# Hubbard-Stratonovich Transformation

Hello, The Hubbard-Stratonovich transformation

$\exp(x^2) = \frac{1}{\sqrt{4 \pi}} \int_{-\infty}^{+\infty} du \exp(-\frac{u^2}{4} - xu)$

allows one to wirte the exponential of a the square of a number $x$ as an integral over a Gaussian variable $u$. Is there a transformation analogous to the Hubbard-Stratonovich transformation to write the exponential of a product of two numbers $x,y$

$\exp(xy)$

?

Thank you

Michele

• Write $2xy = -x^2 - y^2 + (x+y)^2$. Jun 12, 2013 at 8:53
• Just to clarify Peter Michor's suggestion: $2xy = (x~y) M (x~y)^T$, where $M$ is a symmetric $2\times 2$ matrix, which you can easily construct. The Hubbard-Stratonovich transformation works for multiple variables as well. Except that your Gaussian $u$-integral will be multi-dimensional (2-dimensional in this case). Jun 12, 2013 at 9:46
• Dear Igor, Following your suggestion $M_{11} = M_{22} =0$ and $M_{12} = M_{21} = 1$, so the eigenvalues of $M$ are $\lambda_1 = -1, \lambda_2 = +1$ and $M$ is not positive definite. So the $u$-integral is not convergent in this case. Jun 12, 2013 at 12:54
• That's a good point, but recall the identity $\exp(-x^2) = \frac{1}{\sqrt{4\pi}} \int_{-\infty}^{\infty} du \, \exp(-u^2/4+iux)$. Jun 12, 2013 at 19:47