Microcausality ($[\varphi (x),\varphi (y)]=0$ for spacelike $x-y$)

Microcausality requires local bosonic observables at spacelike separation to commute. For a free real scalar field, the field commutator is the difference of its two Wightman functions; it vanishes at equal time and, by Lorentz invariance, at every spacelike separation. Nonzero spacelike vacuum correlations do not violate this condition, since a correlation is different from the response to a local intervention. Odd fermionic fields satisfy a corresponding spacelike anticommutation condition, while physical even local observables commute.

For a free complex scalar field, independent particle annihilators and antiparticle creators giveThe invariant mass shell integral vanishes at spacelike separation: transform to equal time, then reverse the spatial momentum in one term. Retaining only the particle-annihilation term would leave a Wightman function, which is generally nonzero even at spacelike separation. Thus both frequency branches are needed for a local relativistic field. For a real scalar field, the creation term supplies the same cancellation but creates the same self-conjugate particle species.