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gaussian [2025/12/02 14:55] – [The Wick formula] spencergaussian [2025/12/02 15:04] (current) – [The Wick formula] spencer
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 A consequence of Wick's formula is that the Gaussian integral of a monomial is a sum of products of matrix coefficients, and thus Gaussian integration against a monomial is a kind of a permanent-type operation applied to $A^{-1}$; thus Cramer's rule implies that Gaussian integrals of polynomials are always rational functions in the entries of $A$. A consequence of Wick's formula is that the Gaussian integral of a monomial is a sum of products of matrix coefficients, and thus Gaussian integration against a monomial is a kind of a permanent-type operation applied to $A^{-1}$; thus Cramer's rule implies that Gaussian integrals of polynomials are always rational functions in the entries of $A$.
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 +The proof of the Wick formula is simple; assume that the input polynomials are all monomial, and after an orthogonal change of coordinates assume that $A$ is diagonal. Then integrate by parts until you're done.
gaussian.1764705335.txt.gz · Last modified: by spencer