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33 votes
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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
20 votes
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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
19 votes

What do the poles of a Green function mean, physically?

Let me expand a little more on what Craig Thone just said : Consider the energy/frequency-dependent Green function : $$ \tilde{G}(\omega)=\frac{1}{\omega-(a-\mathrm{i}b)} $$ with one single pole in $\...
dolun's user avatar
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18 votes

What do the poles of a Green function mean, physically?

This is a somewhat broad question, because there are a number of different Green's functions in quantum physics. Perhaps the simplest one is the resolvent Green's function for a single-particle system....
David Ruiz-Tijerina's user avatar
17 votes
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Why do people care so much about 'linear response theory'?

You are largely right. The mathematics you've been presented could indeed be packaged under several other names, including solutions via Fourier transforms or via Green functions. However, that doesn'...
Emilio Pisanty's user avatar
16 votes
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Particle/Pole correspondence in QFT Green's functions

For simplicity, take $\hbar=1$ and consider a Hermitian scalar, renormalized field $\phi(x)$; other fields are treated analogously. Then (for simplicity ignoring the necessary smearing since the field ...
Arnold Neumaier'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
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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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13 votes
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Is $\delta(r-ct)/4\pi r$, the 3D wave equation elementary solution, a transverse or longitudinal wave?

The question doesn't make sense, since the terms "transverse" and "longitudinal" don't apply to scalar fields. They refer to the relationship between the polarization of a wave and its propagation ...
knzhou's user avatar
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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
12 votes
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Calculating Green's function from Dyson's series without normal ordering

Normal Ordering The trick is to realise that a normal ordered string of fields is just an Hermite polynomial (ref. $[1]$): $$ \colon \phi^n\colon=H_n(\phi)\tag{1} $$ so that, for example, $$ \colon \...
AccidentalFourierTransform's user avatar
12 votes

Green's function on torus

When both "space" ($x$) and "time" ($y$) directions are periodic, the Laplacian on torus with coodinate $z=x+iy$ has a normalized zero mode $$ \varphi_0(z) = \frac 1 {\sqrt{{\rm Im}(\tau)}} $$ (Here $\...
mike stone'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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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

Why is the Yukawa potential equal to Green's function for free space?

Helmholtz equation is identical with the coordinate part of Klein-Gordon equation, which is the wave equation for a massive particle, so no wonder that they produce identical solutions. Note also that ...
Roger V.'s user avatar
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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
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11 votes
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Is Kirchhoff's scalar theory of diffraction mathematically inconsistent?

Kirchhoff's scalar diffraction theory is mathematically inconsistent. The reason is as follows. Kirchhoff proved that any function $U$ satisfying the homogeneous wave equation $\nabla^2U+k^2U=0$ also ...
hyportnex's user avatar
  • 19.3k
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

Where are the poles of the one-particle Green's function located in the complex plane?

It's a good question, and it has a beautiful answer. It is true that for any finite sum (referring to your first expression), one cannot have a complex pole. So the question is: how can the complex ...
Ruben Verresen's user avatar
9 votes
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Truncated $N$-point functions

It is basically a matter of being consistent with definitions/notation. The $n$-point function, denoted by $$ G^{(n)}(x_1,\dots,x_n) $$ is given by the sum over all Feynman diagrams with $n$ external ...
AccidentalFourierTransform's user avatar
9 votes
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Why the imaginary part of green function is the spectral function?

To take things in order How is the third Green's function derived? I ask for detailed derivation process. To reduce clutter and focus on ideas, I will drop a lot of subscripts and such. Note: the ...
bRost03's user avatar
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9 votes
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What happens in the Hartree and Fock diagrams?

In Hartree term the time and the spatial position of the ends of circle line (representing Green's function) coincide. i.e. equal to $\langle \psi^\dagger (t,\mathbf{r}) \psi (t,\mathbf{r})=n(t,\...
Alex's user avatar
  • 106
9 votes

Green's function on torus

The constant term is needed because on a compact manifold with periodic boundary conditions there is a zero mode in the spectrum of the laplacian. This is easier to see on the circle, where $\frac{d^2}...
nox's user avatar
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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
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8 votes
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Callan-Symanzik Equation

I think it goes more or less as follows: As written in the book the dependence of the two-point function on $p$ and $M$ reduces to $$G^{(2)}(p)=\frac{i}{p^2}g(-p^2/M^2).$$ Therefore one has $$p\...
DelCrosB's user avatar
  • 903
8 votes
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Propagator of a real scalar field does not give an unambiguous result

When we integrate the propagator with respect to $k^0$ (i.e. the energy), we encounter two poles: one at $\omega_{\mathbf{k}}=\sqrt{\mathbf{k}^2+m^2}$ and one at $-\omega_{\mathbf{k}}=-\sqrt{\mathbf{k}...
probably_someone's user avatar
8 votes

Spectral function and bound states in condensed matter

The poles of the spectral density give you the particles of the system, whether they are fundamental or bound states. There is no fundamental difference between these two types: they behave exactly ...
AccidentalFourierTransform's user avatar
8 votes
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Cauchy boundary conditions and Greens functions with Fourier transform

Your problem is not specific to d'Alembert's PDE, but can be traced back to ODE's, even the simple harmonic oscillator. Say you want to solve: $$ \frac{d^2G}{dt^2}+G = \delta $$ Going to Fourier ...
LPZ's user avatar
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8 votes
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Why do different contours give different answers in the limit $\epsilon \rightarrow 0$ when calculating propagators?

Here is a cute toy example from What is a Quantum Field Theory? A First Introduction for Mathematicians. M. Talagrand. CUP. Appendix N: Feynman Propagator and Klein-Gordon Equation, which hopefully ...
Tobias Fünke's user avatar

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