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Quantum Mechanics

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.

Perturbative coefficients and saddle sectors

Section titled “Perturbative coefficients and saddle sectors”

Perturbation theory often gives a formal expansion such as

E(g)∼∑n=0∞angn,g→0.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−Sinst/g,e^{-S_{\mathrm{inst}}/g},

which are invisible to every finite order in powers of gg.

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

Section titled “Borel Summation Preview”

Given a formal series

E(g)∼∑n=0∞angn,E(g) \sim \sum_{n=0}^{\infty} a_n g^n,

define its Borel transform by

BE(t)=∑n=0∞ann!tn.\mathcal B E(t) = \sum_{n=0}^{\infty} \frac{a_n}{n!} t^n.

For g>0g>0, assume the Borel transform has analytic continuation along the positive ray with sufficiently slow exponential growth for the following integral to converge. The normalization compatible with the stated transform is

SE(g)=1g∫0∞e−t/gBE(t) dt.\mathcal S E(g) = \frac{1}{g} \int_0^\infty e^{-t/g} \mathcal B E(t) \,dt.

The factor 1/g1/g is essential: integrating tn/n!t^n/n! must return gng^n, not gn+1g^{n+1}. For example, BE=1\mathcal B E=1 reconstructs E=1E=1.

Singularities of BE(t)\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

Section titled “Transseries Sectors”

A simple transseries has the schematic structure

E(g,σ)∼∑k=0∞σke−kSinst/ggβk∑n=0∞ak,ngn.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.

This one-action ansatz is schematic. Resonant sectors can contain powers of log⁡g\log g, and several independent or complex saddle actions require multiple exponential labels. The sector k=0k=0 is the perturbative sector around a reference saddle. The sectors k≥1k\ge1 describe instantons, multi-instantons, or related complex saddles, depending on the problem and boundary conditions.

The parameter σ\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

Section titled “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

Section titled “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

Section titled “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.

For the perturbative vacuum energy of a symmetric double well, the leading positive-axis ambiguity is associated with an instanton–anti-instanton pair of action 2SI2S_I, whereas the parity splitting begins at e−SI/ge^{-S_I/g}. The relevant Borel singularity depends on the sector and its boundary conditions; it is not always at the single-instanton action. In more complicated theories, the semiclassical saddle inventory can be harder to define and may depend on compactification, boundary conditions, or analytic continuation.

Exercises

Section titled “Exercises”
  1. Sum the alternating model series ∑n≥0(−1)nn!gn\sum_{n\geq0}(-1)^n n!g^n by Borel transformation, and compare it with the integral in asymptotic series.
Solution

Its Borel transform is 1/(1+t)1/(1+t). The positive ray has no pole, and changing variables t=gxt=gx gives

1g∫0∞e−t/g1+t dt=∫0∞e−x1+gx dx.\frac1g\int_0^\infty\frac{e^{-t/g}}{1+t}\,dt =\int_0^\infty\frac{e^{-x}}{1+gx}\,dx.

The factorial series diverges, but its Borel sum on this ray is unambiguous.

  1. For the nonalternating series ∑n≥0n!gn\sum_{n\geq0}n!g^n, let S+\mathcal S_+ and S−\mathcal S_- pass above and below the pole of 1/(1−t)1/(1-t). Compute their difference with both paths directed from zero to positive infinity.
Solution

The upper detour runs clockwise about t=1t=1 and gives +iπe−1/g/g+i\pi e^{-1/g}/g, since the integrand’s residue is −e−1/g/g-e^{-1/g}/g. The lower detour gives the opposite sign. Hence

S+−S−=2πige−1/g.\mathcal S_+-\mathcal S_-=\frac{2\pi i}{g}e^{-1/g}.

The principal value is their real average. Choosing that average alone does not specify a physical nonperturbative completion: a real term proportional to e−1/g/ge^{-1/g}/g remains invisible to the power expansion unless boundary data fix it.

References

Section titled “References”