A Why is Quantum Field Theory Local?

[Tendex]

QFT is not deterministic, because the state provides probabilities for the outcome of measurements not determined values of all observables of the quantum system

You mean any theory that uses probabilities for causal predictions can't be deterministic? This is not mathematically true. I think you are confusing the "quantum indeterminacy" of quantum theories(their use of probabilities for precise predictions) with a mathematical theory not being able to be logically deterministic, which is the determinism relevant for a mathematical theory.
In any case quantum relativistic causality is logically deterministic as shown by the time-reversing properties of its operators.

 

[vanhees71]

I use the word "deterministic" in the usual sense of physics: It means that any observable takes a well-defined value at any time. That's not the case in quantum theory. I don't know, what mathematics has to do with determinism or indeterminism.

 

[Tendex]

I use the word "deterministic" in the usual sense of physics: It means that any observable takes a well-defined value at any time. That's not the case in quantum theory. I don't know, what mathematics has to do with determinism or indeterminism.

I see, the thing is that Bell's theorem is supposed to be constructed mathematically and thus its premises, more specifically the concepts of deterministic(hidden variables) or local(as discussed in the previous posts) theory must have some content having to do with mathematics.
For instance "that any observable takes a well-defined value at any time" is compatible with the probabilities given in scattering matrix predictions of QFT depending on what one means by well defined.

 

[Nugatory]

its premises, more specifically the concepts of deterministic(hidden variables)

Hidden variables are not necessarily deterministic and deterministic theories are not necessarily hidden variable theories. Bell's proof works with probability distributions and neither assumes nor requires that the mechanism that leads to these distributions is deterministic; it precludes local non-deterministic hidden-variables theories as well local visible-variable theories (which are already excluded because if there were a viable visible-variable theory we'd see it) as well as the local hidden-variables that everyone is talking about.

 

[vanhees71]

I see, the thing is that Bell's theorem is supposed to be constructed mathematically and thus its premises, more specifically the concepts of deterministic(hidden variables) or local(as discussed in the previous posts) theory must have some content having to do with mathematics.
For instance "that any observable takes a well-defined value at any time" is compatible with the probabilities given in scattering matrix predictions of QFT depending on what one means by well defined.

The point is that the probabilities predicted by QFT (or any other type of QT) have properties different from local deterministic HV theories. To figure this out was the great achievement by Bell. It made the question, whether QT is compatible with the assumption that all observables of a system always have determined values as some local HV theory. Bell found out that while the local HV theories necessarily fulfill Bell's inequality that's not the case for QT. Particularly maximally entangled states show correlations that violate Bell's inequality. This made the question whether the predictions of any local HV theory or QT deliver the correct predictions of probabilities are correct, decidable by experiment. As is well known today, all such "Bell tests" falsify the predictions of the HV theories and confirm those of QT (including local relativsitic QFTs). So at least local HV theories are ruled out.

For non-relativistic QM Bohmian mechanics is an example for a non-local HV theory that delivers the same predictions as standard QT.

 

[vanhees71]

Hidden variables are not necessarily deterministic and deterministic theories are not necessarily hidden variable theories. Bell's proof works with probability distributions and neither assumes nor requires that the mechanism that leads to these distributions is deterministic; it precludes local non-deterministic hidden-variables theories and well as local visible-variable theories (which are already excluded because if there were a viable visible-variable theory we'd see it) as well as the local hidden-variables that everyone is talking about.

As I understand it "HV theory" stands for Einstein's idea that the probabilities of QT are of the same nature as the probabilities in classical statistical physics, i.e., there are some observables not taken into account yet by QT (the thus "hidden variables" (HV)) and are thus "ignored" and treated statistically. That's analogoes to, e.g., classical statistical mechanics: In classical statistical physics for a gas in stead of describing the complete deterministic system, i.e., the motion of the point in $\sim 10^{24}$-dimensional phase space (which is of course impossible in practice) and the corresponding full phase-space distribution function one considers only very "coarse-grained" observables like a one-particle phase-space distribution function and in the dynamics, derived from the full Liouville equation, truncates the corresponding BBGKY hierarchy at the one-particle level by the "molecular-chaos assumption". The corresponding probabilities are just due to our inability to fully resolve all the "microscopic" details but "in reality" the observables of the gas in the full picture always take determined values (determinism) and knowing their initial values at one point in time given the Hamiltonian of the system you precisely know them at any later time.

What Bell has shown is that no local deterministic HV theory can lead to all statistical properties predicted by QT, i.e., QT violates his famous inequalities and thus you can experimentally decide whether Nature behaves as described by such a local deterministic HV theory or according to QT. Of course we know today that all "Bell tests" confirm very precisely the predictions of QT.

 

[bhobba]

What Bell has shown is that no local deterministic HV theory can lead to all statistical properties predicted by QT, i.e., QT violates his famous inequalities and thus you can experimentally decide whether Nature behaves as described by such a local deterministic HV theory or according to QT. Of course we know today that all "Bell tests" confirm very precisely the predictions of QT.

See:
https://cds.cern.ch/record/372369/files/9811072.pdf

Strictly speaking, what Bell showed was assuming the Kolmogorov axioms of probability and locality it is incompatible with counterfactual definiteness. If we relax the Kolmogorov axioms requirement, i.e. assume from the start QM is a Generalised Probability Theory, then the whole 'issue' is bypassed. The generalised probability view of QM is fascinating in its own right:
https://en.wikipedia.org/wiki/Generalized_probabilistic_theory

It shows such theories, as a class, allow for many features of QM, with QM perhaps the simplest. This has been my view for a long time. We also have discussed many times on this forum its compatibility with the cluster decomposition property as expressed by Wienberg. I these days side with Peter Donis on that; it is a semantic thing.

Thanks
Bill

 

[A. Neumaier]

QFT includes entanglement, since it includes non-relativistic QM as a special case and makes all of the same predictions for that case.

This is not correct. Local QFT is by definition relativistic QFT and does not include non-relativistic QM as a special case. Indeed, nonrelativistic quantum fields are not local in the sense of local QFT.

Non-relativistic QM is only an approximation of local QFT. It is obtained by forcing time to be instantaneous in the observer frame. In interacting local QFT, time must be smeared to produce valid operators rather than operator distributions; thus instantaneous operators are necessarily approximate. Without instantaneous operators there is also no interacting particle picture; particles make sense only approximately - namely asymptotically at microscopically long times before or after collisions.

 

[PeterDonis]

Local QFT is by definition relativistic QFT and does not include non-relativistic QM as a special case.

By "special case" I meant "approximation":

Non-relativistic QM is only an approximation of local QFT.

 

[A. Neumaier]

Not in the sense you are using the term "local", since you are saying that entanglement means "nonlocal", and QFT includes entanglement, since it includes non-relativistic QM as a special case and makes all of the same predictions for that case.

By "special case" I meant "approximation":

So local QFT makes only approximately the same predictions. The quality of the approximations in case of long-distance Bell experiments is difficult to assess and has never been discussed. This makes your claim invalid, even with your new (nonstandard) semantics.

QFT is "local" in the sense that spacelike separated measurements, including those on entangled particles, must commute--their results must not depend on the order in which they are made (since the ordering of spacelike separated measurements is not invariant).

I recommend reading the book 'Local Quantum Physics' by Rudolf Haag, the originator of Haag's theorem on the lack of an interaction picture in relativistic QFT. This book gives precise definitions of causal locality in quantum physics, in particular quantum field theory.

Local interacting QFT in Minkowski space means that every open and bounded region $O$ (the region accessible to an observer with a finite lifetime) has its associated algebra $A(O)$ of observables local to $O$ (i.e., vanishing on states with zero support on $O$), the local observables of spacelike separated regions commute. The dynamics is given by the time shift of $O$ in Minkowski space. Thus in the Heisenberg picture, the dynamics inside two causally separated regions is completely independent - independent of any measurement issues and of interaction specifics. (Causal locality issues become very obscured in the Schrödinger picture since the latter is noncovariant as it singles out a particular observer frame.)

Coincidence counting experiments (related to Bell nonlocality) consider instead what happens when two causally separated regions merge. Causal locality (i.e., locality in the QFT sense) says nothing at all about this situation - here everything depends on the details of the interactions. To my knowledge there has been no analysis of Bell nonlocality in terms of local QFT.

 

[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.

 

[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.)