The general structure of relativistic QFTs

[vanhees71]

Well, at the first glance, your preprint is in strict opposition to what I think about the issues discussed, particularly the claim that the Aharonov-Bohm effect would admit the measurement of a gauge-dependent quantity, which would in fact be a desaster for the standard treatment of the electromagnetic interaction in terms of a massless spin-1 field (which then necessarily is a gauge field too), because then the dynamical equations (aka the Maxwell equations in the classical case) were incomplete, because if gauge-dependent quantities were observable, you'd have to find a way to specify the correct gauge to make the equations describe observables uniquely. Of course, in the standard treatment of electrodynamics and also of course in case of the AB effect, what's observable are the shifts of interference patterns when switching on a magnetic field, and these shifts depend on the magnetic flux through an area and are thus indeed NOT gauge dependent, i.e., the gauge dependence of the four-potential cannot be observed using the AB effect. It's only "difficult to derive" without the use of the four-potential (if it is not impossible at all), because QT relies on Hamilton's principle and the canonical formalism, and thus to describe em. interactions of charged particles with the em. field within QT needs to use the potentials.

 

[WernerQH]

How do you come to the conclusiion that Bell used the term in a different way I did? In the quote he explicitly spoke about "local quantum field theory":

That he believes that local quantum field theory was not a satisfactory description of physics, is another point. I'd be keen to learn, which alternatives Bell had to offer. Unfortunately I don't have the quoted book at hand.

I only have the first edition of his book (without the extra chapter), and it seems to me that Bell was using the term "locally causal" in at least two different ways. Section 3 of the essay "The theory of local beables" is entitled "Quantum mechanics is not locally causal". On the other hand, section 7 ("Messages") concludes with the sentence

In this human sense [no faster than light signalling] relativistic quantum mechanics is locally causal.

It is probably not useful to engage too much in the exegesis of these essays. But in section 3 of that essay he does turn to the same question that I asked you about in my post #9: a single radioactive nucleus surrounded by several detectors. How do you explain that only one of the detectors can register the decay, without assuming some backward influence (to avoid the word "cause") that serves as a kind of veto for the other detectors? For me the "quantum field" is a mathematical device that appears to establish some kind of continuity and locality, but it is hard to say how it relates to the real world. What negotiations are going on between the detectors while the alpha-particla is on its way? What is your answer, if it's not propagators reaching backwards in time?

 

[A. Neumaier]

How do you explain that only one of the detectors can register the decay, without assuming some backward influence (to avoid the word "cause") that serves as a kind of veto for the other detectors?

The correlations and the conservation of energy forbid this. To analyze this in detail one would need to take the entire past preparation of the surrounding detectors into account, which gives plenty of room for forward causation, as one cannot prepare things detailed enough to avoid the latter.

 

[WernerQH]

The correlations and the conservation of energy forbid this. To analyze this in detail one would need to take the entire past preparation of the surrounding detectors into account, which gives plenty of room for forward causation, as one cannot prepare things detailed enough to avoid the latter.

Of course. I have no doubt that QFT is fundamentally a non-local theory. If a global analysis ("the entire past preparation of the surrounding detectors") is essential, why is "locality" so important, other than placing mathematical restrictions on the possible correlations? Its only physical basis is a leftover from classical, macroscopic theories.

The state of a system at a particular instant of time, represented by a quantum field, is a chimera. To apply energy conservation one needs to consider the evolution for a significant finite interval of time.

 

[PeterDonis]

So now, I'd like to see one for the claim there is another meaning of "locality" in the physics (!) literature.

Bell's original papers, which have been referenced many times in these discussions, do not use "locality" to mean what you use it to mean.

 

[PeterDonis]

In the quote he explicitly spoke about "local quantum field theory":

Yes, and the reason he had a problem with "local quantum field theory" is that this "local" theory violates the Bell inequalities, and the whole point of his original papers on the Bell inequalities was that a "local" theory with his preferred definition of "local" (which is not the same as the definition of "local" in "local quantum field theory") cannot violate the Bell inequalities--that's the theorem he proved.

 

[vanhees71]

I only have the first edition of his book (without the extra chapter), and it seems to me that Bell was using the term "locally causal" in at least two different ways. Section 3 of the essay "The theory of local beables" is entitled "Quantum mechanics is not locally causal". On the other hand, section 7 ("Messages") concludes with the sentenceIt is probably not useful to engage too much in the exegesis of these essays. But in section 3 of that essay he does turn to the same question that I asked you about in my post #9: a single radioactive nucleus surrounded by several detectors. How do you explain that only one of the detectors can register the decay, without assuming some backward influence (to avoid the word "cause") that serves as a kind of veto for the other detectors? For me the "quantum field" is a mathematical device that appears to establish some kind of continuity and locality, but it is hard to say how it relates to the real world. What negotiations are going on between the detectors while the alpha-particla is on its way? What is your answer, if it's not propagators reaching backwards in time?

What you have here is simply the fact that in $\alpha$ decay an $\alpha$ nucleus, which is preformed in the decaying nucleus tunnels out the potential barrier with some probability and then is detected. You can detect only the whole $\alpha$ particle or nothing in this setup. So only one detector can register the $\alpha$ particle, because once this one particle is absorbed by one detector, it cannot be detected by any other detector anymore. I don't think that this is a very exciting "quantum-interpretational issue", is it?

 

[WernerQH]

You can detect only the whole $\alpha$ particle or nothing in this setup.

Thanks. We all know the "simple facts".

[...] once this one particle is absorbed by one detector, it cannot be detected by any other detector anymore.

And what happens physically at that instant when the particle is detected? There is no wave function collapse in your view, is there? You didn't answer my question, and it is easy to dismiss it as meaningless. But if you insist on calling QFT a local theory, you should be able to say how the quantum field of the alpha-particle (continuously) evolves from point to point. Otherwise I'd call this use of the word "locality" empty jargon. It would apply only to some mathematical fiction. Perhaps we can agree to call the quantum field a useful book-keeping device that has no direct correspondence to the real world.

 

[A. Neumaier]

To apply energy conservation one needs to consider the evolution for a significant finite interval of time.

To perform a measurement, too. So what?

 

[A. Neumaier]

If a global analysis ("the entire past preparation of the surrounding detectors") is essential, why is "locality" so important

Because it means that fields can be independently prepared at mutually spacelike distances. This has nothing to do with measurements but with what happens before we measure.

 

[bob012345]

Thanks. We all know the "simple facts".

And what happens physically at that instant when the particle is detected? There is no wave function collapse in your view, is there? You didn't answer my question, and it is easy to dismiss it as meaningless. But if you insist on calling QFT a local theory, you should be able to say how the quantum field of the alpha-particle (continuously) evolves from point to point. Otherwise I'd call this use of the word "locality" empty jargon. It would apply only to some mathematical fiction. Perhaps we can agree to call the quantum field a useful book-keeping device that has no direct correspondence to the real world.

I always wondered, if a wavefunction is a mathematical description of something, how can it be said to collapse? What collapses?

 

[dextercioby]

I always wondered, if a wavefunction is a mathematical description of something, how can it be said to collapse? What collapses?

"Collapse" is just a word given a metaphorical significance. This "collapse" is nothing but a projection of a vector onto some linear subspace of the vector space, i.e. a mathematical operation. I do not know who was the first English speaking physicist to invent this use.

 

[PeterDonis]

what happens physically at that instant when the particle is detected?

This is interpretation dependent; basic QM, independent of any interpretation, does not even attempt to answer this question. It only predicts probabilities for measurement results.

Discussions of QM interpretations are off limits in this forum; they belong in a separate thread in the interpretations subforum.

 

[CoolMint]

Is the concept of 'wave' and wave-particle duality still useful in some sense in QFT?

 

[WernerQH]

I always wondered, if a wavefunction is a mathematical description of something, how can it be said to collapse? What collapses?

Good question! I'd say the picture of a quantum field evolving continuously and deterministically is in conflict with the discontinuous and random nature of the processes occurring in the real world. The wave function can't be the "whole truth".

 

[vanhees71]

Thanks. We all know the "simple facts".

And what happens physically at that instant when the particle is detected? There is no wave function collapse in your view, is there? You didn't answer my question, and it is easy to dismiss it as meaningless. But if you insist on calling QFT a local theory, you should be able to say how the quantum field of the alpha-particle (continuously) evolves from point to point. Otherwise I'd call this use of the word "locality" empty jargon. It would apply only to some mathematical fiction. Perhaps we can agree to call the quantum field a useful book-keeping device that has no direct correspondence to the real world.

A particle by definition is an asymptotic free state. In your case you start with a nucleus ("asymptotic free state") which decays due to $\alpha$ decay. All you can describe and so far also all that has ever been observed is that after the decay you have another nucleus and an $\alpha$ particle. This $\alpha$ particle interacts with the material in the detector and with some probability this leads to a signal from the detector.

The description of interactions is local in a very specific sense, indicated in my original posting. Locality is indeed a property of the mathematical description that is sufficient to guarantee consistency of this description with relativistic causality (no causal effects between space-like separated events) and Poincare invariance of observables as well as unitarity of the time evolution and thus also for the S-matrix.

 

[WernerQH]

Locality is indeed a property of the mathematical description [...]

Agreed.

This $\alpha$ particle interacts with the material in the detector and with some probability this leads to a signal from the detector.

How does probability enter your description? The $\alpha$ particle could interact with any detector. This requires a global viewpoint (considering the whole experimental situation), and this makes actually applying QFT a decidedly non-local business! I think the probabilistic/stochastic aspect of QFT is important, and it's misleading to call it a causally local theory.

 

[vanhees71]

Fields exactly provide what you call "a global viewpoint". They describe quantities as a function of time and space. The probabilities enter the description in QFT as in any formulation of QT via Born's rule. How else?

 

[WernerQH]

Yes, I also believe that Born's rule is an integral part of Q(F)T. But for most people it doesn't enjoy the same significance as fields and their equations of motion.

Fields [...] describe quantities as a function of time and space.

In the analysis of Bell-type experiments you work with "local" fields and interactions, but in the end you use a non-local operator to derive the joint probabilities of the photon polarizations. I don't object to this procedure at all (after all it produces the correct results). For me it is a remarkable coincidence that "local" field equations can be used at all, as if there were real waves spreading into space towards the detectors. What we know for sure is only that there must be two emission events at the source, and two absorption events at the detectors a few nanoseconds later. QFT let's us calculate correlations between events; on what happens "in between" it remains remarkably ambiguous. For me fields are a mathematical artifact, and the emphasis on locality obscures the fact (in my mind) that QFT is a fundamentally non-local theory.

 

[vanhees71]

Which non-local operator are you referring too?

Now, again, comes up the issue about "words"! In physics there is one (and imho only one) clear definition of locality, and that's what's meant by this word in the relativistic local (sic!) QFTs as summarized in #1 and as is most clearly written in Weinberg's QT of Fields Vol. 1.

You have to say, what you mean by a "fundamentally non-local theory", when you talk about standard QFT, because it must be another meaning of the word "locality" than the one (and imho only one) meaning it has in the standard literature.

 

[Lord Jestocost]

For me fields are a mathematical artifact, and the emphasis on locality obscures the fact (in my mind) that QFT is a fundamentally non-local theory.

In the book „Elegance and Enigma The Quantum Interviews” (edited by Maximilian Schlosshauer), some of the key terms appearing in the book are defined in the glossary list. Under the entry 'nonlocality', one finds:

"In the context of quantum mechanics, this term chiefly has two meanings.
(1) The impossibility of describing correlations between outcomes of local measurements, performed at two different locations, in terms of a local hidden-variables model.
(2) Actual physical action-at-a-distance, where the physical situation in one region instantaneously influences the physical situation in another, arbitrarily distant region.

 

[vanhees71]

Ok, and the standard meaning as used in relativistic QFT is not even mentioned... Then it's no surprise that there's always a lot of discussion without mutual understanding: People use the same words with different not well defined meanings. It's a "great" way to get into unsubstantial discussions rather than to a concise understanding!

 

[martinbn]

Ok, and the standard meaning as used in relativistic QFT is not even mentioned... Then it's no surprise that there's always a lot of discussion without mutual understanding: People use the same words with different not well defined meanings. It's a "great" way to get into unsubstantial discussions rather than to a concise understanding!

What is the standard meaning of nonlocality in QFT? I would have guessed that this term is not used in QFT.

 

[vanhees71]

I'd understand it as a theory that violates locality, i.e., the fields don't transform locally under Poincare transformations and/or the microcausality property is violated.

It seems in the quoted book it is more about the understanding of EPR, and then (1935) the understanding of relativistic QFTs was not that advanced as it is today. I'm not sure, whether at this time they were aware of the microcausality constraint and its implication for the consistency of the theory. Also the idea of the S-matrix was in its infancy at best. So for me it was always clear that EPR could not know that relativistic QFTs provide a local (though not "realistic") and causal description of relativistic QT.

It's clear that any non-relativistic theory is "non-local" in the sense of the meaning 2, because in Newtonian physics interactions are usually described as "actions at a distance", and there's also nothing in conflict with anything within this theory, because there's no other constraint on the possibility of causal connections between any events other than time ordering, and time is absolute in Newtonian (quantum) mechanics.

 

[WernerQH]

Which non-local operator are you referring too?

I thought it was obvious that an operator describing photon polarization at two detectors (at different locations!) must be a non-local operator. Is it not?

You have to say, what you mean by a "fundamentally non-local theory", when you talk about standard QFT, because it must be another meaning of the word "locality" than the one (and imho only one) meaning it has in the standard literature.

Let's not waste our time on the "standard" or "proper" meaning of words. Obviously I should have added more details. I think that QFT is better viewed as a theory of isolated events, points in spacetime, and what is classically called a field is correlations between (microscopic) events. This idea should feel quite natural for someone who applies QFT in condensed matter physics. It is not a priori necessary to assume something physical to exist between those events. They need not be "immersed" in some kind of medium (aether, vacuum, field, ...) that is continuous. There's a theory of point processes, or random point fields, describing this. It would give physical meaning to the actual graininess of quantum fields.

 

[vanhees71]

I thought it was obvious that an operator describing photon polarization at two detectors (at different locations!) must be a non-local operator. Is it not?

Let's not waste our time on the "standard" or "proper" meaning of words. Obviously I should have added more details. I think that QFT is better viewed as a theory of isolated events, points in spacetime, and what is classically called a field is correlations between (microscopic) events. This idea should feel quite natural for someone who applies QFT in condensed matter physics. It is not a priori necessary to assume something physical to exist between those events. They need not be "immersed" in some kind of medium (aether, vacuum, field, ...) that is continuous. There's a theory of point processes, or random point fields, describing this. It would give physical meaning to the actual graininess of quantum fields.

I still do not know, what you mean. To describe the coincidence experiments when measuring photon polarization states on different places you need, of course, the two-point correlation (two-photon Green's) function of the (electromagnetic) field. Of course, it's all about correlations. What else?

 

[Lord Jestocost]

In the book „Elegance and Enigma The Quantum Interviews” (edited by Maximilian Schlosshauer), some of the key terms appearing in the book are defined in the glossary list. Under the entry 'nonlocality', one finds:

"In the context of quantum mechanics, this term chiefly has two meanings.
(1) The impossibility of describing correlations between outcomes of local measurements, performed at two different locations, in terms of a local hidden-variables model.
(2) Actual physical action-at-a-distance, where the physical situation in one region instantaneously influences the physical situation in another, arbitrarily distant region.

I hope there is no misunderstanding. To my mind, these definitions of nonlocality simply mean that it would be an abuse of language if the adjective “nonlocal” is used as a description of quantum or quantum field theory. Quantum or quantum field theory are non-separable, local theories.

 

[PeterDonis]

it would be an abuse of language if the adjective “nonlocal” is used as a description of quantum or quantum field theory.

Not with definition (1) that you quoted, since that definition is equivalent to saying that the Bell inequalities are violated, and QFT does violate the Bell inequalities.

 

[Lord Jestocost]

Bell's inequality is derived from the assumption of separability. Separability, as Don Howard puts in his paper “Einstein on locality and separability”, means:

“The separability principle operates on a more basic level as, in effect, a principle of individuation for physical systems, a principle whereby we determine whether in a given situation we have only one system or two. If two systems are not separable, then there can be no interaction between them, because they are not really two systems at all.”

[Edit]You can use definition (1) in the context of conversation about quantum mechanics, but when using the term 'nonlocal' as an adjective for the noun 'quantum mechanics', one only muddies the waters.

 

[PeterDonis]

Bell's inequality is derived from the assumption of separability.

Haven't I asked someone (possibly you) to point out exactly where in Bell's papers the "separability" assumption is made?

 

[Lord Jestocost]

My point is merely the following:

Even if observed quantum-mechanical phenomena appear to us as human beings to be ‘nonlocal’, physicists should avoid to use the label ‘nonlocal’ as an adjectival term in regard to quantum theory. The impossibility of describing correlations between outcomes of local measurements in terms of a local hidden-variables model, has at the end nothing to do with quantum theory. In discussions about quantum theory, vague terminology is indeed a big problem.

 

[PeterDonis]

physicists should avoid to use the label ‘nonlocal’ as an adjectival term in regard to quantum theory.

I would say physicists should avoid both "nonlocal" and "local" as terms when discussing quantum theory. Instead, as I have posted before, they should state explicitly what specific criterion they are talking about. For example, if it's spacelike separated measurements commuting, say that.

 

[gentzen]

Nicolas Gisin’s short book Quantum Chance nicely explains how the paradox arises that quantum mechanics is local and nonlocal at the same time: The randomness itself is nonlocal, and it must be really random, because otherwise this non-locality could be used for instantaneous signal transmission. At the same time, however, the randomness also becomes less problematic, because it is now clear in the sense of which idealization it must be perfect. After all, there is probably no such thing as mathematically perfect true randomness.

I don't think that the word "nonlocal" here is problematic. You might disagree that there is a paradox at all, and if you replace "nonlocal" by "inseparable", it might get easier to argue against there being a paradox. Or you might find the quote incomprehensible. But I doubt that the word "nonlocal" is responsible for that.

 

[vanhees71]

Not with definition (1) that you quoted, since that definition is equivalent to saying that the Bell inequalities are violated, and QFT does violate the Bell inequalities.

True, but it's not due to a violation of locality but of "reality", i.e., the concept that all observables take determined values, which are only unknown to us in full detail, so that we describe them statistically, i.e., via probability distributions of some hidden variable(s). That's of course what is not fulfilled according to any QT, including local relativistic QFT. Of course, I use the physical notion of "locality" in the sense of standard relativistic QFTs.

 

[vanhees71]

I don't think that the word "nonlocal" here is problematic. You might disagree that there is a paradox at all, and if you replace "nonlocal" by "inseparable", it might get easier to argue against there being a paradox. Or you might find the quote incomprehensible. But I doubt that the word "nonlocal" is responsible for that.

It is, in my opinion, mandatory to use "inseparable" rather than "nonlocal", because locality and thus also its negation have a precise mathematical meaning in relativistic local (sic!) QFTs, namely that the Hamilton density commutes with all local observable operators at space-like separated arguments, which excludes faster-than-light propagation of (causal) effects, i.e., space-like separated events cannot be causally connected also within relativistic local QFTs by construction.