nLab asymptotic expansion

Idea

General

An asymptotic expansion of a function is a formal power series that may not converge, but whose terms decrease fast enough such that the truncation of the series at any finite order still provides a controlled approximation to a given function.

A key class of examples of asymptotic expansions are the Taylor series of smooth functions (example below) around any point. Beware that by Borel's theorem this means that every formal power series is the asymptotic expansion of some smooth function and of more than one smooth function (remark below).

In resurgence theory one tries to re-identify from an asymptotic expansion the corresponding non-analytic contributions.

In perturbative quantum field theory

The concept of asymptotic expansions plays a key role in the interpretation of perturbative quantum field theory (pQFT): This computes quantum observables as formal power series (in the coupling constant and in Planck's constant) whose radius of convergence necessarily vanishes in cases of interest (Dyson 52).

Nevertheless, for examples such as quantum electrodynamics and quantum chromodynamics as in the standard model of particle physics, the truncation of these series to the first handful of loop orders happens to agree with experiment (such as at the LHC collider) to high precision (for QED) or at least good precision (for QCD). Therefore one interprets the scattering matrix in perturbative quantum field theory as an asymptotic expansion of what should be the true non-perturbative result.

With resurgence theory one may try to deduce from the Feynman perturbation series regarded as an asymptotic expansion the hidden non-perturbative effects.

Definition

Definition

Given a function $f \colon \mathbb{R} \to \mathbb{R}$, a formal power series $\sum_{n = 0}^\infty a_n x^n$ is an asymptotic expansion of $f$ at $x = 0$ if for each $n \in \mathbb{N}$ the limit of the difference between $f$ and the sum of the first $n$ terms of the series divided by $x^n$ is zero as $x$ tends to 0:

$$\underset{x \to 0}{\lim} \left( \frac{1}{x^n} \left( f(x) - \sum_{k = 0}^n a_k x^k \right) \right) \;=\; 0 \,.$$

Remark

This definition makes no statement about the behaviour as $n \to \infty$. In particular an asymptotic expansion may have vanishing radius of convergence (and nevertheless provide useful approximate information).

Examples

Example

(Taylor series of smooth function is asymptotic series)

The Taylor series of a smooth function $f \colon \mathbb{R} \to \mathbb{R}$ at any point is always an asymptotic expansion of $f$ around that point, regardless of whether its radius of convergence vanishes or not.

Proof

This follows from the Hadamard lemma, which says that for each $n \in \mathbb{N}$ and each expansion point $x_0 \in \mathbb{R}$ (which we may without restrict of generality assume to be $x_0 = 0$) there exists a smooth function $h_n \colon \mathbb{R} \to \mathbb{R}$ such that

$$f(x) = f(0) + x f^{(0)}(0) + \frac{1}{2} x^2 f^{(2)}(0) + \cdots + \frac{1}{n!} x^n f^{n}(0) + \frac{1}{(n+1)!} x^{n+1} h_n(x) \,,$$

where $f^{(k)} \colon \mathbb{R} \to \mathbb{R}$ denotes the $k$th derivative of $f$.

Therefore with

$$(a_k)_{k \in \mathbb{N}} \coloneqq \left( \frac{1}{k!} f^{(k)}(0) \right)_{k \in \mathbb{N}}$$

the coefficients of the Taylor series of $f$ at $x_0 = 0$, we have

$$\begin{aligned} \underset{x \to 0}{\lim} \left( \frac{1}{x^n} \left( f(x) - \sum_{k = 0}^n a_k x^k \right) \right) & = \underset{x \to 0}{\lim} \left( \frac{1}{x^n} \left( \frac{1}{(n+1)!} x^{n+1} h_n(x) \right) \right) \\ & = \underset{x \to 0}{\lim} \left( x \frac{1}{(n+1)!} h_n(x) \right) \\ & = 0 \cdot \frac{1}{(n+1)!} h_n(0) \\ & = 0 \end{aligned} \,.$$

Here in taking the limit we used from Hadamard's lemma that $h_n(x)$ and hence also $x h_n(x)$ is a smooth function, hence in particular a continuous function, on all of $\mathbb{R}$, hence that its limit as $x \to 0$ is just the value of the function at $x = 0$.

Remark

(Borel's theorem)

Beware that by Borel's theorem, every formal power series is the Taylor series of some smooth function, and of more than one smooth function; hence by example every formal power series is the asymptotic expansion of some smooth function, and of more than one smooth function.

Properties

Optimal truncation and superasymptotics

A rule-of-thumb for where to truncate an asymptotic series so that the resulting finite sum is as close as possible to the “actual” value is to truncate at the term that gives the smallest contribution. This rule-of-thumb is called optimal truncation, or superasymptotics (Berry-Howls 90, see Berry 91).

For some classes of asymptotic series there are proofs that “optimal truncation” indeed works, see the references below.

History

From Suslov 05:

Classical books on diagrammatic techniques $[$in perturbative quantum field theory$]$ describe the construction of diagram series as if they were well defined. However, almost all important perturbation series are hopelessly divergent since they have zero radii of convergence. The first argument to this effect was given by Dyson with regard to quantum electrodynamics.

$[$…$]$

Even though Dyson’s argument is unquestionable, it was hushed up or decried for many years: the scientific community was not ready to face the problem of the hopeless divergency of perturbation series.

$[$…$]$

The modern status of divergent series suggests that techniques for manipulating them should be included in a minimum syllabus for graduate students in theoretical physics. However, the theory of divergent series is almost unknown to physicists, because the corresponding parts of standard university courses in calculus date back to the mid-nineteenth century, when divergent series were virtually banished from mathematics.

Related concepts

• series, divergent series

• Taylor series, Puiseux series

• asymptotic series, transseries

• resummation, regular summation method, Borel summability

• perturbation series, Stokes phenomenon

• Feynman perturbation series

References

General

An original article is

• G. Watson, A theory of asymptotic series, Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character Vol. 211, (1912), pp. 279-313 (JSTOR)

Basic introductions include

• Joel Feldman, Taylor series and asymptotic expansions lecture notes pdf

• R. Shankar Subramanian, An Introduction to Asymptotic Expansions (pdf)

• Richard Chapling, Asymptotic methods, 2016 (pdf)

• Gerald Dunne, Introduction to Resurgence, Trans-series and

  Non-perturbative Physics_, 2018 (pdf)

Non-convergence of the Feynman perturbation series

The argument that the S-matrix formal power series in all perturbative quantum field theories of interest is necessarily divergent (and hence at best an asymptotic series) is due to

• Freeman Dyson, Divergence of perturbation theory in quantum electrodynamics, Phys. Rev. 85, 631, 1952 (spire)

made more precise in

• Lev Lipatov, Divergence of the Perturbation Theory Series and the Quasiclassical Theory, Sov.Phys.JETP 45 (1977) 216–223 (pdf)

recalled for instance in

• Igor Suslov, section 1 of Divergent perturbation series, Zh.Eksp.Teor.Fiz. 127 (2005) 1350; J.Exp.Theor.Phys. 100 (2005) 1188 (arXiv:hep-ph/0510142)

• Justin Bond, last section of Perturbative QFT is Asymptotic; is Divergent; is Problematic in Principle (pdf)

• Franco Strocchi, §2.2 of: An Introduction to Non-Perturbative Foundations of Quantum Field Theory, Oxford University Press (2013) [doi:10.1093/acprof:oso/9780199671571.001.0001]

• Stefan Hollands, Robert Wald, section 4.1 of Quantum fields in curved spacetime, Physics Reports Volume 574, 16 April 2015, Pages 1-35 (arXiv:1401.2026)

• Marco Serone, from 2:46 on in A look at $\phi^4_2$ using perturbation theory (recording)

In the example of phi^4 theory this non-convergence of the perturbation series is discussed in

• Robert Helling, p. 4 of Solving classical field equations (pdf)

Optimal truncation and superasymptotics

Discussion of “optimal truncation” of asymptotic series and of “superasymptotics” includes the following:

• Michael Berry, C. J. Howls, Hyperasymptotics, Proceedings: Mathematical and Physical Sciences Vol. 430, No. 1880 (Sep. 8, 1990), pp. 653-668 (jstor:79960)

• Michael Berry, Asymptotics, superasymptotics, hyperasymptotics, in H. Segur, S. Tanveer, and H. Levine, (eds.) Asymptotics Beyond All Orders, Plenum, Amsterdam, 1991, pp. 1-14 (doi:10.1007/978-1-4757-0435-8_1)

• O. Costin, M. D. Kruskal, On optimal truncation of divergent series solutions of nonlinear differential systems; Berry smoothing, Proc. R. Soc. Lond. A 455, 1931-1956 (1999) (arXiv:math/0608410)