Questions tagged [perturbation-theory]

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

248 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
11
votes
0answers
479 views

Capturing (perturbatively) non-equilibrium field theory effects using “elementary” methods

I am considering a system of two interacting scalar fields: $\psi$, and $\phi$. The Lagrangian is given by: \begin{equation} \mathcal{L}[\psi]=\frac{1}{2}\partial_\mu\psi\partial^\mu\psi+\frac{1}{2}\...
9
votes
0answers
226 views

Basic questions on the PPN formalism in General Relativity

I'm trying to learn about testing modified gravity using the PPN formalism. I have several textbooks that I am reading through (including Clifford Will's book), and have some basic questions on the ...
7
votes
0answers
164 views

Kallen-Lehmann representation and branch cuts at threshold masses

Let us consider the Kallen-Lehmann representation for the two-point function of scalar fields $$ \langle \Omega | T\left\{\phi(x) \phi(y)\right\}|\Omega\rangle = \int \frac{d^4 p}{(2\pi)^4} e^{ip\...
7
votes
0answers
218 views

Renormalization group approach to renormalization

Given a $n$-point bare Green function in a massless asymptotically free theory, we have that the following limit exists and is finite \begin{equation} \lim_{\Lambda\rightarrow\infty} Z^{-n/2}(g_0,\...
7
votes
0answers
257 views

Path Integral on Feynman Hibbs: Interaction of EM field and matter, how can we get to equation (9.68) from (9.67)?

On Feynman Hibbs "Quantum Mechanics and Path Integrals", the equation (9.67) describe the transition amplitude of the matter (for example an atom) to go from the state $M$ to the state $M$ when it ...
6
votes
0answers
69 views

Using relativistic QFT to compute energy levels

I've taken a year of QFT so far, and although there seems to be a lot of attention paid to scattering amplitudes and decay rates and perhaps bound states, I view computing energy spectra as certainly ...
5
votes
0answers
79 views

Break down of time independent perturbation formula of quantum mechanics in quantum field theory

The following paragraph is from Schwartz Sec 4.2.1 Using OFPT we would calculate the energy shift using $$\Delta E_n = \langle\psi_n\rvert H_{int} \rvert \psi_n\rangle +\sum_{m,m \ne n} \frac{...
5
votes
0answers
130 views

Tadpoles in sigma models

In some QFTs, the tadpoles are not taken into account, since they vanish due to certain symmetries of the theory. Peskin and Schröder address this issue in QED around Equation (10.5) of their book; ...
5
votes
0answers
717 views

Cubic perturbation to coupled quantum harmonic oscillators

I recently came across this two-dimensional problem of a particle in a potential of the form $$V = \displaystyle{\frac{1}{2}m \omega^2} \big(y^2 + x^2y \big) - \alpha y,$$ where $x$ and $y$ are known ...
5
votes
0answers
116 views

No mixing in light cone perturbation theory

In hep-ph/0609090, Triumvirate of Running Couplings in Small-x Evolution, Kovchegov et. al. calculated the running coupling correction to the Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov and ...
4
votes
1answer
101 views

Hydrodynamic interaction between two spheres in $Re\ll 1$ flow

I am studying the interaction between two spherical particles of radius $a$ in a low Reynolds number flow. Because of linearity, I know that their respective velocities will be linear in the forces ...
4
votes
0answers
124 views

Symmetry of interaction lagrangian and symmetry of full lagrangian

Suppose we have lagrangian $$ \tag 1 L = \frac{\theta}{f_{\gamma}}F_{EM}\tilde{F}_{EM} +\frac{1}{2}(\partial_{\mu}\theta)^2 - \frac{1}{2}m_{\theta}^2\theta^2 + L_{SM}, $$ where $\tilde{F}_{EM}$ ...
4
votes
0answers
228 views

Integrability of the many body problem

In classical canonical perturbation theory of many degrees of freedom we encounter the problem of small divisors when attempting to find a solution for the generating function of the canonical ...
4
votes
0answers
441 views

Question about the perturbative renormalization group

I'm currently learning about the renormalization group (RG) in condensed matter physics and just want to clarify a couple of things: When doing the RG transformation, there's a flow to a fixed point. ...
3
votes
0answers
107 views

How do physicists use the Feynman path integral practically?

I somewhat understand how the path integral works in simple examples, I just don't get how someone can add up an infinite amount of paths. How is that even calculated? If it's got something to do ...
3
votes
0answers
58 views

Schrieffer-Wolff Transformation for conventional superconductors

I was trying to follow the discussion in Radi A. Jishi's book (Feynman Diagrams in Condensed Matter Physics), Chapter 12 on superconductors. They basically have a Hamiltonian that comprises of a ...
3
votes
1answer
490 views

Srednicki Eqs. (6.22) and (9.6). How to get rid of $i\epsilon$ in the interaction term?

I'm studying qft from Srednicki's book. If one writes down the full $i\epsilon$ terms, passing from Eq. (6.21) (non-perturbative definition) to Eq. (6.22) (perturbative definition) yields $$\left<0|...
3
votes
1answer
209 views

Optical theorem in $\phi^4$: which poles contribute to discontinuity in Feynman amplitude?

Section 7.3 ("The Optical Theorem") in Peskin and Schroeder's QFT text contains a leading order verification of the optical theorem in $\phi^4$ theory by calculating the (discontinuity across the ...
3
votes
0answers
182 views

Feynman diagrams included in Hartree-Fock approximation

Given a hamiltonian, I compute the Hartree-Fock self-energy. Let's say I now compute the second order self-energy with diagrams. Some of them are just like the Hartree or Fock diagrams of first order, ...
3
votes
0answers
192 views

Procedure for Effective Hamiltonian using Perturbation Theory? (Bilayer Graphene model)

Sorry if this is a dumb question as I'm just starting out, but in this paper https://arxiv.org/pdf/1803.08057.pdf on Twisted Bilayer Graphene, the authors claim to use "standard perturbation theory" ...
3
votes
0answers
97 views

Perturbative Techniques In Finding Electric Field of Symmetric Distributions

Lets say we have a uniform sphere of charges at the origin (at retarded time = 0) with some velocity and we are interested in the field at a point along the x-axis (normal to the surface of the sphere)...
3
votes
0answers
582 views

Proof non-convergence of perturbation in QED

This is an attempt to ask separately about aspects of my previous question, which was closed as too broad. Note that I strongly prefer results that are or can be made mathematically completely ...
3
votes
0answers
346 views

Adiabatic fluctuations

In Baumann's cosmology lecture,http://www.damtp.cam.ac.uk/user/db275/Cosmology/Chapter4.pdf , chapter 4, page 89, he defines adiabatic perturbation as: Adiabatic perturbations have the property ...
3
votes
2answers
291 views

Pauli- Villars regularization in the Electron Vertex Function: Evaluation

I'm studying one loop contribution for electron vertex function form Peskin and Schroeder's book " An introduction to quantum field theory " Section: 6.3. I have some troubles with Pauli- Villars ...
3
votes
0answers
138 views

Perturbation theory and the accuracy of physics in real life scenarios

Back when I studying the time independent perturbation theory of the Hamiltonian in quantum mechanics, I remember reading that there are only three problems in physics with exact solutions: the free ...
3
votes
0answers
331 views

Degenerate perturbations: why is it not necessary that $ [H_0,H']=0$?

Suppose we are doing a degenerate Rayleigh-Schodinger perturbation problem. Let's say the Hamiltonian $H_0$ is perturbed by a small perturbation $H'$, and we want corrections to the energy eigenstates/...
3
votes
0answers
625 views

Solving the quantum an-harmonic oscillator pertubatively?

Background Generally while solving the quantum an-harmonic oscillator: $$ -\frac{d^2 y}{dx^2} + k_1 x^4 y + k_2 x^2 y= E y $$ Most people (I've googled) on the internet always solve this using: ...
3
votes
0answers
145 views

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 ...
3
votes
0answers
253 views

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 ...
3
votes
0answers
179 views

Gauge invariance of Fermi's golden rule

I am having some issues with gauge invariance of Fermi's golden rule. Say we have a system Hamiltonian for a particle in an electric field and some additional potential $V$ with \begin{equation}H=(p-A(...
3
votes
0answers
111 views

Why does the time-independent perturbation theory become no longer useful when its order gets larger?

In Griffith's Introduction to Quantum Mechanics p. 256, after figuring out $$E_n^2=\sum_{m\neq n} \frac{|\langle\psi_m^0|H'|\psi_n^0\rangle|^2}{E_n^0-E_m^0}$$ he says We could go on to calculate ...
3
votes
0answers
2k views

“Good” States In Degenerate Perturbation Theory

During the section on Degenerate Perturbation Theory, Griffiths (Introduction to Quantum Mechanics 2ed) starts with a general linear combination of two orthogonal eigenfunctions of $H_0$. He walks ...
3
votes
0answers
214 views

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 ...
3
votes
0answers
611 views

What does it mean to expand a Hamiltonian using perturbation theory?

On UC Davis chemwiki website, the Hamiltonian for quadrupolar coupling in NMR is analyzed. (The details of this aren't important.) It is said in the analysis that: The expansion of the ...
3
votes
1answer
214 views

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}(...
3
votes
0answers
500 views

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 $H_{0}$: $$ \left|b\right&...
3
votes
0answers
885 views

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 ...
3
votes
0answers
265 views

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 ...
2
votes
0answers
23 views

How to calculate the total cross section of a QCD process? (qq->gg)

I want to calculate the total cross section of various QCD processes (let's go with $q\bar{q}\rightarrow gg$ for this question) at tree level for some colliding hadrons at a certain energy (let's go ...
2
votes
0answers
76 views

When does the Post-Newtonian expansion break down?

In the book Gravitational waves Vol.1: theory and experiment by M.Maggiore, in chapter 5, page 236, the author discusses the Post-Newtonian (PN) expansion and says that it is valid for small speed and ...
2
votes
1answer
52 views

Jeans Instability in an Expanding Universe, Understanding the perturbation

I am trying to derive the equation for a case, where we have a dust(zero-pressure) in an expanding universe. There are 4 equations but I think exercising on one of them would be helpful for me. So ...
2
votes
0answers
39 views

Keldysh Field Theory: Self Energy Structure in RAK basis

Consider the Keldysh formulation of electrons interacting with each other through the Coulomb potential. Suppose that we've switched to the RAK (Retarded-Advanced-Keldysh) basis of Larkin & ...
2
votes
0answers
24 views

Error estimation for field theories

I am looking for resources on error estimation for field theories, both the error due to perturbation theory and measurement error. In other words, consider a field theory of a field $\phi$, with some ...
2
votes
0answers
71 views

What is a reason for the energy to not conserve in QM perturbation theory?

Consider the transition rate of the evolution of the state $|i\rangle$ to a state $|f\rangle$ - the Fermi golden rule: $$ d \omega_{if} = 2\pi |\mathcal{M}_{fi}|^{2} \delta(E_{f} - E_{i})d\nu, \quad \...
2
votes
0answers
30 views

Rigorous adiabatic elimination for $N$-state quantum system under harmonic perturbation

I am interested in a rigorous derivation of the coherent evolution of a quantum system with $N$ states under the application of a harmonic perturbation. I have read several articles about the use of ...
2
votes
1answer
71 views

Perturbation Theory - Exact Solutions and Good Quantum States

I'm having a problem with the following question: Problem: Consider the unperturbed, degenerate Hamiltonian $H_0=\bigg(\begin{matrix} E &0\\ 0& E\end{matrix}\bigg)$. Add the perturbation $...
2
votes
0answers
41 views

What happens to Hubbard chain when perturbation theory blows up? Singular energy for complex interaction

Consider the spinful Hubbard chain: $$ H = - t \sum_{i,\sigma} \left( c^\dagger_{i,\sigma} c^{\vphantom \dagger}_{i+1,\sigma} + h.c. \right) + U \sum_n \left( n_{i,\uparrow} - \frac{1}{2} \right)\left(...
2
votes
0answers
35 views

The stark effect of Hydrogen

I have read the stark effect of Hydrogen (calculating energy levels of the n=2 states of a Hydrogen atom placed in an external uniform electric field along the positive z-direction) from Quantum ...
2
votes
0answers
31 views

Learning resources for the PPN formalism

I'm trying to learn more about the PPN formalism, and I was hoping to find some papers that show how the various parameters are calculated. Basically, I want to go through the usual calculations of ...
2
votes
0answers
45 views

Horizon entry, Meszaros suppression and start of perturbation growth

I thought that the onset of perturbation growth was determined by horizon entry of the perturbation (because there won't be a gravitational collapse of an over dense region not causally connected to ...

1 2 3 4 5