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

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How was this one probability amplitude derived by Mattuck?

I'm reading A Guide to Feynman Diagrams in the Many-Body Problem by Richard D. Mattuck (2nd edition). You can look at the relevant pages here. On page 45, he presents a formula for $D_t c_p(t)$. ...
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Perturbation theory in second quantization

I am dealing with electron/phonon interaction in QM. In particular, given the Hamiltonian of a solid, $$H=H_{el}+H_{ion}+H_{el-ion}$$ we have that the el-phonon Hamiltonian is treatened ...
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How is translational symmetry related to Fourier decomposition?

The book (The Cosmic Microwave Background By Ruth Durrer) about cosmological perturbations says that because of translational symmetry of the background at a constant time, we can decompose our ...
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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 ...
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Corrections in Perturbation theory

Is there a way to construct a bound on the perturbative corrections to a problem in perturbation theory? For example, if I have the standard 1st order correction to the eigensolutions of a problem ...
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How to decide whether one can use perturbation theory in QM?

In QM, it is said that perturbation theory can be used in the case in which the total Hamiltonian is a sum of two parts, one whose exact solution is known and an extra term that contains a small ...
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Rigorous Proof of General Relativity's Non-renormalizability?

The answer to this question and the comments on it implies that general relativity has not been rigorously shown to be non-renormalizable for all loop diagrams -- only shown for two loops. However, ...
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Is WKB really applicable for the ground state?

It seems that WKB is applicable for a given $E$ if and only if $\hbar$ is sufficiently small. Or in other words, WKB is applicable if and only if the quantum number is large enough. Is this ...
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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: ...
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Perturbation in Linear Response Theory (classical formalism)

A N-particles system is described by the following Hamiltonian $$H=H_0+H'(t)$$ where $H_0$ is the unperturbated Hamiltonian and $H'(t)$ is the perturbation $$H'(f)=-A\cdot\mathcal{F}(t)$$ written as ...
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Estimating volume of moduli space of genus-g Riemann surface with n marked points

I wanted to know how can I estimate the volume of the moduli space of a Riemann surface of genus $g$ and having $n$ marked points. I am reading some old string theory papers which discuss divergences ...
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Time evolution of two orthogonal states in Time Dependent Perturbation Theory

Given the two orthogonal states for $H_0$ , $|n(t)>_I, |m(t)>_I$, in the interaction picture, we want to find the probability of transforming from one to the other after time t, aka: $ \ (1) \ ...
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How do the renormalization enter the actual amplitude calculation in QFT?

I have studied QFT from Peskin and Schroeder and from a few other books and lectures and I think I understand the procedure of renormalizing various parameters in the Lagrangian like mass, coupling ...
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Hartree-Fock correction to $e$-$e$ interaction

The corrections to the energy per electron in a jellium model (uniform distribution of positive ion charge approximation to the regulated long range order ionic array) is given by (in units of Ry) ...
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Kubo formula for general observables

In the wiki page about Kubo formula, the expectation of some observable under weak time-dependent perturbation is derived. However, from my point of view, some crucial steps are missing. I did the ...
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Expectation value in spin-orbit coupling

So I was just trying a question where it asked to find the Energy shift due to a spin-orbit coupling Hamiltonian to first order using perturbation theory. The Hamiltonian is $$H_{LS} = ...
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What is a diffusionless fluid?

I'm taking a course in astrophysical fluid dynamics and have come across a problem involving "small diffusionless disturbances" of a fluid. Based on the nature of the course I expect the examiner to ...
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What really are perturbation expansions?

I'm unsure if this question belongs here or at Math.SE, but since I've got to it by reading some articles about Physics I'm going to post it here anyway. In this particular article (Theoretical ...
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Perturbations in linear response theory

I've been working on applications of linear response theory to condensed matter systems, and I've got quite far into the literature on the subject. However, there is an identity which seems to be ...
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How should I interpret degenerate $\pm m_j$ states under the Stark effect?

I'm thinking about the Stark effect in Alkalis where fine structure is important (Cs, Rb, etc). The Stark effect doesn't lift the degeneracy of the $\pm m_j$ states. So should I interpret a state ...
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Post-Minkowskian expansion of some quantities in Post-Newtonian theory

I'm studying Post-Newtonian theory on the book "Gravity" by Poisson and Will and I found a few formulas that I can't obtain by myself. I'm pretty sure it must be quite simple, but can't find the right ...
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Scalar QED perturbation theory [closed]

Where can I find material about scalar QED? Renormalization, scattering matrix, perturbation theory, etc...
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LSZ reduction vs adiabatic hypothesis in perburbative calculation of interacting fields

As far as I know, there are two ways of constructing the computational rules in perturbative field theory. The first one (in Mandl and Shaw's QFT book) is to pretend in and out states as free ...
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Fermi Golden Rule

First order time dependent perturbation theory tells us that under the influence of a perturbation $Ve^{i\omega t}$, a system that started in the state $|n\rangle$ at time $t=0$ has probability ...
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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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Variational principle perturbation

I am trying to learn 2D SHO and free particle variational principles. However, I came across a perturbation like so: $Ax^2y^2$. The particle is simple harmonic in $x$ and a free particle in $y$? I'm ...
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Second-order correction in Quantum-Confined Stark effect

In the wikipedia article, there is a second-order correction in the Quantum-Confined Stark Effect. I could not understand how it was solved. I did not understand the meaning of 2(0) and 1(0) and how I ...
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Perturbation theory in quantum harmonic oscillator [closed]

This question concerns the quantum harmonic oscillator: (a)Express the operator $\hat B = \hat x \hat p + \hat p \hat x + \hbar$ in terms of $\hat a_{\pm}$ and $\hbar$ (b)Write the matrix ...
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How to tell the order of a Feynman diagram?

How can we know the order of a Feynman diagram just from the pictorial representation? Is it the number of vertices divided by 2? For example, I know that electnro-positron annihilaiton is first ...
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Perturbation theory : quadratic external field

I'm trying to derive the explicit form of S-matrix of an interaction Hamiltonian $$H' = \frac{1}{2} \lambda \left[ \int d^3 x \rho({\vec x}) \phi({\vec x}, t)\right]^2\tag{1}$$ Even though the ...
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Kramers-Kronig relations for the electron Self-Energy Σ

I'm currently studying an article by Maslov, in particular the first section about higher corrections to Fermi-liquid behavior of interacting electron systems. Unfortunately, I've hit a snag when ...
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Linear Perturbation theory in General Relativity, what to do with products of derivatives?

I'm trying to do a problem in which I am given the Einstein tensor for the following metric: $$ ds^2 = -e^{2\Phi}(d{x^0})^2 + e^{2\Psi}\delta_{ij}dx^idx^j, $$ And then asked to find the Einstein ...
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Definition of “nonlinear” in the context of perturbation of gravity

What exactly is the definition of a nonlinear perturbation when applied to a background spacetime metric? I have seen so called "linear perturbations" which look like $$ds^2 = -(1+2\Phi)dt^2 ...
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Applicability of perturbation theory

Consider some system in some initial state $|k^{(0)}\rangle$. The probability that such a state makes a transition to some other state $|m^{(0)}\rangle$ can be computed to various orders in time ...
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Gauge invariant quantities [closed]

In the context of cosmological perturbation one write the most general perturbed metric as $$ ...
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Energy levels in close-proximity of each other in time-independent degenerate perturbation theory

I've applied second order time-independent degenerate perturbation theory corrections to the energy with the method presented in Modern Quantum Mechanics by J.J. Sakurai. I shortly summaries this ...
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Performing one-loop “triangle” integral

Let's have integral $$ \tag 1 I_{\alpha \beta \gamma \delta} = \int d^{4}qD^{V}_{\alpha \beta}(k + q)D^{V}_{\gamma \delta}(q + k - p)D^{f}(q), $$ where $D^{V}_{\alpha \beta}$ corresponds to massive ...
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Scattering states of Hydrogen atom in non-relativistic perturbation theory

In doing second order time-independent perturbation theory in non-relativistic quantum mechanics one has to calculate the overlap between states $$E^{(2)}_n ~=~ \sum_{m \neq n}\frac{|\langle m | H' ...
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Non-degenerate or degenerate perturbation theory for a non-degenerate level of a system with other levels degenerate?

To decide whether I have to use non-degenerate or degenerate perturbation theory, I have to look only on whether the energy level I am calculating corrections to is degenerate, the degree of ...
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Ambiguity in Asymptotic Perturbative Series and Instantons

I know there are a number of questions about the asymptoticity of perturbative series and about instantons on StackExchange (e.g. Instantons and Non Perturbative Amplitudes in Gravity from user566, ...
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Why are the zeroth order terms in degenerate perturbation theory the eigenstates of the perturbing Hamiltonian?

I have for quite some time now tried to find a satisfactory answer to this, but I haven't yet. In perturbation theory, with small parameter $\lambda$, we expand the eigenstate as $$| E \rangle=| ...
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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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Decay of some particle involved quarks vs mesons as outgoing states

Let's have decay width of some mother particle into the state which involves hadrons. For simplicity, let's assume that creation of hadrons (on diagram) is possible only through electroweak vertices ...
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Stationary Perturbation Theory

We write first order correction in the wavefunction as a linear sum of eigenstates of unperturbed hamiltonian. Why is this possible?
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Quadratic order perturbation terms in the expansion of Ricci tensor [closed]

I want to expand Einstein-Hilbert action for the metric $$ g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu} $$ up to quadratic order in $h_{\mu \nu}$. For this purpose I need to calculate the Ricci tensor ...
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What is the sum over the transition rates?

I was looking at the solution to an exercise, and I came over this expression: $$P_{i\to f} = \sum \limits_{f} {2 \pi \over \hbar }\; |\langle f |\hat V | i \rangle |^2 \delta(E_{fi}-E),$$ where ...
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Final States in time dependet pertubation theory

I am trying to understand how to get over a common contradiction in time dependent perturbation theory. In time dependent perturbation theory we assume, that the perturbation only changes the ...
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Does the fact that we cannot exactly solve the Standard Model undermine the validity of QFT?

I have seen discusstions of this types before: there is a question about photons or virtual particles or vaccuum, etc. And there is usually a good and clear explanation from the point of view of ...
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Is the energy expectation value comparable to the equation from power series ansatz?

The Hamiltonian is given by $$ H = H_0 + \lambda H_1 $$ where $H_0$ is the unperturbed Hamiltonian, which solves the Schrödinger Equation $$ H_0 \left |n^{(0)} \right \rangle = E_n^{(0)} \left ...