Does Causality in QFT Require Commuting Field Operators Outside the Light Cone?

[rbphysics]

TL;DR

Klein Gordon field. Non zero propagator outside the light cone in QFT.

The Causality in QFT is said not be violated by showing field operators in Klein Gordon field commute outside the light cone. This is necessary because If I here measure a observable A and my friend there at mars measure observable B that does not commute with A then he can influence my measurement by just measuring his observable. This is why operators defined at space separated points must commute. This is what has been described by Sidney Coleman in his QFT lectures. However, I am worried about the non zero propagator i.e probability amplitude for particle to propagate from y to x and it does not vanish outside the light cone. This no zero probability amplitude means particle created at space point can be detected outside the light cone. Won't it violate causality. Several books on QFT says it won't. Can anyone clarify the same?

 

[PeterDonis]

I am worried about the non zero propagator i.e probability amplitude for particle to propagate from y to x and it does not vanish outside the light cone.

That's correct, but it doesn't violate causality because the operators commute (in this case the operators are "a particle is measured to be at spacetime event x" and "a particle is measured to be at spacetime event y"). In other words, the results of both measurements are the same regardless of the order in which they occur--which is good because the order in which they occur is frame-dependent.

 

[Demystifier]

This is discussed in some detail in the QFT book by Padmanabhan, Sec. 1.5. Fundamentally, relativistic quantum field theory is not a theory of particles. It is a theory of fields. The propagator of particles non-vanishing outside the light-cone only shows that particles are not the true physical objects. When you interpret the so called "propagator" in terms of fields, it is no longer interpreted as a "propagator", instead it is a vacuum expectation value of a product of fields. The fact that it does not vanish outside the light-cone means that the fields are correlated across the space-like distances. Ultimately, this correlation is a consequence of entanglement in the vacuum state. But it is well known that entanglement cannot be used for sending signals faster than light, in this sense relativistic causality is not violated.

 

[rbphysics]

Is there any other book that discuss the same like Padmanabhan. Given that he starts with path integral and lots of maths. It's somewhat difficult to follow.

 

[rbphysics]

I request anyone to clarify the same. Causality in QFT and need of antiparticles. And, Non vanishing Propagator outside the lightcone.

 

[PeterDonis]

I request anyone to clarify the same. Causality in QFT and need of antiparticles. And, Non vanishing Propagator outside the lightcone.

What do you need clarified? We've already discussed what causality means in QFT and what the propagator not vanishing outside the light cone does and doesn't mean.

What does the need for antiparticles have to do with the topic?

 

[rbphysics]

What do you need clarified? We've already discussed what causality means in QFT and what the propagator not vanishing outside the light cone does and doesn't mean.

What does the need for antiparticles have to do with the topic?

This is often stressed in QFT books that concept of antiparticle is required to preserve causality in QFT. I don't understand it. Moreover, It is also mention that the propagator G(x1,x2) for timelike separtaed events give probability amplitude for particle to travel from x1 to x2 if t2>t1 and probability of antiparticle to travel from x2 to x1 if t1>t2. I want to know what actually propagates? Does the sign of changes particle or antiparticle to be created? I am asking this because let say I have a some kind of particle creator, will it create particle and antiparticle both, particle in future and antiparticle in past? Moreover it said that for space like separated points amplitude fot particle to travel from x1 to x2 cancels probability for antiparticle to travel from x2 to x1 and they cancels each other and causality is preserved. No books seens to discuss this good way in intuitive way

 

[bhobba]

It is often stressed in QFT books that the concept of the antiparticle is required to preserve causality in QFT. I don't understand it.

Medium has the details:



It may be behind a paywall, but because I am a member, I can't tell. If so, you can join for a month at $5.00

Here is a precis:

The momentum integral f(t) only has positive frequencies when Fourier transformed.

A theorem then says f(t) cannot be zero for any finite range of time t=t₂- t₁ unless it vanishes identically for all times.

First, fix the space coordinates x₁ and x₂ and consider the ω-dependent version of the momentum integral. But for a fixed x₁, the momentum integral cannot be equal to zero when the second coordinate x₂ lies outside the lightcone of x₁. We conclude that it must include nonzero amplitudes containing spacelike intervals (particles moving faster than light).

Spacelike intervals have frame-dependent order of events. Hence, particles are seen in another reference frame as “propagating back in time.” Let us add an electric charge to the particle. According to Feynman–Stueckelberg's interpretation, particles moving backward in time are equivalent to antiparticles moving forward in time!

Suppose we change into a new (primed) reference frame where t₂< t₁. Let us see what happens. In the new reference frame, until t=t₂, there is just one moving particle ψᵢ. However, at t₂ suppose some potential creates two particles, and one seems to be moving backward in time. Then at t₁ the original particle and the backward-moving particle meet and vanish. In other words, the two particles annihilate each other in the new reference frame.

The following explains it at an even more basic level:


Thanks
Bill

 

[PeterDonis]

This is often stressed in QFT books that concept of antiparticle is required to preserve causality in QFT.

A specific reference and quote would help. I have not seen it put quite this way.

I want to know what actually propagates?

There is no "right" answer to this question, because "propagation" is not a fundamental concept in QFT (despite the use of the term "propagator" to describe an important item in the math). The fundamental concept is quantum fields, and while particular quantum field states and processes are often called "particles" and described as "propagation of particles", that's really a crutch for our intuitions, which we haven't fully retrained to deal with quantum fields. The only statements we can really rely on in this context are statements about actual measurement events: we measured a particle to be at a certain spacetime event. In general, if we measure a particle at spacetime event A, and a particle of the same type (an electron, say) at spacetime event B, we can't even say that the two electrons are "the same electron". All we can say is that an electron was detected at each of the two events.

No books seens to discuss this good way in intuitive way

That's because there is no "intuitive way" to discuss this. QFT is not "intuitive". It describes things that have no counterparts in our common experience. That's why we need math to do it right, and experiments to confirm that the math makes correct predictions. Nature doesn't care whether what it does is 'intuitive" to you. It just does it.

 

[.Scott]

I saw @rbphysics question soon after he posted it - and knew right away that it would require some of this forums "real Physicists" (such as @PeterDonis, @Demystifier, @bhobba, @DrChinese). So, my participation here is to see if I can't get a better grasp of it myself.

I will start by building a physically optimal radar system - limited only by the Physics. The transceiver transmits a single photon, it reflects off a first surface mirror, and then returns to be detected by the transceiver. So the amplitude in the field of the reflected signal photon will be non-zero before its light-cone arrives. But certainly I am not allowed to register that mirror as "detected" until that reflected light-cone crosses my detector. How is it that this non-zero amplitude can never be of any use to me in detecting the photon early?

 

[renormalize]

But certainly I am not allowed to register that mirror as "detected" until that reflected light-cone crosses my detector. How is it that this non-zero amplitude can never be of any use to me in detecting the photon early?

By what logic are you concluding this? Are you arguing that your hypothetical experiment requires the Feynman propagator for the photon? Why not use instead the retarded (causal) propagator?

 

[Demystifier]

So the amplitude in the field of the reflected signal photon will be non-zero before its light-cone arrives.

Why do you think so?

 

[.Scott]

By what logic are you concluding this? Are you arguing that your hypothetical experiment requires the Feynman propagator for the photon? Why not use instead the retarded (causal) propagator?

If you mean how am I concluding that I will not be allowed to detect the mirror early, it is simply that QFT never creates an exception to FTL communications. That mirror's albedo could be controlled by an event that is still spacelike to my detector.
I am not arguing that it requires a propagator, but shouldn't it still be valid to use it? If not, can the experiment be change to allow a propagator? The experiment is intended to be as vanilla as possible while still proving an experimental case-in-point. So, if I need an electron or a neutrino, then use an electron or a neutrino.

 

[.Scott]

[me: So the amplitude in the field of the reflected signal photon will be non-zero before its light-cone arrives.]

Why do you think so?

Probably because I don't understand the discussion that precedes this. I thought that was the premis.

To expand on what I just mentioned to @renormalize , I am looking for a case-in-point where:
1) A particle has amplitude outside of the light cone;
2) an attempt is made to detect the particle early because of that "errant" amplitude; and
3) the method of detection would demonstrate that the particle was detected early.

And the result of the experiment might be something like: It fails because ... whatever.
In other words: At the moment, the phrases "amplitude outside the light cone" and "potentially detectable too early" are fairly synonymous to me. I am looking for the distinction.

I am thinking something along the Bell experiment where FTL isn't violated because no actual message can be sent until the results of two separated detectors are compared.

 

[Nugatory]

potentially detectable too early

The difficulty here comes from an assumption hidden in the “too early” part. As with two-particle entanglement, our classically-trained intuition chokes on the idea that there can be correlation between two measurements without one having affected (“affected” implying a causal influence) the second - but it is only our inability to accept that possibility that makes a measurement “too early”.

 

[.Scott]

The difficulty here comes from an assumption hidden in the “too early” part. As with two-particle entanglement, our classically-trained intuition chokes on the idea that there can be correlation between two measurements without one having affected (“affected” implying a causal influence) the second - but it is only our inability to accept that possibility that makes a measurement “too early”.

I may not be understanding the set-up here. I am assuming that there is a single particle emission event (event A) and a particle detection event (event B) and that the "light cone" is associated with that emission event (event A). If that is the case, "too early" simply means that the event A and B are space-like.

 

[PeterDonis]

I am assuming that there is a single particle emission event (event A) and a particle detection event (event B) and that the "light cone" is associated with that emission event (event A). If that is the case, "too early" simply means that the event A and B are space-like.

Yes, and you are assuming that, if we ensure that there is only one photon in the experiment, ever (note that I am sweeping a lot under the rug here, and assuming that you've done all the right things to actually ensure this, which are not at all easy to do, certainly not as easy as setting your Acme laser photon source to "one photon" and pulling the trigger), then there will be a nonzero probability of detecting a photon at both event A and event B. But that is not the case.

What is the case is that it is possible to have photon detections at both event A and event B which are correlated in ways that violate the Bell inequalities, i.e., in ways that can't be explained by any "classical" model. But there is no claim made in such cases that the photons detected at event A and event B are "the same photon". Indeed, if, as above, you try to ensure that they would have to be "the same photon", by enforcing the constraint that only one photon is ever inside the experiment, then you can no longer observe photon detections at two spacelike separated events inside your experiment.

 

[.Scott]

@PeterDonis :
I was only going to transmit a photon at event A - not try to detect it. So, I know when event "A" happens because I triggered it.

There seems to be a persistent drift in this thread using two detectors.
Are we talking about an effect can only exist with a Bell-type experiment?


If I start over:
We have two photons, perhaps entangled, and one is received as event X, the other as event Y.
I will reserve "event Z" as the emission event of those photons.

So, in this set-up, when we talk about "commuting beyond the light cone", would this be the light cone for event X, Y, Z, or something else?

 

[PeterDonis]

@PeterDonis :
I was only going to transmit a photon at event A - not try to detect it. So, I know when event "A" happens because I triggered it.

That's fine, for "detection" read "triggering of transmission" at event A.

There seems to be a persistent drift in this thread using two detectors.
Are we talking about an effect can only exist with a Bell-type experiment?

I don't know. You talked about a scenario with just one photon. Now you're talking about one with two (see below). Those are different scenarios, and you can't just carry over answers from one to the other.

We have two photons, perhaps entangled, and one is received as event X, the other as event Y.
I will reserve "event Z" as the emission event of those photons.

Ok.

So, in this set-up, when we talk about "commuting beyond the light cone", would this be the light cone for event X, Y, Z, or something else?

If events X and Y are spacelike separated, the measurements at those two events must commute (i.e., their results must be independent of the order in which they happen).

Both events X and Y will be on the future light cone of event Z. So neither will be spacelike separated from event Z; they will both be lightlike (null) separated from event Z.