Matsubara Field Theory - what does imaginary time $\tau$ in $G(\tau,\mathbf{x})$ mean?

Question

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.

Answer 1

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