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So, I once watched a video (I think it was by Andrew Dotson) that I can't find any more about techniques used to deal with singularities in path integrals. The presenter mentioned how the integral formula:

$$\int x^{n} \; \mathrm{d} x = \frac{1}{n+1}x^{n+1} + C$$

cannot be used for $n=-1$. However, to deal with the singularity at $n=-1$, the presenter said to choose $C = - \frac{1}{n+1}$, and taking the following limit:

$$\lim_{n \to -1} \left( \frac{1}{n+1}x^{n+1} - \frac{1}{n+1} \right)$$

You approach the natural log function as $n$ approaches $-1$. I think the presenter said later that a similar idea is used to deal with path integrals with singularities, but I forget what. How does this relate to getting rid of singularities, and what do these types of integrals with singularities represent in physics? I know that path integrals are used in Quantum Mechanics, so how must a particle behave that necessitates getting rid of a singularity in an integral?

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  • $\begingroup$ This is called dimensional regularization. In practice one has to evaluate various integrals in 4 dimensions, but these integrals are rarely convergent. So instead you can do the integral in d dimensions, and plug in $d=4-\epsilon$ and get rid of the infinities by subtracting the appropriate terms that depend on $epsilon$. $\endgroup$ Oct 12, 2020 at 0:56
  • $\begingroup$ @AdolfoHolguin Ok, that's actually really helpful and I'll look into it. What are these integrals for though, like what do they represent? $\endgroup$
    – Eliot
    Oct 12, 2020 at 1:34
  • $\begingroup$ @AdolfoHolguin Note that answering in comments is strongly discouraged. We need people to write proper answers so that they can be voted on, act as a permanent record for searches by other people and so the OP can, if they wish, accept one as a preferred answer. Also note you get more reputation points for an upvote on an answer than on a comment and there is the potential to gain badges for good answers. $\endgroup$ Oct 12, 2020 at 9:03

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I'm glad to see someone was interested in my video! As commenters deduced, the point of the video was that some divergent integrals (not necessarily in the path integral formalism) in QFT may still be manipulated in a systematic way (through regularization and renormalization) to give a finite result. Part of the Feynman rules for evaluating Feynman diagrams include imposing momentum conservation and integrating over undetermined momentum. The latter is especially characteristic of Feynman diagrams which contain loops. The integral you shared was an analogy I gave for one method of removing the divergence through the addition of a divergent counter term, which, in the case of the analogy, was given by $$-\frac{1}{n+1}$$ See Collins book on renormalization for more info than you can stomach on counter terms. By the way, the clip you're referring to is in the following video from Flammable Maths that I took part in (around 23:29): https://www.youtube.com/watch?v=UdnDjJIkkmo&t=1440s&ab_channel=FlammableMaths

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