What is Propagator: Definition and 200 Discussions
In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. In Feynman diagrams, which serve to calculate the rate of collisions in quantum field theory, virtual particles contribute their propagator to the rate of the scattering event described by the respective diagram. These may also be viewed as the inverse of the wave operator appropriate to the particle, and are, therefore, often called (causal) Green's functions (called "causal" to distinguish it from the elliptic Laplacian Green's function).
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
$$
\begin{align}
H_D S_F(x-y) &= (i \hbar...
I was reading Diagrammatica by Veltman and he treats the photon field as a massive vector boson in which gauge invariance is disappeared and the propagator has a different expression than in massless photon. After some googling, I found that this is one way to formulate QED which has the...
If I look at the photon propagator <A_mu (x) A^nu(0) > in momentum space, as I understand it I am to compute this by summing up all the self-energy diagrams of the photon, which look like:
photon -> stuff -> photon
In particular, since the photon shares the same quantum numbers as the Z, you...
I asked this from a number of people but no one knew what to do about this exponential with iota infinity in the power,in Eqtn 5.7
The textbook seems to imply that it is zero but cos and sine are undefined at infinity.
Also,all the exponentials seem to vanish from the final result of Eqtn 5.8...
Let ##\phi## be a scalar field and ##g_{\mu \nu} = \eta_{\mu \nu}+h_{\mu \nu}/M_p## where ##M_p## is the Planck mass (so we assume we deal with perturbations). Let ##\Lambda_2,\Lambda_3## be energy scales such that ##\Lambda_2 \gg \Lambda_3##. These are defined by ##\Lambda^2_2 = M_p H_0## and...
I am a little bit confused about the definition of the propagator.
We start with the evolution equation for our state vector.
$$ \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...
I know in the Heisenburg picture,
$$\Phi(\vec{x},t)=U^{\dagger}(t,t_0)\Phi_{0}(\vec{x},t)U(t,t_0)$$
where $$\Phi_{0}$$ is the free field solution, and
$$U(t,t_0)=T(e^{i\int d^4x \mathcal{L_{int}}})$$. Is there a way I could solve this using contractions or Feynman diagrams?
Because otherwise, it...
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...
The first thing I have to consider is that, since ##\Delta t \rightarrow 0##, the potential ##U## is not going to contribute and we can consider it to be ##0##.
Next thing I did was calculate ##\left<xt|x_1t_1\right>## directly with the definition considering what I said before, and I got...
Well, this calculation is straightforward in the Heisenberg picture. After finding the eigen values and eigen vectors of the total Hamiltonian, I found the explicit form for the exponential of the integral of the matrix and then did the matrix multiplication and calculated its expectation value...
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
$$\epsilon...
I don't understand a step in the derivation of the propagator of a scalar field as presented in page 291 of Peskin and Schroeder. How do we go from:
$$-\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...
Nieuwenhuizen uses a method for calculating the propagator by decomposing the field ## h_{\mu\nu}, ## first into symmetric part ## \varphi_{\mu\nu} ## and antisymmetric part ## \psi_{\mu\nu} ##, and then by a spin decomposition using projector operators. Using this he writes the dynamical...
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
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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
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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}$$
2. Loops of particles can produce effective...
Homework Statement
Calculate the classical propagator for a massive spin 1 particle by inverting the equations of motion to the form $$A_\mu=\Pi_{\mu\nu}J_\nu$$
Homework EquationsThe Attempt at a Solution
By solving the lagrangian for a massive spin 1 particle one gets $$(\Box +...
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...
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...
Homework Statement
Question:
To find/ explain why there exists a continuous lorentz transformation that flips the sign for space-like separation but not time-like.
Homework Equations
Signature ## (-,+,+...) ##
Definition of lorentz transformation:
##x^u=\lambda^u_v x^v ##...
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...