Why is Quantum Field Theory Local?

[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.

 

[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.

Here is a yet another point of view. Bell inequality is derived neither from QM nor from QFT. Bell inequality is derived from some general principles of scientific reasoning (macroscopic realism, statistical independence of the choice of parameters, Reichenbach principle, Kolmogorov probability axioms, no causation backwards in time, ...) and from the
assumption of (Bell) locality. Experiments with photons show violation of Bell inequality. Hence, if we take those general principles of scientific reasoning for granted, then we can conclude that photons violate (Bell) locality. In this argument it doesn't matter whether photons are described by QM, continuum QFT, lattice truncated QFT, string theory or unicorn theory.

What does it tell us about QFT? If QFT can explain the experiments, then either QFT violates (Bell) locality or QFT violates some of those general scientific principles.

 

[vanhees71]

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.

Isn't the physics the other way around? Within non-relativistic QT there's a priori no fundamental limit to localize particles. Here the HUR just says $\Delta x \Delta p \geq \hbar/2$ and you can localize particles as pecise as you like (at the expense of accuracy of there momenta).

This is not true in relativistic QT. If you try to localize a particle the uncertainty relation together with the maximum relative speed of $c$ leads to the conclusion that the accuracy of particle localization is maximally of the order of the Compton wave length of the particle $\Delta q \geq \hbar/(m c)$. Of you try to squeeze the particle in even smaller volumes you rather create particle-antiparticle pairs than really localizing the particles better. That's why the naive particle picture and the naive first-quantization approach to relativistic QT fails. Historically that came clear when Dirac was forced to invent his hole theory to reinterpret his first-quantization formulation of the Dirac equation after all as a many-body description, making the theory pretty hard to comprehend since on the one hand you argue with single-particle concepts from non-relativistic QM but then reinterpret them in terms of a many-body theory with a Dirac sea that is just unobservable by declaration (where is the infinite amount of negative charge being present according to the hole theory to occupy the "negative-energy states"?).

At the end the conclusion is that one better starts from a many-body approach from the very beginning and that leads to the use of quantum field theory. One must not forget that also in classical relativistic theory the "point particle is a stranger" as Sommerfeld said concerning the trouble with the point-like electron in Lorentz's electron theory. Even in the classical theory continuum-mechanical descriptions make much less trouble. So in this sense the field concept is rather more consistent with relativity already in the classical realm, and indeed, as you say, a particle interpretation (or should one rather say a "particle metaphor"?) only works out in the sense of "asymptotic free states"...

 

[A. Neumaier]

photons violate (Bell) locality. In this argument it doesn't matter whether photons are described by QM, continuum QFT, lattice truncated QFT, string theory or unicorn theory.

It matters. Bell inequalities also need the concept of photons as particles moving along trajectories. The photon concept of QFT is quite different, hence Bell's reasoning does not apply. The violation of Bell's inequality just emphasize this fact.

 

[Demystifier]

It matters. Bell inequalities also need the concept of photons as particles moving along trajectories. The photon concept of QFT is quite different, hence Bell's reasoning does not apply. The violation of Bell's inequality just emphasize this fact.

I disagree. Bell inequalities can be rephrased just in terms of macroscopic measurement outcomes, without referring to photons.

 

[A. Neumaier]

Isn't the physics the other way around? Within non-relativistic QT there's a priori no fundamental limit to localize particles. Here the HUR just says $\Delta x \Delta p \geq \hbar/2$ and you can localize particles as precise as you like (at the expense of accuracy of their momenta).

Only the history of physics is the other way around. But clearly, field theory is more fundamental than particle theory (which arises in the approximation of geometric optics). Thus QFT is more fundamental than QM.

the field concept is rather more consistent with relativity already in the classical realm, and indeed, as you say, a particle interpretation (or should one rather say a "particle metaphor"?) only works out in the sense of "asymptotic free states"...

There is not even a relativistic classical theory of multiple point particles - one can even prove a corresponding no-go theorem!

 

[A. Neumaier]

I disagree. Bell inequalities can be rephrased just in terms of macroscopic measurement outcomes, without referring to photons.

The inequalities only refer to mathematical symbols. But their interpretation in terms of nonlocality requires particles moving along paths! Without paths, the inequalities cannot be applied to argue anything about nonlocality!

 

[Demystifier]

The inequalities only refer to mathematical symbols. But their interpretation in terms of nonlocality requires particles moving along paths! Without paths, the inequalities cannot be applied to argue anything about nonlocality!

Are you saying that QFT offers a local interpretation of Bell inequality violation in a way that cannot be achieved just by QM?

 

[A. Neumaier]

Are you saying that QFT offers a local interpretation of Bell inequality violation in a way that cannot be achieved just by QM?

Yes. For sufficiently nonclassical states, a nonclassical concindence count statistics is predicted from 2-point correlations of causally local, locally propagating field operators, in agreement with experiment. See, e.g. Section 14.6 of the quantum optics book by Mandel and Wolf 1995, and Section 12.14 for a discussion of Bell inequalities. Thus instead of problematic Bell nonlocality (resting on essentially classical particle and hidden variable assumptions) one just has unproblematic nonclassical quantum states of fields.

Note also that Bell's argument interpreting the mathematics of the Bell inequalities do not apply if the hidden variables are fields.

 

[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. 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).

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.

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.

Actually, this is more or less done in the book by Mandel and Wolf cited in post #154.

 

[Demystifier]

Yes. For sufficiently nonclassical states, a nonclassical concindence count statistics is predicted from 2-point correlations of causally local, locally propagating field operators, in agreement with experiment. See, e.g. Section 14.6 of the quantum optics book by Mandel and Wolf 1995, and Section 12.14 for a discussion of Bell inequalities. Thus instead of problematic Bell nonlocality (resting on essentially classical particle and hidden variable assumptions) one just has unproblematic nonclassical quantum states of fields.

From Sec. 12.14.5 it is evident that they avoid the reasoning resulting in the Bell inequality by allowing "not true probability density" which is not necessarily positive. First, it is not an exclusive property of QFT because Wigner distributions (and coherent states) appear in QM as well. Second, the GHZ proof of nonlocality does not depend on probabilistic reasoning at all, so their argument is not really a strong argument for locality. Presumably, at the time of writing the book they were not aware of the GHZ (1993) proof.

Note also that Bell's argument interpreting the mathematics of the Bell inequalities do not apply if the hidden variables are fields.

I disagree. The Bell's argument is applicable to any local beables, namely variables defined on spacetime positions. This includes both pointlike particles and fields. (But it excludes multi-local beables that appear in your thermal interpretation.)

 

[stevendaryl]

Note also that Bell's argument interpreting the mathematics of the Bell inequalities do not apply if the hidden variables are fields.

Bell explicitly said that the hidden variable can be absolutely anything. I'm pretty sure that he meant it to include fields. He said that the hidden variable could be some nonlocal information.

 

[A. Neumaier]

Bell explicitly said that the hidden variable can be absolutely anything. I'm pretty sure that he meant it to include fields. He said that the hidden variable could be some nonlocal information.

I'd be interested in a specialization of Bell's (or similar) arguments to the case where the hidden variables are local fields. That it cannot work in general can be seen from a paper that I wrote a long time ago,

A. Neumaier, A simple hidden variable experiment, 2007. arXiv:0706.0155

Abstract: An experiment is described which proves, using single photons only, that the standard hidden variables assumptions (commonly used to derive Bell inequalities) are inconsistent with quantum mechanics. The analysis is very simple and transparent. In particular, it demonstrates that a classical wave model for quantum mechanics is not ruled out by experiments demonstrating the violation of the traditional hidden variable assumptions.

 

[stevendaryl]

I don't agree that Bell's proof has anything specifically to do with particles.

Let's assume in an EPR-type experiment that Alice conducts her measurement in some small region of spacetime $A$, and that Bob conducts his measurement in a spacelike separated region of spacetime $B$. Let $\bar{A}$ be the backwards lightcone of spacetime points in $A$, and let $\bar{B}$ be the backwards lightcone of spacetime points in $B$.

Define a few more regions: Let $C$ be those points in $\bar{A}$ that are not in $\bar{B}$, and let $D$ be those points in $\bar{B}$ that are not in $\bar{A}$, and let $E$ be the intersection of $\bar{A}$ and $\bar{B}$. In EPR, a twin pair of particles is produced in region $E$, and then one particle travels to region $A$ while the other travels to region $B$.

Bell is assuming that:

1. Alice's results in region $A$ depend only on facts about regions $C$ and $E$.
2. Bob's results in region $B$ depend only on facts about regions $D$ and $E$.

He is also assuming that Alice's and Bob's settings (their choice of which orientation to measure spins relative to, for example) are NOT determined by the common backwards lightcone $E$. For example, Alice might make her choice based on information about region $C$ and Bob might make his choice based on information about region $C$. Bell is basically assuming that there are facts about those two regions that are not deducible from facts about region $E$.

So in this setup, the "hidden variables" are just facts about region $E$ that causally affect regions $A$ and $B$. Any facts about region $E$ are fair game. Maybe it's the values of fields in region $E$, or maybe it's facts about the particles. Bell's theorem doesn't depend on the nature of those facts, only what region of spacetime they are about.

 

[A. Neumaier]

I don't agree that Bell's proof has anything specifically to do with particles.

Let's assume in an EPR-type experiment that Alice conducts her measurement in some small region of spacetime $A$, and that Bob conducts his measurement in a spacelike separated region of spacetime $B$. Let $\bar{A}$ be the backwards lightcone of spacetime points in $A$, and let $\bar{B}$ be the backwards lightcone of spacetime points in $B$.

Define a few more regions: Let $C$ be those points in $\bar{A}$ that are not in $\bar{B}$, and let $D$ be those points in $\bar{B}$ that are not in $\bar{A}$, and let $E$ be the intersection of $\bar{A}$ and $\bar{B}$. In EPR, a twin pair of particles is produced in region $E$, and then one particle travels to region $A$ while the other travels to region $B$.

Note that you still have particles traveling, not fields!

Bell is assuming that:

1. Alice's results in region $A$ depend only on facts about regions $C$ and $E$.
2. Bob's results in region $B$ depend only on facts about regions $D$ and $E$.

He is also assuming that Alice's and Bob's settings (their choice of which orientation to measure spins relative to, for example) are NOT determined by the common backwards lightcone $E$. For example, Alice might make her choice based on information about region $C$ and Bob might make his choice based on information about region $C$. Bell is basically assuming that there are facts about those two regions that are not deducible from facts about region $E$.

So in this setup, the "hidden variables" are just facts about region $E$ that causally affect regions $A$ and $B$. Any facts about region $E$ are fair game. Maybe it's the values of fields in region $E$, or maybe it's facts about the particles. Bell's theorem doesn't depend on the nature of those facts, only what region of spacetime they are about.

Then why does my single photon experiment demonstrate apparent Bell nonlocality though it is explained by Maxwell's classical local field equations?

 

[stevendaryl]

Note that you still have particles traveling, not fields!

Then why does my single photon experiment demonstrate apparent Bell nonlocality though it is explained by Maxwell's classical local field equations?

For many claims, there are proofs of both the claim and the negation. You have to take such things with a grain of salt. You're claiming to have done something that others have proved can't be done. Obviously, either someone has made a mistake, or there are subtle differences in the interpretations of key concepts.