Why [f*(x),f(y)]=0 for spacelike intervals but <0|f*(x)f(y)|0>≠0

[LearningDG]

Hi,

Can anyone explain why in field theory we require [f*(x),f(y)] = 0 for space-lkie intervals x,y; but not <0|f*(x)f(y)|0> = 0?

Thanks!

 

[Bill_K]

LearningDG, You're right to call attention to this. It's the miracle that makes quantum field theory work! <0|f*(x)f(y)|0> ≠ 0 expresses the fact that a particle cannot be confined to a single point, that the Green's function extends outside the light cone a distance given by the Compton wavelength. And yet [f*(x),f(y)] = 0 expresses the fact that this apparent nonlocality does not destroy causality, saying that influences cannot propagate faster than light. And it's all due to the existence of antiparticles. The influence caused by emitting a particle is exactly canceled by the influence caused by the absorption of an antiparticle.