= Antiparticle modes and spacelike commutativity
For a free <complex scalar field>, independent particle annihilators and <antiparticle> creators give
$$
[\Phi(x),\Phi^\dagger(y)]=\int\frac{d^3p}{(2\pi)^3\,2E_p}\left(e^{-ip(x-y)}-e^{ip(x-y)}\right).
$$
The 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.
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