Propagator Definition and 180 Threads

Hi everyone How can i prove that the propagator is And then

I have this Lagrangian for a free massless left Weyl spinor, so it’s just the kinetic term, that can be written embedding the field into a larger Dirac spinor and then taking the left projector in this way: $$i \bar{\psi} \cancel{\partial} P_L \psi$$ Srednicki says that the momentum space...

I have a confusion about how the heavy quark propagators are constructed in HQET and how the loops (in the included figure) are constructed. A standard sort of introduction and motivation to HQET (as in reviews and texts like Manohar & Wise and M.D Schwartz) is as follows : The momentum of a...

Greentings, I've dealt with Quantum Theory a lot, but there's one thing I don't quite understand. When deriving the Fermion-Propagator, say ##S_F##, all the authors I've read from made use of the Fourier-Transform. Basically, it always goes like $$ \begin{align} H_D S_F(x-y) &= (i \hbar...

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I was reading Diagrammatica by Veltman and he treats the photon field as a massive vector boson in which gauge invariance is disappeared and the propagator has a different expression than in massless photon. After some googling, I found that this is one way to formulate QED which has the...

If I look at the photon propagator <A_mu (x) A^nu(0) > in momentum space, as I understand it I am to compute this by summing up all the self-energy diagrams of the photon, which look like: photon -> stuff -> photon In particular, since the photon shares the same quantum numbers as the Z, you...

I asked this from a number of people but no one knew what to do about this exponential with iota infinity in the power,in Eqtn 5.7 The textbook seems to imply that it is zero but cos and sine are undefined at infinity. Also,all the exponentials seem to vanish from the final result of Eqtn 5.8...

This question arises from Carroll's Appendix I on the parallel propagator where he shows that, in matrix notation, it is given...

I am a little bit confused about the definition of the propagator. We start with the evolution equation for our state vector. $$ \ket{\Psi(t)} = U(t,t_0)\ket{\Psi(t_0)} $$ Now, I would expect $$ \Psi(x, t) = \bra{x}U(t,t_0)\ket{\Psi(t_0)} = \int \delta(x'-x) U(t,t_0) \Psi(x',t_0) dx' $$...

To show that when ##[J^2, H]=0 ## the propagator vanishes unless ##j_1 = j_2## , I did (##\hbar =1##) $$ K(j_1, m_1, j_2 m_2; t) = [jm, e^{-iHt}]= e^{iHt} (e^{iHt} jm e^{-iHt}) - e^{-iHt} jm $$ $$ = e^{iHt}[jm_H - jm] $$ So we have $$ \langle j_1 m_1 | [jm, e^{-iHt} ] | j_2 m_2 \rangle $$ $$ =...

In this case, the lagrangian density would be $$\mathcal{L}=\frac{1}{2}((\partial_{\mu}\Phi)^2-m^2\Phi^2)-\frac{\lambda}{4!}\Phi^4$$ whe $$\Phi$$ is the scalar field in the Heisenburg picture and $$\ket{\Omega}$$ is the interacting ground state. Was just curious if there were ways to do Feynman...

First off let me say I am a bit confused by this question. Searching for some references I found the following related to the KG propagator, given by (P&S, chapter 2 pages 29, 30) Then they Fourier-transformed the KG propagator Is this what is aimed with this exercise? If yes, could you...

We know that the non-relativistic propagator describes the probability for a particle to go from one spatial point at certain time to a different one at a later time. I came across an expression (lecture notes) relating ##\Psi(x,t)##, an initial wave function and the propagator. Applying the...

I am following a lecture note on the QFT. But am a little confused about some parts related to the vacuum bubbles. We define the Feynman propagator, ##D_{F}(x-y)##, as giving the amplitude for a particle emitted at ##x## to propagate to ##y## (where it can be measured). After following the...

I'm guessing that there must be some nuance that I do not quite understand in the difference between ##|p\rangle## and ##|E\rangle##? Like, later in the book even ##dk## is used as a variable of integration, where ##k = p/\hbar.## Surely this has effects on the value of the integral - wouldn't...

This is section 16.3 of QFT for the Gifted Amateur. I understand the concept of the spacetime propagator ##G^+(x, t, x', t')##, but the following propagator is introduced without any explanation I can see: $$G^+(x, y, E) = \sum_n \frac{i\phi_n(x)\phi_n^*(y)}{E - E_n}$$ It would be good to have...

The following exercise was proposed by samalkhaiat here. The given Lorentz identities were proven here. We first note that ##d^4 k = d^3 \vec k dk_0##, the ##k_0## integration is over ##-\infty < k_0 < \infty## and ##\epsilon (k_0)## is the sign function, which is defined as $$\epsilon...

wanna know how to get to 8.2

I don't understand a step in the derivation of the propagator of a scalar field as presented in page 291 of Peskin and Schroeder. How do we go from: $$-\frac{\delta}{\delta J(x_1)} \frac{\delta}{\delta J(x_2)} \text{exp}[-\frac{1}{2} \int d^4 x \; d^4 y \; J(x) D_F (x-y) J(y)]|_{J=0}$$ To...

I am currently starting with my first qft lectures and i am trying to see for the free particle that the propagator $$ <x_i | e^{-i\frac{p}{2m} T|x_f}>$$ will equal to one if x_f = 1, x_i=0 m=1 u=1 p=1, T=1 and $$\hbar \rightarrow 0$$ or 0 otherwise. I understand that this limit will result in...

My fundamental issue with this exercise is that I don't really know what it means to "show that X is a propagator".. Up until know I encountered only propagators of the from ##\langle 0\vert [\phi(x),\phi(y)] \vert 0\rangle##, which in the end is a transition amplitude and can be interpreted as...

I would like to get some information on this topic. It is not discussed in many places, so if any members here know about it, i would be interested in a brief explanation. Or any books or online documents where it is discussed. D is the "invariant propagation function" or the "propagator". I...

Summary: how to cure infrared divergences in fermion propagator in QED? In calculating the fermion propagator in QED, we identify Ultraviolet and Infrared divergences. the Ultraviolet divergences solved by regularization, but I don't understand how to treat the Infrared divergences. Infrared...

I'm reading through Lancaster & Blundell's Quantum Field Theory for the Gifted Amateur and have got to Chapter 17 on calculating propagataors. In their equation 17.23 they derive the expression for the free Feynman propagator for a scalar field to be...

I'm having trouble understanding a specific line in my lecturers notes about the path integral approach to deriving the Klein Gordon propagator. I've attached the notes as an image to this post. In particular my main issue comes with (6.9). I can see that at some point he integrates over x to...

Homework Statement [/B] Hi in the first attachment I am stuck on the sign change argument used to get from line 2 to 3 , see below Homework Equationsabove The Attempt at a Solution [/B] Q1) please correct me if I'm wrong but : ##d^3 p \neq d\vec{p} ## since ##d^3 p = dp_x dp_y dp_z ## and...

Homework Statement I want to show that the propagator of Proca Lagrangian: \mathcal{L}=-\frac{1}{4}F_{\mu \nu}F^{\mu \nu}+\frac{1}{2}M^2A_\mu A^\mu Is given by: \widetilde{D}_{\mu \nu}(k)=\frac{i}{k^2-M^2+i\epsilon}[-g_{\mu\nu}+\frac{k_\mu k_\nu}{M^2}]Homework Equations Remember that...

Homework Statement (This is part of a problem from Schwarz book on QFT). 1. Show that a propagator only has an imaginary part if it goes on-shell. Explicitly, show that $$Im(M)=-\pi\delta(p^2-m^2)$$ when $$iM=\frac{i}{p^2-m^2+i\epsilon}$$ 2. Loops of particles can produce effective...

Homework Statement I have the Lagrangian $$L=-\frac{1}{2}\phi\Box \phi-\frac{1}{2}m^2\phi^2$$ and I need to show that the propagator in the momentum space I obtain using this lagrangian (considering no interaction) is the same as if I consider the free Lagrangian to be...

The logic of the Feynman Propagator is confusing to me. Written in integral form as it is below $$\Delta _ { F } ( x - y ) = \int \frac { d ^ { 4 } p } { ( 2 \pi ) ^ { 4 } } \frac { i } { p ^ { 2 } - m ^ { 2 } } e ^ { - i p \cdot ( x - y ) },$$ there are poles on the real axis. I have seen...

I am thinking about a problem. Consider the forward Compton scattering process e(p)+γ(k) -> e(p)+γ(k), as shown in the following figure. If we consider the initial red photon is emitted by another electron which then goes to anything, then how can we write down the whole amplitude for this...

Homework Statement I am following a textbook "Seismic Wave Propagation in Stratified Media" by Kennet, I was greeted by the fact that he decided to use cylindrical coordinates to compute the Stress and Strain tensor, so given these two relations, that I believed to be constitutive given an...

I'm trying to make sense of the derivation of the Klein-Gordon propagator in Peskin and Schroeder using contour integration. It seems the main step in the argument is that ## e^{-i p^0(x^0-y^0)} ## tends to zero (in the ##r\rightarrow\infty## limit) along a semicircular contour below (resp...

I'm studying QFT by David Tong's lecture notes. When he discusses causility with real scalar fields, he defines the propagator as (p.38) $$D(x-y)=\left\langle0\right| \phi(x)\phi(y)\left|0\right\rangle=\int\frac{d^3p}{(2\pi)^3}\frac{1}{2E_{\vec{p}}}e^{-ip\cdot(x-y)},$$ then he shows that the...

I was reading about the classical Klein-Gordon propagator here: https://en.wikipedia.org/wiki/Propagator#Relativistic_propagators Basically they are looking for ##G##, that solves the equation $$(\square _{x}+m^{2})G(x,y)=-\delta (x-y).$$ So they take the Fourier transform to get...

I note the following: \begin{equation} \begin{split} \langle\vec{x}_n|e^{-i \frac{\mathcal{H}_n}{\hbar} (t_n-t_0)}|\vec{x}_{0}\rangle &=\delta(\vec{x}_n-\vec{x}_0)e^{-i \frac{\mathcal{H}_n}{\hbar} (t_n-t_0)} \end{split} \end{equation}I divide the time interval as follows...

I note the following: \begin{equation} \begin{split} \langle \vec{x}| \hat{U}(t-t_0) | \vec{x}_0 \rangle&=\langle \vec{x}| e^{-2 \pi i \frac{\mathcal{H}}{\hbar} (t-t_0)} | \vec{x}_0 \rangle \\ &=e^{-2 \pi i \frac{\mathcal{H}}{\hbar} (t-t_0)} \delta(\vec{x}-\vec{x}_0)...

My aim is to derive the photon propagator in an Coulomb gauge following Pokorski's book method. In this book the photon propagator in Lorenz gauge was obtained as follows: 1. Lorenz gauge: ##\partial_{\mu}A^{\mu}=0## 2. It's proved that ##\delta_{\mu}A^{\mu}_T=0##, where...

My aim is to derive the photon propagator in an Coulomb gauge following Pokorski's book method. In this book the photon propagator in Lorenz gauge was obtained as follows: Lorenz gauge: ##\partial_{\mu}A^{\mu}=0## It's proved that ##\delta_{\mu}A^{\mu}_T=0##, where...

Hello! I am reading Peskin's book on QFT and in the first chapter (pg. 30) he introduces this: ##<0|[\phi(x),\phi(y)]|0> = \int\frac{d^3p}{(2\pi)^3}\int\frac{dp^0}{2\pi i}\frac{-1}{p^2-m^2}e^{-ip(x-y)}## and then he spends 2 pages explaining the importance of choosing the right contour integral...

Hello! I am reading about Klein Gordon operator from Peskin book and he reaches at a point the integral ##\int_0^\infty \frac{1}{p^2-m^2}e^{-ip(x-y)}dp^0##. He then explains the different approaches of doing this integral, depending on how you pick the contour around the 2 poles. Why does the...

Hello! I read that in Heisenberg picture the propagator from x to y is given by ##<0|\phi(x)\phi(y)|0>##, where ##\phi## is the Klein-Gordon field. I am not sure I understand why. I tried to prove it like this: ##|x>=\phi(x,0)|0>## and after applying the time evolution operator we have...

Hi everyone! I'm having a problem with calculating the fermionic propagator for the quantized Dirac field as in the attached pdf. The step that puzzles me is the one performed at 5.27 to get 5.28. Why can I take outside (iγ⋅∂+m) if the second term in 5.27 has (iγ⋅∂-m)? And why there's a...

The momentum-space fermion propagator in the free Dirac theory is given by The arrow on the fermion propagator is said to represent the flow of charge. How can we derive this statement quantitatively from the Dirac Lagrangian? What is the quantitative form of the charge being referred to here?

Hello! I have a question about the photon propagator in Feynman diagrams. I am looking over a brief derivation (probably there are some details missing), so basically it starts from Proca equation for a mass 0 particle, then it assumes Lorentz condition and in the end it obtains...

Hi all, I'm studying ## \mu \to e~ \gamma ## decay from cheng & Lie' book " gauge theory of elementary particles ". In Equation (13.84), he wrote the neutrino propagator ## \sum_i \Big ( \frac{U^{*}_{ei} U_{\mu i}}{(p+k)^2-m_i^2} \Big), ## (where the sum taken over neutrinos flavors) in the...

Hi, I read in a QFT book that the free massive vector boson Lagrangian is ## \mathcal{L}_W = - \frac{1}{4} (\partial_\mu W^\dagger_\nu - \partial_\nu W_\mu^\dagger ) (\partial^\mu W^\nu - \partial^\nu W^\mu ) + M^2_W W^\dagger_\mu W^\mu ## gives the propagator in momentum space by: ## i...

I am wanting to get the expression up to ##O(\epsilon^{2}) ## : To show that ##\frac{1}{2w_{k}} (\frac{1}{w_{k}-k_{0}-i\epsilon} + \frac{1}{w_{k}+k_{0}-i\epsilon})## ##=## ## \frac{1}{k_{v}k^{v} + m^{2} - i\epsilon}##, [2] where ##w_{k}^{2}=k^{2}+m^{2}##, ##k## the variable, and (this seemed...

Homework Statement Compute ##\displaystyle{\int\ \frac{d^{4}p}{(2\pi)^{4}}} \frac{i}{p^{2}-m^{2}+i\epsilon}e^{-ip \cdot{(x-y)}}## in terms of the invariant interval. Interpret your answer in the limit of small and large invariant intervals and for zero mass. Homework Equations The Attempt...