Let $(M,g)$ be a globally hyperbolic. Let $P = \Box - m^2$ be the Klein-Gordon differential operator. Following Fewster's notes, we may define the retarded/advanced propagators $$E^\pm : C^\infty_0(M)\to C^\infty(M)$$ as follows. First let $f\in C^\infty_0(M)$ and consider the problem $$P\phi = f$$

with two conditions imposed on the solution:

  1. $\operatorname{supp} \phi \subset J^\pm (\operatorname{supp}f)$;
  2. $\operatorname{supp}\phi \cap J^\mp (\operatorname{supp}f)$ is compact;

One shows that the problem has unique solution and define $E^\pm(f)$ to be the solution to the corresponding problem.

I'm trying to gain intuition on this.

First consider the $E^+$ case. Condition (1) seems to mean that "to the past of when $f$ is turned on the solution vanishes". In that sense, it seems it allows us to say that the solution is created by the source $f$.

Now for the $E^-$ case. Now condition (1) seems to mean that "to the future of when $f$ ceases to exist the solution vanishes". In that sense, it seems that the solution is in fact what creates $f$.

Condition (2), on the other hand, I can't see how to interpret.

So, is my intuition on condition (1) correct? What is the intuition for demanding condition (2) when defining $E^\pm$?


Let's consider the $E^+$ case. The set $J^+(\text{supp} f)$ is the region of $M$ that can be reached causally starting from the support of $f$, either by light rays or by massive particles. So it's a causality condition: it means that $\phi(x)$ has to vanish if $x$ cannot be reached by a light ray starting from the support of $f$.

The second condition is less intuitive. The set $J^{-}$ is obtained from the support of $f$ by evolving backwards in time. There is bound to be some overlap between $J^+$ and $J^-$: there are points in $M$ that can be reached going in both directions. However, the support of $\phi$ should not reach back in time indefinitely: $\phi$ is generated by the source $f$ which only kicks in at some definite time, say $t=0$. So the support of $\phi$ can have a little bit of overlap with $J^{-}(f)$, but $\phi$ should completely vanish for $t < t_0$ for some time $t_0$.

Above I've introduced a time coordinate $t$, and since you're working on some abstract manifold $M$ you probably want to avoid that. But it's the gist of the idea.


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