[XMP/QFT-04001] Short Example --- Microcausality

If we quantize a real scalar fields with equal time anti commutators, then it can be shown that the commutators $\big[\phi(x), \phi(y)\big]$ does not vanish for space like \(x,y\). What about products bilinear in fields? One may argue that like fermions we may try restricting observable quantities to bilinear in fields.It turns out that even the bilinears do not commute for space like separations.

See for example:
Voja Radovanovi\'{c}, Problem Book in Quantum Field Theory, Second Edition Springer (2008)