Microcausality in field theory

[RedX]

In field theory, the commutator of two fields vanishes at space-like separations. The explanation given is microcausality, which means that things separated farther than light can travel, cannot influence each other.

However, the Green's function does not vanish at space-like separations. This would imply that a source located at a space-like separation from a point should contribute to the field at that point. Doesn't this too violate microcausality? The Green's function seems to decay off (at least for a massless spin 0 particle) as 1/r² where r is the space-time separation, so things separated really far practically contribute zero to the field, but still it's not exactly zero.

Why is it that the commutator must vanish at space-like separations, but not the propagator?

Also, for field theory, we use the Feynman propagator, which contains both the advanced and retarded propagators. Should we literally take that to mean things in the future can affect the present? Because a source in the future would contribute to the field at the present.

 

[A. Neumaier]

Why is it that the commutator must vanish at space-like separations, but not the propagator?

Because there are clear arguments for the former (collected in Weinberg's book, Chapter 3 and 4 see also the current thread https://www.physicsforums.com/showthread.php?t=388556 ), while there hasn't been any argument for the latter, and it is not even the case for the simplest case of free fields.

Also, for field theory, we use the Feynman propagator, which contains both the advanced and retarded propagators. Should we literally take that to mean things in the future can affect the present? Because a source in the future would contribute to the field at the present.

No. The Feynman propagator is only used to compute scattering probabilities, which are time-symmetric (at least for the most important processes).

But dynamics is governed by the retarded propagator. Thus only the past affects the present.

 

[RedX]

No. The Feynman propagator is only used to compute scattering probabilities, which are time-symmetric (at least for the most important processes).

But dynamics is governed by the retarded propagator. Thus only the past affects the present.

The propagator/Green's function itself is time symmetric,

$$\Delta(t)=\Delta(-t)$$

But I think interactions can violate t-symmetry (but not the propagator) through the CKM matrix.

Obviously just the retarded or just the advanced propagator can never be time symmetric. It is only the combination of both of them (so that they transform into each other under time reversal) that works.

How about classical electrodynamics where one only uses the retarded propagator? Are processes that take place there not time symmetric? If light can go from point A to point B, can't it just as easily go from B to A? And yet there are no advanced green's functions there.

 

[A. Neumaier]

The propagator/Green's function itself is time symmetric,
$$\Delta(t)=\Delta(-t)$$

Yes, that's why it must use the Feynman propagator. But observed dynamics has an arrow of time, and this involves the retarded propagator. (Look at derivations of kinetic equations from QFT using CTP.)

But I think interactions can violate t-symmetry (but not the propagator) through the CKM matrix.

That why I had added ''at least for the most common processes''.

How about classical electrodynamics where one only uses the retarded propagator? Are processes that take place there not time symmetric?

Have you ever heard of a spherical wave being absorbed by an antenna? it only works the other way round.