58
votes
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 ...
- 850
45
votes
Accepted
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-...
30
votes
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 ...
- 88.9k
19
votes
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 ...
- 1,485
19
votes
Accepted
Dimensional Regularization and Massless Integrals
You have to be careful when performing certain operations in dimensional regularization, and you always have to check the convergence.
What meaning does an integral like
$$
\int\frac{d^Dk}{(2\pi)^D}\...
- 1,834
17
votes
What does mathematical consistency in QFT mean?
I have never heard people talk about mathematical "consistency" of QFT in any other sense than that of rigor. Your mention of "Wightman axioms" supports this.
This isn't about the ...
- 127k
16
votes
Accepted
Gelfand-Yaglom theorem for functional determinants
User Simon has already given a good answer. Here we sketch a derivation of the Gelfand-Yaglom formula.
Let there be given a self-adjoint Hamiltonian operator
$$H~=~H^{(0)}+V, \tag{1}$$
with non-...
- 209k
14
votes
Accepted
Dirac delta function defined in Zee's Quantum Field Theory book
When considering a nascent delta function $\delta_{\epsilon}:\mathbb{R}\to \mathbb{C}$ with a regularization parameter$^1$ $\epsilon>0$, it is not necessary that (the Lebeque measure of) the ...
- 209k
14
votes
If quantum fields are operator valued distributions, why aren't they always smeared?
Yes, the quantum fields must be smeared in order to become well-behaved (symmetric, densely defined) operators (in the Hilbert space of the theory). In mathematically-minded textbooks it is the ...
- 75.6k
13
votes
Accepted
What is a logarithmic divergence?
The term 'logarithmic divergence' is normally used for integrals of the type
$$
F(x) = \int_{x_0}^x \frac{1}{\xi}\mathrm d\xi
$$
(or possibly of the form $F(x) = \int_{x_0}^x \frac{1}{\xi}f(\xi)\...
- 135k
13
votes
Accepted
Why is there no anomaly when particle mechanics is quantized?
Quantum mechanics can also become anomalous. An example is a charged particle moving in a uniform magnetic field. On the classical level, the system is translation invariant in both x- and y-direction....
- 12k
13
votes
Accepted
Water wave analog of Casimir effect — Does it involve the zeta function? If not, why do QED calculations involve the zeta function?
At some point in the calculation of the Casimir effect, you have to sum over the energy contained in each of mode of the cavity
\begin{equation}
E = \sum_n E_n
\end{equation}
where $n$ labels the mode ...
- 52.3k
12
votes
I'm missing the point of renormalization in QFT
Here's some complementary perspective to the excellent answer by AccidentalFourierTransform. This turned out very long, but this is a huge topic which can't be entirely summarized in one answer. An ...
- 3,750
11
votes
Accepted
Why can consistent QFTs only arise from CFTs?
Although I disagree with the definition of well-behaved QFT (why are these people always insisting on taking the continuum limit ?), the reason is the following.
If one wants to take the continuum ...
- 12k
11
votes
Accepted
Wick rotation vs. Feynman $i\varepsilon$-prescription
Starting in the Minkowski formulation, the Feynman $i\varepsilon$-prescription is just the first infinitesimal angle $\theta=\varepsilon$ of a Wick rotation
$$\begin{align} t(\theta) ~=~& e^{i\...
- 209k
10
votes
Accepted
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- "...
- 12k
10
votes
Accepted
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 ...
- 6,856
10
votes
Accepted
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 ...
- 52.3k
10
votes
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 ...
- 16.2k
10
votes
Accepted
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 ...
- 127k
9
votes
Accepted
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 ...
- 45.4k
9
votes
Accepted
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 ...
- 6,856
9
votes
Accepted
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 ...
- 2,053
9
votes
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 ...
- 209k
9
votes
Euler-Maclaurin formula for Casimir Effect
Trick 1: Rewrite the integral so it has the bounds you want.
\begin{equation}
\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{...
- 52.3k
9
votes
Accepted
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 ...
- 27k
9
votes
Accepted
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 ...
- 8,375
8
votes
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 ...
- 14.1k
8
votes
Accepted
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 ...
- 6,167
8
votes
Accepted
"Dimensional analysis" arguments in quantum field theory
A good way to do dimension analysis in computing amplitudes relies on a good power-counting of the action. Let me explain how it works before answering your question. For simplicity, in the following, ...
- 2,197
Only top scored, non community-wiki answers of a minimum length are eligible