Timeline for Why does perturbation theory involve a Taylor series rather than a Laurent series?
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13 events
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| Nov 20, 2021 at 12:00 | history | tweeted | twitter.com/StackPhysics/status/1462027952804220941 | ||
| Nov 19, 2021 at 23:42 | comment | added | printf | 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. | |
| Nov 19, 2021 at 21:58 | history | became hot network question | |||
| Nov 19, 2021 at 18:27 | comment | added | ShoutOutAndCalculate | @ACuriousMind I see. Thank you. So technically the "perturbation" by definition worked with the restrict domain to which the Laurent series could not obtain the useful results, (and then sometimes renormalization patched the different domains together?) But was there possible a situation(experience) for which the Laurent series that's necessarily be realized than the Taylor's series? For example, if there's a case to predicatively remove an infinity at lower order? | |
| Nov 19, 2021 at 15:26 | history | edited | Qmechanic♦ |
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| Nov 19, 2021 at 15:15 | answer | added | picop | timeline score: 2 | |
| Nov 19, 2021 at 15:14 | comment | added | ACuriousMind♦ | 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. | |
| Nov 19, 2021 at 14:27 | comment | added | ShoutOutAndCalculate | @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. | |
| Nov 19, 2021 at 14:16 | comment | added | Javier | 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. | |
| Nov 19, 2021 at 14:11 | answer | added | Henrique Calazans Prates | timeline score: 6 | |
| Nov 19, 2021 at 14:07 | comment | added | ACuriousMind♦ | 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"? | |
| Nov 19, 2021 at 14:00 | history | edited | Nihar Karve | CC BY-SA 4.0 |
added 21 characters in body; edited tags; edited title
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| Nov 19, 2021 at 13:55 | history | asked | ShoutOutAndCalculate | CC BY-SA 4.0 |