I'm a physicist and I'm working in a problem in general relativity where I end up with N-point functions that have to satisfy equations like this
\begin{equation} \Big[\nabla_\mu \nabla^\mu +V(x_1)\Big]G(x_1,x_2,x_3)=f(x_1,x_2,x_3) \end{equation}
where $\nabla_\mu$ is a covariant derivative acting on $x_1$, $V(x)$ is some potential, $G(x_1,x_2,x_3)$ in this case is a 3-point function evaluated at three different points $x_1$, $x_2$ and $x_3$ and $f$ is a source. There might be some properties that we can ask for. For example, $f(x_1,x_2,x_3)$ might already be symmetric in the last two indices. I would even be interested in a partial result for the case where $f=0$. The 3-point function must satisfy
- It must be symmetric in all arguments.
- It must vanish when at least two pairs of points are time-like separated.
In physics, time-like separated just means that using the metric tensor, the distance between them is less than zero (Which is possible in semi Riemannian manifolds such as spacetime in General Relaivity).
My question is: Are there any sources or literature about building these sort of generalized Green's functions with such properties?
Any tip, help or advice in the right direction will be appreciated, thanks!