Saddle Point and Steepest Descent

Many physical quantities are integrals dominated by a single point where the integrand is largest — the saddle point. Deforming the contour through it, along the direction of steepest descent, extracts the leading asymptotics. This is the mathematics of the semiclassical ($\hslash \to 0$) limit, the large-$N$ expansion, and instanton dominance. This page builds on analytic-continuation.md (contour deformation).

Laplace's method (real)

For a real integral $I(\lambda )=\int g(x)e^{\lambda \varphi (x)}dx$ with $\lambda \to +\infty$, the integrand concentrates at the maximum $x_0$ of $\varphi$. Expanding $\varphi (x)\approx \varphi (x_0)-\frac{1}{2}|{\varphi }^{\prime \prime }(x_0)|(x-x_0{)}^2$ and doing the Gaussian, $$I(\lambda )\sim g(x_0)e^{\lambda \varphi (x_0)}\sqrt{\frac{2\pi }{\lambda |{\varphi }^{\prime \prime }(x_0)|}}(\lambda \to \infty ).$$ The exponential factor $e^{\lambda \varphi (x_0)}$ dominates; the square root is the leading fluctuation correction. Stirling's formula $n!\sim \sqrt{2\pi n}(n/e{)}^n$ is the canonical example (applied to $\Gamma (n+1)=\int t^n{e}^{-t}dt$).

Steepest descent (complex)

For a complex integral $\int g(z)e^{\lambda f(z)}dz$, one uses holomorphicity to deform the contour (freely, by Cauchy's theorem — cauchy-integral.md) to pass through a saddle point $z_0$ where $f^{\prime }(z_0)=0$. Near a saddle the real part $\mathrm{Re}f$ has a mountain-pass geometry; the optimal contour crosses along the path of steepest descent, where $\mathrm{Im}f$ is constant (so the integrand does not oscillate) and $\mathrm{Re}f$ falls off fastest. The result mirrors Laplace's: $$\int g{e}^{\lambda f}dz\sim g(z_0)e^{\lambda f(z_0)}\sqrt{\frac{2\pi }{-\lambda f^{\prime \prime }(z_0)}},$$ with the branch of the square root fixed by the descent direction. When several saddles contribute, one sums them — and which saddles are "reached" by deforming the original contour is itself a subtle analytic question (Stokes phenomenon).

Stationary phase

The purely oscillatory case $\int g(x)e^{i\lambda \psi (x)}dx$ ($\lambda \to \infty$) is dominated by the stationary points ${\psi }^{\prime }(x_0)=0$, where neighbouring phases add coherently instead of cancelling: $$\int g{e}^{i\lambda \psi }dx\sim g(x_0)e^{i\lambda \psi (x_0)}\sqrt{\frac{2\pi }{\lambda |{\psi }^{\prime \prime }(x_0)|}}{e}^{\pm i\pi /4}.$$ This is the stationary-phase approximation — steepest descent rotated onto the imaginary axis, with the extra $e^{\pm i\pi /4}$ Maslov phase.

Asymptotic series and Watson's lemma

These methods generally produce asymptotic series — divergent series whose truncations nonetheless approximate the integral with error smaller than the last kept term. Watson's lemma systematizes the expansion of Laplace-type integrals in inverse powers of $\lambda$. Asymptotic (not convergent) series are the norm in physics — the perturbation series of QFT is one — and their divergence is often tied to the very instanton contributions steepest descent reveals.

The semiclassical limit

The physics payoff: the path integral $$Z=\int \mathcal{D}\varphi e^{iS[\varphi ]/\hslash }$$ is a steepest-descent problem with $\lambda =1/\hslash$. As $\hslash \to 0$ it is dominated by the saddle points of the action — the classical solutions $\delta S=0$ (the principle of stationary action is the stationary-phase condition). The Gaussian fluctuation factor gives the one-loop determinant, and subleading saddles give instanton contributions ($e^{-S_E/\hslash }$, nonperturbative) — the Euclidean saddles of advanced/topological.md. Thus "classical physics is the saddle point of the quantum path integral" is literally the steepest-descent approximation.

References

• Bender & Orszag, Advanced Mathematical Methods for Scientists and Engineers, Ch. 6.
• Wong, Asymptotic Approximations of Integrals.
• Zinn-Justin, Quantum Field Theory and Critical Phenomena (instantons via steepest descent).