field theory - Differentiating Propagator, Green's function, Correlation function, etc

For the following quantities respectively, could someone write down the common definitions, their meaning, the field of study in which one would typically find these under their actual name, and most foremost the associated abuse of language as well as difference and correlation (no pun intended):

Maybe including side notes regarding the distinction between Covariance, Covariance function and Cross-Covariance, the pair correlation function for different observables, relations to the autocorrelation function, the $n$-point function, the Schwinger function, the relation to transition amplitudes, retardation and related adjectives for Greens functions and/or propagators, the Heat-Kernel and its seemingly privileged position, the spectral density, spectra and the resolvent.

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Edit: I'd still like to hear about the "Correlation fuction interpretation" of the quantum field theoretical framework. Can transition amplitudes be seen as a sort of auto-correlation? Like... such that the QFT dynamics at hand just determine the structure of the temporal and spatial overlaps?

Answer

The main distinction you want to make is between the Green function and the kernel. (I prefer the terminology "Green function" without the 's. Imagine a different name, say, Feynman. People would definitely say the Feynman function, not the Feynman's function. But I digress...)

Start with a differential operator, call it $L$. E.g., in the case of Laplace's equation, then $L$ is the Laplacian $L = \nabla^2$. Then, the Green function of $L$ is the solution of the inhomogenous differential equation

$$L_x G(x, x^\prime) = \delta(x - x^\prime)\,.$$ We'll talk about its boundary conditions later on. The kernel is a solution of the homogeneous equation $$L_x K(x, x^\prime) = 0\,,$$ subject to a Dirichlet boundary condition $\lim_{x \rightarrow x^\prime}K(x,x^\prime) = \delta (x-x^\prime)$, or Neumann boundary condition $\lim_{x \rightarrow x^\prime} \partial K(x,x^\prime) = \delta(x-x^\prime)$.

So, how do we use them? The Green function solves linear differential equations with driving terms. $L_x u(x) = \rho(x)$ is solved by

$$u(x) = \int G(x,x^\prime)\rho(x^\prime)dx^\prime\,.$$ Whichever boundary conditions we what to impose on the solution $u$ specify the boundary conditions we impose on $G$. For example, a retarded Green function propagates influence strictly forward in time, so that $G(x,x^\prime) = 0$ whenever $x^0 < x^{\prime\,0}$. (The 0 here denotes the time coordinate.) One would use this if the boundary condition on $u$ was that $u(x) = 0$ far in the past, before the source term $\rho$ "turns on."

The kernel solves boundary value problems. Say we're solving the equation $L_x u(x) = 0$ on a manifold $M$, and specify $u$ on the boundary $\partial M$ to be $v$. Then,

$$u(x) = \int_{\partial M} K(x,x^\prime)v(x^\prime)dx^\prime\,.$$ In this case, we're using the kernel with Dirichlet boundary conditions.

For example, the heat kernel is the kernel of the heat equation, in which

$$L = \frac{\partial}{\partial t} - \nabla_{R^d}^2\,.$$ We can see that $$K(x,t; x^\prime, t^\prime) = \frac{1}{[4\pi (t-t^\prime)]^{d/2}}\,e^{-|x-x^\prime|^2/4(t-t^\prime)},$$ solves $L_{x,t} K(x,t;x^\prime,t^\prime) = 0$ and moreover satisfies $$\lim_{t \rightarrow t^\prime} \, K(x,t;x^\prime,t^\prime) = \delta^{(d)}(x-x^\prime)\,.$$ (We must be careful to consider only $t > t^\prime$ and hence also take a directional limit.) Say you're given some shape $v(x)$ at time $0$ and want to "melt" is according to the heat equation. Then later on, this shape has become $$u(x,t) = \int_{R^d} K(x,t;x^\prime,0)v(x^\prime)d^dx^\prime\,.$$ So in this case, the boundary was the time-slice at $t^\prime = 0$.

Now for the rest of them. Propagator is sometimes used to mean Green function, sometimes used to mean kernel. The Klein-Gordon propagator is a Green function, because it satisfies $L_x D(x,x^\prime) = \delta(x-x^\prime)$ for $L_x = \partial_x^2 + m^2$. The boundary conditions specify the difference between the retarded, advanced and Feynman propagators. (See? Not Feynman's propagator) In the case of a Klein-Gordon field, the retarded propagator is defined as

$$D_R(x,x^\prime) = \Theta(x^0 - x^{\prime\,0})\,\langle0| \varphi(x) \varphi(x^\prime) |0\rangle\,$$ where $\Theta(x) = 1$ for $x > 0$ and $= 0$ otherwise. The Wightman function is defined as $$W(x,x^\prime) = \langle0| \varphi(x) \varphi(x^\prime) |0\rangle\,,$$ i.e. without the time ordering constraint. But guess what? It solves $L_x W(x,x^\prime) = 0$. It's a kernel. The difference is that $\Theta$ out front, which becomes a Dirac $\delta$ upon taking one time derivative. If one uses the kernel with Neumann boundary conditions on a time-slice boundary, the relationship $$G_R(x,x^\prime) = \Theta(x^0 - x^{\prime\,0}) K(x,x^\prime)$$ is general.

In quantum mechanics, the evolution operator

$$U(x,t; x^\prime, t^\prime) = \langle x | e^{-i (t-t^\prime) \hat{H}} | x^\prime \rangle$$ is a kernel. It solves the Schroedinger equation and equals $\delta(x - x^\prime)$ for $t = t^\prime$. People sometimes call it the propagator. It can also be written in path integral form.

Linear response and impulse response functions are Green functions.

These are all two-point correlation functions. "Two-point" because they're all functions of two points in space(time). In quantum field theory, statistical field theory, etc. one can also consider correlation functions with more field insertions/random variables. That's where the real work begins!