A note on a certain non-Gaussian integral

Abstract

In this paper we present a general formula for the inhomogeneous non-Gaussian integral $I_d(S_1,S_2)=\int d{x}_1...d{x}_d{e}^{-1/2S_1^2-S_2}$, where $S_1$ and $S_2$ are symmetric quadratic forms. The solution depends on the eigenvalues of the matrix $A=-i{M}_2{M}_1^{-1}$, where $M_1$ and $M_2$ are the matrix representations of $S_1$ and $S_2$ respectively. In the 2-dimensional case we also give a manifestly SO(2)-invariant formulation in terms of invariants of the matrix $A$. An expression for $I(S_1,S_2)$ in the infinite-dimensional case is calculated and the solution depends only on the determinants of $M_1$ and $M_2$. The infinite-dimensional case may be of use in QFT.

9 pages, 3 figures

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