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So in QFT the main way we get results are from objects of the form: $$ \langle \phi...\rangle. $$

Why are these sometimes referred to as greens functions? Do they solve a differential equation like: $$ D\langle...\rangle=\delta ~? $$

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  • $\begingroup$ Tbh I've never seen those to be referred to as Greens functions. Could you provide a source? $\endgroup$
    – nwolijin
    Jan 29, 2021 at 10:11
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    $\begingroup$ Yes, they do solve an equation like $D\langle \cdots\rangle=\delta$. It is called the Dyson-Schwinger equation, see e.g. equation (6) here. $\endgroup$ Jan 29, 2021 at 13:53
  • $\begingroup$ It's also worth mentioning that if you write the partition function as a functional power series in the sources, then the coefficients are the $n$-pt functions, so they are also like a non-linear generalization of the usual idea behind Green's functions to non-linear systems. $\endgroup$ Jan 29, 2021 at 18:54
  • $\begingroup$ Notation is sometimes not correctly chosen and confusing. Why do you care about how the thing is called? $\endgroup$ Jan 29, 2021 at 23:49

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A more explicit example is the time-ordered correlation between two free scalar fields: $$ \langle0|\mathcal{T}[\phi(x_1)\phi(x_2)]|0\rangle $$ In momentum space, this is the Feynman propagator: $$ \langle0|\mathcal{T}[\phi(x_1)\phi(x_2)]|0\rangle =i\int \frac{\mathrm d^4 k}{(2\pi)^4}\frac{e^{ik(x_1-x_2)}}{k^2-m^2+i\varepsilon} \equiv i\Delta_{12} $$ In the limit $\varepsilon \to 0$, this is exactly the Green's function for the Klein-Gordon operator $\partial^2+m^2$ with boundary conditions added in: $$ (\partial^2+m^2)\Delta_{12}=-\delta^{(4)}(x_1-x_2) $$

This is very easy to see for two-point correlators in free theories. The more general "Green's function" describes n-point correlators in the interacting theory, i.e. $\langle\Omega|\mathcal{T}[\phi(x_1)...\phi(x_n)]|\Omega\rangle$ - however, these should be regarded as generalisations of correlators from two fields to an arbitrary number: they do not correspond to Green's functions in the strict mathematical sense since they are not, in general, inversions of a linear (differential) operator.

However, the relationship is not as tenuous as it seems: the Schwinger-Dyson equation for the partition function is $$ \left(-i\frac{\delta S}{\delta\phi}\bigg|_{\frac{\delta}{\delta J}} + J(x)\right)Z[J] = 0 $$

By expanding the partition function as a functional series in $J$, or equivalently by applying consecutive functional derivatives with respect to the source, one obtains the Schwinger-Dyson equations for the $n$-point functions, which ostensibly look like a kernel equation for a linear operator.

These work in the non-perturbative regime, but while they may look pretty, they are highly non-linear, forming an ever-rising chain of higher-order integral equations for each $n$ (the $n$-point Schwinger-Dyson equation will typically depend on several $m$-point functions, for $m > n$). As such they must usually be analysed asymptotically or perturbatively, by truncating this chain. An explicit example of such an equation for Lagrangians of the form $\mathcal{L} = -\frac12\phi(\partial^2+m^2)\phi+\mathcal{L}_\text{int}[\phi]$ is:

$$ (\partial^2+m^2)\langle\phi(x)\phi(x_1)\dots\phi(x_n)\rangle=\langle\mathcal{L}_\text{int}[\phi(x)]\phi(x_1)\dots\phi(x_n)\rangle - i\hbar\sum_j\delta^4(x-x_j)\langle\phi(x_1)\dots\phi(x_{j-1})\phi(x_{j+1})\dots\phi(x_n)\rangle $$

[Schwartz, Quantum Field Theory and the Standard Model, equation $(7.12)$]

which is a suitable non-linear generalisation of the garden-variety Green's functions.

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