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. E

    Lorentz Invariance of Propagator for Complex Scalar Field

    Homework Statement Show that [\hat{\phi}(x_1),\hat{\phi}^\dagger(x_2)] = 0 for (x_1 - x_2)^2 < 0 where \phi is a complex scalar field Homework Equations \hat{\phi}=\int\frac{d^3 \mathbf{k}}{(2\pi)^3 \sqrt{2\omega}}[\hat{a}(k)e^{-ik\cdot x} + b^\dagger(k)e^{ik\cdot x}]...
  2. G

    Why does the Klein-Gordon propagator have a negative i in the exponential?

    Hello, I'm looking at the following computation from the Peskin and Schroeder's book: See file attached In the second page, the second term that's being integrated, I don't understand why it has a negative i in the exponential, that'll keep the energy term the same, but will swap the sign...
  3. G

    Why does the propagator converge?

    The propagator $$\frac{1}{k^2+m^2} $$ diverges in the ultraviolet when integrated over all dkn, with n>=2. However, when you throw in an exponential, $$\Delta(x)=\int d^nk \frac{e^{ikx}}{k^2+m^2} $$ is convergent for x≠0 Intuitively adding the oscillating exponential decreases the...
  4. C

    Can the 'mass' of bound states show up full propagator?

    The result of the Kallen-Lehmann spectral representation is that the two point correlation (and thus also the full propagator) has a pole in the physical mass of the particle. In Peskin and Schroeder it is also argued that multiparticle states show up as a cut, but bound states can also show up...
  5. C

    Relation between bare and full scalar booson propagator.

    One can show that at around ##p \approx m## where m is the physical mass the full propagator ##D_F## is something like $$D_F = \frac{Z}{p^2 - m^2}.$$ Where ##Z = (1 - \Sigma '(m))^{-1}##, ##\Sigma## is the self energy and m is the physical mass of the particle. If i were now to write a...
  6. O

    Complex scalar field propagator

    i am trying to understand how to express contractions of field operators via propagators. we are talking about an interacting theory of 2 complex scalar fields, lets call them ψ1 and ψ2. the interaction term is: Lint=λ(ψ2)^3(ψ1) i have found the free propagator defined as...
  7. jfy4

    Propagator using Functional QFT

    Hi, I am trying to write down the propagator for a scalar field theory, but I want to try and get it in the functional representation. My plan is to compute the following: \langle \psi (x', t') | \psi (x,t) \rangle which gives the amplitude to go from x' to x. Now I guess I have to...
  8. L

    About propagator and poles in QFT

    Hi all, I am studying QFT using John Preskill's notes. I have a question about the propagator and poles. On page 2.91, at the bottom, he said that there is a s-channel pole, which is the pole of the exact propagator. Then he claimed that by the argument about unitarity in page 2.70, the pole...
  9. S

    Feynman Factors & Relativistic Scalar Propagator

    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...
  10. C

    Using Path Integral to calculate propagator

    Hi, It's great to find this forum. I'm teaching myself QM using Shankar, it's a great book, I've covered 8 chapters so far. My question is about the notion of using Path Integral method to calculate the propagator. The recipe given by Shankar says the propagator is U(x,t;x')=A\int...
  11. G

    Bound states in propagator

    Why must it be true that a system that has a bound state must have its scattering amplitude have a pole in the upper half of the complex wave-number plane? For example, if the scattering amplitude as a function of the initial wave number magnitude |k| is: A=\frac{1}{|k|-iB} with B>0, then...
  12. D

    Structure of the Phonon Free Propagator

    Hey Everyone, I've a quick a question regarding the make-up of bosonic Green's functions, taking the free propagator for phonons as example. According to Mahan, 3rd ed. it is given by: D(q, \lambda, t-t')=-i\langle0|TA_{q}(t)A_{-q}(t')|0\rangle with A_{q}=a_{q}+a^{+}_{q} [ Eqs...
  13. BWV

    Deriving the Propagator: Understanding Zee's Book and the Fourier Transform

    This is more of a math question than a physics one, but following the discussion of the propagator in Zee's book: -(∂2+m2)D(x-y)=δ(x-y) he then gets, by taking the Fourier transform of the Dirac delta and dividing through: D(x-y) = \int\frac{d^4k}{2π^4} \frac{e^{ik(x-y)}}{k^2-m^2+iε}...
  14. I

    Contour integration in propagator

    Hi all, I've been playing around with (2.54) of Peskin and Schroeder, and I've some quick questions about it. Firstly, the part I'm stuck on: $$\int\frac{d^3p}{(2\pi)^3}\left[\left.\frac{1}{2E_p}e^{-ip\cdot (x - y)}\right|_{p^0=E_p} - \left.\frac{1}{-2E_p}e^{-ip\cdot (x -...
  15. F

    Propagator for inverted harmonic potential.

    Hello. I was trying to find out the propagator for the inverted SHO (something like tachyon oscilltor) and turns out that it remains unitary only for very short times. Which didnt make much sense to me. I tried looking at the usual SHO propagator, and that too seems to be not Unitary! ( I...
  16. E

    Understanding the Role of Poles in the Propagator for Massive Vector Fields

    Hi! From "Le Bellac, Quantum and statistical field theory, 10.5.2-Massive vector field": "The longitudinal part of the propagator k_{\mu}D^{\mu\nu} has no pole at k^2=m^2, so the longitudinal part doesn't constitute a dynamical degree of freedom." I have two questions: 1) Why the propagator...
  17. D

    The propagator divergence in weak theory

    So I am wondering about one thing. The charged propagators in weak theory are W+- bosons. The mathematical expression for them, while drawing the Feynman diagrams is: -i\frac{g_{\mu\nu}-\frac{q_\mu q_\nu}{m_W^2}}{q^2-m_w^2}. The problems that are usually given to me are simple and involve...
  18. P

    Calculating Quantum Massless Photon Propagator

    So I want to calculate the quantum massless photon propagator. To do this, I write A_\mu(x) = \sum\limits_{i=1}^2 \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2\omega_p}} \left( \epsilon_\mu^i (p) a_{p,i} e^{-i p \cdot x} + { \epsilon_\mu^i} ^* (p) a_{p,i}^\dagger e^{i p \cdot x} \right)...
  19. I

    Parallel propagator and covariant derivative of vector

    Hi all, I'm trying to figure out the link between the connection coefficients (Christoffel symbols), the propagator, and the coordinate description of the covariant derivative with the connection coefficients. As in...
  20. I

    Connection coefficients as derivatives of parallel propagator

    Hi all, I've been fiddling around with this problem for a while. I intuitively understand that the parallel propagator is the path integral of the connection. I would like to be able to show the converse (connection is derivative of parallel propagator) mathematically, and I am having a...
  21. N

    Graviton Propagator and energy-momentum tensor

    Dear PF, I am a little bit confused could you pls help me ... Suppose I a have a scatering or conversion of two particles via graviton propagator. Graviton propagator couples with energy-momentum tensor of matter fields. So can i assume that vertex to which graviton propagator is coupled...
  22. N

    Time integrals of free particle propagator

    Homework Statement As part of an assignment on matter wave diffraction I'm to calculate the following integrals I_1=\int_0^{\infty}G(\vec r_2,\vec r_1;\tau)e^{i\omega\tau}d\tau,\quad I_2=\int_0^{\infty}G(\vec r_2,\vec r_1;\tau)e^{i\omega\tau}\frac{d\tau}{\tau} Homework Equations To do so...
  23. J

    Propagator Operator: Clarifying H Acting & Function of (t-t')

    So in the attachment, in fact (6), the formula for the propagator rectangled in red... is the Hamiltonian ACTING on (t-t')? is the Hamiltonian a function of (t-t')? or should it be (this is what I think), to be more clear U(t,t') = exp((-i/h)(t-t')H), so that when acting on a state...
  24. M

    Finding Feynman's propagator

    Well I am doing an assignment concerning methods of finding Feynman's propagator. I understand pretty well how everything's coming up. I also get that the form: <x'',t'|x',0> defined as Feynman's propagator gives the amplitude of a system initially being in state |x',0> to be in |x'',t> after...
  25. R

    Majorana Propagator: Dirac vs. Majorana Equations

    The Dirac propagator (e.g. for an electron) is given by the inverse of the field equation in momentum space i.e. (\displaystyle{\not} p - m)\psi = 0, which gives: \frac{i}{(\displaystyle{\not} p - m)} = \frac{i(\displaystyle{\not} p + m)}{(p^2-m^2)}. So is the propagator for a Majorana...
  26. I

    What is the explanation for the sequence in the Mandl QFT textbook (p.53)?

    Could anyone please explain the sequence below taken from Mandl QFT textbook (p.53)? 1. i\hbar c\Delta^+(x-x')=[\phi^+(x),\phi^-(x')] 2. i\hbar c\Delta^+(x-x')=\langle 0|[\phi^+(x),\phi^-(x')]|0\rangle 3. i\hbar c\Delta^+(x-x')=\langle 0|\phi^+(x)\phi^-(x')|0\rangle 4. i\hbar...
  27. L

    Propagator for matrix fields (based on Srednicki ch80, p490)

    Hi, If I have a matrix valued field B(x)_i^{..j}=B^a (x) (T^a)_i^{..j} and the relevant part of my Lagrangian is L=Tr(-\tfrac{1}{2}\partial^{\mu}B\partial_{\mu}B+..) then how can I see that the propagator for the matrix field is...
  28. Z

    Can someone explain what a propagator is and does?

    So, Shankar in his famous book started to use the propagator a bunch of times in his Simple Problems in One Dimension chapter, and I have been confused to what it is and does.
  29. 1

    Propagator with Pauli matrices

    Homework Statement Consider a 1/2-spin particle. Its time evolution is ruled by operator U(t)=e^{-i\Omega t} with \Omega=A({\vec{\sigma}}\cdot {\vec{L}})^{2}. A is a constant. If the state at t=0 is described by quantum number of {\vec{L}}^2, L_{z} and S_{z}, l=0, m=0 and s_{z}={1/2}...
  30. T

    Spin-1 Propagator and polarization vectors

    Hi. I am stuck. By inverting the spin-1 differential operator I was able to derive (quite easily) the propagator for the spin-1 field (in a spontaneously broken gauge theory) in the R_\xi gauges for the arbitrary gauge parameter \xi. The result is...
  31. N

    Should the quark propagator vanish because of confinement?

    Hi everyone! Something has been bothering me lately. Consider the quark propagator: \langle 0|\psi_a(x)\psi_a(0) |0\rangle For a given color a. Now let's say we insert 1 = \sum |n \rangle \langle n| between the two quark fields, where the sum is over a complete set of energy...
  32. michael879

    Deriving Graviton Propagator from Linearized Lagrangian

    Hi, I'm working with the free space linearized gravitation lagrangian and trying to derive the proper propagator for it. I have no problem doing this, the only problem is that my QFT makes a quick note of what form this should take and I'm off by a factor of 4. The flatspace metric terms...
  33. T

    QCD gluon propagator in axial gauge, polarization sum

    Hi! I have a process with multiple feynman diagrams where gluon propagators occur. When I use an axial gauge for the gluon propagator, do I have to use the same n-vector for every propagator? Following this I wonder whether I can use the same n-vector for every polarization sum in axial gauge...
  34. R

    Binomial expansion of massive spin 0 propagator

    I've seen written the following manipulation of the KG Feynman propagator: \frac{1}{p^2-m^2+i\epsilon}=\frac{1}{p^2+i\epsilon} \frac{1}{(1-\frac{m^2}{p^2+i\epsilon})}= \frac{1}{p^2+i\epsilon} (1+\frac{m^2}{p^2+i\epsilon}+\left(\frac{m^2}{p^2+i\epsilon}\right)^2+...) I don't think this...
  35. R

    Propagator at time-like separations

    On page 68, equation (8.13) of Srednicki's QFT book is the equation for the scalar propagator: \Delta(x-x')=i\theta(t-t') \int \frac{d^3k}{2(2\pi)^3E_k}e^{ik(x-x')} +i\theta(t'-t) \int \frac{d^3k}{2(2\pi)^3E_k}e^{-ik(x-x')} where the exponential is the product of 4-vectors and k is...
  36. R

    Trace of momentum-space propagator

    The integral: \int \Pi_k d\phi_k e^{-\phi_i A_{ij} \phi_j} is a Gaussian and is equal to: (\pi)^{n/2}\sqrt{det(A^{-1})}= (\pi)^{n/2} e^{\frac{1}{2}Tr ln A^{-1}} Now usually A is a diagonal matrix that represents the Lagrangian (so that the sum over i and j collapses to a sum just over i...
  37. E

    Calculating Self-Energy Correction to Electron Propagator

    If one wants, to calculate the self energy correction to the electron propagator(using the approach where one introduces a photon mass \mu to deal with IR divergences), one gets after some work an integral like this (this is from the Itzykson Zuber book equ. 7-34): \int_ 0 ^ 1 d\beta \beta...
  38. M

    Massive vector boson propagator - Definition

    I'm reading Zee's QFT textbook and I'm stuck trying to understand why the \delta^\mu_\lambda appears when he defines the propagator of a massive spin-1 boson as the inverse of a differential operator: [(\partial^2 + m^2)g^{\mu\nu}-\partial^\mu\partial^\nu]D_{\nu\lambda} = \delta_\lambda^\mu...
  39. Peeter

    Propagator for quantum linear potential

    Homework Statement Attempting a problem related to a quantum particle in free fall due to constant force (old exam question) \begin{align*}H = \frac{1}{{2m}} P^2 + m g X\end{align*} The last part of the question asks to verify that the position space propagator has the following form...
  40. N

    Computing harmonic oscillator propagator via path integral

    Homework Statement Show that G(q_2,q_1;t)=\mathcal{N}\frac{e^{iS_{lc}}}{\sqrt{\det A}} where \mathcal{N} is a normalization factor independent of q1, q2, t, and w. Using the known case of w=0, write a formula for G such that there is no unknown normalization factor. Homework Equations I...
  41. Rasalhague

    Integration in Sean Carroll's parallel propagator derivation

    Integration in Sean Carroll's "parallel propagator" derivation Reading Chapter 3 of Sean Carroll's General Relativity Lecture Notes, I've followed it up to and including eq. 3.38. \frac{d}{d\lambda} P^\mu_{\;\;\; \rho}(\lambda,\lambda_0) = A^\mu_{\;\;\; \sigma} P^\sigma_{\;\;\...
  42. O

    How Does the Propagator Relate to the Heat Kernel on a Manifold?

    Hi, I have a question about the relation between the propagator of a scalar field and the heat kernel. I'm not sure wether I should rather put this question into the math section: Given a Laplacian D on some manifold M, what I mean by heat kernel is just K(x,y;s) = \langle x | \exp(-sD)...
  43. G

    Propagator using cauchy integral

    hi I don't understand this bit about the derivation of propagator expressions. Bjorken and Drell describe the step function as: \theta(\tau)=lim_{\epsilon \to 0}\frac{-1}{2\pi i}\oint_{-\infty}^{\infty}\frac{d\omega e^{-i\omega r}}{\omega + i \epsilon } the singularity is at -i \omega...
  44. I

    Why does the propagator has a cut starting from EXACTLY 4m^2?

    Consider the usual phi^4 theory, when we derive the Lehmann-Kallen representation of the propagator, by inserting a complete set we know that the propagator has a branch cut starting from 4m^2, where the m is the physical mass. My question is: what's the construction of these...
  45. N

    What is the role of the i in the propagator of Feynman rules?

    I'm probably missing something small but I haven't been able to figure this out. In the Feynman rules (for a scalar field that obeys the Klein-Gordon equation), you write a propagator for internal lines as \frac{i}{k^2 - m^2 + i \epsilon}. The propagator integrand is originally...
  46. N

    Proving that a propagator decays exponentially for spacelike separations

    I'm trying to show that the propagator for spacelike separation decays like e^{-m r} and I'm stuck. At some point I hit the integral \int_{-\infty}^{\infty} \frac{dk}{\sqrt{k^2 + m^2}} e^{i k r}. Integration of complex functions not being my forte, I only managed to get to this point...
  47. N

    Propagator D for a particle is basically the Green's function

    The propagator D for a particle is basically the Green's function of the differential operator that describes that particle, e.g. (\partial^2 + m^2) D(x-y) = \delta^4 (x-y). This propagator is supposed to give the probability of the particle propagating from x to y. Why does this make...
  48. Š

    Klein-Gordon propagator ill-defined?

    Hi, I've had the following problem in elementary quantum field theory. The propagator for the Klein-Gordon scalar field takes the form D(x-y)=\int\frac{\textrm{d}^3\mathbf{p}}{(2\pi)^3}\frac{1}{2\sqrt{|\mathbf{p}|^2+m^2}}e^{-ip\cdot(x-y)} I was interested what the propagator looks like for...
  49. N

    Integral for the free propagator

    This may be more of a maths question, but because I may actually just be interpreting the expression wrong, I think I'd better post it here. I'm reading Quantum Field Theory in a Nutshell by A. Zee and I'm stuck on a bit of maths he does. He provides an expression for the free propagator for a...
  50. R

    How does the renormalization factor affect the propagator?

    In Kaku's book, the self-energy in a \phi^4 scalar theory is expanded in a Taylor series as: \Sigma(p^2)=\Sigma (m^2)+\Sigma'(m^2)(p^2-m^2)+\tilde_{\Sigma}(p^2) where \tilde_{\Sigma}(p^2) is finite and m is arbitrary (but finite). The full propagator is then...
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