Questions tagged [asymptotics]

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18 votes
1 answer
2k views

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 ...
1 vote
0 answers
23 views

Energy gap of mean field model for transverse ising chain

Polynomials of spin operators with real coefficients appear not infrequently in Hamiltonians and in mean field theory, and there are often tricks to find their eigenvalues. For example, the polynomial ...
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4 votes
1 answer
99 views

In what sense a path integral can be approximated by the classical contribution $\exp{[\frac{\mathrm{i}}{\hbar}S_{\text{cl}}}]$?

People often say that the amplitude $K(b,a)$ to go from $a$ to $b$ can be approximated by $$K(b,a) \sim \exp{\left[\frac{\mathrm{i}}{\hbar}S_{\text{cl}}(b,a)\right]},\tag{1}$$ where $S_{\text{cl}}(b,a)...
2 votes
1 answer
58 views

Rewriting an asymptotic series as a convergent integral [closed]

I am given the function $$ f(x) = \sum_{n=1}^{\infty} \frac{\Gamma(2n)}{\Gamma(n)}(-x)^n $$ and I need to show that it can be rewritten as an integral that is convergent for a range of values of x. I ...
0 votes
1 answer
48 views

Question about asymptotic assumption in LSZ reduction formula derivation

I have a silly question in derivation of LSZ reduction formular, I can go directly with the derivation until I found a assumption that I can't convince myself. In the book Quantum Field Theory and the ...
3 votes
1 answer
81 views

What is the asymptotic charge for a two-form theory in Lorenz gauge?

I'm trying to derive the generating charge of the asymptotic symmetries for a two-form field in Lorenz gauge at future null infinity. I'm working in retarded Bondi coordinates $(u,r,x^A, x^B)$. First ...
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1 vote
1 answer
64 views

On the asymptotic condition

The text that I am following is John Taylor's Scattering Theory. This relates specifically to page 28 and 29 where we discuss the asymptotic condition in quantum mechanics. What I am ultimately ...
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4 votes
1 answer
153 views

Kalb-Ramond current fall-offs at future null infinity

I can couple the electromagnetic field to a current generated by the complex scalar field for example: $S=- \int d^4x \frac{1}{4} F_{\mu\nu} F^{\mu\nu} + A_\mu J^\mu$ with $J_\mu = i(\partial_\mu \phi^...
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3 votes
1 answer
195 views

Can residual gauge symmetries have compact support?

I have been reading this review about asymptotic symmetries, and one definition that is used is apparently due to Penrose: $$ G = \frac{\mbox{gauge symmetries preserving boundary conditions}}{\mbox{...
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0 votes
1 answer
70 views

How is the Virasoro symmetry realised on $AdS_3$?

In the context of the Holographic correspondence, $AdS_{n}/CFT_{n-1}$, it is often cited as a "confirmation" that the two symmetry groups of the theories correspond. Indeed, in dimensions $n&...
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1 vote
1 answer
59 views

Are all asymptotic symmetries and their meaning known?

Beyond the Standard Model and the General relativity invariant groups, recently we have met (again) the BSM groups of asymptotic symmetries given by the Bondi-Metzner-Sachs (BMS) or the extended BSM ...
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2 votes
1 answer
85 views

What are the Maxwell equations of motion in retarded Bondi coordinates?

I'm reading a paper about asymptotic symmetries at null infinity in electrodynamics. There, they had the following calculation: The Maxwell equations $\nabla^{\nu} F_{\mu\nu} = J_{\nu}$ written in ...
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11 votes
2 answers
188 views

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 $\sim n^\beta$ ...
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2 votes
2 answers
98 views

Why is it that in gauge theories the assumption "all fields decay sufficiently rapidly at infinity" not justified anymore?

I read that in gauge theories the assumption that "all fields decay sufficiently rapidly at infinity" is not justified anymore and therefore, one needs to consider boundary terms that ...
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0 votes
1 answer
55 views

How can can we show that a metric is asymptotically AdS?

Given any metric, for example $$ ds^2=d\tau^2+L^2\cosh(H\tau)d\vec{x}^2 $$ how can we show that this metric is asymptotically Euclidean AdS? Specicifally, when $\tau\rightarrow\pm\infty$ is it ...
0 votes
1 answer
94 views

Fokker-Planck: uniqueness and convergence to stationary distribution

Consider the Langevin equation ($N$-dimensional) with nonlinear drift term, but expressible as a gradient of a function $U(\vec{x})$. Namely, consider the stochastic process described by the set of ...
3 votes
1 answer
115 views

Question about field configurations on the boundary of $\mathcal{I}^+$

I am reading Strominger's lecture notes "The infrared structure of gravity and gauge theory" (https://arxiv.org/abs/1703.05448). In chapter two, while trying to derive an expression about ...
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3 votes
1 answer
166 views

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 ...
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1 vote
0 answers
54 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
103 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 ...
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1 vote
0 answers
55 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
188 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
55 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 $...
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1 vote
0 answers
74 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
99 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 $...
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0 votes
2 answers
88 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?
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0 votes
0 answers
46 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 ...
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0 votes
0 answers
57 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{\...
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0 votes
0 answers
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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
46 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.
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1 vote
2 answers
95 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, $\delta$ ...
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2 votes
1 answer
168 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
124 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
53 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 ...
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1 vote
0 answers
197 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
120 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) = ...
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7 votes
1 answer
263 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
105 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{...
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1 vote
2 answers
126 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). (...
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6 votes
0 answers
155 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 "...
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0 votes
1 answer
276 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
103 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
114 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 ...
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0 votes
1 answer
219 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 ...
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1 vote
1 answer
209 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
227 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
107 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 ...
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6 votes
1 answer
197 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 $(u,x^A)$ ...
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3 votes
0 answers
106 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 $4d$ ($\omega \to 2$): ...
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3 votes
1 answer
57 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 ...
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