## Gaussian quadratic form integrals and multivariable approximation of exponential integrals

Posted by peeterjoot on January 26, 2014

# Motivation

In [1] 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.

# Guts

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

This gives

Putting these together, we have to second order in is

or

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

# References

[1] A. Zee. *Quantum field theory in a nutshell*. Universities Press, 2005.

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