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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?

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Why do you want to give the same thing two interpretations? I'd argue that one can motivate $\vec p$ via $m\vec v$ and doesn't have to introduce another interpretation after introducing new variables. Also, I wouldn't view it as only one averaged plane wave but a summation of many, but whatever. Also the formulation with a relativistically invariant measure is nice and visually to look at, I'd use that for another orientation, especially regarding the $1$ over $E$. –  NiftyKitty95 May 24 '12 at 14:15
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up vote 4 down vote accepted

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.

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