Questions tagged [asymptotics]

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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?
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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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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 ...
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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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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 $$...
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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 ...
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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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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{...
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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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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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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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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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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 ...
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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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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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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}(\...
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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}}_{...
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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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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 ...
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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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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 (...
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2 votes
1 answer
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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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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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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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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?
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3 votes
1 answer
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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)...
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3 votes
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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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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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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 & \...
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5 votes
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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. ...
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2 votes
1 answer
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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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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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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
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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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7 votes
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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 ...
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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 ...
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4 votes
2 answers
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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}) = -...
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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 ...
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2 votes
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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_{\...
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0 votes
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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 ...
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1 vote
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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 ...
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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....
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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 ...
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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.
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1 vote
1 answer
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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\...
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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{\...
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2 answers
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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+\...
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3 votes
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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 ...
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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 ...
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1 vote
1 answer
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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 ...
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