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 $\zeta \to \mathrm{\infty }$:

$$ℐ=\underset{\zeta \to \mathrm{\infty }}{lim}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\text{d}x{\text{e}}^{-\zeta f(x)}.$$(1)

The saddle point approximation can be applied if the function $f(x)$ satisfies certain conditions. Assume that $f(x)$ has a global minimum $f(x_0)=y_{min}$ at $x=x_0$, which is sufficiently separated from other local minima and whose value is sufficiently smaller than the value of those.Consider the Taylor expansion of $f(x)$ about the point $x_0$:

$$f(x)=f(x_0)+{{\partial }_{x}f(x)|}_{x=x_0}(x-x_0)+{\frac{1}{2}\partial _x{}^{2}f(x)|}_{x=x_0}{(x-x_0)}^2+O(x^3).$$(2)

Since $f(x_0)$ is a (global) minimum, it is clear that $f^{\prime }(x_0)=0$. Therefore $f(x)$ may be approximated to quadratic order as

$$f(x)\approx f(x_0)+\frac{1}{2}{f}^{\prime \prime }(x_0){(x-x_0)}^2.$$(3)

The above assumptions on the minima of $f(x)$ ensure that the dominant contribution to (1) in the limit $\zeta \to \mathrm{\infty }$ willcome from the region of integration around $x_0$:

$ℐ$$\approx \underset{\zeta \to \mathrm{\infty }}{lim}{\text{e}}^{-\zeta f(x_0)}{\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}}\text{d}x{\text{e}}^{-\frac{\zeta }{2}{f}^{\prime \prime }(x_0){(x-x_0)}^2}$(4)

$\approx \underset{\zeta \to \mathrm{\infty }}{lim}{\text{e}}^{-\zeta f(x_0)}{\left({\displaystyle \frac{2\pi }{\zeta f^{\prime \prime }(x_0)}}\right)}^{1/2}.$

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):

$$ℐ\approx \underset{\zeta \to \mathrm{\infty }}{lim}{\text{e}}^{-\zeta f(x_0)}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\text{d}x{\text{e}}^{-\frac{\zeta }{2}{f}^{\prime \prime }(x_0){(x-x_0)}^2}\left(1-{\frac{\zeta }{4!}({\partial }_x^{4}f(x))|}_{x=x_0}{(x-x_0)}^4\right).$$(5)

…to be continued with applications to physics…