saddle point approximation

The saddle point approximation (SPA), a.k.a. stationary phase approximation, is a widely used method in quantum field theory (QFT) and related fields. Suppose we want to evaluate the following integralDlmfPlanetmath in the limit ζ:

=limζ-dxe-ζf(x). (1)

The saddle point approximation can be applied if the functionMathworldPlanetmath f(x) satisfies certain conditions. Assume that f(x) has a global minimumMathworldPlanetmath f(x0)=ymin at x=x0, which is sufficiently separated from other local minima and whose value is sufficiently smaller than the value of those. Consider the Taylor expansionMathworldPlanetmath of f(x) about the point x0:

f(x)=f(x0)+xf(x)|x=x0(x-x0)+12x2f(x)|x=x0(x-x0)2+O(x3). (2)

Since f(x0) is a (global) minimum, it is clear that f(x0)=0. Therefore f(x) may be approximated to quadratic order as

f(x)f(x0)+12f′′(x0)(x-x0)2. (3)

The above assumptions on the minima of f(x) ensure that the dominant contribution to (1) in the limit ζ will come from the region of integration around x0:

limζe-ζf(x0)-dxe-ζ2f′′(x0)(x-x0)2 (4)

In the last step we have performed the Gaußian integral. The next nonvanishing higher order correction to (4) stems from the quartic term of the expansion (2). This correction may be incorporated into (4) to yield (after expanding part of the exponentialMathworldPlanetmathPlanetmath):

limζe-ζf(x0)-dxe-ζ2f′′(x0)(x-x0)2(1-ζ4!(x4f(x))|x=x0(x-x0)4). (5)

…to be continued with applications to physics…

Title saddle point approximation
Canonical name SaddlePointApproximation
Date of creation 2013-03-22 13:38:07
Last modified on 2013-03-22 13:38:07
Owner msihl (2134)
Last modified by msihl (2134)
Numerical id 5
Author msihl (2134)
Entry type Topic
Classification msc 00A05
Classification msc 00A79
Synonym stationary phase method