Resurgence Preview

Resurgence is a framework for relating perturbative expansions to nonperturbative saddle sectors. In quantum mechanics, it appears most concretely in systems where ordinary perturbation theory produces divergent asymptotic series and instantons produce exponentially small corrections.

This is an advanced preview, not a complete research review. Asymptotic Series and Nonperturbative Corrections supplies the prerequisite physics of factorial growth, optimal truncation, and beyond-all-orders terms. The basic resurgent phenomenon is well established in many quantum-mechanical models, while its scope and best formulation in more general quantum field theories remain active research topics.

The Basic Question

Perturbation theory often gives a formal expansion such as

$$E(g) \sim \sum_{n=0}^{\infty} a_n g^n, \qquad g\to0.$$

In many quantum systems this series is asymptotic rather than convergent. The coefficients eventually grow so fast that the series has zero radius of convergence. At the same time, tunneling and instanton effects can produce terms like

$$e^{-S_{\mathrm{inst}}/g},$$

which are invisible to every finite order in powers of $g$.

Resurgence asks whether these two facts are independent. In favorable problems, the large-order behavior of the perturbative coefficients contains information about the nonperturbative saddles, and the nonperturbative sectors contain information about the ambiguities of perturbation theory.

Borel Summation Preview

Given a formal series

$$E(g) \sim \sum_{n=0}^{\infty} a_n g^n,$$

define its Borel transform by

$$\mathcal B E(t) = \sum_{n=0}^{\infty} \frac{a_n}{n!} t^n.$$

If the Borel transform can be analytically continued and the integral is well defined, one tries to reconstruct a function by

$$\mathcal S E(g) = \int_0^\infty e^{-t/g} \mathcal B E(t) \,dt.$$

Singularities of $\mathcal B E(t)$ often occur at actions associated with other saddles, such as instantons or instanton-anti-instanton configurations. When a singularity lies on the integration contour, the Borel sum can be ambiguous. In a resurgent transseries, that ambiguity is canceled by an ambiguity in a nonperturbative sector.

Transseries Sectors

A simple transseries has the schematic structure

$$E(g,\sigma) \sim \sum_{k=0}^{\infty} \sigma^k e^{-kS_{\mathrm{inst}}/g} g^{\beta_k} \sum_{n=0}^{\infty} a_{k,n}g^n.$$

The sector $k=0$ is the perturbative sector around a reference saddle. The sectors $k\ge1$ describe instantons, multi-instantons, or related complex saddles, depending on the problem and boundary conditions.

The parameter $\sigma$ is not a new elementary constant of nature. It encodes the choice of solution, boundary condition, contour, or lateral summation prescription. Different physical problems can require different transseries sectors even when the local perturbative expansion looks similar.

Quantum Mechanics as a Laboratory

One-dimensional quantum mechanics is a useful laboratory because many pieces can be checked against exact spectra or high-precision numerics. Standard examples include:

• the anharmonic oscillator,
• the symmetric double well,
• periodic cosine potentials,
• metastable wells with bounce saddles.

In the double well, perturbation theory around one minimum does not by itself produce the exponentially small splitting between even and odd states. Instanton sectors supply the missing exponential scale. Resurgence refines this story by relating the large-order behavior of perturbation theory to the structure of the nonperturbative sectors.

What Is Standard and What Is Active

Standard:

• Many perturbative series in quantum mechanics are asymptotic.
• Instanton effects are nonanalytic in the small parameter.
• Borel summation is a powerful way to assign functions to some divergent series.
• In well-studied models, perturbative and instanton sectors are linked by precise large-order relations.

Active:

• how best to formulate resurgence in broad classes of quantum field theories,
• which saddle sets are complete in strongly coupled problems,
• how complex saddles, thimbles, and renormalons fit together,
• how to turn formal transseries into practical nonperturbative definitions in general settings.

This page should be read as a map of ideas, not as a claim that every quantum theory is solved by identifying instantons.

Relation to Instantons

Instantons are one source of nonperturbative sectors. They are not the only possible source. Complex saddles, renormalon-like singularities, boundary effects, and topology can also matter depending on the theory.

In simple quantum-mechanical tunneling problems, the instanton action sets the location of characteristic Borel-plane singularities and the exponential scale of level splittings. In more complicated theories, the semiclassical saddle inventory can be harder to define and may depend on compactification, boundary conditions, or analytic continuation.

Common Mistakes

• Saying that a divergent perturbation series is automatically useless.
• Saying that Borel summation alone always determines the physical answer.
• Treating resurgence as a settled universal solution to all nonperturbative problems.
• Forgetting that boundary conditions and contours are part of the problem.
• Assuming that instantons are the only nonperturbative saddles.

Exercises

1. Why can no ordinary power series in $g$ reproduce $e^{-S/g}$ near $g=0^+$?

Solution

For any fixed positive integer $N$,

$$\lim_{g\to0^+} \frac{e^{-S/g}}{g^N} = 0$$

when $S\gt0$. The exponential is smaller than every power of $g$. Therefore no finite truncation of a power series in $g$ can reproduce it, and it is not visible in the ordinary perturbative expansion.

2. Compute the Borel transform of the formal series $\sum_{n=0}^{\infty}n!g^n$.

Solution

The Borel transform divides each coefficient by $n!$:

$$\mathcal B(t) = \sum_{n=0}^{\infty} \frac{n!}{n!}t^n = \sum_{n=0}^{\infty}t^n = \frac{1}{1-t},$$

for $|t|\lt1$, with analytic continuation away from $t=1$. The singularity at $t=1$ obstructs the positive real Borel integral.

3. Explain why a Borel-plane singularity on the positive real axis can signal nonperturbative ambiguity.

Solution

The Borel reconstruction integral runs over positive real $t$. If the analytically continued Borel transform has a singularity on that contour, the integral must be defined by deforming the contour above or below the singularity. These lateral choices can differ by an imaginary or exponentially small ambiguity. In resurgent problems, a corresponding ambiguity in a nonperturbative sector cancels it in the physical quantity.

Cross-Links

• Higher-Order Structure
• Anharmonic Oscillator
• Asymptotic Series and Nonperturbative Corrections
• Instantons in Quantum Mechanics
• Fluctuation Determinants Preview

References

• J. Zinn-Justin and U. D. Jentschura, “Multi-instantons and exact results I: conjectures, WKB expansions, and instanton interactions,” Annals of Physics 313, 197-267, 2004.
• J. Zinn-Justin and U. D. Jentschura, “Multi-instantons and exact results II: specific cases, higher-order effects, and numerical calculations,” Annals of Physics 313, 269-325, 2004.
• G. V. Dunne and M. Unsal, “Resurgence and trans-series in quantum field theory: the CP(N-1) model,” Journal of High Energy Physics 2012, 170, 2012.
• J. Zinn-Justin, Path Integrals in Quantum Mechanics, Oxford University Press, 2005.