The Hubbard-Stratonovich Identity

The Hubbard version of the Hubbard-Stratonovich identity is¹

$$ \begin{align} \exp{\left( a^2 \right)} =& \frac{1}{\sqrt{\pi}} \int_{-\infty}^\infty \mathrm dx\, \exp{ \left( - x^2 - 2 a x \right)}\\ =& \frac{1}{\sqrt{\pi}} \int_{\infty}^{-\infty} \mathrm dx'\, \exp{ \left( - x'^2 + 2 a x' \right)}, \end{align} $$

where we changed the sign of $x$, i.e., $x’ = -x$.

In many partition functions, we have expressions like $\exp{\left( a^2/2\right)}$, using the identity, we have

$$ \begin{align} \exp{\left( \frac{a^2}{2} \right)} =& \frac{1}{\sqrt{\pi}} \int_{\infty}^{-\infty} \mathrm dx\, \exp{ \left( - x^2 + \sqrt{2} a x \right)} \\ =& \frac{1}{\sqrt{2\pi}} \int_{\infty}^{-\infty} \mathrm dx'\, \exp{ \left( - \frac{x'^2}{2} + a x' \right)}, \end{align} $$

where we define $x’=\sqrt{2}x$. We perform this transformation to get a kinetic energy like term.