Questions tagged [perturbation-theory]

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

Filter by
Sorted by
Tagged with
11
votes
2answers
894 views

Why do the counterterms in renormalized $\phi^4$-theory with power two in fields give vertices and not propagators?

I am reading Peskin and Schroeder, chapter ten, and my Lagrangian is $$ \mathcal{L}=\frac{1}{2}(\partial_\mu\phi_r)^2-\frac{1}{2}m^2\phi_r^2-\frac{\lambda}{4!}z^2\phi^4+\frac{1}{2}\delta_Z(\partial_\...
21
votes
1answer
884 views

Asymptoticity of Pertubative Expansion of QFT

It seems to be lore that the perturbative expansion of quantum field theories is generally asymptotic. I have seen two arguments. i)There is the Dyson instability argument as in QED, that is showing ...
37
votes
5answers
6k views

Tree level QFT and classical fields/particles

It is well known that scattering cross-sections computed at tree level correspond to cross-sections in the classical theory. For example the tree level cross-section for electron-electron scattering ...
19
votes
2answers
4k views

Why is the second order perturbative correction to the ground state energy always down?

What is the physical/deeper reason for the second order shift of the ground state energy in time independent perturbation theory to be always down? I know that it follows from the formula quite ...
7
votes
2answers
2k views

Finding the effective Hamiltonian in a certain subspace

In order to find the effective Hamiltonian in a subspace which is energetically well separated from the rest of the Hilbert space people try to find a unitary transformation which makes the ...
26
votes
3answers
4k views

A pedestrian explanation of Renormalization Groups - from QED to classical field theories

shortly after the invention of quantum electrodynamics, one discovered that the theory had some very bad properties. It took twenty years to discover that certain infinities could be overcome by a ...
27
votes
6answers
2k views

Why Does Renormalized Perturbation Theory Work?

I've read about renormalization of $\phi^4$ theory, ie. $\mathcal{L}=\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-m^2\phi^2-\frac{\lambda}{4!}\phi^4\,,$ particularly from Ryder's book. But I am ...
23
votes
2answers
456 views

Can we get full non-perturbative information of interacting system by computing perturbation to all order?

As we know perturbative expansion of interacting QFT or QM is not a convergent series but an asymptotic series which generally is divergent. So we can't get arbitrary precision of an interacting ...
21
votes
3answers
7k views

Perturbation theory with degeneracy even after 1st order

Most textbooks on basic quantum mechanics tell you that when your initial Hamiltonian $H_0$ has degenerate states, then before you can do (time independent) perturbation theory with a perturbation ...
7
votes
2answers
1k views

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=| E^{(...
5
votes
3answers
962 views

References for ADM formalism and cosmological perturbation theory [closed]

What would you consider the best online resources for learning the 3+1 ADM formalism and gauge invariant perturbation theory in cosmology? (Assuming intermediate level GR and QFT familiarity)
5
votes
1answer
286 views

Eigenkets of degenerate perturbation theory

Suppose the original Hamiltonian is $H$ and we perturb it by a small potential $V$. The basis kets of the original hamiltonian $H$ contains some degeneracy. Since there's some degeneracy, we take ...
4
votes
2answers
1k views

proof of radius of convergence of perturbation series in quantum electrodynamics zero

Can anyone show detailed proof of why radius of convergence of perturbation series in quantum electrodynamics is zero? And how is perturbation series constructed? So, as this argument requires ...
6
votes
2answers
1k views

What are non-perturbative effects and how do we handle them?

Schwartz's QFT book contains the following passage. To be precise, total derivatives do not contribute to matrix elements in perturbation theory. The term $$\epsilon^{\mu\nu\alpha\beta} F_{\mu\...
18
votes
5answers
2k views

Can Feynman diagrams be used to represent any perturbation theory?

In Quantum Field Theory and Particle Physics we use Feynman diagrams. But e.g. in Schwartz's textbook and here it is shown that it applies to more general cases like general perturbation theory for ...
12
votes
3answers
747 views

How can perturbativity survive renormalization?

The most usual way to renormalize quantum field theories is by re-writing the Lagrangian in terms of physical (finite) parameters plus counter-terms. Take $\lambda \phi^4$ theory for instance: $$ {\...
12
votes
2answers
4k views

What does a non-perturbative theory mean?

I'm a science writer and I'm having difficulty understanding what a non-perturbative approach means. I thought I understood what perturbative meant, but in looking for explanations of non-perturbative,...
22
votes
3answers
4k views

Does QED really break down at the Landau pole?

In QED, the fine structure constant $\alpha$ runs upwards in the UV, with a loop calculation (involving a geometric series of the vacuum polarisation diagram) indicating a divergence in $\alpha$ at $\...
14
votes
2answers
2k views

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,...
8
votes
3answers
688 views

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 ...
11
votes
2answers
1k views

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' |...
10
votes
3answers
617 views

Use my example to explain why loop diagram will not occur in classical equation of motion?

We always say that tree levels are classical but loop diagrams are quantum. Let's talk about a concrete example: $$\mathcal{L}=\partial_a \phi\partial^a \phi-\frac{g}{4}\phi^4+\phi J$$ where $J$ is ...
7
votes
1answer
647 views

How can an asymptotic expansion give an extremely accurate predication, as in QED?

What is the meaning of "twenty digits accuracy" of certain QED calculations? If I take too little loops, or too many of them, the result won't be as accurate, so do people stop adding loops when the ...
1
vote
1answer
211 views

Understanding degenerate vacua in a Quantum field theory with sponatneously broken symmetry

$\bullet$ Consider the following Hamiltonian (density) $$\mathscr{H}=\frac{1}{2}(\partial^\mu\phi)(\partial_\mu\phi)+\mathcal{V}(\phi) \hspace{0.4cm}\text{where}\hspace{0.4cm}\mathcal{V}(\phi)=\frac{\...
5
votes
2answers
729 views

Why are derivatives in interaction terms treated differently from derivatives in the kinetic term?

I know that derivative couplings in a Lagrangian interaction, such as $$\mathcal{L}_{int} = \lambda \phi (\partial_{\mu}\phi)(\partial^{\mu}\phi)$$ bring down two momentum factors into the matrix ...
4
votes
1answer
2k views

Scattering Processes in Scalar Yukawa Theory

I'm trying to compute nucleon-nucleon scattering in scalar Yukawa theory. Here we view a nucleon as a complex scalar field $\psi$ and a meson as a real scalar field $\phi$. They interact through $H_I=...
11
votes
1answer
633 views

Details of Newtonian Prediction for Mercury's Precession

Could anyone point me to a book or outline the methods used to actually calculate the 532 arcseconds per century that Newtonian theory apparently predicts for Mercury's precession. I am completely ...
7
votes
3answers
2k views

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. ...
7
votes
1answer
2k views

Where can a good treatment of the 'sudden' perturbation approximation be found?

Where can a good treatment of the 'sudden' perturbation approximation be found? A lot of quantum mechanics books have very brief discussions of it but I want to see it in some detail and preferably ...
3
votes
2answers
671 views

Higher orders in perturbation theory

I would like to compute an energy level up to many orders in perturbation theory. My difficulty right now is not in the calculation itself but in understanding the algebraic structure of the higher ...
17
votes
4answers
1k views

Staying in orbit - but doesn't any perturbation start a positive feedback?

I am not a physicist; I am a software engineer. While trying to fall asleep recently, I started thinking about the following. There are many explanations online of how any object stays in orbit. The ...
5
votes
4answers
528 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^\...
4
votes
1answer
387 views

Srednicki's Path Integrals: Ground-State to Ground-State Transition Amplitude in the Presence of a Perturbation

Srednicki's Quantum Field Theory mentions the following at the end of the unit on path integrals in non-relativistic quantum mechanics: Assume that the total Hamiltonian is of the form, $$ H = ...
1
vote
0answers
145 views

Perturbation theory in quantum mechanics: assumptions on eigenvectors [duplicate]

In a course that I follow, we use the perturbation method to find the eigenvectors and energies to an Hamiltonian written $ H_0 + V $ where $V$ is a weak perturbation. It is written that as $V$ is a ...
0
votes
1answer
53 views

Force on two level system in standing wave electric field

Given external radiation, produced from an electric field travelling from the left and an electric field travelling from the right, producing a standing wave $$E_0e^{i(kx - \omega t)} + E_0e^{-i(kx - ...
1
vote
1answer
278 views

Sakurai QM section 5.8

If anyone is familiar with Sakurai's book, specifically section 5.8 on energy shift and decay width, I am stuck and could use some help. I can't see how he derives 5.8.9 (in the revised edition). He ...
0
votes
1answer
123 views

Solution of the coupled non-linear oscillators by using perturbation theory [closed]

The integration shown here, $$∫_{-\infty}^{+∞}x^r\mathrm{Exp}[−x^2]\mathrm{H_n}^2[x]\mathrm{d}x,$$ appears when we try to calculate the spectrum of the perturbed non-linear oscillators by using ...
-1
votes
2answers
68 views

What does the term $\mathcal O(\epsilon^2)$ mean?

In the highest upvoted answer to Where does the $i$ come from in the Schrödinger equation? the author writes the following equation: $$ U^\dagger U=(\mathbb I+\epsilon^* A^\dagger)(\mathbb I+\...
18
votes
1answer
683 views

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 states,...
8
votes
1answer
2k views

Physical Interpretation of Relationship Between Hall Conductivity and Berry Curvature?

Why is the Hall conductivity in a 2D material $$\tag{1} \sigma_{xy}=\frac{e^2}{2\pi h} \int dk_x dk_y F_{xy}(k)$$ where the integral is taken over the Brillouin Zone and $F_{xy}(k)$ is the Berry ...
18
votes
1answer
1k 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 ...
15
votes
1answer
453 views

Why do we care about old-style, counterterm renormalizability?

There are a few different definitions of renormalizability that are standard in quantum field theory textbooks. They're all called the same thing, but I'll make up names to make the distinctions clear....
16
votes
2answers
2k views

Quantum Field Theory in position space instead of momentum space?

What are the reasons why we usually treat Quantum Field Theory in momentum space instead of position space? Are the computations (e.g. of Feynman diagrams) generally easier and are there other ...
4
votes
1answer
156 views

How to path-integrate over the half-line?

Consider the path-integral over a scalar field $\varphi$: $$ Z=\int_{\mathcal S}\ \mathrm e^{iS[\varphi]}\mathrm d\varphi $$ where $\mathcal S$ is some function space (say, Schwartz or its dual). How ...
4
votes
1answer
952 views

A question about the asymptotic series in perturbative expansion in QFT

Related post I heard about the argument that the perturbative expansion in QFT must be asymptotic, such as http://ncatlab.org/nlab/show/perturbation+theory#DivergenceConvergence Roughly this can ...
8
votes
1answer
487 views

Calculating Green's function from Dyson's series without normal ordering

I'm reading the derivation of the QED generating functional on Mandl&Shaw, "Quantum Field Theory" 2nd ed., 12.5.2.. The authors start from the expression (schematically): $$Z[j]=\frac{\langle 0\...
5
votes
2answers
389 views

Why does the branch cut of the self-energy begin at $2m$?

Consider a scalar field $\phi(x)$, and let its two-point function be $$ \frac{1}{p^2-m^2-\Pi(p^2)}=\int_0^\infty\mathrm d\mu^2\ \rho(\mu^2)\ \frac{1}{p^2-\mu^2} $$ We usually have $\Pi(m^2)=0$, which ...
4
votes
2answers
452 views

Relation between perturbation theory and Taylor expansion in QM

So I am looking at non-degenerate perturbation theory. The idea is that the perturbing term in the Hamiltonian is small so you somehow expand the energies and wave functions in this small term and ...
3
votes
2answers
4k views

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 ...
3
votes
1answer
169 views

Are these two definitions of the effective Hamiltonian equivalent?

Consider a Hamiltonian $H$ on a Hilbert space $\mathcal H_A \oplus\mathcal H_B$ and let $P$ (and $Q$) denote the projection operator onto $A$ (and $B$). There are two common definitions of an ...