Questions tagged [asymptotics]
The asymptotics tag has no usage guidance.
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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 $2+1$ ...
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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 ...
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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-...
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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 ...
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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+...
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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 ...
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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:
$$Z[J, \hbar] = \...
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Riemann-Lebesgue lemma in Faddeev-Kulish approach
I am learning about the established formalism used in the literature of IR divergences and dressed states, and I invariably come across an argument of the following form when evaluating a (photon) ...
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Why do we use perturbative series if they don't converge?
My course instructor mentioned that the Perturbative Series are not convergent but diverge as we consider more and more terms in the expansion. He then briefly mentioned that the Perturbative Series ...
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Energy gap of mean field model for transverse ising chain
Polynomials of spin operators with real coefficients appear not infrequently in Hamiltonians and in mean field theory, and there are often tricks to find their eigenvalues.
For example, the polynomial ...
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In what sense a path integral can be approximated by the classical contribution $\exp{[\frac{\mathrm{i}}{\hbar}S_{\text{cl}}}]$?
People often say that the amplitude $K(b,a)$ to go from $a$ to $b$ can be approximated by $$K(b,a) \sim \exp{\left[\frac{\mathrm{i}}{\hbar}S_{\text{cl}}(b,a)\right]},\tag{1}$$
where $S_{\text{cl}}(b,a)...
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Rewriting an asymptotic series as a convergent integral [closed]
I am given the function
$$
f(x) = \sum_{n=1}^{\infty} \frac{\Gamma(2n)}{\Gamma(n)}(-x)^n
$$
and I need to show that it can be rewritten as an integral that is convergent for a range of values of x. I ...
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Question about asymptotic assumption in LSZ reduction formula derivation
I have a silly question in derivation of LSZ reduction formular, I can go directly with the derivation until I found a assumption that I can't convince myself.
In the book Quantum Field Theory and the ...
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What is the asymptotic charge for a two-form theory in Lorenz gauge?
I'm trying to derive the generating charge of the asymptotic symmetries for a two-form field in Lorenz gauge at future null infinity.
I'm working in retarded Bondi coordinates $(u,r,x^A, x^B)$.
First ...
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On the asymptotic condition
The text that I am following is John Taylor's Scattering Theory. This relates specifically to page 28 and 29 where we discuss the asymptotic condition in quantum mechanics. What I am ultimately ...
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Kalb-Ramond current fall-offs at future null infinity
I can couple the electromagnetic field to a current generated by the complex scalar field for example:
$S=- \int d^4x \frac{1}{4} F_{\mu\nu} F^{\mu\nu} + A_\mu J^\mu$
with $J_\mu = i(\partial_\mu \phi^...
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Can residual gauge symmetries have compact support?
I have been reading this review about asymptotic symmetries, and one definition that is used is apparently due to Penrose:
$$
G = \frac{\mbox{gauge symmetries preserving boundary conditions}}{\mbox{...
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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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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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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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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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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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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 ...
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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 ...
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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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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:
$$
-\frac{a}{2}\phi''(\xi) + \left(\frac1{2a}\sinh^2(2\xi) + (2b-1)\cosh(2\xi)\right)\phi(\xi) = ...
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Asymptotic Series in QFT: What to do when all "trustworthy" terms are known?
In my Introduction to QFT lecture, we quantized a Klein-Gordon Field and as a toy model we looked at $\phi^3$ theory. For this toy model we expanded the $S = U(-\infty, \infty)$ operator in a series (...
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Asymptotic form of a Coulomb-like integral
I need to evaluate or work out the asymptotic scaling of the following integral:
\begin{equation}
I~=~\int_{\mathbb{R}^3} dq d^2p \frac{e^{i\vec{p}\cdot \vec{r}}e^{iq z}}{p^2 + \frac{1}{g^2}q^4}
\end{...
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What parameter can you expand QFT in for a convergent series?
It is known that the expansion in terms of Feynman diagrams with a series in terms of the coupling constant is an asymptotic series (i.e. it starts off a good approximation but eventually diverges). (...
user84158
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How can QED by predictive if it diverges?
One of the tests of Quantum Electrodynamics is the value of the "Anomalous magnetic dipole moment".
The theoretical value is:
$$a_e = 0.001\ 159\ 652\ 181....$$
We say that QED "...
user84158
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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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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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