Resummation Techniques for Divergent Series

Welcome to the final lesson of our module on asymptotic and semiclassical methods. In our previous lessons, we've seen how to construct perturbative expansions for physical systems. A recurring theme, from the WKB method to singular perturbations, is that the resulting series solutions are often not convergent. They are asymptotic series. Today, we'll confront this head-on and learn how to extract precise, meaningful physical information from these divergent series.

You might instinctively think a divergent series is useless. However, in physics, the opposite is often true. A divergent series can carry profound information about the non-perturbative aspects of a theory—effects that are invisible to any finite order of perturbation theory. Our goal today is to learn the art of resummation, which transforms these seemingly pathological series into powerful predictive tools. We will explore the deep conceptual framework of resurgence and a highly practical computational tool known as the Padé approximant. These techniques are at the forefront of modern theoretical physics, with crucial applications in quantum field theory, string theory, and condensed matter theory.

1. The Meaning of Divergence

For centuries, mathematicians and physicists were bewildered by divergent series. The conventional wisdom, articulated forcefully by mathematicians like Abel, was that they were "an invention of the devil." Yet, physicists kept finding them indispensable. The modern understanding, championed by figures like Sir Michael Berry, is that divergence is not a failure but a feature, encoding a richer physical reality.

To get a sense of this intellectual journey and the conceptual beauty of the subject, let's begin with a lecture from Michael Berry.

Michael Berry - Divergent series: Thomas Bayes’s bewilderment to today’s resurgence (Oct 18, 2022)

This lecture, "Divergent series: Thomas Bayes’s bewilderment to today’s resurgence" from the Simons Foundation, provides a masterful overview of the history and physics of divergent series.

Watch the first thirteen and a half minutes to appreciate the context. Start with the general remarks on why most series in physics are divergent. Continue through the historical tour, from Bayes's discovery of the divergence of Stirling's formula to Airy's work on the rainbow. Finally, watch the section on Stokes's contribution and his discovery of the "Stokes phenomenon"—the sudden appearance of a small exponential term. This is the key physical insight.

The crucial idea introduced by Stokes is that a single analytic function can have different asymptotic expansions in different regions of the complex plane. The transition between these regions, across what we now call a Stokes line, involves the birth of a new, exponentially small term. For example, the asymptotic series for the Airy function on the "dark side" of a rainbow (exponential decay) is related to the series on the "bright side" (oscillation, which is a sum of two exponentials). The divergence of the series is what allows one to hide information about the subdominant exponential, which then "reappears" as you cross a Stokes line. This is the seed of the modern theory of resurgence.

2. From Optimal Truncation to Borel Resummation

The simplest way to handle a divergent series $\sum a_{n} g^{n}$ , where the terms $∣ a_{n} g^{n} ∣$ initially decrease and then grow, is optimal truncation. You simply sum the series up to its smallest term and discard the rest. The error in this approximation is on the order of the first neglected term. Since the series comes from a perturbation expansion in a small parameter $g$ , this error is typically exponentially small, like $e^{- A /∣ g ∣}$ . This is often a good first estimate, but it's philosophically unsatisfying as it throws away the infinite amount of information contained in the tail of the series.

A much more powerful approach is Borel resummation, which aims to use the entire series. The procedure consists of three main steps:

The Borel Transform: Given a formal power series $ϕ (g) = \sum_{n = 0}^{\infty} a_{n} g^{n}$ , its Borel transform is defined as:
$B [ϕ] (ζ) = n = 0 \sum \infty \frac{a _{n}}{n !} ζ^{n}$
Since perturbative series in physics often have coefficients $a_{n}$ that grow factorially, like $n!$ , dividing by $n!$ "tames" this growth. The resulting series for $B [ϕ] (ζ)$ often has a finite radius of convergence.
Analytic Continuation: The function $B [ϕ] (ζ)$ , initially defined by its convergent series near $ζ = 0$ , is analytically continued to the entire complex $ζ$ -plane. The function may have singularities (poles or branch cuts) in the plane. As we saw from Berry's lecture, these singularities are where the physics is hidden.
The Laplace Transform: The resummed function, $s (ϕ) (g)$ , is then recovered by taking the Laplace transform of the analytically continued Borel transform:
$s (ϕ) (g) = \frac{1}{g} \int_{0}^{\infty} e^{- ζ / g} B [ϕ] (ζ) d ζ$
If the integral exists, it provides an unambiguous, numerical value from the original divergent series.

The problem, of course, is when the analytic continuation of $B [ϕ] (ζ)$ has singularities on the path of integration (the positive real axis for $g > 0$ ). This signals that the series is not Borel summable and often points to a physical instability. The beauty of resurgence theory is that the structure of these singularities tells you exactly what non-perturbative information you are missing.

3. A Practical Resummation Tool: Padé Approximants

The main practical obstacle in Borel resummation is the analytic continuation. We typically only know a finite number of coefficients $a_{n}$ . How can we approximate a function over the whole complex plane from its behavior near the origin?

A remarkably effective tool for this is the Padé approximant. The idea is to approximate a function not with a polynomial (like a Taylor series), but with a rational function (a ratio of two polynomials). An $[N / M]$ Padé approximant is a function of the form:

R_{N, M} (g) = \frac{P _{N} ( g )}{Q _{M} ( g )} = \frac{p _{0} + p _{1} g + \dots + p _{N} g ^{N}}{1 + q _{1} g + \dots + q _{M} g ^{M}}

The $N + M + 1$ unknown coefficients $p_{i}$ and $q_{j}$ are determined by requiring that the Taylor expansion of $R_{N, M} (g)$ matches the known Taylor series of the function being approximated up to order $g^{N + M}$ .

To see how this construction works, watch the following video.

Padé Approximants

This video, "Padé Approximants" by Dr. Will Wood, provides a clear, step-by-step guide to constructing and understanding these rational approximations.

First, watch the introduction from the beginning to see why Padé approximants can be superior to Taylor series. Then, follow the detailed construction of a [2/2] Padé approximant for \sin(x). This is the core "how-to" part. The explanation from 05:02 to 05:43 gives the intuition for why they work so well. Finally, the recap summarizes the general procedure.

As the video explains, Taylor series are polynomials and must diverge to $\pm \infty$ for large arguments. Padé approximants, being rational functions, can exhibit much more varied and physically realistic asymptotic behavior. They can also have poles, which makes them excellent for approximating functions with singularities, a key advantage for analytic continuation.

A comparison of the function $f(x)=e^x$ (black line) with its 5th-order Taylor polynomial $P_5(x)$ (red dashed line) and its [3/2] Padé approximant $R_{3,2}(x)$ (green dash-dotted line). Even with the same number of input coefficients, the Padé approximant provides a much better approximation over a wider range.

The strategy for resummation, known as the Padé-Borel method, is therefore:

Compute the first $K$ terms of the Borel transform $B [ϕ] (ζ)$ .
Construct a diagonal or near-diagonal Padé approximant, e.g., $[K /2, K /2]$ , for $B [ϕ] (ζ)$ . This provides an approximate analytic continuation.
Compute the Laplace transform of this rational function. Since the integral of $e^{- ζ / g} \times (rational function)$ can often be calculated exactly using techniques like the residue theorem, this yields a numerical prediction.

4. Resurgence in Quantum Mechanics and Field Theory

Let's now apply these powerful ideas to the domains you are most interested in: quantum mechanics and QFT. Perturbation series are the bread and butter of these fields, and they are almost always divergent asymptotic series.

The lecture notes below by Marcos Mariño provide a superb introduction written for physicists. They connect the formal machinery of resurgence directly to canonical problems.

[PDF] An introduction to resurgence in quantum theory

These notes are an excellent entry point into the modern application of resurgence in theoretical physics.

First, read Section 2, "Elementary resurgent technology" (pp. 2-6). This section formalizes the concepts of asymptotic series, optimal truncation, and the Borel transform. Crucially, it introduces the core idea of resurgence: the singularities of the Borel transform generate new power series corresponding to other sectors of the theory. Next, read about Borel resummation (pp. 10-11). The example of the quartic integral is a toy model for \lambda\phi^4 theory and shows how Borel resummation can exactly reconstruct the full function from its divergent perturbation series. Then, jump to Section 3.1, "Resurgence and perturbation theory" (pp. 16-21). Don't worry about every equation; focus on the physics for two key examples: The Quartic Oscillator: The perturbation series for the ground state energy is divergent. The large-order behavior is governed by a singularity in the Borel plane at \zeta = -1/3. This value is precisely the action of an instanton (a non-trivial classical solution in imaginary time) for the theory with a "wrong-sign" coupling g<0. The perturbative series for g>0 knows about the non-perturbative tunneling physics at g<0. The Double-Well Potential: Here, the Borel singularity lies on the positive real axis, so the series is not Borel summable. This reflects the physical fact that the perturbative ground state is unstable; quantum tunneling lifts the degeneracy of the two classical minima. To get the true energy, one needs a trans-series, which combines the perturbative series with exponentially small terms coming from instanton effects. Finally, read Appendix A, "Resurgence in practice" (pp. 45-48). This shows how the methods we've discussed are used in practice, including using Padé approximants for Borel resummation.

These examples reveal a deep and beautiful structure. The divergent tail of a perturbative expansion around one vacuum (e.g., the trivial vacuum in the quartic oscillator) contains the full information about the perturbative expansions around other, non-perturbative vacua (the instanton solutions). All these different perturbative sectors are interconnected in a single, coherent mathematical object—this is the essence of resurgence.

The power of these techniques is demonstrated in modern research. The image below shows a comparison between an exact result and approximations using resurgence theory.

A comparison of an exact quantity (black dots) with different resummation approximations. The dashed red line shows a simple approximation. The solid blue line, which is nearly indistinguishable from the exact result, is obtained using a more sophisticated resummation technique ("transseries"). These methods can achieve extraordinary precision. From [PhysRevD.97.105027](https://doi.org/10.1103/PhysRevD.97.105027).

For a final set of examples showing the raw power of Padé approximants, you can refer to the following text.

[PDF] Summation Techniques, Padé Approximants, and Continued Fractions

This document by K.A. Milton provides several compelling examples from field theory.

Read the subsection "Field Theory Examples" (pp. 52-53). Look at Tables 5.3, 5.4, and 5.5. Notice how Padé approximants provide remarkably accurate results far outside the radius of convergence of the Taylor series, and can even work when the Taylor series converges nowhere.

Conclusion

Today we have journeyed from the historical confusion surrounding divergent series to the modern, powerful framework of resurgence and resummation. You now possess the conceptual and practical tools to understand that the perturbation series ubiquitous in theoretical physics are not pathologies to be discarded, but rich archives of information to be decoded.

Key Takeaways:

Divergent series in physics are the norm, not the exception. They are asymptotic series that encode information about non-perturbative phenomena.
Borel resummation is a formal procedure to assign a unique, finite value to a divergent series by transforming it, performing an analytic continuation, and transforming it back.
The theory of resurgence states that the singularities of the Borel transform of a perturbative series are related to other contributions, such as instantons. The various perturbative and non-perturbative sectors of a theory are all mathematically linked.
Padé approximants are rational function approximations that are far more powerful than Taylor series for analytic continuation. They are a key practical tool for numerically implementing Borel resummation.

This lesson concludes our module on asymptotic and semiclassical methods. You have built a formidable toolkit for approximating solutions to complex physical problems, from identifying small parameters and performing expansions, to navigating the subtleties of singular perturbations and divergent series.

In our next module, we will shift our focus to the collective behavior of many-particle systems. We will begin with Mean-Field and Landau Theories, which provide a first, powerful approximation for understanding phase transitions and spontaneous symmetry breaking, themes that connect directly to your studies in condensed matter and quantum field theory.

Recommended resources

1. The Meaning of Divergence

2. From Optimal Truncation to Borel Resummation

3. A Practical Resummation Tool: Padé Approximants

4. Resurgence in Quantum Mechanics and Field Theory

Conclusion