Questions tagged [asymptotics]
The asymptotics tag has no usage guidance.
112
questions
1
vote
0
answers
14
views
How do asymptotic symmetries help to get information about black holes?
I've heard that asymptotic symmetries (like BMS) are used to get information about black holes just by looking at their fields. How does that work?
3
votes
1
answer
113
views
+100
In and out states of scattering in Asymptotically flat spacetimes
I am reading a paper called "New symmetries of massless QED", written by Temple He, Prahar Mitra, Achilleas P. Porfyriadis and Andrew Strominger (https://arxiv.org/abs/1407.3789). At some ...
0
votes
0
answers
39
views
Was asymptotic expansion also a form of symmetry?
Consider the infinitesimal expansion, which was used to describe the behavior of of the expression when taking the parameter to be small.
The infinitesimal expansion was usually used to describe the ...
0
votes
1
answer
52
views
A calculation of microstates
Pathria, Statistical mechanics pg 11,4ed
In order to find the number of microstates $\Omega(N,V,E$) author writes
" In other words, we have to determine the total number of (independent) ways of ...
1
vote
0
answers
51
views
Help to evaluate an integral given in appendix of Quantum Field Theory in a Nutshell [duplicate]
On p. 16 in appendix 3 in section I.2 of Quantum Field Theory in a Nutshell by Zee the integral to be evaluated is $$I = \int_{-\infty}^{+\infty}dqe^{-(1/\hbar)f(q)}.$$
Where $f(q)$ is expanded as
$$...
-1
votes
2
answers
179
views
Thought experiment on boundary condition of galaxies
A thought experiment: Let's assume that there is only one single galaxy in the whole universe.
How would it look like regarding the curvature of spacetime?
Would the spacetime be flat in the infinity ...
0
votes
0
answers
47
views
Evolution operator as a Laurent series of coupling constant
Let the Hamiltonian be $H_{0}+gV$, where $g$ is the coupling constant. In the interaction picture, the equation for the evolution operator is $i\frac{dU}{dt}=gV_{I}U$. What I am going to do is assume $...
1
vote
0
answers
66
views
Integral partition function of a cubic anharmonic oscillator Energy complex values [closed]
I am interesting in the following integral
$$\int_{-\infty }^{\infty } e^{-\frac{g z^3}{6}-\frac{z^2}{2}} \, dz.$$ Mathematica does not provide any result nor maple either I try to used$$ \text{...
0
votes
1
answer
73
views
Should the $S$-matrix always analytic in coupling constant?
If we use Dyson series, the $S$-matrix is always an analytic function of the coupling constant. However, if that is the case, how can non-perturbative effects arise in QFT? My question is, should the $...
0
votes
2
answers
71
views
Perturbation theory and size of the perturbation
In quantum field theory, we usually perturb the free field by a little bit. What would be so bad about using a large perturbation to the free field?
0
votes
0
answers
42
views
Confusion about this explanation of the radial hydrogen Schrodinger solution
I was reading this page about solving the radial part of hydrogen's energy eigenstates. They explain how to solve for the asymptotic behavior $R_{\infty}(r)$ first by taking the limit of the ...
0
votes
0
answers
47
views
Radius of convergence of beta function of fine structure constant
I'm looking at the beta function of the fine structure constant $\alpha=\frac{e_R^2}{4\pi}$
\begin{equation}
\beta(\alpha)=\mu \frac{d\alpha}{d\mu}=-2 \alpha \left[ \frac{\epsilon}{2}+ \left(\frac{\...
0
votes
0
answers
30
views
Fall-off of Klein-Gordon massless field in flat spacetime (proof from Wald)
In Wald's General Relativity (1984) he devotes one of the last chapters to asymptotic flatness. He starts by showing how the conformal compactification of Minkowski spacetime can be used to determine ...
1
vote
2
answers
43
views
Why $E$ is neglected at large and small $r$ of quantum harmonic oscillator?
In obtaining radial solution of quantum oscillator why E is neglected?
Radial equation:
Resource: nouredine zettili.
1
vote
2
answers
73
views
Finiteness of Maxwell gauge field symplectic form?
The symplectic form for a Maxwell $U(1)$ gauge field is
$$
\omega = \int_\Sigma d \Sigma^\mu \delta F_{\mu \nu} \wedge \delta A^\nu
$$
where $\wedge$ acts on field space. (In this notation, $\delta$ ...
2
votes
1
answer
125
views
Integration by parts in derivation of LSZ reduction formula
This is something that every text book or notes skips to explain in the derivation of the LSZ reduction formula
Suppose we have $$
a_{1}^{\dagger} \equiv \int d^{3} k f_{1}(\mathbf{k}) a^{\dagger}(\...
2
votes
0
answers
119
views
Deriving the large $E$ expansion for geodesic boundary time from paper arXiv:2004.01192
In equation (14) of the paper "Holographic flows from CFT to the Kasner universe" https://arxiv.org/abs/2004.01192, they express the boundary time as
$$\label{1} t(0) = -P \int^{r_{\star}}_{...
0
votes
0
answers
35
views
Asymptotic behaviour of Green's function in real space
It is well known that in $k$-space, a Green's function usually has asymptotic behaviour $\frac{1}{\omega}$ as $\omega \to \infty$. Are there any similar result in position space? This idea is inspired ...
1
vote
0
answers
109
views
Difference between asymptotically AdS and asymptotically locally AdS spacetime
In the literature, there is often a distinction made between spacetimes that are asymptotically or asymptotically locally some other spacetime. For example, in holography, referring to some spaces ...
2
votes
0
answers
76
views
Schrodinger equation with parameters
I need to know the ground state energy $E_0$ defined by the following stationary Schrodinger equation:
$$
-\frac{a}{2}\phi''(\xi) + \left(\frac1{2a}\sinh^2(2\xi) + (2b-1)\cosh(2\xi)\right)\phi(\xi) = ...
7
votes
1
answer
196
views
Asymptotic Series in QFT: What to do when all "trustworthy" terms are known?
In my Introduction to QFT lecture, we quantized a Klein-Gordon Field and as a toy model we looked at $\phi^3$ theory. For this toy model we expanded the $S = U(-\infty, \infty)$ operator in a series (...
2
votes
1
answer
97
views
Asymptotic form of a Coulomb-like integral
I need to evaluate or work out the asymptotic scaling of the following integral:
\begin{equation}
I~=~\int_{\mathbb{R}^3} dq d^2p \frac{e^{i\vec{p}\cdot \vec{r}}e^{iq z}}{p^2 + \frac{1}{g^2}q^4}
\end{...
1
vote
2
answers
94
views
What parameter can you expand QFT in for a convergent series?
It is known that the expansion in terms of Feynman diagrams with a series in terms of the coupling constant is an asymptotic series (i.e. it starts off a good approximation but eventually diverges). (...
6
votes
0
answers
135
views
How can QED by predictive if it diverges?
One of the tests of Quantum Electrodynamics is the value of the "Anomalous magnetic dipole moment".
The theoretical value is:
$$a_e = 0.001\ 159\ 652\ 181....$$
We say that QED "...
0
votes
1
answer
171
views
Maxwell-Boltzmann distribution and area under the curve
If the curve of MBDist is asymptotic on the x-axis does this mean the area under the curve (no. of particles) is infinite?
3
votes
1
answer
94
views
Why $\int_{\mathcal I_+^+} \varepsilon*F=0$ for any $\varepsilon$ when there are no massive charges?
My problem is really simple. I was reading the Strominger lectures where he defines the future charges $Q_\varepsilon^+$, and he does something that I don't understand. He says on the equation $(2.5.4)...
3
votes
0
answers
71
views
Free massive propagator as an OPE?
Consider a free massive propagator $$G(p)\equiv\frac{1}{p^2+m^2}$$
There is a 'naive' expansion in terms of the mass
$$G(p)\sim\sum^\infty_{n=0} (-1)^n \frac{m^{2n}}{p^{2(n+1)}}$$
This expansion might ...
0
votes
1
answer
124
views
How and why the state of free particle in quantum physics is represented by plane wave packet? [closed]
In Quantum Mechanics (Cohen Tannoudji) Topic: "Asymptotic Form Of Stationary Scattering States"
It is written that for large negative values of $t$, the incident particle is free and it's ...
1
vote
1
answer
113
views
Analysis of the eigenvalues of the particle in a finite square well
The eigenstates of the particle in a 1D finite square well Hamiltonian:
\begin{align}
H = \frac{\hat{p}^2}{2m} + V(x)
\end{align}
\begin{align}
V(x) =
\begin{cases}
-V_0 & \...
5
votes
1
answer
191
views
Divergent Energies and Analytical Continuation - Two questions on the inverted harmonic oscillator and the inverted double well
I have two questions on the general topic of energy potentials that diverge at infinity.
First of all, the inverted harmonic oscillator. I found this post on Physics SE, Inverted Harmonic oscillator.
...
2
votes
1
answer
89
views
Confusions on expectation value for $\hbar$ going to zero
In Matthew D. Schwartz's QFT book, Chapter 28, the author claims when $\hbar \rightarrow 0$, the following equality (eq 28.4) holds:
So how can I see the second "$=$" holds? It seems the ...
6
votes
1
answer
183
views
Past boundary of $\mathcal{I}^+$ and future boundary of the hyperboloid resolving $i^0$
Let us consider Minkowski spacetime. Let $(u,r,x^A)$ be retarded coordinates with $x^A$ coordinates on the sphere. Future null infinity is described here as the $r\to \infty$ limit with $(u,x^A)$ ...
3
votes
0
answers
104
views
Why can I not asymptotically expand a Feynman integral this way?
I would like to asymptotically expand a series of Feynman diagrams in Euclidean space, and as a toy I started with the following integral, for which I know the full solution in $4d$ ($\omega \to 2$):
...
3
votes
1
answer
50
views
How are the authors obtaining the asymptotic form of the sympletic form for the Maxwell + massive field system?
I've been studying the paper "Asymptotic symmetries of QED and Weinberg’s soft photon theorem" by Campiglia & Laddha and there is one step in their analysis I'm being unable to understand. I shall ...
7
votes
1
answer
217
views
How can QED have degenerate vacua without superselection?
This question is based on Andrew Strominger's lecture on the IR structure of field theories, in particular Section 2.11
The usual story with spontaneous symmetry breaking is a follows. You have ...
3
votes
1
answer
312
views
How is Large Gauge Transformation a kind of global symmetry if it varies from point to point?
In "Lectures on the Infrared Structure of Gravity and Gauge Theories", Strominger considers the so-called asymptotic symmetries. If I got it right, the basic idea is that one chooses a set of falloff ...
4
votes
2
answers
630
views
The use of $a^\dagger(\mathbf{k}) = -i \int d^3x e^{ikx}\stackrel{\leftrightarrow}{\partial}_0 \phi(x)$ in the derivation of the LSZ-formula
I noticed that in Srednicki's derivation of the LSZ-formula the expression (chapter 5) for the creation (and also later for the annihilation) operator by the field operator:
$$a^\dagger(\mathbf{k}) = -...
2
votes
0
answers
37
views
The radiative modes of an asymptotically flat spacetime and the symmetries
In Ashtekar's paper, the radiative degrees of freedom of an asymptotic flat spacetime in general relativity are obtained. These degrees of freedom live on the null infinity $\mathscr I$, given by the ...
2
votes
0
answers
80
views
Why do we require that the gauge condition $\alpha(x)$ falls off at infinity?
Let's say we are working in QED. The lagrangian is
\begin{equation}
\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\bar{\psi}\big(i\not{D}-m\big)\psi
\end{equation}
where $F_{\mu\nu}=\partial_{\mu}A_{\...
0
votes
1
answer
112
views
QFT in in the asymptotic region
Let $\phi(x)$ be a scalar field operator. It often postulate in text books that in the asymptotic region we have
$$\lim_{x_0\to-\infty} \phi(x)=\sqrt Z \phi_{in}(x)$$
where $Z$ is a constant.
The ...
1
vote
0
answers
24
views
The variation of the flux of the linkage for the asymptotically flat spacetime
The concept of the linkage for the asymptotically flat spacetime is defined and dicussed in this paper by Geroch and Winicour. This is a nice paper, but I came across one problem.
The linkage is an ...
0
votes
0
answers
99
views
Why is the modified spherical Bessel function an asymptotic solution of this ODE?
I am trying to solve the radial equation with $R = u/r$
$$
\frac{d^2}{dr^2}u - \frac{l(l+1)}{r^2}u + (E-V)u = 0; \qquad V(r) = -\frac{2Z}{\alpha}\frac{e^{-r}}{1 - e^{-r}},
$$
using the shooting method....
1
vote
1
answer
101
views
Relation Asymptotic Series and perturbative effects
Perturbative expansions of a function $f(x)$ around say $x=0$ cannot determine contributions from a function such as $e^{-1/x}$ since its Taylor series vanishes to all orders. This kind of ...
1
vote
1
answer
76
views
Asympototic analysis for the following series sum
I am wondering is there one way to extract the asymptotic behavior of $x$ in the following expression near $x=0$?
$$\sum_{n=1}^{\infty} n\log(1-\exp(-n x))$$ where $x $ is real.
1
vote
1
answer
123
views
Quantum Harmonic Oscillator- Solving the Differential Equation at a Limit
The eigenvalue equation for the quantum harmonic oscillator is $$\langle y | E\rangle '' +(2\epsilon-y^2)\langle y| E \rangle=0$$ where $\epsilon = \frac{E}{\hbar\omega}$ and $y=\sqrt{\frac{\hbar}{m\...
0
votes
1
answer
45
views
What does the notation $\mathcal{O}\left(\frac{1}{r^2}\right)$ mean? [duplicate]
I was reading a text about quantum scattering, and I faced a notation I don't understand. The equation is the following:
$$
\nabla \psi_{\text{scattered}} = \frac{i k f(\theta) e^{ikr}}{r} \mathbf{\...
-1
votes
2
answers
268
views
What does the term $\mathcal O(\epsilon^2)$ mean?
In the highest upvoted answer to Where does the $i$ come from in the Schrödinger equation?
the author writes the following equation:
$$ U^\dagger U=(\mathbb I+\epsilon^* A^\dagger)(\mathbb I+\...
3
votes
0
answers
91
views
Pathria's number of states asymptotic estimation
I'm seeking for a mathematical grounding behind one trick employed in "Statistical Mechanics" book by R. Pathria and P. Beale to calculate the entropy $S(N,V,E)$ of an ideal monoatomic gas by counting ...
1
vote
1
answer
298
views
I really wonder about the time derivative of creation and annihilation operators in the derivation of LSZ
On p. 71 below eq. (6.12) in Schwartz book, they assume that
$$\lim_{t \to \pm\infty}\partial_0 a_p(t)=0.\tag{1}$$
But I thought that this is just an assumption. So we have to construct the ...
1
vote
1
answer
65
views
Heuristic for large $x$ behavior from small $q$ behavior of Fourier Transform
If I have a function $h(\mathbf x)$ which may be written
$$h(\mathbf x)= \int \frac{\text{d}^d\mathbf q}{(2\pi)^d} \, h(\mathbf q) e^{-i \mathbf q \cdot \mathbf x}$$
and assume spherical symmetry, is ...