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34 votes

How can the Feynman rules be read off the Lagrangian?

Keep in mind that it is nearly impossible to explain how perturbative QFT calculations follow from Lagrangians such that the answer is both relatively short and detailed. So I am going to write an ...
Prof. Legolasov's user avatar
33 votes
Accepted

How to interpret correlation functions in QFT?

Yes, in scalar field theory, $\langle 0 | T\{\phi(y) \phi(x)\} | 0 \rangle$ is the amplitude for a particle to propagate from $x$ to $y$. There are caveats to this, because not all QFTs admit ...
user1504's user avatar
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31 votes

Differentiating Propagator, Green's function, Correlation function, etc

It has been many years since you asked this question. I assume that over time you have compiled meaning definitions and distinctions for the other terms in your list. However, there are terms not ...
ThomasTuna's user avatar
27 votes

The position-representation matrix elements of the propagator for a particle in a ring

You wouldn't think it, from how easy it is to pose this question, but it is ridiculously nontrivial. As it happens, it is entirely impossible to find the position-basis matrix elements of this ...
Emilio Pisanty's user avatar
20 votes
Accepted

Dirac Delta in definition of Green function

Your question has been answered again and again, and again, albeit indirectly and elliptically--I'll just be more direct and specific. The point is you skipped variables: in this case, t, and so the ...
Cosmas Zachos's user avatar
15 votes

How to interpret correlation functions in QFT?

No, $⟨0|T{ϕ(y)ϕ(x)}|0⟩$ is NOT the probability amplitude for a particle to propagate from $x$ to $y$, even for a free scalar field. It seems to be a common false belief that it is. There is one ...
Mikhail Skopenkov's user avatar
14 votes
Accepted

How do I Derive the Green's Function for $-\nabla^2 + m^2$ in $d$ dimensions?

The first step is to recognize that equation is invariant under $d$-dimensional rotations around $\mathbf{x} - \mathbf{x}' = \mathbf{0}$ and simultaneous identical translations of $\mathbf{x}$ and $\...
Sean E. Lake's user avatar
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14 votes
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What actually means to compute things at tree level?

Suppose you want to compute a correlation say in Euclidean signature $$ \frac{1}{Z}\int D\phi\ \prod_i \phi(x_i)\ \exp\left(-\frac{1}{\hbar}S(\phi)\right) $$ with $$ S(\phi)=\frac{1}{2}(\phi,A\phi)+g\...
Abdelmalek Abdesselam's user avatar
13 votes
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Bessel function representation of spacelike KG propagator

This can be seen by partial integration $$\frac{\partial}{\partial \rho}\sqrt{\rho^2-m^2}=\frac{\rho}{\sqrt{\rho^2-m^2}}$$ OP edit: More explicitly, we use this to write $(3)$ as \begin{align} D(...
PascExchange's user avatar
13 votes

Is the Green function of electromagnetism a scalar or a tensor?

Here's the gist of it: If your field lives in a vector space $V$, then the propagator is a map $V\to V$, i.e., it lives in $V\otimes V^*$. In more down-to-earth terms, if your field has a certain ...
AccidentalFourierTransform's user avatar
13 votes
Accepted

Period of the propagator of quantum harmonic oscillators

Good observation. OP's eq. (2) should be amended with a metaplectic correction/Maslov index: There is a caustic at every half-period, which leads to a phase factor $\exp\left(-\frac{i\pi}{2}\right)$ ...
Qmechanic's user avatar
  • 205k
12 votes
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Fermion propagator as derivative of scalar propagator

The free scalar and fermion propagator is $$ G_\psi(x,y) = \int \frac{d^dp}{(2\pi)^d} \frac{-i(\gamma^\mu p_\mu + m)}{ p^2 + m^2 - i \epsilon} e^{- i p \cdot ( x - y ) } $$ The scalar propagator is $$...
Prahar's user avatar
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11 votes
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Physical interpretation of the retarded vs. Feynman propagators?

The convolution $G_{ret}*f$ of the retarded propagator $G_{ret}$ with a source term $f$ which vanishes sufficiently far in the past is the unique solution of the inhomogeneous Klein-Gordon equation ...
Pedro Lauridsen Ribeiro's user avatar
11 votes
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Wick rotation vs. Feynman $i\varepsilon$-prescription

Starting in the Minkowski formulation, the Feynman $i\varepsilon$-prescription is just the first infinitesimal angle $\theta=\varepsilon$ of a Wick rotation $$\begin{align} t(\theta) ~=~& e^{i\...
Qmechanic's user avatar
  • 205k
11 votes
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What information is contained in spectral density function in QFT?

The two-point function is not nearly enough to determine a general QFT. You need the whole set of correlators, which is infinite. The formal statement is Wightman's reconstruction theorem, namely that ...
AccidentalFourierTransform's user avatar
11 votes

What is the string equivalent of the Feynman Propagator?

I will discuss the closed string propagator because this case is pictorially closer to the scalar propagator in quantum field theory case. The closed string analog of the (two-leg amputated) line of ...
Ramiro Hum-Sah's user avatar
11 votes
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Why does causality lead to different contours when calculating propagators?

The integral $$G(x,y)=\int \dfrac{d^4p}{(2\pi)^4}\dfrac{e^{-ip(x-y)}}{p^2+m^2}\tag{1}$$ is undefined because of the fact that the denominator $p^2+m^2$ vanishes when $p$ goes on shell. This ...
Gold's user avatar
  • 36.3k
10 votes

The analytical result for free massless fermion propagator

Method One: \begin{eqnarray*} & & \int\frac{d^{4}k}{(2\pi)^{4}}\frac{i}{k^{2}+i\epsilon}e^{-ik\cdot x}\\ & = & \frac{i}{(2\pi)^{4}}\int d^{3}ke^{i\mathbf{k}\cdot\mathbf{x}}\int dk_{...
Ren-Hong Fang's user avatar
10 votes
Accepted

Kallen-Lehmann representation versus perturbation theory

Kallen-Lehmann representation originates from the decomposition of $\hat{\phi}(x)|\Omega\rangle$ into the Hamiltonian eigenstates $|\lambda_{\mathbf{p}}\rangle$. If $|\langle\Omega|\hat{\phi}(x)|\...
OON's user avatar
  • 8,374
10 votes
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Why do the counterterms in renormalized $\phi^4$-theory with power two in fields give vertices and not propagators?

Consider $\phi^4$ theory: $$ \mathcal L=\frac12 Z_1(\partial\phi)^2-\frac12 Z_m m^2\phi^2-\frac{1}{4!}\lambda_0\phi^4 $$ There are two approaches to perturbation theory: First The propagator is ...
AccidentalFourierTransform's user avatar
10 votes
Accepted

Propagator of gauge boson

The propagator of an arbitrary vector field is [ref.1] \begin{equation} \langle A_\mu A_\nu\rangle=\frac{-\eta_{\mu\nu}+p_\mu p_\nu/m_1^2}{p^2-m_1^2}-\frac{p_\mu p_\nu/m_1^2}{p^2-m_0^2}\tag1 \end{...
AccidentalFourierTransform's user avatar
9 votes
Accepted

Deriving the photon Propagator

As the original poster of that question I think I might be able to help. I will use the same notation as in the original question. Starting from the equation $$\left(-k^2g_{\mu\nu}+(1-\frac{1}{\xi})...
Apogee's user avatar
  • 1,286
9 votes

$i\epsilon$ in the expression of Feynman Propagator

The $i\epsilon$ is used as a prescription to tell you how to integrate in the complex $p^{0}$ plane. In a sense, it enforces the boundary conditions of your propagator, and uniquely fixes the contour ...
Cynthia's Light's user avatar
9 votes
Accepted

Why is the $i\epsilon$-prescription necessary in the Klein-Gordon propagator?

Note that the original integral you are trying to compute is over the real line, not over a closed contour, so the Cauchy theorem does not apply until you find a suitable way to close the contour. Due ...
kaylimekay's user avatar
  • 2,053
9 votes
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Quantum fields can leak out of the light cone?

The posts linked in the comments give a good explanation of why the microcausality condition $[\phi(x),\phi^\dagger(y)]=0$ for spacelike $x-y$ is sufficient to make sure that only causal signals can ...
11zaq's user avatar
  • 896
8 votes

Differentiating Propagator, Green's function, Correlation function, etc

josh's answer is good, but I think there are two points that require clarification. First, his sentence defining the kernel makes no sense, because as written the dummy limit variable appears on both ...
tparker's user avatar
  • 48.1k
8 votes
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An integral in Peskin & Schroeder's Quantum Field Theory p. 27

In those paragraphs, P&S want to obtain the behaviour of the amplitude $D(x-y)$ in the limit $r \to \infty$. Starting from the integral $$ \tag{1} \label{int} \frac{-i}{2(2\pi)^2 r} \int_{-\infty}^...
ric_n's user avatar
  • 122
8 votes
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Arrow and flow of charge in fermion propagator

The arrow is not really about charge. Neutrinos are neutral, and you need arrows in their propagators anyway. The more correct statement is that arrows represent the flow of fermion number. Moreover, ...
AccidentalFourierTransform's user avatar
8 votes
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Understanding renormalization conditions in the $\phi^4-$theory

The logic is that we want the exact propagator $$\Delta(p^2) = \frac{1}{p^2 - m^2 - M(p^2)} $$ to behave like the free propagator $1/(p^2 - m^2)$ near the pole $p^2 = m^2$. This is because the ...
Javier's user avatar
  • 28.3k
8 votes
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Propagator Correction in $\phi^4$ theory - why doesn't this secular growth break perturbation theory?

Some comments: For some reason, you chose to include the $n$-th order tadpole but not the rest of diagrams that contribute to the same order. This is in general a meaningless operation: you either ...
AccidentalFourierTransform's user avatar

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