Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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

share|improve this question
2  
Write $2xy = -x^2 - y^2 + (x+y)^2$. –  Peter Michor Jun 12 '13 at 8:53
1  
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). –  Igor Khavkine Jun 12 '13 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. –  Michele Jun 12 '13 at 12:54
    
That's a good point, but recall the identity $\exp(-x^2) = \frac{1}{\sqrt{4\pi}} \int_{-\infty}^{\infty} du \, \exp(-u^2/4+iux)$. –  Igor Khavkine Jun 12 '13 at 19:47

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.