Physical interpratation of propagator

Question

Consider the space-time domain Klein-Gordon propagator:

$$G_F(x)=\int\frac{d^4p}{(2\pi)^4}e^{ipx}\frac{1}{p^2-m^2+i\epsilon}$$

I understand this as the amplitude at location $x$ created by a source located at spacetime event $(0,0)$. I also see it as plane waves propagating with momentum $p$ weighted by $\frac{1}{p^2-m^2}$ : the density if sharply peaked at on-shell particles by the pole in the denominator.

Now consider the same propagator after integration on $p^0$ : $$G_F(x)=\frac{-i}{2}\int\frac{d^3p}{(2\pi)^3}e^{i\bf{p.x}}\frac{e^{-iEt}}{E}$$

What is the physical interpretation of this? What happened to the off-shell modes?

Answer 1

Let me first please correct your expression after the integration $p_0$ component, the result should be

$G_F(\mathbf{x}) = \int \frac{d^3p}{2(2\pi)^3E}e^{i\mathbf{p}.\mathbf{x}}(e^{iEt}-e^{-iEt})$

This is because there are two poles corresponding to positive and negative energies.

Now please observe that the propagator can be written as:

$G_F(\mathbf{x}) = \int \frac{d^3p}{2i(2\pi)^3}\int_{-\infty}^{\infty}d\tau e^{i\mathbf{p}.\mathbf{x}}(\Theta(t-\tau)e^{iE\tau}+\Theta(\tau-t)e^{-iE\tau})$

In this representation the (relativistic) causality is manifest as only "earlier" sources effect the point for positive energies and "later" sources for negative energies.

On the other hand in the original representation the propagator is analytic in the $p_0$ plane except for the two poles.

This exercise therefore shows that the causality in the time domain is a consequence of the analyticity in the energy domain, and only one of them is manifest in a given form.