I saw an interesting comment in my notes about the [itex]E = \pm \sqrt{\vec{p}^2 + m^2}[/itex] eingenvalues plane wave solution to the free particle Klein Gordon equation. The propagator(adsbygoogle = window.adsbygoogle || []).push({});

[itex]U = \langle \vec{x} \mid e^{-i H t} \mid \vec{x_0} \rangle = \langle \vec{x} \mid e^{-i t \sqrt{\vec{p}^2 + m^2} } \mid \vec{x_0} \rangle \propto[/itex]

[itex]\propto e^{-m \sqrt{ (\vec{x} - \vec{x_0})^2 - t^2}} \neq 0 \quad \text{also for} \quad (\vec{x} - \vec{x_0})^2 > c^2 t^2 [/itex]

that is outside the light cone. About this there's even a whole section on Wikipedia's propagator page, but it lacks the math. My questions are:

Thanks.

- how do you get to the above result, after the proportionality symbol?
- The imaginary exponent becomes real, how does that happen? Also like this it seems that inside the light cone the exponent is imaginary again, what does that mean?

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# The propagator is nonzero outside of the light cone.

Can you offer guidance or do you also need help?

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