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Perturbation theory in the QM and the QFT is usually explained in terms of a small parameter expansion $\epsilon$ and expanded in a Taylor series.

$$O(t)=O_0(t)+\epsilon O_1(t)+\epsilon^2 O_t(t)+...$$

However, if we recall the course in transition from the real calculus to complex analysis, there is an emphasized transition from a Taylor series to Laurent series, where the terms of the divergent order were included, i.e.

$$\epsilon^{-1} O_{-1}(t)+\epsilon^{-2} O_{-2}(t)+...$$

But though complex numbers are popularly used, the perturbation approach is built from the Taylor series (at least in the QM and the QFT), and rarely were Laurent series used (except perhaps special branches like strings). This seems to be a bit counter intuitive from a mathematical perspective, and seemingly indicates some structures or assumptions built in the derivation of the theories.

Why does perturbation theoy involve a Taylor series rather than a Laurent series in QM and QFT?

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    $\begingroup$ What would be the point of doing perturbation theory where the perturbation parameter $\epsilon$ is small and then having terms that grow larger as $\epsilon$ gets smaller? The whole point of "perturbation" is the effect goes to zero as $\epsilon$ goes to 0. Can you point to any specific thing in standard perturbation theory where you think the expansion of something in a Taylor series instead of a Laurent series is wrong? Perturbation theory is, after all, not based on "let's use some Taylor series because we love Taylor series", it's based on "what happens if this parameter here is small"? $\endgroup$
    – ACuriousMind ♦
    Commented Nov 19, 2021 at 14:07
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    $\begingroup$ Laurent series are used all the time in conformal field theory, but the parameter is distance, and you expect things to diverge at small distances. A perturbation should, by definition, converge to something as the parameter goes to zero. $\endgroup$
    – Javier
    Commented Nov 19, 2021 at 14:16
  • $\begingroup$ @ACuriousMind The one point correlation function in a bosonic string was divergent and still "useful", and especially there's lots of poles in the complex analysis, but they were also useful through contour and residuals. Or may be $O_{-1}$ converged faster than $\epsilon$(through fitting) approach to $0$. In some cases the Laurent series were most useful when thing got small because most of the functions were asymptotically zero so that some path might be zeros. $\epsilon$ might be small but not "small enough", as with many practical cases except the fine tuning. $\endgroup$
    – ShoutOutAndCalculate
    Commented Nov 19, 2021 at 14:27
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    $\begingroup$ I'm not saying Laurent series can't be useful - I'm saying that by the very notion of what a "perturbation" is that whatever you're doing when you're using Laurent series is not perturbation theory. $\endgroup$
    – ACuriousMind ♦
    Commented Nov 19, 2021 at 15:14
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    $\begingroup$ Mathematically, if a function $f(x)$ is regular (analytic) at $x=0$, it can be expanded as a Taylor series (not Laurent) in $x$ around $x=0$. For most physical results, the result is regular (i.e. does not blow up) at $\epsilon=0$, where $\epsilon$ is your small parameter, so a Taylor series in $\epsilon$ is appropriate. $\endgroup$
    – printf
    Commented Nov 19, 2021 at 23:42

2 Answers 2

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This is a nice question. However, notice that the idea behind perturbation theory is being able to write $O(t)$ as $O_0(t) \ +$ higher order terms, that should be much less relevant, i.e., \begin{equation} |\varepsilon^k O_k(t)|\ll |O_0(t)|, \forall k>0 \end{equation} Thus you are interested mostly in the case $\varepsilon \to 0$. Then it doesn't make much sense to consider Laurent series: you are just introducing unnecessary divergences.

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Generically, as has been said, it is nice if the observable you're trying to compute is well defined as $\epsilon\to0$.

However, there are certain phenomena which can happen which go in the direction you mention. Firstly, when $\epsilon$ is the dimensional regularisation parameter (a regularisation scheme when we deform away from integer 4 dimensions to $4+\epsilon$), infrared divergences of theories which suffer from them like Yang-Mills do show up as higher order terms in $1/\epsilon$, but this is the signal that something goes bad with the model.

Furthermore, a very physical phenomenon which generalises the Taylor expansion you mention, and occur in QM and QFT, is the presence of non-perturbative effects (instantons, etc.) which show up as $\exp(-1/g)$ effects. They are thus invisible in the Taylor expansion near weak coupling $g=0$ but do affect the summability of the perturbative series.

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