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Summary:

I want to understand why the invariance of ##\Delta(xy)## implies that
$$[\phi(x), \phi(y)]=i \hbar c \Delta(xy)=0, \ \ \ \ \forall \ \ (xy)^2<0$$
Which means that any two points x and y, with spacelike separation, commute
Main Question or Discussion Point
I stumbled upon the concept of microcausality while studying how to covert the (equal time) commutation relations of the (given as an example) Real Klein Gordon Field (QFT Mandl & Shaw second edition, page 40) ##[\phi(\vec x, t), \phi(\vec x', t)]=0## into its manifestly covariant form (QFT Mandl & Shaw second edition, page 47):
$$[\phi(\vec x, t), \phi(\vec y, t)]=i \hbar c \Delta(\vec x  \vec y,0)=0$$
Then Mandl & Shaw asserted that 'The invariance of ##\Delta(xy)## implies that
$$[\phi(x), \phi(y)]=i \hbar c \Delta(xy)=0, \ \ \ \ \forall \ \ (xy)^2<0$$
Which means that any two points x and y, with spacelike separation, commute' (i.e. , as vanhees71 wrote in (4.1), ##[\mathscr{O}_1 (x), \mathscr{O}_2 (y)]=0##). Then microcausality means that 'measurements of the fields at two points with spacelike separation must not interfere with each other'.
I didn't really understand Mandl & Shaw's point so I checked Tong's notes (2.6.1) together with his lecture (at 1:06:46 he starts explaining what I do not get). There he explains microcausality based on the light cone but I still do not get what he means... Could you please explain it to me?
Thank you
$$[\phi(\vec x, t), \phi(\vec y, t)]=i \hbar c \Delta(\vec x  \vec y,0)=0$$
Then Mandl & Shaw asserted that 'The invariance of ##\Delta(xy)## implies that
$$[\phi(x), \phi(y)]=i \hbar c \Delta(xy)=0, \ \ \ \ \forall \ \ (xy)^2<0$$
Which means that any two points x and y, with spacelike separation, commute' (i.e. , as vanhees71 wrote in (4.1), ##[\mathscr{O}_1 (x), \mathscr{O}_2 (y)]=0##). Then microcausality means that 'measurements of the fields at two points with spacelike separation must not interfere with each other'.
I didn't really understand Mandl & Shaw's point so I checked Tong's notes (2.6.1) together with his lecture (at 1:06:46 he starts explaining what I do not get). There he explains microcausality based on the light cone but I still do not get what he means... Could you please explain it to me?
Thank you