A Why is Quantum Field Theory Local?

[vanhees71]

But all Bell experiments are compatible with local relativistic QFT. From your explanation it's very clear that as long as the outcome of the Bell experiments can be explained within local relativistic QFT you must conclude that there are no causal influences betwween the corresponding space-like separated "measurement events" (e.g., the clicks of two far separated photon detectors when you do polarization measurements on entangled two-photon states). Also this is most explicitly seen in the Heisenberg picture, where the states are represented by the time-independent statistical operator, defined by the initial conditions, while what you measure are local observables, i.e., the probabilities for detector clicks at spatially separated detector positions, i.e., precisely what you describe within the formalism above.

So what you prove with the Bell experiments is not "non-locality" but "inseparability", i.e., the correlations due to the preparation in the entangled state and not due to superluminal interactions due to the measurements, i.e., local relativistic QFT is compatible with both "no spooky interactions" (i.e., no violation of Einstein causality) and the correlations described by entanglement which are "stronger" than within any local deterministic HV theory indicated by the violation of Bell's inequality.

 

[A. Neumaier]

all Bell experiments are compatible with local relativistic QFT.

I think this has not been demonstrated anywhere in the literature.

What my arguments show is only that the apparent conflict is due to mixing two different notions of locality - Bell locality (a purely classical concept defined in terms of hidden variables) and causal locality (a quantum concept relevant for QFT).

 

[vanhees71]

That's right. In my opinion one should not call "Bell locality" "locality" but "inseparability". Einstein was much more aware of these subtlties than usually is attributed to him!

I don't understand what you mean by saying that the compatibility of Bell experiments with local relativistic QFt hasn't been demonstrated. As far as I know all the Bell experiments, particularly those with photons, are described by standard QFT (aka the Standard Model). There's no hint that the local photon detections in the lab in any way contradict QED. After all it's based on some photoeffect in the detector material and the standard theoretical treatment using 1st-order perturbation theory in the dipole approximation shows that the detection probability is proportional to the energy density of the em. field, which is a local observable.

 

[A. Neumaier]

I don't understand what you mean by saying that the compatibility of Bell experiments with local relativistic QFt hasn't been demonstrated. As far as I know all the Bell experiments, particularly those with photons, are described by standard QFT (aka the Standard Model). There's no hint that the local photon detections in the lab in any way contradict QED. After all it's based on some photoeffect in the detector material and the standard theoretical treatment using 1st-order perturbation theory in the dipole approximation shows that the detection probability is proportional to the energy density of the em. field, which is a local observable.

All you say involves the individual analysis of the photodetectors, not an analysis of their joint statistics. There would not be a sustained tension in the interpretation of the results if this were settled without doubt. I don't think one will find a discrepancy; I just point out that there is no theoretical analysis of this in terms of QED.

the detection probability is proportional to the energy density of the em. field, which is a local observable.

But the joint detection probability of a common prepared source by two far away detectors is governed by noncommuting observables, and this needs further analysis.

Thus while I believe that Bell nonlocality and causal locality are fully compatible, I haven't seen yet a convincing proof of it. Can you point to one?

 

[Demystifier]

To my knowledge there has been no analysis of Bell nonlocality in terms of local QFT.

I'm not sure what do you mean by "Bell nonlocality in terms of local QFT". There certainly has been analysis of Bell nonlocality in terms of quantum optics. Quantum optics is a branch of QED, which, in turn, is an example of local QFT.

 

[vanhees71]

I still don't see what is necessary to be proven. Of course to get the joint probability of the two detectors you need to compare the local measurement protocols and this you can only do "later" via a classical channel exchanging the information on the two measurement protocols. The measurements themselves to get these protocols are due to local interactions between photons (the em. field) with the detector (atoms/molecules making up the detector material).

 

[Demystifier]

Thus while I believe that Bell nonlocality and causal locality are fully compatible, I haven't seen yet a convincing proof of it.

But Bell nonlocality is derived from quantum theory (e.g. quantum optics). What exactly is not convincing?

 

[Demystifier]

What my arguments show is only that the apparent conflict is due to mixing two different notions of locality - Bell locality (a purely classical concept defined in terms of hidden variables) and causal locality (a quantum concept relevant for QFT).

I think we all agree on that, but what is not clear is why do you still have some reservations on that.

 

[A. Neumaier]

I still don't see what is necessary to be proven. Of course to get the joint probability of the two detectors you need to compare the local measurement protocols and this you can only do "later" via a classical channel exchanging the information on the two measurement protocols. The measurements themselves to get these protocols are due to local interactions between photons (the em. field) with the detector (atoms/molecules making up the detector material).

I don't understand the origin of the nonlocal correlations in certain experiments where choices are made after the signal was sent but before any measurement was made.

But Bell nonlocality is derived from quantum theory (e.g. quantum optics). What exactly is not convincing?

Bell nonlocality is derived solely by proving that Schrödinger picture quantum mechanics in a finite-dimensional Hilbert space predicts violations of Bell inequalities. No quantum optics or quantum field theory is involved at all, not even relativity. Interacting relativistic QFT has not even a consistent particle picture at finite times. Hence there is a large gap between QFT and Bell nonlocality.

 

[A. Neumaier]

I still don't see what is necessary to be proven. Of course to get the joint probability of the two detectors you need to compare the local measurement protocols and this you can only do "later" via a classical channel exchanging the information on the two measurement protocols. The measurements themselves to get these protocols are due to local interactions between photons (the em. field) with the detector (atoms/molecules making up the detector material).

The problem is not in the comparison of the protocols. In the Heisenberg picture, one has a joint measurement of two noncommuting observables, since these were created by a common past preparation. Measurement theory for this is not governed by Born's rule since the latter assumes commuting variables.

 

[vanhees71]

I don't understand the origin of the nonlocal correlations in certain experiments where choices are made after the signal was sent but before any measurement was made.

Can you give a concrete example, where you don't understand this? I don't see any problems with that when intepreting the state within the ensemble interpretation. Then all these "nonlocal correlations" are just due to the preparation in the entangled state (or by (post)-selection of partial ensembles as in the case of the quantum-erasure experiment or entanglement swapping).

Bell nonlocality is derived solely by proving that Schrödinger picture quantum mechanics in a finite-dimensional Hilbert space predicts violations of Bell inequalities. No quantum optics or quantum field theory is involved at all, not even relativity. Interacting relativistic QFT has not even a consistent particle picture at finite times. Hence there is a large gap between QFT and Bell nonlocality.

But the violation of Bell's inequality holds in any QT not only in non-relativistic QM. You cannot describe photons with non-relativistic QM but must Bell tests are made with photons.

Further, observable prediction of any QT also can depend on the choice of the picture of time evolution since by construction observable predictions like the probability for the outcome of measurements are independent of that choice.

 

[A. Neumaier]

In my opinion one should not call "Bell locality" "locality" but "inseparability".

It is impossible to change thoroughly entrenched terminology. Thus one must clarify instead the usage of the terms.

Can you give a concrete example, where you don't understand this?

I don't want to go again into the lengthy discussions we had on this some years ago. Concrete examples do not matter for the present discussion.

What matters is that in relativistic QFT, coincidence measurements are joint measurements of noncommuting observables. This is outside the scope of traditional QFT, which discusses measurement only via Born's rule for asymptotic particle states. But Born's rule assumes in its very formulation (e.g., on p.20 of your lecture notes, version of July 22, 2019) observables with a joint spectrum, hence does not apply to coincidence measurements.

You cannot describe photons with non-relativistic QM but must Bell tests are made with photons.

For the purposes of Bell tests, entangled photons are just tensor products of nonrelativistic 2-state systems, since the motion is always treated classically. The real problems are swept under the carpet by this approximation.

 

[PeterDonis]

the joint detection probability of a common prepared source by two far away detectors is governed by noncommuting observables

Which noncommuting observables? If the two detection events are spacelike separated, their observables commute.

 

[A. Neumaier]

Which noncommuting observables? If the two detection events are spacelike separated, their observables commute.

This is an illusion caused by the traditional simplified discussions, which treat the dynamics classically and analyze each detector separately.

The joint observation of commuting observables leads to classical statistics satisfying the Bell inequalities, since there is a basis in which both observables are diagonal, hence can be classically interpreted by hidden variables. The very fact that the Bell inequalities are violated in experiments thus disproves your statement.

 

[PeterDonis]

This is an illusion caused by the traditional simplified discussions, which treat the dynamics classically and analyze each detector separately.

I don't understand. You yourself said that, even in Haag's algebraic approach to QFT, observables in spacelike separated regions commute. So I'm still confused about which non-commuting observables you are talking about.

 

[A. Neumaier]

I don't understand. You yourself said that, even in Haag's algebraic approach to QFT, observables in spacelike separated regions commute.

Only local observables in spacelike separated regions commute. Note that in QFT we work in the Heisenberg picture, where the state is fixed and the preparation is in the operators, not in the state. Observables prepared at the same location in the past are guaranteed to be local only in the future cone of the preparation, not in smaller, spacelike separated regions.

 

[vanhees71]

Again, the choice of the picture of time evolution is irrelevant for any discussion about physics, because any physics is independent of the choice of the picture.

The observable a photodetector measures is the energy density of the electromagnetic field, which is a local operator (i.e., fulfilling the microcauslity condition). The coincidence measurement of two photon detectors is described by a corresponding two-point autocorrelation function of this energy density. Space-like separated detection events thus cannot be causally connected within local relativistic QFT but of course there can be correlations due to entanglement, e.g., when you have an entangled two-photon pair from a parametric-downconversion process (the usual way nowadays to "prepare" such two-photon states).

 

[A. Neumaier]

The observable a photodetector measures is the energy density of the electromagnetic field, which is a local operator (i.e., fulfilling the microcauslity condition). The coincidence measurement of two photon detectors is described by a corresponding two-point autocorrelation function of this energy density.

In the Heisenberg picture, this two-point autocorrelation function is described by a bilocal operator, responsible for the nonlocal effects of local quantum field theory. I'd like to see a discussion of Bell inequality violations in terms of the covariant two-point autocorrelation function. It would be illuminating as it would show the frame dependence of entanglement effects in a covariant way.

Again, the choice of the picture of time evolution is irrelevant for any discussion about physics, because any physics is independent of the choice of the picture.

You could as well say that the choice of coordinates is irrelevant for any discussion about physics, because any physics is independent of the choice of coordinates.

However, good choices make things easy to understand, and are therefore very relevant for the understanding of physics. Discussions are to serve the understanding, hence need good choices of whatever can be freely chosen.

In particular, the quantum mechanical picture is relevant because locality issues are clearly visible only in the Heisenberg picture, whereas in the Schrödinger picture they are very obscure. In the Schrödinger picture, the dynamic two-point autocorrelation function is an exceedingly ugly and unintelligible expression never used, neither in theory nor in practice.

 

[vanhees71]

Ok, so you look for a formal description using a correlation function like $\langle T^{\mu \nu}(x) T^{\rho \sigma}(y) \rangle$. This I haven't seen yet indeed. It's an interesting question.

I also agree that the most "natural" description of quantum theory is the Heisenberg picture, but it doesn't change anything when calculating something in another picture, and indeed it's as with the independence of the physics on the choice of coordinates.

I still don't know how a autocorrelation function can be more ugly in the Schrödinger than in the Heisenberg picture. Both calculations must give the same autocorrelation function. I only think the Schrödinger picture is much more cumbersome to perform the calculation.

 

[A. Neumaier]

I still don't know how a autocorrelation function can be more ugly in the Schrödinger than in the Heisenberg picture. Both calculations must give the same autocorrelation function. I only think the Schrödinger picture is much more cumbersome to perform the calculation.

much more cumbersome = more ugly

In the Schrödinger picture one can easily get equal-time correlation functions, which is done in solid state physics. This suffices for coincidence measurements in a fixed frame. However, to see the frame dependence one needs the spacetime dependence. Already writing down the operator defining this dynamical 2-point correlations in the Heisenberg picture is much more cumbersome.

 

[vanhees71]

I think it's more cumbersome to formulate and evaluate in the Schrödinger picture. Maybe we don't talk about the same quantity?

 

[A. Neumaier]

I think it's more cumbersome to formulate and evaluate in the Schrödinger picture. Maybe we don't talk about the same quantity?

We talk about the same but evaluate it differently.

More cumbersome = more ugly. Understanding comes from beauty.

 

[Demystifier]

Bell nonlocality is derived solely by proving that Schrödinger picture quantum mechanics in a finite-dimensional Hilbert space predicts violations of Bell inequalities. No quantum optics or quantum field theory is involved at all, not even relativity. Interacting relativistic QFT has not even a consistent particle picture at finite times. Hence there is a large gap between QFT and Bell nonlocality.

Let me look at this argument from another angle. Do you actually argue that there is a large gap between QFT and QM?

 

[A. Neumaier]

Do you actually argue that there is a large gap between QFT and QM?

Quantum mechanics is an approximation of quantum field theory in which the field concept at arbitrary spacetime points is replaced by the concept of localizable particles at arbitrary times. In interacting QFT, the latter is only asymptotically realized, not at finite times.

Thus there is a significant gap, and for foundational aspects it must be considered to be quite large.

 

[gentzen]

Let me look at this argument from another angle. Do you actually argue that there is a large gap between QFT and QM?

At least for Bohmian mechanics, there is a large gap between QFT and QM. And for the thermal interpretation? I guess one reason why A. Neumaier restarted this thread was my question and comment about measurability of timelike quantum correlations:

For timelike correlations, there is a preferred order, and the order is important, but for spacelike correlations, there is no preferred order, and the order is irrelevant.
... Therefore it is unclear whether it is even possible in principle to measure timelike quantum correlations in a similar way as it is possible to measure spacelike quantum correlations.

That comment was a bit naive, in that often even for timelike correlations the order will be irrelevant, because often they simply cannot interact with each other (during measurement) for a given preparation and measurement setup.
And there was also the unspoken "non-question" that there can be correlations between macroscopic observations at different times (even if there is interaction during measurement between the different timelike separated parts). That unspoken part might have been the thing that A. Neumaier was unsure and unhappy about, when he wrote: "Measurement theory for this is not governed by Born's rule since the latter assumes commuting variables."

 

[Demystifier]

At least for Bohmian mechanics, there is a large gap between QFT and QM.

I have elaborated my opinion on that in the paper linked in my signature below.

 

[A. Neumaier]

I have elaborated my opinion on that in the paper linked in my signature below.

Not everyone sees your signature...

 

[Demystifier]

Not everyone sees your signature...

I think seeing signature is default and I believe that not many people change it. In any case, those who do not see it can always tell me so in which case I will give them the link by other means.

 

[gentzen]

I have elaborated my opinion on that in the paper linked in my signature below.

I have browsed that paper before, and I can see your signature. However, the mirror de.arxiv.org doesn't seem to work anymore since quite some time.

With respect to the argument itself, ... maybe I should open a new thread if I wanted to discuss it.

 

[Demystifier]

However, the mirror de.arxiv.org doesn't seem to work anymore since quite some time.

Thanks for pointing this out! Now I have changed the link accordingly.