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Asymptotic states and physical states in QFT scattering theory

Context In the scattering theory of QFT, one may impose the asymptotic conditions on the field: \begin{align} \lim_{t\to\pm\infty} \langle \alpha | \hat{\phi}(t,\mathbf{x}) | \beta \rangle = \sqrt{Z} \...
Steven Chang's user avatar
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How to state that a function has a certain andament in a limit? [migrated]

Assuming we have a function $f(r)$ that has the following limit $$ \lim_{r\to0} f(r) = \frac{5}{3 r^2} \,.$$ What is the correct symbol to express that the denominator goes like $r^2$? Is the ...
Aleph12345's user avatar
5 votes
2 answers
113 views

Long-range approximations of the Uehling interaction

A common approximation to the \begin{equation} U(\vec{r})=-m\frac{\alpha(Z\alpha)}{\pi} \int_1^\infty\mathrm{d}u\frac{\sqrt{u^2-1}\left(2u^2+1\right)}{3u^4}\frac{\exp(-2mur)}{mr} \tag{$\star$} \end{...
dennismoore94's user avatar
2 votes
0 answers
64 views

Calculating LSZ reduction for higher order in fields terms

Consider a theory with only a single massless scalar field $\phi(x)$ and a current $J^\mu(x)$ which can be polynomially expanded as fields and their derivatives and spacetime \begin{align} J^\mu(x) = ...
Mmmao 's user avatar
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1 answer
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What goes wrong with strongly coupled theories?

Let $\lambda$ be the coupling constant of a quantum field theory. It is said that Perturbation theory is only valid when the theory is weakly coupled ($\lambda \ll 1$). In most cases, the series of ...
CBBAM's user avatar
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3 votes
1 answer
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Asymptotic form of solution to biased random walk

(Cross post from math.stackexchange) Consider a continuous time biased random walk on a 1D lattice. The random walker walks with rate $k_\mathrm{R}$ to the right and with rate $k_\mathrm{L}$ to the ...
Caesar.tcl's user avatar
1 vote
0 answers
34 views

Asymptotics of 2D Ising model transfer matrix eigenvalues

The NN 2D Ising Model at inverse temperature $\beta$ and external magnetic field $h$ on an $L_1\times L_2$ sized box within $\mathbb{Z}^2$ with periodic boundary conditions $$\sigma_{L_{1}+1,x_{2}} = \...
PPR's user avatar
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How does the asymptotic metric fluctuation in $n \to m$ scattering relates to the soft factor in Weinberg's soft graviton theorem?

I'm reading arXiv: 1411.5745 [hep-th]. In Sec. 5, the authors show how the memory effect and Weinberg's soft graviton theorem are two faces of the same coin. I'm interested in understanding a specific ...
Níckolas Alves's user avatar
2 votes
1 answer
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Introductory references on the gravitational memory effect

I'm currently reading Andrew Strominger's Lectures on the Infrared Structure of Gravity and Gauge Theory. While I love the reference, the discussion on the gravitational memory effect feels a bit ...
1 vote
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Expression of $\langle 0 | 0 \rangle _{f,h}$ in the Srednicki's quantum field theory book (eq. (6.21), p.47) [duplicate]

I am reading the Srednicki's quantum field theory book and stuck at some statement. In the book p.46, the author worte that : "Now consider modifying the lagrangian of our theory by including ...
Plantation's user avatar
2 votes
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Gauge symmetries, isometries of spacetime and asymptotic symmetries

I am having a hard time understanding the physical meaning of asymptotic symmetries in General relativity. I think I understand the mathematics, but the meaning eludes me. I'll try to write the things ...
P. C. Spaniel's user avatar
5 votes
1 answer
122 views

Do supertranslations act in a physically nontrivial way?

I'm currently reading arXiv: 1703.05448 [hep-th]. In this question, I'm interested in a statement made on page 67 of the pdf (76 of the printed book, if you prefer to check in it). There, the author ...
Níckolas Alves's user avatar
1 vote
0 answers
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Next-to-leading $1/N$ contributions to Feynman diagrams in large $N$

I want to understand $1/N$ contributions to quark bilinear operators $J(x)$ in large $N$, for instance, operators of the form $q\bar{q}$ or $\bar{q}\gamma^\mu q$. As pointed out by E. Witten, in the ...
Spectree's user avatar
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Examples of spacetimes that are asymptotically flat at future timelike infinity

There are interesting non-trivial examples of spacetimes which are asymptotically flat at null and spacelike infinities. For example, the Kerr family of black holes satisfies these conditions. However,...
Níckolas Alves's user avatar
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What are Supertranslations and Superrotations in General relativity, and how does it inform us about a detector at null infinity?

How did I get here? While drafting my question, I found this very similar question on our site. Three days ago, I happened upon the concept of supertranslations and superrotations in General ...
cows's user avatar
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Definition of asymptotically flat spacetime

Following the definition in Wald's book on general relativity, in page 276 asymptotically flat spacetimes are defined using conformal isometry with conformal factor $Ω$. Then one of the requirements ...
ziv's user avatar
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1 vote
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Why Quantum Field Theory series expansion is called Asymptotic? [duplicate]

I once saw the Freeman Dyson's article, referring the Expansion series occurring in Quantum Field Theory is asymptotic in nature; It will deviate from actual physical phenomena when too many terms are ...
K.R.Park's user avatar
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Asymptotic development of the black-hole metric

In the Kruskal-Szekeres extension of the Schwarzschild metric parallel universes appear. In a couple of questions on this site, for instance: Where does the parallel universe in the Penrose diagram ...
Frederic Thomas's user avatar
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Is QED asymptotically free in lower dimensions?

It is well-known that QED (=quantum electrodynamics) is NOT asymptotically free in spacetime dimension $4$. However, I wonder if it becomes asymptotically free in lower dimensions, such as $2+1$ ...
Keith's user avatar
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3 votes
1 answer
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Leading order of integrals in Field Theory

I am studying Statistical Field Theory from the notes of D. Tong (http://www.damtp.cam.ac.uk/user/tong/sft.html). I have trouble to understand how to estimate the leading order of diverging integrals ...
Ruth Murphy's user avatar
0 votes
1 answer
133 views

How does asymptotic analysis work with quantum mechanics?

I am trying to wrap my head around asymptotic analysis when it is applied to quantum mechanics. The best source I have found so far is from MIT (https://ocw.mit.edu/courses/8-04-quantum-physics-i-...
CodingFryCook's user avatar
2 votes
0 answers
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Definition of Asymptotically Flat at Null, Spacelike, and Timelike Infinities

Most books I've looked into discuss asymptotic flatness in general relativity for the null and spacelike case (e.g. Wald), or for the null and timelike case (e.g. arXiv: 1706.09666 [math-ph]), but I ...
Níckolas Alves's user avatar
3 votes
1 answer
138 views

Massless limit of Dirac fermion correlation functions

In the 2D massless Dirac fermion CFT we have correlation functions like $$\langle J(z,\bar{z})J(0)\rangle \sim \frac{1}{z^2},$$ where in terms of real Euclidean coordinates $x^0,x^1$, we have $z=x^0+...
octonion's user avatar
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3 votes
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Symmetry at spatial infinity

One definition for asymptotic flatness is $$\lim_{r \rightarrow \infty}g_{\mu\nu}=\eta_{\mu \nu}+\mathcal{O}(r^{-1}),$$ where $\eta_{\mu \nu}$ is the Minkowski metric. The asymptotic symmetry group is ...
David Shaw's user avatar
1 vote
1 answer
73 views

Semiclassic limit of a QFT in Zinn-Justin

I am reading the Zinn-Justin book "Quantum Field Theory and Critical Phenomena" and i have come across a perplexing point. Given the partition functional, in Euclidean QFT: $$Z[J, \hbar] = \...
LolloBoldo's user avatar
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2 votes
1 answer
214 views

Riemann-Lebesgue lemma in Faddeev-Kulish approach

I am learning about the established formalism used in the literature of IR divergences and dressed states, and I invariably come across an argument of the following form when evaluating a (photon) ...
N.E.'s user avatar
  • 333
20 votes
2 answers
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 ...
Dev's user avatar
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0 answers
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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 ...
user196574's user avatar
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4 votes
1 answer
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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)...
Xiaosheng Yang's user avatar
2 votes
1 answer
62 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 ...
postscript's user avatar
0 votes
1 answer
86 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 ...
a Fish in Dirac Sea's user avatar
3 votes
1 answer
106 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 ...
Robin's user avatar
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1 vote
1 answer
164 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 ...
Charlie's user avatar
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4 votes
1 answer
192 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^...
Robin's user avatar
  • 63
3 votes
1 answer
282 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{...
Frotaur's user avatar
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0 votes
1 answer
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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&...
Frotaur's user avatar
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2 votes
1 answer
155 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 ...
riemannium's user avatar
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2 votes
0 answers
209 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 ...
Robin's user avatar
  • 63
11 votes
2 answers
214 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 $\sim n^\beta$ ...
FusRoDah's user avatar
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2 votes
2 answers
201 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 ...
Robin's user avatar
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1 vote
1 answer
67 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 ...
twisted manifold's user avatar
1 vote
1 answer
265 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 ...
user1172131's user avatar
3 votes
1 answer
144 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 ...
schris38's user avatar
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3 votes
1 answer
228 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 ...
schris38's user avatar
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1 vote
0 answers
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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 ...
ShoutOutAndCalculate's user avatar
0 votes
1 answer
380 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 ...
Kashmiri's user avatar
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1 vote
0 answers
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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 $$...
Anuj Tanwar's user avatar
-1 votes
2 answers
191 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 ...
BarrierRemoval's user avatar
0 votes
0 answers
65 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 $...
Ken.Wong's user avatar
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1 vote
0 answers
92 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{...
Charlessilva's user avatar