In Eq. (5.7) of his book "Statistical Physics of Fields", M. Kardar proposes the identity
$$
\langle e^{\sum_i a_ix_i} \rangle
=\exp{\left[\sum_{ij}\frac{a_ia_j}{2}\langle x_ix_j \rangle\right]}\tag{5.7}
$$
where the expectation value is taken with respect to a set $\{x_i\}$ of Gaussian distributed variables. I attempted to prove this as follows:
$$
\begin{align}
\langle e^{\sum_i a_ix_i} \rangle
&\propto\prod_i\int dx_i\,e^{a_ix_i}\,e^{-b_ix_i^2}\\
&=\prod_i\int dx_i\,\exp{\left[-b_i\left(x_i^2-\frac{a_i}{b_i}\,x_i+\frac{a_i^2}{4b_i^2}\right)+\frac{a_i^2}{4b_i}\right]}\\
&=\prod_ie^{\frac{a_i^2}{4b_i}}\int dx_i \exp{\left[-b_i\left(x_i-\frac{a_i}{2b_i}\right)^2\right]}\\
&=\prod_i\sqrt{\frac{\pi}{b_i}}\,e^{\frac{a_i^2}{4b_i}}.
\end{align}
$$
The product $\prod_i\sqrt{\frac{\pi}{b_i}}$ will disappear once the distribution is properly normalized; ie.
$$
\langle e^{\sum_i a_ix_i} \rangle
=\prod_i e^{\frac{a_i^2}{4b_i}}
$$
On the other hand
$$
\begin{align}
\langle x_i^2\rangle
&\propto\prod_i\int dx_i\,x_i^2\,e^{-b_ix_i^2}\\
&=\prod_{j\ne i}\sqrt{\frac{\pi}{b_j}}\int dx_i\,x_i^2\,e^{-b_ix_i^2}\\
&=\frac{1}{2b_i}\prod_j\sqrt{\frac{\pi}{b_j}}
\end{align}
$$
and similarly
$$
\langle x_i^2\rangle
=\frac{1}{2b_i}.
$$
Combining these expressions:
$$
\langle e^{\sum_i a_ix_i} \rangle
=\prod_i\exp{\left[\frac{a_i}{2}\,\langle x_i^2\rangle\right]}
=\exp{\left[\sum_i\frac{a_i}{2}\,\langle x_i^2\rangle\right]}
$$
which agrees with Kardar's result, modulo the cross-terms. By the even-odd symmetry of the integrand, I understand that these cross-terms $\langle x_i x_j\rangle$ will vanish for $i\ne j$, and that Kardar is therefore free to add them to his identity with impunity, but I don't understand the point of doing so. Is there a reason Kardar is including these terms, or have I made a mistake in my derivation?
Best Answer
Eq. (5.7) is often given in this form. It is equivalent to Isserlis' theorem, which is one version of Wick's theorem.
Perhaps Kardar is deriving (5.7) from the fact that (i) a linear combination $x=\sum_ia_ix_i$ of Gaussian distributed variables $x_i$ is again a Gaussian distributed variable, and (ii) that $\langle e^x\rangle = e^{\langle x^2\rangle/2}$.