Questions tagged [asymptotics]
167 questions
36
votes
2 answers
Can we get complete non-perturbative information of the interacting system by computing perturbation to all orders?
As we know, the perturbative expansion of interacting QFT or QM is not a convergent series but an asymptotic series that is generally divergent. So we can't get arbitrary precision of an interacting theory by computing higher enough order and adding…
maplemaple
- 2,217
28
votes
1 answer
Asymptoticity of Perturbative Expansion of QFT
It seems to be lore that the perturbative expansion of quantum field theories is generally asymptotic. I have seen two arguments.
There is the Dyson instability argument as in QED, that is showing the partition function is nonanalytic around the…
BebopButUnsteady
- 7,017
26
votes
2 answers
Conformal Compactification of spacetime
I have been reading Penrose's paper titled "Relativistic Symmetry Groups" where the concept of conformal compactification of a space-time is discussed. My other references have been this and this. In case you cannot see the paper above, let me…
Prahar
- 29,157
23
votes
3 answers
Why don't very high order Feynman diagrams contribute significantly?
In a particle physics lecture I had today it was stated that the magnetic moment, $g$, is not quite equal to 2, and the difference is accounted for by QED.
Later it was stated that we can see this because as well as first-order decays, there are…
T. Smith
- 331
20
votes
2 answers
Why do we use perturbative series if they don't converge?
My course instructor mentioned that the Perturbative Series are not convergent but diverge as we consider more and more terms in the expansion. He then briefly mentioned that the Perturbative Series are Asymptotic Series. I have some idea about…
Dev
- 317
18
votes
1 answer
Question about convergence of Dyson Series. Why Dyson series is in general divergent?
Given operator equation like:
$$i{\frac d{dt}}U(t,t_{0}) =V_I(t)U(t,t_{0})\tag{1} $$
The Dyson series solution is
\begin{array}{lcl}U(t,t_{0})&=&1-i\int _{{t_{0}}}^{{t}}{dt_{1}V_I(t_{1})}+(-i)^{2}\int _{{t_{0}}}^{t}{dt_{1}\int…
user168943
16
votes
4 answers
To what extent can one recover plane waves from the Airy eigenfunctions of a linear potential as the field is turned off?
Consider a single massive particle in one dimension under the action of a static linear potential, with the hamiltonian
$$
\hat H=\frac{\hat p^2}{2}+\hat{x}F_0.
$$
The eigenstate at energy $E$ is, with this normalization, given by
$$
\langle…
Emilio Pisanty
- 137,480
15
votes
1 answer
How can an asymptotic expansion give an extremely accurate prediction, as in QED?
What is the meaning of "twenty digits accuracy" of certain QED calculations? If I take too little loops, or too many of them, the result won't be as accurate, so do people stop adding loops when the result of their calculation best agrees with…
yonni
- 151
12
votes
1 answer
Renormalization group resummation
I'm having trouble in understeanding a mathematical feature of RG, namely how it provides a way to resum the perturbation series and how that's defined mathematically.
From a conceptual point of view one applies the RG flow to a 'theory' from a…
Fra
- 2,223
11
votes
2 answers
Can energy levels rise faster than $n^2$?
For a 1D particle in a box, energy levels are exactly proportional to $n^2$.
For the harmonic oscillator, $E_n\sim n$. And for a particle in an $|x|^\alpha$ potential, the energies are $\sim n^\beta$ with $\beta <2$ from the WKB approximation.
I…
FusRoDah
- 1,219
11
votes
1 answer
What is a 'turning point' in WKB and why does it fail at that point?
What is meant by a classical turning point in quantum mechanics and why does the WKB approximation fail at that point?
Roshan Shrestha
- 1,425
10
votes
4 answers
Asymptotic integral in Peskin & Schroeder, Problem 6.3
The question is about P&S QFT Problem 6.3. In question (a), the contribution to $a = \frac{g - 2}{2}$ from Higgs boson is calculated, the result is:
$$
\delta a = \frac{\lambda^2}{16 \pi^2} \int_0^1 \mathrm{d} x \frac{(1-x)^2(1+x)}{(1-x)^2 + x…
Jason Chen
- 647
10
votes
1 answer
In quantum field theory with a mass gap, why do states in the asymptotic future/past turn out to have a Fock space structure?
In quantum field theory with a mass gap, why do states in the asymptotic future/past turn out to have a Fock space structure? For a free quantum field theory, that's trivial, but why is that the case for interacting theories? In fact, the more one…
9
votes
0 answers
Questions on Penrose's paper - Conformal Treatment of Infinity
I have several questions. Perhaps it would be better to separate them into different posts. However, given their relative closeness to each other, I think putting it all in one place would be better. On suggestion, I will modify this post.
I am…
Prahar
- 29,157
9
votes
0 answers
Asymptotic Invariants in General Relativity
I was trying to understand Witten's proof of the Positive Energy Theorem in General Relativity by reading the original argument given by Witten. I am comfortable with the overall argument, but I would like to understand the following statement, made…
Rodrigo Barbosa
- 307