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Questions tagged [asymptotics]

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1answer
34 views

Connection formulas: Why do we assume asymptotic behavior of the Airy functions?

The derivation is obtained from Introduction to Quantum Mechanics by Griffiths Let assume the turning point occurs at $x=0$, then the WKB solutions right and left to the turning point are: $$ \psi=\...
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1answer
77 views

Asymptotic LSZ reduction formula (Peskin & Schroeder)

Peskin & Schroeder, An Introduction to Quantum Field Theory, write at page 224 $$\int d^{4} x e^{i p \cdot x}\left\langle\Omega\left|T\left\{\phi(x) \phi\left(z_{1}\right) \cdots\right\}\right| ...
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1answer
107 views

The fields of Liénard and Wiechert and Poynting vector

EDIT: I know that the electric and magnetic fields depend not only on speed but also on acceleration and can both be expressed as the sum of two contributions: \begin{equation} \overline{E} (\bar{r},...
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WKB connection formulae from the path integral

The semiclassical, or WKB, approximation is one that is far more natural in the path integral formalism than it is when derived from the Schrodinger equation directly. Furthermore, the connection ...
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0answers
57 views

Occurances of integrals of the form $Z(\lambda) = \int g(x)e^{-\frac{f(x)}{\lambda}}dx$ (and perturbation techniques) [closed]

I am writing a review on perturbation techniques (actually hyperasymptotic techniques) for integrals of the form $$Z(\lambda) = \int g(x)e^{-\frac{f(x)}{\lambda}}dx,$$ where the interest is in the ...
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1answer
86 views

Hydrodynamic interaction between two spheres in $Re\ll 1$ flow

I am studying the interaction between two spherical particles of radius $a$ in a low Reynolds number flow. Because of linearity, I know that their respective velocities will be linear in the forces ...
2
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1answer
111 views

How to make sense of $\mathcal{I}^-$ as a Cauchy surface rigorously?

In some references, like Hawking's derivation of black hole radiation, one considers that $\mathcal{I}^-$ is a Cauchy surface. One recent reference with such a claim is the paper "Soft Hair on Black ...
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1answer
61 views

How many elements can the set of asymptotic states of a reduced dynamics have?

Given a Hilbert space $\mathbb{C}^N$ and the reduced dynamics $\Lambda(t)$ of the open quantum system, we can define the set of asymptotic states as $$ \mathcal{A}=\left\{\tilde\rho \in \mathcal{S}(\...
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35 views

Show the conformal transformation of the components of the Schouten tensor at the Spatial Infinity in an asymptotically flat spacetime

In Ashtekar & Hansen, the authors discussed a unified treatment of null and spatial infinity in general relativity. In Section 5.D., they derived the relation (20). I failed to reproduce it. Let ...
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1answer
109 views

Is there any proof that any result from perturbation theory is necessary an asymptotic series?

I know that almost all the series coming from perturbation theory are divergent, such as those from eigenvalue problems or the S-matrix in quantum field theory. The lore is that the series are ...
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2answers
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Is electic field is always asymptotic to $r^{\alpha}$ for some rational $\alpha$?

Suppose you have an electric field in three dimensions created by some finite (but possibly arbitrarily high) number of point charges, each with charge equal to an integer multiple (positive or ...
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164 views

Classical oscillator in a rotating frame

I would like to understand the behaviour of a simple mass-and-spring system - a classical harmonic oscillator - in the $xy$ plane that is in rotation about $\hat z$ with frequency $\vec \Omega=\Omega\...
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1answer
327 views

Beta function in QFT renormalization group

In order to know how the coupling constant depends on the energy scales,it is necessary to know the Beta function.Normally the Beta function is is calculated perturbatively. Now, my question is this: ...
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0answers
97 views

Gap above ground state equals inverse of temporal correlation length?

Two definitions: $\boxed{\Delta}$ is defined to be the energy gap above the ground state. $\boxed{\xi_t}$ is defined to be the temporal correlation length, i.e. it is the number such that for a ...
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2answers
400 views

What does asymptotically flat solution mean?

Can somebody explain what does it mean when a solution is "asymptotically flat"? like the schwarzschild metric which is asymptotically flat solution to vacuum Einstein equations.
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Renormalization group approach to renormalization

Given a $n$-point bare Green function in a massless asymptotically free theory, we have that the following limit exists and is finite \begin{equation} \lim_{\Lambda\rightarrow\infty} Z^{-n/2}(g_0,\...
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2answers
424 views

Asymptotic series in QFT

In QFT is said that the renormalized Dyson series is only asymptotic. But to be able to say it is necessary to know to what function of $g$ (the coupling constant) the Dyson series is asymptotic. ...
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1answer
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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})}+(-...
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The AdS in AdS/CFT correspondence is really a class of spacetimes which asymptotes to a subclass of spacetimes with the same causal structure as AdS

On page 131 of these notes, a precise formulation of the AdS/CFT correspondence is given by the GKPW dictionary $$Z_{\text{grav}}[\phi_{0}^{i};\partial M] = \langle \exp \left( - \frac{1}{\hbar} \...
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Finite mass excitations of $AdS_{3}$

Consider the following extract from page 2 of this paper. AdS3 is the $SL(2, \mathbb{R})$ group manifold and accordingly has an $SL(2, \mathbb{R})_{L} \otimes SL(2, \mathbb{R})_{R}$ isometry ...
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1answer
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Gauge transformation and large gauge transformation

Recently, Strominger posted his lecture notes on the infrared structure of gravity and gauge theory 1703.05448. In section 2.5, the equation (2.5.16) takes the following form $$e^2\partial_zN=A_z^{(0)...
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148 views

Expansion in $\epsilon$ in dimensional regularisation

I am having trouble with a step in a calculation in quantum field theory, as I believe I do not understand the reasoning underlying it. Often, in the context of dimensional regularisation, we often ...
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1answer
314 views

About asymptotic field assumption in QFT

I'm studying QFT and in a trouble about asymptotic assumption. It states every Heisenberg field converges to free field (asymptotic field) if one takes a limit of $x_0\rightarrow \pm \infty$. $$ \phi(...
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About weak limit of one point polynomial of Heisenberg fields

In QFT it is assumed that every Heisenberg field has free field as weak limit of $x_0 \rightarrow \pm \infty $. $$ \phi(x) \rightarrow \sqrt{Z}\phi^{as}(x) (as = in,out) $$ where $Z$ is ...
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0answers
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Conformal infinity in the Hawking-Hunter-Taylor-Robinson metric

I have been trying to follow some of the computations of this paper: http://arxiv.org/abs/hep-th/0408217 and particularly I couldn't derive the asymptotic form of the Kerr-AdS background (3.27) using ...
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0answers
208 views

Eigenvalue problem $−\psi''(x) − (ix)^ N \psi(x) = E\psi(x)$ in complex plane

To find the eigenvalue in the complex plane of $x$ for one dimensional Schrodinger equation $$ −ψ''(x) − (ix)^ N ψ(x) = Eψ(x). $$ where $N$ can be any real number, the boundary condition $ψ(x) → 0$ ...
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0answers
121 views

Asymptotic expansion with stationary phase

I am reading Rudolf Haag's book on "Local Quantum Physics", and on page 89 he writes that the following function $f(t,\vec{x}) = \int \tilde{f}(\vec{p})\exp(i(\vec{p}\cdot\vec{x}-\epsilon_{\vec{p}}t))...
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248 views

How does the expanding of null hypersurface orthogonal geodesic congruence imply a particular result?

Sorry that I do not know how to summarize my problem in the title. First, please go to the website here (free access, even though it looks otherwise) to download the paper done by R. Sashs on ...
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1answer
974 views

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 ...
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3answers
440 views

Why is spatial conformal infinity a point

One property of spatial infinity is that all spacelike geodesics end at it. Since spacelike geodesics can have different directions, I do not understand why spatial infinity is a point. It looks more ...
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211 views

What is the exact meaning that QED perturbative series is only asymptotic and eventually diverges at very high orders?

When I read paper PRB89, 235431 about the effective field theory of graphene, there is a statement that QED perturbative series is only asymptotic and eventually diverges at very high orders (e. g. ...
9
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1answer
128 views

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, ...
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85 views

Linear KDV eq. asymptotics

The question arises from the book Solitons by P. G. Drazin about the linearized KDV eq. $$ u_t+u_{xxx}=0 $$ My first step was to take a Fourier transform of the equation, find that the dispersion ...
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54 views

Mathieu equation and instabilities

Consider the Mathieu equation $$ \tag 1 y'' + (A -2q\cos(2t))y = 0, $$ How does it provide instabilities for small $q << 1, A > q$? I don't understand this because as the result of ...
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1answer
207 views

Fourier transform & asymptotic expansion of Klein-Gordon equation

I am looking for an approximate analytical solution to the generalized Klein-Gordon equation \begin{equation} \frac{\partial^2{\phi}}{\partial{t^2}}+\frac{\partial^2{\phi}}{\partial{x^2}}+\phi=0 \end{...
2
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1answer
108 views

DOS integral when surface is not closed

According to the density of states (DOS) formula $$\rho(\varepsilon)\propto \int_{\varepsilon=\text{const}}\frac{dS}{|\nabla_k \varepsilon_k|}$$ Since there is an integral on the constant energy ...
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568 views

DOS behavior of Van Hove singularity in a line

When there are some points in momentum space give $|\nabla_k \varepsilon_k|=0$, they are called Van Hove points and give singularity in the desity of states (DOS). But what if $|\nabla_k \...
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DOS of Van Hove singularity in 2D square lattice tight binding model

For the simplest example, 2D square lattice tight binding model gives the energy band as $$\varepsilon_k=-2t(\cos k_x+\cos k_y) \, .$$ We know that $\mathbf{k}=(0,\pi)$ and related momentum points are ...
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0answers
80 views

How to get the asymptotic expression of DOS near Van Hove singularity [duplicate]

For the simplest example, 2D square lattice tight binding model gives the energy band as $$\varepsilon_k=-2t(\cos k_x+\cos k_y) \, .$$ We know that $\vec{k}=(0,\pi)$ and related momentum points are ...
1
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1answer
121 views

Use of termination in solving quantum harmonic oscillator, hydrogen atom etc

I can't seem to understand the use of termination to make the series solutions physically acceptable (when solving the linear harmonic oscillator etc.). So what if the series does not terminate, it's ...
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1answer
7k views

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?
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0answers
176 views

Asymptotic Analysis of 1-D Schrödinger Equation [closed]

I'm looking to do a small personal project regarding the time independent Schrödinger equation in 1-D: $$y'' +V(x)y=Ey$$ $$y''=Q(x)y$$ where $ Q(x):=E-V(x) $. There is obviously nothing stopping ...
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0answers
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Assigning an asymptotic power to the volume form?

I was reading about the covariant theory of asymptotic symmetries in this review: http://arxiv.org/abs/hep-th/0111246 I have a question about eq. (1.8), but before I ask I should describe what the ...
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Do all the spacelike curve terminate at the spatial infinity $i_0$ in the Penrose Diagram of a Schwarzchild black hole?

Let's restrict to the radial direction, so the metric can be expressed as $ds^2=-(1-r_S/r)dt^2+(1-r_S/r)^{-1}dr^2$ with $r_S$ the Schwarzchild radius. Expressed in Kruskal coordinates, the metric is ...
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1answer
238 views

Asymptotic freedom in QCD

From renormalization group equation $$ t \frac{d \bar{g}(t , g)}{dt} = \beta (\bar{g}(t , g)), \quad \bar{g}(1 , g) = g $$ (here $t$ is momentum scale factor, $g$ is initial coupling constant and $\...
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1answer
911 views

A question about the asymptotic series in perturbative expansion in QFT

Related post I heard about the argument that the perturbative expansion in QFT must be asymptotic, such as http://ncatlab.org/nlab/show/perturbation+theory#DivergenceConvergence Roughly this can ...
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1answer
261 views

Derivation of $a_{j}$ coefficients in the quantum harmonic oscillator

In Griffiths' book page 53, when we derive the solution of the quantum harmonic oscillator by using the power series way, we have: $$a_{j+2} = \frac{2j+1-K}{(j+1)(j+2)}\, a_{j} .$$ And for large $j$, ...
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0answers
117 views

Help with deriving an asymptotic expression

Note: I don't know if this is the best place for this question, because it is very specific. However, I'm not sure of a better place to go (apart from one of the other SE's). If you have a ...
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2answers
196 views

Ground State Energy in Euclidean Spacetime

Calculating the transition amplitude in Euclidean spacetime is useful because from it we can extract the ground state energy and ground state wave-functions values. For example, let's assume we are ...
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0answers
115 views

(References) Study of Asymptotically Flat spacetimes

I am interested in studying the asymptotic structure of Minkowski spacetime in General Relativity. I believe most of the work in this area concerns the asymptotic structure of Minkowski space at null ...