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).

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1. I Compton scattering amplitude and propagator

Hi everyone How can i prove that the propagator is And then
2. I Propagator of massless Weyl field

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...
3. A Heavy Quark Propagators in HQET

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...

13. A How Do Feynman Diagrams Work in Phi^4 Theory?

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...
14. Phi 4 Theory Propagator Question

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...
15. Showing that a given propagator is proportional to Green's function

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...
16. Computing a wave function through a (non-relativistic) propagator

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...
17. A Propagator from a space-time point to itself

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...
18. Feynman's Propagator calculation

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...
19. Calculation of a Propagator for a Spin 1/2 system

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...
20. Change of variables in a propagator

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...
21. I Understanding the Role of Space Energy Propagator in Quantum Field Theory

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...

36. What is the closed form for the series S?

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...
37. A What is the "real" Feynman propagator?

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...
38. A What happens if photon propagator goes on shell?

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...
39. Stress and Strain tensors in cylindrical coordinates

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...
40. On deriving the standard form of the Klein-Gordon propagator

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...
41. A What's the idea behind propagators

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...
42. I Klein-Gordon propagator derivation

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...
43. I The propagator and the Lagrangian

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

I note the following: \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)...
45. A QED propagator in Coulomb gauge

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...
46. A Photon propagator in Coulomb gauge

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...
47. Quantum Theory, propagator and causality, commutator

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 ##...
48. I Does the Contour Integral for the Klein-Gordon Propagator Matter?

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...
49. I Why Does the Contour Matter in the Klein Gordon Propagator 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...
50. I Propagator operator in Heinsenberg picture

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...