The Hubbard-Stratonovich Identity

#Linear Algebra #Calculus

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.


  1. Hubbard1959 Hubbard J. Calculation of Partition Functions. Physical Review Letters. 1959. pp. 77–78. doi:10.1103/physrevlett.3.77 ↩︎

Published: by ;

Lei Ma (2021). 'The Hubbard-Stratonovich Identity', Datumorphism, 06 April. Available at: https://datumorphism.leima.is/cards/math/hubbard-stratonovich-identity/.

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