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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 integral in the limit $\zeta \rightarrow \infty$ :

$\displaystyle {\mathcal I} = \lim_{\zeta \rightarrow \infty} \int_{-\infty}^{\infty} {\mbox d}x \; {\mbox e}^{- \zeta f(x)}.$

(1)

The saddle point approximation can be applied if the function

satisfies certain conditions. Assume that

has a global minimum $f(x_0)=y_{min}$ at

, which is sufficiently separated from other local minima and whose value is sufficiently smaller than the value of those. Consider the Taylor expansion of

about the point

$\displaystyle f(x) = f(x_0) + \partial_x f(x){\Big \vert}_{x=x_0}(x-x_0) + \frac{1}{2} {\partial_x}^2 f(x){\Big \vert}_{x=x_0}(x-x_0)^2 + O(x^3).$

(2)

Since

is a (global) minimum, it is clear that

. Therefore

may be approximated to quadratic order as

$\displaystyle f(x) \approx f(x_0) + \frac{1}{2} f''(x_0) (x-x_0)^2.$

(3)

The above assumptions on the minima of

ensure that the dominant contribution to (1) in the limit $\zeta \rightarrow \infty$ will come from the region of integration around

$\displaystyle {\mathcal I}$	$\displaystyle \approx \lim_{\zeta \rightarrow \infty} {\mbox e}^{-\zeta f(x_0)}... ...{-\infty}^{\infty} {\mbox d}x \; {\mbox e}^{-\frac{\zeta}{2} f''(x_0)(x-x_0)^2}$	(4)
	$\displaystyle \approx \lim_{\zeta \rightarrow \infty} {\mbox e}^{-\zeta f(x_0)} \left( \frac{2 \pi}{\zeta f''(x_0)}\right)^{1/2}.$

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 exponential):

$\displaystyle {\mathcal I} \approx \lim_{\zeta \rightarrow \infty} {\mbox e}^{-... ...\left( 1 - \frac{\zeta}{4!} (\partial_x^4 f(x))\vert _{x=x_0}(x-x_0)^4 \right).$

(5)

...to be continued with applications to physics...

1	Zeros And Poles Of Rational Function
1	Polygon
1	Point
1	Basic Polynomial
1	Field
1	Obvious
1	Function
1	Integral
2	Exponential Operation
2	Limit Of Real Number Sequence
2	Taylor Series
3	Approximation
4	Extremum
5	Separated Scheme
5	Foundations Of Quantum Field Theories
5	Stationary Set
8	Dominant Strategy