I've been watching Sidney Coleman's QFT lectures (http://www.physics.harvard.edu/about/Phys253.html). I've gotten up to his discussion of functional integration, and I have some questions.(adsbygoogle = window.adsbygoogle || []).push({});

He starts out by discussing a finite-dimensional integral of a Gaussian function: [itex]\int{\frac{d^n x}{(2\pi)^{n/2}}e^{-\frac{1}{2}xAx}} = (det A)^{-1/2}[/itex], where [itex]x[/itex] is an n-dimensional vector, and [itex]A[/itex] an n-dimensional symmetric matrix. So far, that makes sense--if you diagonalize [itex]A[/itex], it just turns into the product of [itex]n[/itex] Gaussian integrals. He then goes on to discuss the integral of a polynomial times a Gaussian, [itex]\int{\frac{d^n x}{(2\pi)^{n/2}}P(x)e^{-\frac{1}{2}xAx}}[/itex], where [itex]P(x)[/itex] is a polynomial. Seemingly out of nowhere, he gives the result of this integral as [itex]P(-\frac{\partial}{\partial b})(det A)^{-1/2}[/itex]. I have absolutely no idea where this comes from.

Google is turning up bits and pieces of information on this, but nothing I can make a complete picture out of. The best I've been able to work out is that it's in some way related to differentiating under the integration sign, but I can't quite put the pieces together. This is clearly going to become important in the subsequent sections, where we're going to go on to develop the path integral formulation of QFT, so I'd really like to figure this out. Can anybody shed any light on how this works?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Functional integral theorem

**Physics Forums | Science Articles, Homework Help, Discussion**