A Why is Quantum Field Theory Local?

[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.

 

[A. Neumaier]

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.)

My experiment linked to in post #102 predicts violation of Bell inequalities using only classical electrodynamics in vacuum, which is a local and causal relativistic theory.

 

[A. Neumaier]

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.

Bell didn't claim to have done anything not involving particles - you did!

The question is, which grain of salt is needed in each case, and which one is convincing! Of course there are subtle differences in the interpretations of key concepts, but these differences are the key point.

 

[stevendaryl]

Note that you still have particles traveling, not fields!

No, that's not true. The setup doesn't say anything about whether the experiments involve particles or whatever. I was using EPR as an example. The only thing that is relevant is that Alice is performing an experiment in a localized region of spacetime $A$, Bob is performing an experiment in a spacelike separated region of spacetime $B$.

 

[Demystifier]

My experiment linked to in post #148 predicts violation of Bell inequalities

No it doesn't, your paper does not deal with Bell inequalities at all.

 

[stevendaryl]

Bell didn't claim to have done anything not involving particles - you did!

He was using particles to illustrate the concept, which doesn't have anything specifically to do with particles.

 

[A. Neumaier]

No it doesn't, your paper does not deal with Bell inequalities at all.

It is not the traditional Bell inequality, but it is of exactly the same kind. namely a constraint of the values of some linear combination of probabilities, in my case the expression (1), where local hidden variable theories predict (with Bell's arguments) even a constant value rather than an inequality only.

All of Bell's arguments also apply to my classical hidden variable reasoning!

 

[Demystifier]

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

Are you talking about your arXiv:0706.0155? This paper only shows that a classical hidden variable theory that does not involve interference is not compatible with QM. Quite trivial and uninteresting result in my view. And since it is about single photon, it has absolutely nothing to do with Bell nonlocality.

 

[stevendaryl]

Bell's inequality is not about particles.

We have a probability of the form $P(R_A = a \wedge R_B = b | O_A = \alpha \wedge O_B = \beta)$

where $R_A$ is the result of Alice's measurement, $R_B$ is the result of Bob's measurement, $O_A$ is Alice's choice of detector setting, $O_B$ is Bob's choice of detector setting.

Bell assumed that such a probability "factors" once you know the common causal influences of Alice's result and Bob's result. In terms of the spacetime regions I mentioned above, $E$ is the common backwards lightcone of Alice's and Bob's measurements. Bell assumed that, under the assumption that there is no causal influence of Alice's measurement on Bob, nor vice-versa, then there is some fact about region $E$, call it $F(E)$ such that knowing that fact would allow us to factor the probabilities:

$P(R_A = a \wedge R_B = b | O_A = \alpha \wedge O_B = \beta)$
$= \sum_\lambda P_E(F(E) = \lambda) P_A(R_A = a | O_A = \alpha \wedge F(E) = \lambda) P_B(R_B = b | O_B = \beta \wedge F(E) = \lambda)$

where $P_E$ gives the probability of region $E$ having property $\lambda$, $P_A$ is the probability of Alice's results given her setting and the hidden variable $\lambda$, and $P_B$ is the probability of Bob's results given his setting and the hidden variable.

There is nothing about particles in the mathematical derivation.

 

[A. Neumaier]

He was using particles to illustrate the concept, which doesn't have anything specifically to do with particles.

Bell's inequality is not about particles.

We have a probability of the form $P(R_A = a \wedge R_B = b | O_A = \alpha \wedge O_B = \beta)$

The mathematics is independent of any nonlocality issues. Nonlocality only enters through its interpretation, which involves particles - something that moves from one place to another.

 

[A. Neumaier]

Are you talking about your arXiv:0706.0155? This paper only shows that a classical hidden variable theory that does not involve interference is not compatible with QM. Quite trivial and uninteresting result in my view. And since it is about single photon, it has absolutely nothing to do with Bell nonlocality.

Measured are correlations that are exactly as nonlocal (or as little nonlocal) as those in the interpretation of Bell inequalities, though the classical field propagates locally.

 

[Demystifier]

Measured are correlations that are exactly as nonlocal as those in the interpretation of Bell inequalities

In your paper I don't see any nonlocal correlations resembling those of Bell.

 

[stevendaryl]

The mathematics is independent of any nonlocality issues. Nonlocality only enters through its interpretation, which involves particles - something that moves from one place to another.

You're wrong about that. But I know from past discussions in PF to just drop it at this point.

 

[A. Neumaier]

In your paper I don't see any nonlocal correlations resembling those of Bell.

My paper is about nonclassical correlations of a tensor product of two-state systems, precisely the situation considered by Bell in his paper ''On the problem of hidden variables in quantum mechanics''.

If you interpret it in terms of particles (photons) with a local hidden variable description you get wrong predictions - just as in the case Bell considered. If you interpret in terms of classical local fields you get the same predictions as quantum mechanics.

The nonlocality is in the fact that the measurements can be done arbitrarily far apart, precisely as in Bell's arguments.

 

[Demystifier]

My paper is about nonclassical correlations of a tensor product of two-state systems

I don't even know what that means, given that you have only one photon.

 

[A. Neumaier]

The mathematics is independent of any nonlocality issues. Nonlocality only enters through its interpretation, which involves particles - something that moves from one place to another.

One can check this easily by looking at a proof of the inequalities, which is just an exercise in probability theory without any physical content.

You're wrong about that. But I know from past discussions in PF to just drop it at this point.

That's the way to sweep the contradiction under the carpet.