Gaussian quadratic form integrals and multivariable approximation of exponential integrals
Posted by peeterjoot on January 26, 2014
In  eq. I.2.20 is the approximation
where . Here is assumed to be an extremum of . This follows from a generalization of the Gaussian integral result. Let’s derive both in detail.
First, to second order, let’s expand around a min or max at . The usual trick, presuming that one doesn’t remember the form of this generalized Taylor expansion, is to expand around , then evaluate at . We have
The second derivative is
Putting these together, we have to second order in is
We can put the terms up to second order in a nice tidy matrix forms
Note that eq. 1.0.7b is a real symmetric matrix, and can thus be reduced to diagonal form by an orthonormal transformation. Putting the pieces together, we have
Integrating this, we have
Employing an orthonormal change of variables to diagonalizae the matrix
and , or , the volume element after transformation is
Our integral is
We now have products of terms that are of the regular Gaussian form. One such integral is
This is just
Applying this to the integral of interest, writing
This last exponential argument can be put into matrix form
Finally, referring back to eq. 1.0.7, we have
Observe that we can recover eq. 1.1.1 by noting that for that system was assumed (i.e. was an extremum point), and by noting that the determinant scales with since it just contains the second partials.
An afterword on notational sugar:
We didn’t need it, but it seems worth noting that we can write the Taylor expansion of eq. 1.0.8 in operator form as
 A. Zee. Quantum field theory in a nutshell. Universities Press, 2005.