Perturbation theory refers to methods for understanding physical systems by treating them as small modifications to exactly solvable systems.

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How does one expand gravity Lagrangians about an $AdS$ background?

I had previously asked this question. This is kind of a continuation of that. I recently found this expression which seems to be called the "Fierz-Pauli action" which is apparently the quadratic ...
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Perturbation of an operator - Meaning of matrix element [closed]

Let be $B$ an operator and $\left|\Psi\right>$, $\left|\Phi\right>$ two states (not necessarily equals). What is the meaning of a matrix element $\left<\Psi\right| B \left|\Phi\right>\...
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Vanishing diagonal matrix elements of pertubation

In time-dependent pertubation theory we can denote the Schrödinger equation by a set of two equations $$\dot{c_a} = -\frac{i}{\hbar}\Big[c_aH'_{aa}+c_bH'_{ab}e^{-i(E_b-E_a)t/\hbar}\Big] \\ \dot{c_b} =...
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279 views

Quick question on perturbation theory

Suppose we have a particle in an infinite potential well, with $V(x) = 0,\space 0< x < a $ and infinity everywhere else. Now suppose we have a perturbation on the LHS of the well: $V_1(x) = v, ...
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Time-dependent perturbation - details in derivation

I get confused about two things when deriving the time-dependent perturbative approach. We have the Hamiltonian $$H = H_0 + \lambda H^{(1)}$$ and we have solved (from Schroedinger) $$\dot{C_f(t)} ...
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What makes laminar cascade break?

Near my house there is a mall that have a cascade, which has a pratically constant flow, and doesn't seem to have perturbations (at least near the edge where water falls), between its two levels. ...
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74 views

How to connect the dimension of perturbation constant with renormalizability

Let's have the Lagrangian $$ L = L_{0} + \lambda V , \qquad (1) $$ where $\lambda$ is constant which is small in the next senses: if $\lambda$ is dimensionless, it means that $\lambda < 1$; if it ...
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Metric Perturbations in General Relativity and quasi-normal modes?

I am familiar with the tools that appear in (linear) perturbation theory for general relativity, that is namely that one writes: $$g_{\mu \nu} = g^{(0)}_{\mu \nu} + \epsilon g^{(1)}_{\mu \nu} + \...
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Effective Hamiltonian / Perturbation theory for non-degenerate case

Trying to wrap my head around the following situation. Consider first a case that I understand well: Let's assume a three level system where the lowest two levels are degenerate and individually ...
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181 views

Why do we need an old perturbation theory?

There are two types of perturbation theory corresponding to explicit lorentz-covariance of amplitudes. The first one is called Rayleigh-Schrodinger perturbation theory. It is based on following ...
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120 views

Perturbation Theory Problem [closed]

I am self studying for upcoming exams and I am stuck on the end of a problem related to perturbation theory. Here is the problem I can't manage to do the very last part -- showing the exact change ...
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171 views

Does the LSZ reduction method prove gauge-independence in massless gauge theories?

I've been working my way through L. Baulieu's excellent paper [Perturbative gauge theories, Physics Reports, Volume 129, Issue 1, December 1985, Pages 1-74]. Towards the end, he goes on to prove that ...
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Degenerate perturbation theory applied to topological degeneracy?

Consider a quantum system described by a gapped Hamiltonian $H_0$ with degenerate ground states (GS), adding a perturbation term $V$ to $H_0$, then the low-energy physics can be described by an ...
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464 views

First Order Correction to wave function in ground state

I am looking at a spin 1/2 particle in a magnetic field. This has Hamiltonian $$H=-\mu s\cdot B_0$$ For simplicity, assume $B_0=B_0\hat z$ so $H=-\mu B_0$. I then apply a perturbative magnetic field ...
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127 views

Perturbation of coupled spin

I am given a system with Hamiltonian (all 1/2 spins) $$H_0=\alpha(S_1\cdot S_2)$$ I broke it down and found that there were four eigenstates: $|1,[0,\pm1]\rangle$ and $|0,0\rangle$. Each has an ...
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80 views

$\mathrm{d} \Omega_{CM}$ for a $1\rightarrow 2$ particle decay?

The differential solid angle is described in e.g. Srednicki's QFT text but only for the case of scattering. Because in the case of scattering it's defined with respect to the incoming three-momentum ${...
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673 views

Riemann curvature tensor in first order perturbation theory as a Lie derivative of Riemann curvature tensor in zero order [closed]

I am having a difficulty solving my homework so I was hoping I could get some help, so here it is. It is about gravitational waves and first order gravitational perturbation theory, I have to prove ...
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Adiabatic approximation and time-dependent problems

I am an undergraduate physics student. I have a question in approximation methods for time-dependent problems in quantum mechanics. I read the proof of the adiabatic theorem but I didn't understand it....
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Perturbation theory in quantum mechanics

In perturbation theory perturbed eigenstates expanded by unperturbed eigenstates, but we know when the system perturbed its Hilbert space altered and hence its basis changed, then we can't state this ...
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What if UV behaviour of gravity was perturbative?

I understand that the UV behaviour of gravity ought to be dominated by black hole production and that graviton-graviton scattering ought to blow up above the Planck scale. Suppose, however, that ...
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457 views

Perturbative Quantum Mechanics

I am, in full generality, confused about perturbation theory in quantum mechanics. My textbook and Wikipedia have the same general approach to explaining it: given some Hamiltonian $H=H^{(0)} + H^\...
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Do gravitational waves cause time dilatation?

The effect of gravitational waves in transverse traceless gauge on matter is represented by the expansion and contraction of a ring of test particles in the direction of polarization of the wave. ...
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General formula to compute the redshift (first order perturbations)

Consider an expanding universe with the following metric in conformal time/co-moving coordinates: $$ds^2=a^2\left[-c^2\left(1+\frac{2\phi}{c^2}\right)d\eta^2+\left(1-\frac{2\phi}{c^2}\right)\left(dx^...
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Why can we not apply perturbation theory in Born-Oppenheimer approximation

In Weinberg's Lectures on Quantum Mechanics, he mentions Unfortunately, we cannot simply use first-order perturbation theory, with $T_{nuc}$ taken as the perturbation and the state vectors $\Phi_{...
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Difference between positron and electron scattering in Coulomb field

In first order of perturbation theory the S-matrix amplitude for electron scattering in the Coulomb field will be (up to normalization factors) $$ S_{fi} = \frac{iZ q^2}{\sqrt{2E_{f}2E_{i}}}\bar {u}(...
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How do I properly express adding perturbed states to unperturbed states?

I have a problem set due tomorrow, and the last problem is driving me nuts. Been combing through griffiths trying to find similar examples to no avail, so it'd be greatly appreciated if stackexchange ...
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Why do some terms vanish in first-order perturbation theory?

In first order perturbation theory, we usually express the first order perturbation in the eigenket of the perturbed Hamiltonian in the basis of the unperturbed Hamiltonian $H_{0}$: $$|b\rangle=\sum_{...
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Interpreting perturbation theory in general relativity

In quantum mechanics we start with a Hamiltonian $H_0$ for which we know the exact eigenstates and energy eigenvalues. We perturb it by a known term $H$, and then attempt to compute (approximately) ...
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Field Strength Renorm in Peskin&Schroeder

On page 237 in PS we have (the unnumbered equation after eq. 7.58) $$\mathcal{P} \sim \frac{iZ}{p^2-m^2-iZ\,\mathrm{Im}M^2(p^2)}$$ but after deriving it myself I obtained $$\mathcal{P} \sim \frac{...
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250 views

How explain this perturbing equation about the 43 arcseconds?

The planetary orbits have been studied as ellipses but the solar system is in motion in relation to the distant stars. Their path is along the tip of an helix and the ecliptic plane is a convenient ...
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What is the physical meaning of the “decay rate” in Fermis golden rule? [duplicate]

As far as I understood, Fermi's golden rule gives a prediction of the transition rate in a perturbed quantum system $H_0+V$ between two eigenstates of the unperturbed system $H_0$, say from $\left| i\...
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Perturbation of a Hydrogen Atom in a Quadrupole Field

Question: A hydrogen atom is located in a quadrupole field, which gives it a perturbation $$H_1=A(x^2-y^2)$$ where $A$ is some constant. Calculate the quantity $...
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Stationary Perturbation Theory : Estimating higher order corrections for anharmonic oscillator

Note $\hbar = 1$. $$H = H_0 + \lambda V =\frac{p^2}{2m} + m\omega^2x^2 + \lambda m^2\omega^3 x^4$$ Supposedly the perturbation expansion diverges. We are supposed to estimate for what order we have a ...
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Linear Metric Perturbation and Brans-Dicke Theory

Recently, I have been researching about modified gravity theories and one of the theories has been the theory of the graviton. If one starts with the metric tensor $g_{\mu\nu}$ and then performs the ...
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Divergent Series

Why is it that divergent series make sense? Specifically, by basic calculus a sum such as $1 - 1 + 1 ...$ describes a divergent series (where divergent := non-convergent sequence of partial sums) but,...
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Peskin's book page 334 proof of $Z_1=Z_2$ to all orders in QED perturbation theory

Peskin in his QFT page 334 argued that $Z_1=Z_2$ to all orders in QED perturbation theory, but I couldn't understand his argument: ... With a generalization of the argument given there (section 7....
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116 views

Clarify formula in quantum perturbation theory

I'm studying perturbation theory in the context of quantum mechanics. My lecture notes say that in order to calculate the first-order correction of eigenfunction $\psi_n$, that is $\psi_n^{(1)}$, I ...
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The expansion of a function in powers of a parameter

In the perturbation theory for non-degenerate levels, the energy $E_n(\lambda)$ of an eigenstate $|\psi_n(\lambda)\rangle$ of the hamiltonian $\mathcal{H}=\mathcal{H}_0+\lambda \mathcal{H}_1$ (where $\...
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QM : Perturbation theory with multiple operators

When doing perturbation theory in quantum mechanics, if the perturbation hamiltonian is made of three terms : $$W = W_1 +W_2 + W_3,$$ can I treat each term separately and performing perturbation ...
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289 views

Lagrangian density of an interacting real scalar field theory

Srednicki writes the Lagrangian density of an interacting scalar field theory as $$ \mathcal{L} = -\frac{1}{2} Z_\phi \partial^\mu \phi \partial_\mu \phi -\frac{1}{2} Z_m m^2 \phi^2 + \frac{1}{6} Z_g ...
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The nature of theoretical models

Mathematics is exact. It is a beautiful language that allows us to express quantities that aren't possible to be represented physically. We build theoretical models of physical systems that work out ...
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QM perturbation theory : When do I have to use degenerate/non-degenerate perturbation theory?

I am considering a perturbation theory problem in quantum mechanics. The unperturbed hamiltonian is $$H_0 = A_1 \boldsymbol{B} S_{1z} + A_2 \boldsymbol{B} S_{2z}.$$ The eigenstates of the unperturbed ...
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Separation of perturbative and non-perturbative contributions in partition function computation

The following is defined, where $\epsilon \to 0^+$ is a cutoff: $$ \mathcal{F}(Z)=\int_{\epsilon}^\infty \frac{\mathrm{d}s}{s} \frac{1}{\sinh^2 s/2} e^{-sx}. $$ Question: how do we see that $\mathcal{...
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646 views

The Origins of Instantons from Path Integrals

I know that you can come across non-perturbative effects in QFT, particular when the coupling constant lies outside the radius of convergence of the asympototic perturbation series. From the ...
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Second order degenerate perturbation theory

What is a good resource to learn about higher degree degenerate perturbation theory - one that involves mathematics that isn't much more advanced than first order perturbation theory? I've looked ...
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668 views

Small oscillations of heavy string

I'm solving problem in classical field theory and I have some difficulties. I'm trying to study small oscilations of heavy string with fixed points. First of all I wrote down this Lagrangian: $$S=\...
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How does one actually compute the amplituhedron?

I was watching Nima's very popular talk (download if you're using chrome) (also mirrored at youtube here) about the "Amplituhedron", which has suddenly become very popular recently. He talks all ...
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442 views

Perturbation theory

I am puzzled with perturbation theory when studying quantum mechanics and solid theory. What I learn about perturbation is, from my ignorant point of view, just mathematics, or even simpler, matrix ...
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646 views

Coupled Oscillators

This is an exercise of my last exam. Since I couldn't find anybody who solved it or knows how to, it would be really nice if somebody could tell me if my thoughts on it go into the right direction. ...
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Adiabatic theorem in the regime of quantum optics

I am wondering whether there is a version of adiabatic theorem in the regime of quantum optics. My understanding of quantum optics involves with the interaction between photon and atom. This ...