My question(s) concern the interpretation and uniqueness of the propagators / Green's functions for both classical and quantum fields.

It is well known that the Green's function for the Laplace equation
$$ \Delta_x G(x,x') = \delta^{(3)}(x-x') $$
with *boundary conditions*
$$ G(x,x') = 0 \text{ for } x\in\partial D, x'\in D$$
is determined *uniquely*. This is because any harmonic function that vanishes on the boundary of domain must vanish inside as well.

In contrast, I am confused about the uniqueness of the Green's function for the wave equation
$$ (∂_t^2 - \Delta)G(x,t;x',t') = \delta^{(3)}(x-x')\delta(t-t') $$
Without specifying boundary conditions, there exist many homogenuous solutions to the wave equation and thus many different Green's functions.

Physicists usually offer me several canonical choices, like the *retarded*, the *advanced* and also the *Feynman propagator*, but I don't understand what makes each choice unique compared to the others, or which boundary conditions correspond to which choice. Hence, my question is

Which boundary conditions correspond to retarded, the advanced and the Feynman propagator? What other possibilities for boundary conditions and propagators are there?

I am also confused by the situation in quantum field theory, where we have various conventions for propagators and time-ordering like
$$ \langle0|T_t\lbrace a(x,t)a^\dagger(x',t')\rbrace |0\rangle $$
Apparently, the ground state is very important for selecting the right Green's function (see also my previous question), but I still don't understand why that is.

How does the vacuum state act as a boundary condition in the spirit of the previous question?

There is also the issue of imaginary time. The point is that imaginary time turns the wave equation $(\partial_t^2 -\Delta)\phi=0$ into the Laplacian $(\partial_\tau^2 + \Delta)\phi = 0$, but I don't understand how the usual analytic continuation $\tau \to it \pm i\eta$ for various propagators depends on the boundary conditions. For instance, Tsvelik writes in equation (22.4) that the imaginary time Green's function for the Laplacian in 2D is "obviously" the one that vanishes when $\tau$ and $x$ go to infinity, but I don't understand the reasoning behind this choice.

What are the right boundary conditions for Green's functions in imaginary time? Is it clear that the path integral formalism selects the right ones?

This post has been migrated from (A51.SE)