}\ \mathcal{D}_{n}(\mathbf{p}) e^{- i \mathbf{p}\cdot\mathbf{x} - i \frac{2 \pi n}{\beta} \tau }
$$
Where $\mathcal{D}_{n}(\mathbf{p}) = \frac{1}{\left(\frac{2\pi n}{\beta} \right)^{2} + \mathbf{p}^2 + m^2}$ is the Fourier transform of this propagator. 
My questions is: what is the physical meaning of the $\tau_1$ and $\tau_2$ in $G\left(\tau_1, \mathbf{x}_{1};\tau_2, \mathbf{x}_{2}\right)$?
In ordinary QFT, there is a very nice explanation of what the propagator $G$ means physically -  it's that $G(t_1, \mathbf{x}_1;t_2, \mathbf{x}_2)$ corresponds to the probability amplitude that a particle travels from a point $\mathbf{x}_1$ at time $t_1$ to another points $\mathbf{x}_2$ at time $t_2$.
I would like to attach a similar meaning to the propagator here. I initially thought that maybe this was something $G$ correlates two different temperatures $\tau_1$ and $\tau_2$, but I realize this doesn't make any sense.
How to understand the imaginary times $\tau$? Is it not possible to make a statement like this here? I know that we can only have $0<\tau<\beta$ due to the periodic boundary conditions.
 A: Because finite temperature QFT in $d$-dimensions can be translated to classical statistical mechanics in $d+1$ dimensions, I believe you can reinterpret the propagator accordingly. In statistical mechanics, you don't have a time variable, instead the imaginary time variable $\tau$ would be thought of as an additional spatial dimension,  and so you work in periodic boundary conditions with $\beta$ being the system size [1].
Because there is no time axis, you shouldn't interpret the function $G$ as a propagator, but instead of as a statistical correlation function of the classical field in $d+1$ spatial dimensions,  one of which is periodic. Actually, as alluded to by AccidentalFourierTransform, thinking of $G$ in terms of correlation functions is always the more sensible thing to do.
[1] http://galileo.phys.virginia.edu/~pf7a/msm2.pdf
A: "I would like to attach a similar meaning to the propagator here. I initially thought that maybe this was something GG correlates two different temperatures τ1τ1 and τ2τ2, but I realize this doesn't make any sense"
It does make sense. Temperature is a measure of potentials where the emission, T^4, in a point, is relative only to the internal state of the emitter. Heat flows according to these potentials at a rate equal to the difference. 
In the simplest way with average temperatures and solar irradiance, assuming that there is no unknown sources of energy, only universal heat flow and universal force in gravity, a planetary balance in the first law, $TSI=4g^2+4Q$.
TSI=1360.8, g=9.78 and Q=244.5(256K).
It can be treated as charge, $TSI/(4/3)=4/3*8g^2$ and we also have $4g^2=383W/m^2(287K)$ and $4/3*g^2$ is equal to tropopause mean temperature.
Temperature seems to be the only thing that makes sense if globally equivalent to heat and work including gravity.
I just play with calculators for fun and I´m not saying I´m right. But the idea of a quantized heat engine in space in orbit around the sun is intriguing and obvious.
