Covariant commutation relations and spacelike intervals in Mandl & Shaw

[lornstone]

Hi,

In Mandl&Shaw, when we calculate the covaiant commutation relations for a scalar field we obtain :
$$[\phi(x),\phi(y)]= i\Delta(x-y)=0$$
and the last equality stands if x-y is a space-like interval. But I don't understand why. We know that it is zero if the time component is zero and we also know that delta is invariant under proper Lorentz transformation. I don't see why we can't do the correct lorentz transformation (which bring time to zero) with a time-like interval so that it is also zero.

Thank you!

 

[schieghoven]

You can't Lorentz transform a time-like interval into a space-like interval. The Minkowski norm
$$t^2 - x^2 - y^2 - z^2$$
is invariant under Lorentz transformations, so a time-like interval (positive norm) cannot be mapped into a space-like interval (negative norm).

 

[lornstone]

I got it, thank you!