quantum field theory - Interpretation of basic free boson propagator (Euclidean action)

In section 2.3.4 of Di Francesco, Mathieu, Senechal's Conformal Field Theory (and I would guess in many other introductory discussions of QFT) one finds a simple sample computation of a propagator

$$K(x,y) := \langle \varphi(x)\varphi(y)\rangle.$$

In the example, the fields $\varphi:\mathbf{R}^2\to \widehat{u}(1)$ are taken to obey the Euclidean action of a free boson of mass $m$

$$S = \frac{1}{2}\int d^2x \{\partial_\mu \varphi\partial^\mu\varphi + m^2 \varphi^2\}.$$

[I'm assuming I won't ruffle any feathers talking about the fields as operator-valued functions rather than operator-valued distributions. By $\widehat{u}(1)$ I mean the affine Heisenberg algebra (see Conformal Field Theory section 14.4.4).]

In particular, the authors compute

$$K(x,y) = -\frac{1}{2\pi} \ln(\|x-y\|), \quad \textrm{if }\ m=0$$ $$K(x,y) = \frac{1}{2\pi} K_0(m \|x-y\|), \quad \textrm{if }\ m>0$$

where $K_0$ is a modified Bessel function.

I find I am at a loss for how to interpret this result.

For simplicity, let's focus on the case $m=0$. Based on the last few paragraphs of Luboš's answer to this question, it seems $K(x,y) = -\ln(\|x-y\|)/2\pi$ might be interpreted in one of two ways:

1) As something that "knows about the correlation of $\varphi(x)$ and $\varphi(y)$." In this case we would conclude that:

• When $\|x-y\|<1$, it looks like $\varphi(x)$ and $\varphi(y)$ "have the same sign." (What could that mean in this context?)


• When $\|x-y\|>1$, it looks like $\varphi(x)$ and $\varphi(y)$ "have opposite signs." (Ditto)


• When $\|x-y\|=1$, it looks like $\varphi(x)$ and $\varphi(y)$ are independent of each other. (What would it mean for $\varphi(x)$ not to be correlated with $\varphi(y)$ only when the distance between the points is 1?)

2) As a probability amplitude for a particle to go from $x$ to $y$. This perspective doesn't seem to apply here: $K<0$ when $\|x-y\|>1$, and in any event $|K|\to\infty$ as $\|x-y\|\to\infty$. I mention it since this is the interpretation Luboš specifically gives in relation to correlation of operators.

Neither of these possibilities, as I have understood them, looks correct to me.

Any help in clearing up my confusion would be most appreciated. If it is possible to see the above propagators manifested in concrete data (perhaps results gathered in a lab (condensed matter?) or from computer simulations (Ising model?)), I think I would such a treatment most satisfying, but that may be asking too much. I don't know.