Propagators Definition and 37 Threads

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. E

    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...
  2. avnl

    A Calculating the propagator of a Spin-2 field

    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...
  3. A

    Amplitude with Feynman diagrams and gluon propagators

    The term which is relevant for the calculus is: $$ \bar u(p) \gamma^\alpha \frac{1}{\displaystyle{\not}p+\not k} \gamma^\nu \frac{1}{\displaystyle{\not}p'-\not k} \gamma^\beta v(p') \frac{k_\alpha k_\beta}{k^2} $$ $$ \bar u(p) \displaystyle{\not}k \frac{1}{\displaystyle{\not}p+\not k}...
  4. A

    What Are the Conditions for On-Shell Renormalization in QFT?

    Homework Statement Show that, after considering all 1 particle irreducible diagrams, the bare scalar propagator becomes: $$D_F (p)=\frac{i}{p^2-m^2-\Sigma (p^2)}$$ And that the residue of the pole is shifted to a new value, and beomes...
  5. M

    A What does Schwarz's QFT theorem say about poles in Green's functions?

    Hello! In Schwarz's QFT, Chapter 24.3 there is a theorem stating that Green's functions have poles when on-shell intermediate particles can be produced. I am not sure I understand how this works. If we have $$e^+e^- \to \gamma^* \to \mu^+\mu^-$$ we can have a positronium as an intermediate...
  6. T

    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...
  7. Urs Schreiber

    Mathematical Quantum Field Theory - Propagators - Comments

    Greg Bernhardt submitted a new PF Insights post Mathematical Quantum Field Theory - Propagators Continue reading the Original PF Insights Post.
  8. Milsomonk

    A Feynman integral with three propagators

    Hi guys, So I'm trying to compute this Feynman integral: $$ V=\dfrac {-i} {2} \int {\dfrac {d^4 k} {(2\pi)^4}} \dfrac {1} {k^2 - m^2} \dfrac {1} {(k+P_1)^2 -m^2} \dfrac {1} {(k+P_1 +P_2)^2 -m^2}$$ I have introduced the Feynman parameters and now have the integral: $$ V=-i \int...
  9. FrancescoS

    Computing propagators with derivative interaction

    Hi guys, I'm working with this interaction Lagrangian density ##\mathcal{L}_{int} = \mathcal{L}_{int}^{(1)} + \mathcal{L}_{int}^{(2)} + {\mathcal{L}_{int}^{(2)}}^\dagger = ia\bar{\Psi}\gamma^\mu\Psi Z_\mu +ib(\phi^\dagger\partial_\mu \phi - \partial_\mu\phi^\dagger \phi)Z^\mu,## with ##...
  10. C

    How to Compute the Propagator in 2D Euclidean Space?

    Homework Statement Consider the following scalar theory formulated in two-dimensional Euclidean space-time; S=∫d2x ½(∂μφ∂μφ + m2φ2) , a) Determine the equations of motion for the field φ. b) Compute the propagator; G(x,y) = ∫d2k/(2π)2 eik(x-y)G(k). Homework Equations Euler-Lagrange equations...
  11. souda64

    Product of two propagators U(-t)U(t) in coord representation

    Here is a mystery I'm trying to understand. Let ##\hat{U}(t) = \exp[-i\hat{H}t]## is an evolution operator (propagator) in atomic units (\hbar=1). I think I'm not crazy assuming that ##\hat{U}(-t)\hat{U}(t)=\hat{I}## (unit operator). Then I would think that the following should hold \left\langle...
  12. T

    Nonrelativistic free particle propagators

    This type of integration is a special case of something that occurs over and over in QM and QFT (it's everywhere in Peskin and Schroeder), but I am having a bit of trouble working out the details. Set \hbar=1 and consider the propagation amplitude for a free, nonrelativistic particle to move...
  13. J

    Born rule and Feynman propagators

    Let us assume that we want to describe the full process of photon emission by electron A and absorption by electron B. Therefore electron B must be on the forward lightcone of electron A. In the normal forwards in time description a virtual photon propagates from A to B depositing a certain...
  14. L

    What Is the Fourier Transform of Propagators in QFT?

    I've been assigned the following homework: I have to compute the spectral density of a QFT and in order to do so I have to compute Fourier tranform of the following quantity (in Minkowsky signature, mostly minus) \rho\left(p\right) = \int \frac{1}{\left(-x^2 + i \epsilon...
  15. K

    How Do Spin-0 and Spin-1 Propagators Differ in Quantum Field Theory?

    Hello! I have worked with Green's functions in electrodynamics and have now started reading qft. First I encountered the spin-0 propagator, D(x-y) = \int \frac{d^4 k}{(2\pi)^4}\frac{e^{ik(x-y)}}{k^2 -m^2}. This seems not so new.. We ahve a blow up around the mass-shell and the wave propagates...
  16. A

    Propagators for time-dependent Hamiltonians

    Suppose I know H \psi(x) = \left( -\frac{1}{2m} \Delta_x + V(x) \right) \psi(x) = E\psi(x). Then \psi(x,t) = e^{-iEt}\psi(x) solves the time-dependent Schrodinger equation \left( i \frac{\partial}{\partial t} + \frac{1}{2m} \Delta_x - V(x) \right)\psi(x,t) = 0. I've done some...
  17. P

    Contour integral trick with propagators

    Hi I am struggling trying to see understand the basic propagator integral trick. \int \frac{d^{3}p}{(2\pi^{3})}\left\lbrace \frac{1}{2E_{p}}e^{-ip.(x-y)}|_{p_{0}=E_{p}}+\frac{1}{-2E_{p}}e^{ip.(x-y)}|_{p_{0}=-E_{p}}\right\rbrace = \int \frac{d^{3}p}{(2\pi^{3})}\int...
  18. S

    Feynman diagrams and relativistic propagators

    Hey again, I have a question on a couple of things related to feynman diagrams but also the relativistic scalar propagator term. First of all, this interaction: The cross represents a self-interaction via the mass and characterised by the term: -im^2, is this just some initial state...
  19. I

    Commutators, propagators, and measurement.

    Hi all, Reading through Peskin and Schroeder, I came across the following statement, with regards to propagators: Could someone explain how the commutator is related to the measurement of the field in this context? Searching online, the only thing that crops up is the usual uncertainty...
  20. Hepth

    Mathematically rigorous way of expanding propagators?

    Such propagators as found in HQET \frac{i}{2 v \cdot k} come about from expanding the full propagator. I'm wondering what the method is to properly Taylor expand denominators that contain 4-dimensional dot products. Lets start with something like : \frac{1}{2 v \cdot k + k^2} If we...
  21. R

    Feynman Propagators: Invariant Amplitude vs Loop Integrals

    Hi, this is probably very simple but what is the difference between these two Feynman propagators: \frac{i}{q^2-m^2} \frac{i(p/+m)}{p^2-m^2} E.g. Is one used for the invariant amplitude and the other for loop integrals? Or is one for a fermion and the other for a boson? =s Thanks!
  22. maverick280857

    Dirac Principle Value Identity applied to Propagators

    Hi, How is \frac{1}{\displaystyle{\not}{P}-m+i\epsilon}-\frac{1}{\displaystyle{\not}{P}-m-i\epsilon} = \frac{2\pi}{i}(\displaystyle{\not}{P}+m)\delta(P^2-m^2) ? This is equation (4-91) of Itzykson and Zuber (page 189). I know that \frac{1}{x\mp i\epsilon} =...
  23. M

    Virtual particle propagators in QFT

    I am reading a nice book (Quarks and Leptons, by Halzen and Martin) about particle physics. It states that the general form of the propagator of a virtual particle is: \dfrac{i\sum_{\text{spins}}}{p^2 - m^2} I see that this is the case for the Dirac propagator...
  24. E

    QED: Electron/Positron Scattering & Bosonic Propagators

    Hi! I'm studying an introduction to QED and I don't understand the bosonic propagators. Consider the electron/positron scattering with the exchange of a virtual photon. According to perturbation theory, the propagator is: T=<f|H|i>+\sum<f|H|n>\frac{1}{E_i-E_n}<n|H|i>+... where f is the final...
  25. E

    Feynman trick for linear propagators?

    Hey guys, say i have some standard propagators then I know how to combine them using Feynman's parameter method. But what do I do if one of these propagators is linear? For instance: \int d^Dk \frac{1}{(k-p)^2(k \cdot q)} where q and p are some momenta. How do I combine them? Does...
  26. N

    Exploring Quantum Mechanics: Propagators and Green's Functions

    Hi I am, at the moment, reading about propagators and Green's functions in QM. An example of a Green's function in some γ-basis is G(\gamma, t, t') = - i\left\langle {c_\gamma \left( t \right),c_\gamma ^ \dagger\left( {t'} \right)} \right\rangle Now, if I expand this in terms of...
  27. V

    Mediators and propagators of particle interactions

    hello I'm studying nuclear physics and I have a lot of questions which I can't figure out. The first thing has been thrown in my mind by a colleague of mine about propagators of particle interactions. It has been said that, for example, photons are the mediators for the electromagnetic...
  28. S

    A question related with propagators in position space

    Dear Colleagues, How to prove Fourier transformation rules of the following function \int k^a * exp[i*k*x] * dk = x^{-1-a} * Gamma[1+a] * sin[a*pi/2] ------------(1) I need this equation to prove some conclusions in translating propagators in momentum space to...
  29. B

    How Do Matrix Valued Propagators Interpret Fermion Amplitudes?

    For scalar fields the propagator is just a number that represents the amplitude for a particle to go from one space time point to another. For fermions, the propagator is matrix valued. What then is the amplitude for a fermion to go from one point to another? How are the elements of the...
  30. J

    Stringy corrections to SM propagators

    First let me express my ignorance about this subject so please forgive me if these questions have well-known answers. The main objections I've heard voiced toward string theory are (1) it's incredible diversity of vacua caused by large number of possible Calabi-Yau compactifications, and (2)...
  31. T

    Gauge Boson Propagators in Spontaneously Broken Gauge Theories

    The propagator for gauge bosons in a spontaneously broken (non-abelian) gauge theory in the R_\xi gauge is (see Peskin and Schroeder eqn. 21.53) \tilde{D}^{\mu\nu}_F(k)^{ab}=\frac{-i}{k^2-M^{ab}}\left[g^{\mu\nu}-(1-\xi)\frac{k^\mu k^\nu}{k^2-\xi M^{ab}}\right]\,, where M^{ab} is the gauge...
  32. V

    Another question on rates, propagators, etc

    Please take a look at the file "Feynman.pdf". The ratio rate for the two decays is shown in the second file, "rate.pdf". I understand it takes that form because the usW vertex is Carribbo supressed relative to the udW vertex. My concern is that the ratio is only squared. Rate is...
  33. V

    The Role of Propagators in Determining Decay Rates

    i was thinking about propagators, and I've got bit of a silly question. The amplitude for any given decay takes the form: f(q^{2})=\frac{g^{2}}{q^{2}+M_{X}^{2}c^{2}}. for EMAG decays, the Mx=0 (as we're dealing with photons), so the amplitude is just: f(q^{2})=\frac{g^{2}}{q^{2}}, Why do...
  34. N

    Is There a Sign Issue in Propagator Integration for QFT?

    Homework Statement I'm trying to show that the general form of the propagator is D(x) = - \int \frac{d^3k}{(2\pi)^32\omega_k}[e^{-i(\omega_k t - \vec{k}\cdot\vec{x})}\theta(x^0) + e^{i(\omega_k - \vec{k}\cdot\vec{x})}\theta(-x^0)] but my answers always seem to differ by a sign. Homework...
  35. E

    Propagators Homework: Understanding K(x,t;x',0) & More

    Homework Statement I am so confused about propagators: K(x,t;x',0) = \int |E\rangle e^{-iEt/\hbar} \langle E| dE I understand the RHS of that equation perfectly: it just decomposes the time-independent state into its eigenstates and then propagates each of the eigenstates individually. I...
  36. K

    Path integrals and propagators

    we know that for the SE equation we find the propagator (i\hbar \partial _{t} - \hbar ^{2} \nabla +V(x,y,z) )K(x,x')=\delta (x-x') with m=1/2 for simplicity then we know that the propagator K(x,x') may be obtained from the evaluation of the Path integral. K(x,x')=C \int \mathcal...
  37. E

    Mathematica Schroedinger, Klein-Gordon & Dirac Propagators

    Let be the propagators for the Schroedinguer,Klein-Gordon and Dirac?...are they hermitian operators?..are their eigenfunctions ortogonal?...
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