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\begin{document}

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\section*{asymptotic expansion}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{formal_geometry}{}\paragraph*{{Formal geometry}}\label{formal_geometry}

[[!include formal geometry -- contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{in_perturbative_quantum_field_theory}{In perturbative quantum field theory}\dotfill \pageref*{in_perturbative_quantum_field_theory} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{OptimalTruncation}{Optimal truncation and superasymptotics}\dotfill \pageref*{OptimalTruncation} \linebreak
\noindent\hyperlink{history}{History}\dotfill \pageref*{history} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general_2}{General}\dotfill \pageref*{general_2} \linebreak
\noindent\hyperlink{nonconvergence_of_the_feynman_perturbation_series}{Non-convergence of the Feynman perturbation series}\dotfill \pageref*{nonconvergence_of_the_feynman_perturbation_series} \linebreak
\noindent\hyperlink{ReferencesOptimalTruncation}{Optimal truncation and superasymptotics}\dotfill \pageref*{ReferencesOptimalTruncation} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

An \emph{asymptotic expansion} of a [[function]] is a [[formal power series]] that may not [[convergence|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 \ref{TaylorSeriesOfSmoothFunctionIsAsymptoticSeries} below) around any point. Beware that by [[Borel's theorem]] this means that \emph{every} [[formal power series]] is the asymptotic expansion of \emph{some} smooth function and of more than one smooth function (remark \ref{TheoremBorel} below).

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

\hypertarget{in_perturbative_quantum_field_theory}{}\subsubsection*{{In perturbative quantum field theory}}\label{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 (\hyperlink{Dyson52}{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 field theory|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]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{}\hypertarget{}{}
Given a [[function]] $f \colon \mathbb{R} \to \mathbb{R}$, a [[formal power series]] $\sum_{n = 0}^\infty a_n x^n$ is an \textbf{asymptotic expansion} of $f$ at $x = 0$ if for each $n \in \mathbb{N}$ the [[limit of a sequence|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:

\begin{displaymath}
\underset{x \to 0}{\lim} 
  \left( 
    \frac{1}{x^n}
    \left(
      f(x)  - \sum_{k = 0}^n a_k x^k
    \right)
  \right)
  \;=\; 
  0
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
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).

\end{remark}
\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

\begin{example}
\label{TaylorSeriesOfSmoothFunctionIsAsymptoticSeries}\hypertarget{TaylorSeriesOfSmoothFunctionIsAsymptoticSeries}{}
\textbf{([[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.

\end{example}
\begin{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

\begin{displaymath}
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)
  \,,
\end{displaymath}
where $f^{(k)} \colon \mathbb{R} \to \mathbb{R}$ denotes the $k$th [[derivative]] of $f$.

Therefore with

\begin{displaymath}
(a_k)_{k \in \mathbb{N}}
  \coloneqq
  \left(
    \frac{1}{k!} f^{(k)}(0)
  \right)_{k \in \mathbb{N}}
\end{displaymath}
the [[coefficients]] of the [[Taylor series]] of $f$ at $x_0 = 0$, we have

\begin{displaymath}
\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}
  \,.
\end{displaymath}
Here in taking the [[limit of a sequence|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 $x = 0$.

\end{proof}
\begin{remark}
\label{TheoremBorel}\hypertarget{TheoremBorel}{}
\textbf{([[Borel's theorem]])}

Beware that by [[Borel's theorem]], \emph{every} [[formal power series]] is the [[Taylor series]] of \emph{some} [[smooth function]], and of more than one smooth function; hence by example \ref{TaylorSeriesOfSmoothFunctionIsAsymptoticSeries} every formal power series is the asymptotic expansion of \emph{some} smooth function, and of more than one smooth function.

\end{remark}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{OptimalTruncation}{}\subsubsection*{{Optimal truncation and superasymptotics}}\label{OptimalTruncation}

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 \emph{smallest} contribution. This rule-of-thumb is called \emph{optimal truncation}, or \emph{superasymptotics} (\hyperlink{BerryHowls90}{Berry-Howls 90}, see \hyperlink{Berry91}{Berry 91}).

For some classes of asymptotic series there are [[proofs]] that ``optimal truncation'' indeed works, see the references \hyperlink{ReferencesOptimalTruncation}{below}.

\hypertarget{history}{}\subsection*{{History}}\label{history}

From \hyperlink{Suslov05}{Suslov 05}:

\begin{quote}%
Classical books on diagrammatic techniques $[$in [[perturbative quantum field theory]]$]$ describe the construction of [[Feynman diagram|diagram]] series as if they were well defined. However, almost all important [[Feynman perturbation series|perturbation series]] are hopelessly [[divergent series|divergent]] since they have zero [[radii of convergence]]. The first argument to this effect was \hyperlink{Dyson52}{given by Dyson} with regard to [[quantum electrodynamics]].

$[$\ldots{}$]$

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.

$[$\ldots{}$]$

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.


\end{quote}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[series]], [[divergent series]]


\item [[Taylor series]], [[Puiseux series]]


\item [[asymptotic series]], [[transseries]]


\item [[resummation]], [[regular summation method]], [[Borel summability]]


\item [[perturbation series]], [[Stokes phenomenon]]


\item [[Feynman perturbation series]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{general_2}{}\subsubsection*{{General}}\label{general_2}

An original article is

\begin{itemize}%
\item G. Watson, \emph{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 (\href{http://www.jstor.org/stable/91005}{JSTOR})

\end{itemize}
Basic introductions include

\begin{itemize}%
\item Joel Feldman, \emph{Taylor series and asymptotic expansions} lecture notes \href{http://www.math.ubc.ca/~feldman/m321/asymptotic.pdf}{pdf}


\item R. Shankar Subramanian, \emph{An Introduction to Asymptotic Expansions} (\href{https://web2.clarkson.edu/projects/subramanian/ch561/notes/Asymptotic%20Expansions.pdf}{pdf})


\item Richard Chapling, \emph{Asymptotic methods}, 2016 (\href{https://rc476.user.srcf.net/asymptoticmethods/am_notes.pdf}{pdf})


\item Gerald Dunne, \emph{Introduction to Resurgence, Trans-series and Non-perturbative Physics}, 2018 (\href{https://www.icts.res.in/sites/default/files/NUMSTRINGS2018-2018-01-27-Gerald-Dunne.pdf}{pdf})



\end{itemize}
\hypertarget{nonconvergence_of_the_feynman_perturbation_series}{}\subsubsection*{{Non-convergence of the Feynman perturbation series}}\label{nonconvergence_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

\begin{itemize}%
\item [[Freeman Dyson]], \emph{Divergence of perturbation theory in quantum electrodynamics}, Phys. Rev. 85, 631, 1952 (\href{http://inspirehep.net/record/29799?ln=en}{spire})

\end{itemize}
made more precise in

\begin{itemize}%
\item [[Lev Lipatov]], \emph{Divergence of the Perturbation Theory Series and the Quasiclassical Theory}, Sov.Phys.JETP 45 (1977) 216--223 (\href{http://jetp.ac.ru/cgi-bin/dn/e_045_02_0216.pdf}{pdf})

\end{itemize}
recalled for instance in

\begin{itemize}%
\item [[Igor Suslov]], section 1 of \emph{Divergent perturbation series}, Zh.Eksp.Teor.Fiz. 127 (2005) 1350; J.Exp.Theor.Phys. 100 (2005) 1188 (\href{https://arxiv.org/abs/hep-ph/0510142}{arXiv:hep-ph/0510142})


\item Justin Bond, last section of \emph{Perturbative QFT is Asymptotic; is Divergent; is Problematic in Principle} (\href{https://mcgreevy.physics.ucsd.edu/s13/final-papers/2013S-215C-Bond-Justin.pdf}{pdf})


\item [[Stefan Hollands]], [[Robert Wald]], section 4.1 of \emph{Quantum fields in curved spacetime}, Physics Reports Volume 574, 16 April 2015, Pages 1-35 (\href{https://arxiv.org/abs/1401.2026}{arXiv:1401.2026})


\item Marco Serone, from 2:46 on in \emph{A look at $\phi^4_2$ using perturbation theory} (\href{https://www.youtube.com/watch?v=J4nxvY1rOhI}{recording})



\end{itemize}
In the example of [[phi{\tt \symbol{94}}4 theory]] this non-convergence of the perturbation series is discussed in

\begin{itemize}%
\item Robert Helling, p. 4 of \emph{Solving classical field equations} (\href{http://homepages.physik.uni-muenchen.de/~helling/classical_fields.pdf}{pdf})

\end{itemize}
\hypertarget{ReferencesOptimalTruncation}{}\subsubsection*{{Optimal truncation and superasymptotics}}\label{ReferencesOptimalTruncation}

Discussion of ``optimal truncation'' of asymptotic series and of ``superasymptotics'' includes the following:

\begin{itemize}%
\item [[Michael Berry]], C. J. Howls, \emph{Hyperasymptotics}, Proceedings: Mathematical and Physical Sciences Vol. 430, No. 1880 (Sep. 8, 1990), pp. 653-668 (\href{https://www.jstor.org/stable/79960}{jstor:79960})


\item [[Michael Berry]], \emph{Asymptotics, superasymptotics, hyperasymptotics}, in H. Segur, S. Tanveer, and H. Levine, (eds.) \emph{Asymptotics Beyond All Orders}, Plenum, Amsterdam, 1991, pp. 1-14 (\href{https://doi.org/10.1007/978-1-4757-0435-8_1}{doi:10.1007/978-1-4757-0435-8\_1})


\item O. Costin, M. D. Kruskal, \emph{On optimal truncation of divergent series solutions of nonlinear differential systems; Berry smoothing}, Proc. R. Soc. Lond. A 455, 1931-1956 (1999) (\href{https://arxiv.org/abs/math/0608410}{arXiv:math/0608410})



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