Microcausality in field theory

In summary: Thus there is a preferred direction in time.In summary, field theory uses the concept of microcausality, where the commutator of two fields vanishes at space-like separations, meaning that things cannot influence each other if they are separated by a distance greater than light can travel. However, the Green's function, which represents the contribution of a source at a space-like separation to the field at a point, does not vanish. This may seem to violate microcausality, but the Green's function decays off and practically contributes zero to the field. The reason for this discrepancy is that there are clear arguments for the commutator vanishing, but not for the Green's function. Additionally, for field theory, the Fey
  • #1
RedX
970
3
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.
 
Physics news on Phys.org
  • #2
RedX said:
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.

RedX said:
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.
 
  • #3
A. Neumaier said:
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,

[tex]\Delta(t)=\Delta(-t)[/tex]

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.
 
  • #4
RedX said:
The propagator/Green's function itself is time symmetric,
[tex]\Delta(t)=\Delta(-t)[/tex]
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.)

RedX said:
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''.

RedX said:
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.
 

FAQ: Microcausality in field theory

1. What is microcausality in field theory?

Microcausality in field theory is the principle that states that the effects of a field at one point in spacetime cannot be influenced by the cause of the field at another point outside its light cone. In other words, no information can travel faster than the speed of light and cause and effect must be localized in spacetime.

2. Why is microcausality important in field theory?

Microcausality is important because it ensures that physical laws and equations are consistent and well-defined. It prevents the occurrence of paradoxes and violations of causality, which are necessary for the predictability and understanding of physical phenomena.

3. How is microcausality mathematically described in field theory?

Mathematically, microcausality is described by the commutation relations between field operators at spacetime points. For a field theory to satisfy microcausality, these commutation relations must vanish for spacelike separated points, indicating that there is no direct causal connection between them.

4. Can microcausality be violated in any field theory?

No, microcausality is a fundamental principle in all field theories and cannot be violated. If a field theory were to violate microcausality, it would lead to inconsistencies and contradictions in the physical predictions of the theory.

5. How does microcausality relate to other principles in physics, such as causality and locality?

Microcausality is a more specific and mathematical formulation of the principles of causality and locality in physics. It ensures that these principles are preserved in field theories and prevents the occurrence of paradoxes and violations of these principles.

Similar threads

Replies
4
Views
1K
Replies
87
Views
6K
Replies
3
Views
1K
Replies
4
Views
2K
Replies
16
Views
2K
Replies
12
Views
4K
Replies
36
Views
4K
Back
Top