 Antiparticle modes and spacelike commutativity

ID: antiparticle-modes-and-spacelike-commutativity

For a free complex scalar field, independent particle annihilators and antiparticle creators give

[Φ (x), Φ^{†} (y)] = \int \frac{d ^{3} p}{( 2 π ) ^{3} 2 E _{p}} (e^{- i p (x - y)} - e^{i p (x - y)}) .

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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