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Questions tagged [propagator]

propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum.

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Gauge fixing, invertibility and Green's functional

consider the photon in QED and the corresponding EOM of its Green's functional in k-space: $$(k^\mu k^\nu-k^2g^{\mu\nu})\Delta_{\nu\rho}(k)=i\delta^\mu_\rho.$$ Now, I understand that $U^{\mu\nu}(k):=...
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Computation of the self-energy term of the exact propagator for $\varphi^3$ theory in Srednicki

In M. Srednicki "Quantum field theory", Section 14 -Loop corrections to the propagator-, the exact propagator $\mathbf {\tilde \Delta} (k^2)$ is stated as $$\frac{1}{i} \mathbf {\tilde \Delta} (k^2)...
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Interpretation of the propagator

In quantum mechanics, it is clear that $\langle x|y\rangle = 0$ for $x\ne y$, where $|x\rangle$ is the state with the particle at position $x$. (Notice that this $|x\rangle$ is different from the ...
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Normalization of the integration measure of the Feynman's formula to combine denominators

In Mark Srednicki "Quantum field theory", section 14 -Loop corrections to the propagator-, it is presented the Feynman's formula to combine denominators: $\frac{1}{A_1 ... A_n} = \int dF_n (x_1 A_1 + ....
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Units of the Klein-Gordon Propagator in SI Units

What are the SI units of the momentum-space propagator of the Klein-Gordon equation for a free particle?
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Can we do one-loop integrals in the unitary gauge?

$\hspace{5cm}$ Imagine we want ot compute one of the diagrams for the self-energy of the quark $u$, with external momentum $p$. Inside the loop, we would have a $W^+$ and a $d$-quark propagator, with ...
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Supergravity and Gamma Matrices

On page 101 of Freedman and Van Proeyen's book on Supergravity they find the propagator of the gravitino, however I'm not sure how to work through the steps in (5.30), and hints or answers would be ...
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Relationship between Dyson equations from different problems

Recently, I noticed that the Dyson equation $$G=G_0+G_0\Sigma G$$ is used not only in quantum field theory but in some other branches of physics. For instance: 1. Wave equation From the wave ...
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Get the transition amplitudes from a wavefunction?

Given a wavefunction $\psi(x,t)$ a transition from time $t_1$ to $t_2$ might be written: $$\psi(x,t_1) = \int \Delta(x,y,t_1-t_2) \psi(y,t_2) d^3y.$$ But can we solve this to get $\Delta$ in terms ...
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Regularising the Green's function in 2D

The Green's function for the 2D Helmholtz equation satisfies the following equation: $$(\nabla^2+k_0^2+\mathrm{i}\eta)\,{\mathsf{G}}_{2\mathrm{D}}(\mathbf{r}-\mathbf{r}',k_o)=\delta^{(2)}(\mathbf{r}-\...
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LSZ reduction, momentum diagram, QFT

I was initially confused about which way to choose the sign of the momentum, since it gives rise to different exponential momentum combinations and thus different deltas for momentum conservation. I ...
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Complex integration in Peskin and Schroeder

In Peskin and Schroeder, I have a problem with a claim in equation (2.54), which I will rewrite more concisely here. He claims that we have the following equality : $$ \frac{1}{2E_p}e^{-iE_p(x_0)}-\...
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What is the Quantum Mechanical analogue of the Bethe-Salpeter equation?

For studying the bound states of quantum fields theories (e.g. studying excitons or mesons), the Bethe-Salpeter equation is often used as the starting point. Quoting Wikipedia the equation is: $$\...
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Any connected diagram is a tree of full propagators

In P. Etingof, Geometry & QFT, MIT 2002 online lecture notes; Lemma 3.11 (https://physics.stackexchange.com/users/7266/abdelmalek-abdesselam).) He says that any connected diagram is a tree of ...
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Combinatorics geometric series two-point function

In this answer Proof of geometric series two-point function it is said: Now what about the coefficients in front of each Feynman diagram? Due to the combinatorics/factorization involved it ...
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How to understand the transition amplitude in the Copenhagen interpretation

In Chapter 8 of Townsend's A Modern Approach to Quantum Mechanics, he states that the expression $\langle x', t' | x_0, t_0 \rangle$ gives the amplitude for a particle that is at position $x_0$ to at ...
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Why does it matter that the propagator is related to the Green's function for the Schrodinger equation?

If $L = i \hbar \hat{H} - \dfrac{d}{dt}$, then $ L \psi(x,t) = 0$ is the Schrodinger equation. It is well known that we can solve the Schrodinger equation with initial condition $\psi(x,0) = f(x)$ ...
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Relation between standard and Kubo-transformed quantum correlations

Via path integral molecular dynamics it is possible to measure the Kubo transformed correlation function between two operators $\hat A$ and $\hat B$ \begin{equation*} K_{\hat A\hat B} = \frac 1 {Z_\...
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Relation between the propagator and probability for the infinite well

This may be an easy question, but I am really confused about it. For the infinite square well, the (time-dependent) energy eigenfunctions are (inside the well):$$\psi_n(x,t) = \sqrt{2/L}\:e^{-iE_nt/\...
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Derivation of the QFT Propagator

I don't understand how we get from the RHS to the last line. \begin{eqnarray} \left[ \hat{H}_x - i \frac{\partial}{\partial t_x} \right] G^+(x,t_x,y,t_y) &=& -i \delta (t_x - t_y) \sum_n{\...
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How does a propagator act on a wave function in $x$-space?

$\newcommand{\ket}[1]{|#1\rangle}$$\newcommand{\bra}[1]{\langle#1|}$In Principles of Quantum Mechanics (2nd edition) by Shankar, Exercise 5.1.3 asks to find the wave function of the free particle by ...
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Kallen-Lehmann representation and branch cuts at threshold masses

Let us consider the Kallen-Lehmann representation for the two-point function of scalar fields $$ \langle \Omega | T\left\{\phi(x) \phi(y)\right\}|\Omega\rangle = \int \frac{d^4 p}{(2\pi)^4} e^{ip\...
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Two-point correlation function of a scalar field $\langle 0 | \phi(x) \phi(0)| 0 \rangle$

I'm trying to find the two point correlation function for a massless scalar field obeying $\square \phi = 0$. I can write $$\langle 0 | \phi(x) \phi(0)| 0 \rangle = \int \frac{d^dk}{(2\pi)^d} \delta(...
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Summation of an exponential operator on quantum amplitude

For a quantum Dirac field interacting with a classical EM field, one can (through the Quantum Dynamical Principle) write the vacuum transition amplitude as $$\langle0_+|0_-\rangle=\exp\left[ie_0\int ...
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1-loop correction to photon propagator

(May be it is a duplicate). I do not understand clearly how should I write down 1-loop correction to photon propagator. I know what is $i\Pi_{\mu\nu}(k^2)$ (I need only this specific correction) and ...
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No bound states if propagator is everywhere infinite?

Assuming the energy spectrum is discrete, the propagator for the time independent Schrodinger equation can be represented as $$G(x,y,E)=\sum_n\frac{\psi_n(x)\psi_n^*(y)}{E-E_n}.$$ The propagator's ...
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How does one calculate Fourier transform of Feynman propagator?

I am struggling with calculating the following integral on Sredinicki: How did he get the second line of (10.6)? That is, how did he calculate the Fourier transform of Feynman propagator?
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Confusion about functional derivative in path integral

If we act a functional derivative $$\frac{\delta}{\delta J(z)}$$On the expression$$\int\int d^4x d^4y \space J(x)\Delta(x-y)J(y)$$ where $\Delta(x-y)$ is Feynman propagator. What one should get is ...
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When can you simplify the $W$ boson propagator?

I have seen in several sources that the propagator of the $W$ boson is: $$\frac{- i \left( g^{\mu\nu} - \frac{P^\mu P^\nu}{m_W^2} \right)}{p^2 - m_W^2} . $$ But then in some calculations (usually ...
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How to grasp the limits of these two integrals? [duplicate]

I find some difficulty in understanding the limits of the two integral below (on Page 27 of Peskin & Schroeder's Quantum Field Theory): $$D(x-y)=\frac{1}{4\pi^2}\int_m^\infty d E \sqrt{E^2-m^2}e^{...
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Why is the imaginary part of the Breit-Wigner propagator given by the total decay width?

The optical theorem links the imaginary part of the forward scattering amplitude to the total decay width of a particle: $\mathrm{Im}\,M_{i\to i} = m\Gamma_{tot}$. Here $\Gamma_{tot} = \frac{1}{2m} \...
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Are powers of the harmonic oscillator semiclassically exact?

The Duistermaat-Heckman theorem, although too complex for me to completely grasp, states that under some conditions, the partition function for a special class of Hamiltonians is semiclassically exact....
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Wick rotation vs. Feynman $i\varepsilon$-prescription

The generating functional $Z[J]$ of some scalar field theory is \begin{equation} Z[J(t,\vec{x})]=\int \mathcal{D}\phi e^{i\int (\mathcal{L}+J\phi)d^4x} \end{equation} This integral is not well ...
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Propagators in interaction with derivatives

Given a Lagrangian density containing an interaction with derivates, it's easy how to guess the Feynman rules for vertexes. However i was wondering about propagators: in S-matrix expansion it's ...
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Calcluating the photon propagator with gauge fixing parameter

I'm trying to calculate the photon propagator via the functional integral, with lagrangian (plus source) $L = -\frac{1}{4}F^{ab}F_{ab} - \frac{\lambda}{2}\left(\partial^aA_a\right)^2 + J^aA_a $ ...
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Propagators for polarised photons

Photons can be thought to come in two types. Depending on which way they are spinning. The propagator for a photon (sum of both helices) in a certain gauge is $\frac{\eta^{\mu\nu}}{|k|^2}$. I read ...
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Are vacuum-fluctuations a consequence of causality?

I'n new to QFT, and recently lerned about the propagator of a free scalar field theory in Minkowski-space, which according to our lecture notes looks like $$G(p, q) = \frac{1}{q^\mu q_\mu + M^2} \...
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Propagator for W boson

I've found in different literature that some write the propagator for the W boson as $\frac{g_{\mu\nu}-\frac{k_\mu k_\nu}{M^2_W}}{k^2-M^2_W+iM_W\Gamma_W}$ and others like $\frac{g_{\mu\nu}-\frac{k_\mu ...
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Is one allowed to split path integrals in the Feynman-Vernon Influence theory

In QFT the propagator $J(t,t_0,x_f,x_i) = \langle x_f | U(t,t_0) | x_i \rangle$ fulfills the property $$ J(t,t_0,x_f,x_i) = \int_{-\infty}^{\infty}dx' J(t,t',x_f,x')J(t',t_0,x',x_i) $$ and can be ...
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Geometric series of two point function and self energy

This question is related to this question Proof of geometric series two-point function. Suppose we have a graph $A$ with a symmetry factor $s_1$. According to Srednicki (chapter 9, eq. (9.13)) for a ...
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Proof of geometric series two-point function

In deriving the expression for the exact propagator $$G_c^{(2)}(x_1,x_2)=[p^2-m^2+\Pi(p)]^{-1}$$ for $\phi^4$ theory all books that i know use the following argument: $$G_c^{(2)}(x_1,x_2)=G_0^{(2)}...
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Definition of one particle irreducible diagrams

Text books often defines One-Particle Irreducible diagram (1PI diagram) as a connected diagram which does not fall into two pieces if you cut one internal line. Is this internal line the full ...
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How many counterterms does QED have?

I have read the statement that QED has four counterterms to cancel divergences. However, I have learnt that there are only three counterterms (vertex, electron propagator, photon propagator), which is ...
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Proof of 1-particle irreducible (1PI) diagrams

If we split the effective action into $$Γ[Φ] =\frac{1}2ΦiG_0^{-1}Φ + Γ^{int} [Φ]$$ we can show that the full propagator is given by $$G= i[iG − Σ]^{-1}$$ With $$Σ=-Γ_{ΦΦ}^{int} [Φ]$$ Here $Γ_{...
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Conformal transformation of a vertex operator before normal ordering

Let us consider a free scalar boson $\varphi(z,\bar{z})$ on the complex plane and assume the following two-point correlation function \begin{eqnarray} \langle\varphi(z,\bar{z})\varphi(w,\bar{w})\...
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Resonances in QFT

I am a bit confused about resonances in QFT. I am reading Schwarz's QFT book and as far as I understand, if in a reaction the mass of the particle acting as a propagator is bigger than the sum of the ...
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Loop-loop propagator in bosonic string theory

I have been searching for information on how to compute (or at least partially compute) the propagator of a string given an initial and a final loop in bosonic string theory. I have found several ...
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Propagator for two spins with time-dependent field?

I'm looking for a closed-form solution for a problem that SEEMS to me to be so simple and basic it HAS to have a solution - but I can't find it. I was wondering if the great minds of this august ...
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Dirac propagator causality

I was studying the Dirac propagator and came across an excelent article which includes all the derivation, and interestingly we can conclude that the anticommutator is zero for space-like intervals. ...
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How does the Lorenz gauge eliminate the scalar component of the vector field?

Wikipedia states that by using the Lorenz gauge, $\partial_\mu A^\mu=0$, we eliminate the scalar part (spin-0) of the vector potential that previously had spin-1 and spin-0 components${}^1$. However,...