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Fourier transform of the Coulomb potential

I really appreciate the physical explanations made in other answers, but I want to add that Fourier transform of the Coulomb potential makes mathematical sense, too. This answer is meant to clarify ...
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I'm missing the point of renormalization in QFT

Here's the thing: renormalization and divergences have nothing to do with each other. They are conceptually unrelated notions. Renormalization Simply put, renormalization is a consequence of non-...

Why are we scared of singularities?

"To me it seems like negative or complex numbers. We used to hate these things but now they are more generally accepted. " Indeed. And in a general context, the infinite answer that some equations ...
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Fourier transform of the Coulomb potential

I am personally fond of this short and sweet argument. If you believe the relatively easy to prove fact that in 3 dimensions $$\nabla^2\frac{1}{r} = -4\pi \delta^{(3)}(\bf{r})$$ then taking the ...
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What's the relation between Wilson Renormalization Group (RG) in Statistical Mechanics and QFT RG?

This is of course a very subtle problem, and I will only scratch the surface here. I think that the best reference for this question is the discussion by B. Delamotte in arxiv:0702.365, Section 2.6- "...
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On scheme dependence in QFT renormalization

If you look at the PDG, equation ($10.7d$), you'll see that they define the electromagnetic constant $\alpha$ at a precise scale (the mass of the muon in this case). Later on they also give the value ...
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How to show Pauli-Villars regularization introduce a momentum cut-off?

When $p^2\gg \Lambda$, as you can see directly from your "modified" expression, the propagator scales as $\sim 1/p^4$ instead of $\sim 1/p^2$. If you go through arguments about the ...
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The underlying cause of ill-defined loop-integrals in Quantum Field Theory

$$\intop_{-\infty}^{\infty} d\omega\, e^{i \omega x} = 2\pi \delta(x)$$ How is it possible that starting with a non-distributional anzatz I arrived at a distribution (a Dirac delta function)? There ...
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Why does non-perturbative QCD need to be regularized and renormalized?

The Wilsonian viewpoint of renormalization (see this excellent answer by Abdelmalek Abdesselam) is not conceptually tied to perturbative expansions at all. Rather, it conceives of a quantum field ...
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How do we know that analytic continuation agrees with UV regulators?

Your example has no counterterms to cancel the 1/r singularity! This explains the discrepancy. In a correctly regularized expression, the counterterms are not introduced in an ad hoc way but by ...
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Showing $I=\int d^3k\int dk^0\frac{1}{k^4}$ to be logarithmically divergent

The correct definition of the integral is $$I = \int_{\mathbb{R}^3} d^3\mathbf{k}\int_{-\infty}^{\infty} d k^0 \,\frac{1}{(|\mathbf{k}|^2 - (k^0)^2 - i\varepsilon)^2}\,.$$ The "$+i\varepsilon$" is ...
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Why is the $i\epsilon$-prescription necessary in the Klein-Gordon propagator?

Note that the original integral you are trying to compute is over the real line, not over a closed contour, so the Cauchy theorem does not apply until you find a suitable way to close the contour. Due ...
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Are constraint forces infinite?

OP already seems to have thought long and hard about this and makes good points. In this answer we will review the argument for why constraint forces could be infinite. We will assume that OP talks ...
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Euler-Maclaurin formula for Casimir Effect

Trick 1: Rewrite the integral so it has the bounds you want. \int_0^\infty \nu d \nu = \int_0^1\nu d \nu + \int_1^\infty \nu d \nu = \frac{1}{2} + \int_1^\infty \nu d \nu \end{...
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One-loop Integral for a Tensor Quantity

Define the tensor, $$A_{\mu\nu} = \int \frac{d^dk}{(2\pi)^d} \frac{ \eta_{\mu\nu} + \frac{k_\mu k_\nu }{M^2 } }{ k^2 + M^2 - i \epsilon }$$ The important thing to note is that this tensor is ...
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Why do different contours give different answers in the limit $\epsilon \rightarrow 0$ when calculating propagators?

Here is a cute toy example from What is a Quantum Field Theory? A First Introduction for Mathematicians. M. Talagrand. CUP. Appendix N: Feynman Propagator and Klein-Gordon Equation, which hopefully ...
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Why are we scared of singularities?

Those infinities can mean the breakdown of the theory. Blackbody radiation predictions using classical theory says that a blackbody will radiate an infinite amount of energy. It meant the theory ...
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What is a point-split?

A point-splitting procedure is one way to make sense of composite fields in QFT. As an easy example, take a free scalar field $\phi(x)$ in Euclidean signature, in $d$ dimensions. Consider the problem ...