In their book Condensed Matter Field Theory, Altland and Simons often use the following formula for calculating thermal expectation values of exponentials of a real field $\theta$:

$$ \langle e^{i(\theta(x,\tau)-\theta(0,0))} \rangle = e^{-\frac12 \langle (\theta(x,\tau)-\theta(0,0))^2 \rangle} $$

An example can be found in chapter 4.5, problem "Boson-fermion duality", part c). (This refers to the second edition of the book, page 185.)

In other words, expectation values of exponentials can be cast as exponentials of expectation values under certain conditions. Unfortunately, I seem to be unable to find an explanation of why this can be done and what the conditions on the Lagrangian of $\theta$ are.

Hence, my question is:

How to derive the above formula? What do we need to know about $\theta$ for it to be valid in the first place?

Ideally, I am looking for a derivation using the path integral formalism. (I managed to rederive a very special case in terms of operators and the Baker-Campbell-Hausdorff formula, but I'd like to gain a more thorough understanding.)

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