Why is Quantum Field Theory Local?

[Demystifier]

No, I just want an operator.

Since you are never satisfied with my answers, I would suggest you to look at the literature. See e.g. https://www.amazon.com/dp/0201360799/?tag=pfamazon01-20

 

[martinbn]

Since you are never satisfied with my answers, I would suggest you to look at the literature. See e.g. https://www.amazon.com/dp/0201360799/?tag=pfamazon01-20

OK, where is the definition of non-local operator?

 

[Demystifier]

OK, where is the definition of non-local operator?

See e.g.
https://en.wikipedia.org/wiki/Nonlocal_operator
https://en.wikipedia.org/wiki/Nonlocal_Lagrangian

 

[martinbn]

See e.g.
https://en.wikipedia.org/wiki/Nonlocal_operator
https://en.wikipedia.org/wiki/Nonlocal_Lagrangian

Thanks!

Just point out that this is yet another sense in which local/nonlocal is used. For example all differential operators are local, in this way, so Newtonian gravity can be considered local, because the Laplace operator that appears in the Poisson equation is local.

 

[Demystifier]

Thanks!

Just point out that this is yet another sense in which local/nonlocal is used. For example all differential operators are local, in this way, so Newtonian gravity can be considered local, because the Laplace operator that appears in the Poisson equation is local.

Never thought that way, but yes, in that sense Newtonian gravity is local.

 

[vanhees71]

Thanks!

Just point out that this is yet another sense in which local/nonlocal is used. For example all differential operators are local, in this way, so Newtonian gravity can be considered local, because the Laplace operator that appears in the Poisson equation is local.

That's the usual "no-nonsense" definition of a non-local operator.

Newtonian gravity is of course local when interpreted as a (non-relativistic!) field theory but also instantaneous in its action between far-distant objects. That's no problem, because in Newtonian physics there's no "speed limit" for causal effects. It's nevertheless amazing that Newton already felt pretty uneasy with such an "action at a distance"! Nevertheless it fitted all the known facts about gravity, i.e., about the motion of celestial bodies that he didn't ponder this issue too much further.

 

[LarryS]

Thank you all for your replies. I originally expected a simple, one-sentence answer and instead got a very rich discussion.

 

[Lynch101]

The statement that "QFT is local" is also a very common statement, and, as has been posted previously in this thread, in order to reconcile this statement with the statement that Bell inequality violations mean "nonlocality", one has to recognize that the term "local" is being used in two different senses.

Sabine Hossenfelder addresses this point in her video:


Please feel free to delete this if it's deemed unhelpful - this is partly to check that my own understanding is correct (or at least more correct than it was), but I also hope it might be helpful to the OP (assuming I have understood correctly).

The term "non-local" is used in two different, but related, ways. The use of the two terms are related in that they both make reference to the correlations observed in quantum experiments, correlations that violate Bell's inequality.

Bell's inequality essentially represents the experimental predictions of a (any??) local hidden variables theory, following the assumptions of Einstein, Podolsky, and Rosen (EPR). If the Universe were local, in the sense that EPR assumed, then Bell's inequality would not be violated by quantum experiments.

The fact that the observed correlations, of entanglement experiments, do violate Bell's inequality tells us that the Universe is not EPR local.

The other way in which the term is used is as a proposed explanation for those observed correlations. While the [here] first use of the term refers to correlations that violate those predicted according to EPR locality, the [here] second use of the term refers to some [undefined/unexplained] FTL causal mechanism, where an action performed on one particle has an instantaneous effect on a spatially separated entangled particle.Is that in the right ball park? Hopefully the video will, at least, be helpful.

 

[DrChinese]

While the [here] first use of the term refers to correlations that violate those predicted according to EPR locality, the [here] second use of the term refers to some [undefined/unexplained] FTL causal mechanism, where an action performed on one particle has an instantaneous effect on a spatially separated entangled particle.

I'd say that's pretty fair. We know there is "quantum nonlocality" per Bell; but we do not know if there are any FTL causal mechanisms. Nature could be otherwise "local", and in fact appears to be so.

 

[vanhees71]

To be brief I'd state it simply as:

"Relativistic QFTs are local" means that the interactions are local, i.e., the Hamilton density commutes with any local observable when the spacetime arguments of the corresponding operators are space-like separated. So a more concise formulation is:

Locality in QFT means that there are field operators realizing a unitary representation of the proper orthochronous Poincare transformations such that these field operators transform locally as their classical analogues and that the Hamilton density commutes with all local operators representing observables at space-like separated space-time arguments.

The locality of the unitary transformation representing Poincare trafos means that, e.g., for a vector field
$$\hat{U}(\Lambda) \hat{A}^{\mu}(x) \hat{U}^{\dagger}(\Lambda)={\Lambda^{\mu}}_{\nu} \hat{A}^{\mu}(\Lambda^{-1}x), \quad \Lambda \in \text{SO(1,3)}^{\uparrow}.$$
These properties lead to (a) a unitary Poincare covariant S-matrix and (b) the corresponding transition-probality rates obey the linked cluster principle.

The second meaning of (non-)locality does not refer to causal interactions but to correlations, i.e., as any quantum theory also a "local relativistic QFT" admits the description of "non-local correlations", described by entanglement. That means that if you prepare a quantum system in an entangled state like a momentum-polarization entangled photon pair, prepared in the state
$$|\Psi \rangle=\frac{1}{2} [\hat{a}^{\dagger}(\vec{k}_1,h=1) \hat{a}^{\dagger}(\vec{k}_2,h=-1)-\hat{a}^{\dagger}(\vec{k}_1,h=-1) \hat{a}^{\dagger}(\vec{k}_2,h=1)]|\Omega \rangle,$$
you can register the two photons at very far-distant places A and B and you have a 100% correlation for the polarization states, i.e., if the observer at A finds his photon having $h=1$, then the observer at B finds his photon having $h=-1$ and vice versa, although both photons are completely unpolarized before the measurement. It doesn't matter who measures his photon first, the 100% correlation of the polarizations is observed although the polarizations before the measurement are completely indetermined.

This together with the fact that a local relativistic QFT cannot describe any faster-than-light signal propagation (due to the microcausality built in this kind of relativistic QFTs) one must conclude that the correlation is not caused by the local measurements on each photon at far distant places but it is due to the preparation in the entangled state.

I'd prefer to call the "non-locality of correlations" rather "inseparability", as Einstein formulated it. Then a lot of misunderstanding were avoided by using different words for the different two meanings of locality vs. non-locality.

 

[atyy]

To be brief I'd state it simply as:

"Relativistic QFTs are local" means that the interactions are local, i.e., the Hamilton density commutes with any local observable when the spacetime arguments of the corresponding operators are space-like separated. So a more concise formulation is:

Locality in QFT means that there are field operators realizing a unitary representation of the proper orthochronous Poincare transformations such that these field operators transform locally as their classical analogues and that the Hamilton density commutes with all local operators representing observables at space-like separated space-time arguments.

The locality of the unitary transformation representing Poincare trafos means that, e.g., for a vector field
$$\hat{U}(\Lambda) \hat{A}^{\mu}(x) \hat{U}^{\dagger}(\Lambda)={\Lambda^{\mu}}_{\nu} \hat{A}^{\mu}(\Lambda^{-1}x), \quad \Lambda \in \text{SO(1,3)}^{\uparrow}.$$
These properties lead to (a) a unitary Poincare covariant S-matrix and (b) the corresponding transition-probality rates obey the linked cluster principle.

The second meaning of (non-)locality does not refer to causal interactions but to correlations, i.e., as any quantum theory also a "local relativistic QFT" admits the description of "non-local correlations", described by entanglement. That means that if you prepare a quantum system in an entangled state like a momentum-polarization entangled photon pair, prepared in the state
$$|\Psi \rangle=\frac{1}{2} [\hat{a}^{\dagger}(\vec{k}_1,h=1) \hat{a}^{\dagger}(\vec{k}_2,h=-1)-\hat{a}^{\dagger}(\vec{k}_1,h=-1) \hat{a}^{\dagger}(\vec{k}_2,h=1)]|\Omega \rangle,$$
you can register the two photons at very far-distant places A and B and you have a 100% correlation for the polarization states, i.e., if the observer at A finds his photon having $h=1$, then the observer at B finds his photon having $h=-1$ and vice versa, although both photons are completely unpolarized before the measurement. It doesn't matter who measures his photon first, the 100% correlation of the polarizations is observed although the polarizations before the measurement are completely indetermined.

This together with the fact that a local relativistic QFT cannot describe any faster-than-light signal propagation (due to the microcausality built in this kind of relativistic QFTs) one must conclude that the correlation is not caused by the local measurements on each photon at far distant places but it is due to the preparation in the entangled state.

I'd prefer to call the "non-locality of correlations" rather "inseparability", as Einstein formulated it. Then a lot of misunderstanding were avoided by using different words for the different two meanings of locality vs. non-locality.

Just as there are different definition of "local", there are different definitions of "cause". In one definition relativistic causality alone does not imply local causality (see Fig. 5 of https://arxiv.org/abs/1503.06413).

 

[Demystifier]

Just as there are different definition of "local", there are different definitions of "cause". In one definition relativistic causality alone does not imply local causality (see Fig. 5 of https://arxiv.org/abs/1503.06413).

The paper is very deep, but not easy to read.

 

[vanhees71]

Fortunately I've some holidays and time to read ;-).

 

[mattt]

The paper is very deep, but not easy to read.

I like it so far...(I'm halfway through)

 

[bhobba]

So what does "signal" mean?

The ability to send information, in particular information that can be used to sync clocks. These days the modern view of SR is as a geometry implied by the symmetries of the POR with a constant that needs to be determined by experiment - it turns out to be the speed of light. It was not always presented in such an elegant and transparent way, but instead how Einstein did it initially using thought experiments about syncing clocks. His arguments break down if you can sync clocks with a signal faster than the speed of light. Of course the modern method breaks down as well if it can be done. In fact things get really bad because the constant c can be determined by means having nothing to do with speeds and sending signals. Yet the equations imply you can't do it ie have speeds faster than that c. The whole edifice of SR would not only be wrong but a logical mess. A good exercise in understanding SR is working it out and why it is of no concern if that speed can't be used to send information. If you are like me and would like to see some detail about it before embarking on the journey, you can see the paper I often reference:
http://physics.umd.edu/~yakovenk/teaching/Lorentz.pdf

Then read the account given in Rindler:
https://www.amazon.com/dp/0198539525/?tag=pfamazon01-20

Happy contemplating . Just kidding - it will require your attention but is certainly not what I would call a mind boggling issue.

Dr Chinese is correct in saying 'Virtually the entirety of the physics community calls the correlations evidence of "quantum non-locality". No one understands the mechanism enough to say with certainty anything is happening that is FTL; certainly I don't.' I am the 'odd man out', in that I believe there is no non-locality involved . We have had many long threads about it - but as became clear it really is a semantic issue, and semantics is one of the silliest things there is to argue about.

I used to argue about it a lot here and on other forums, but not so much now because I understand its semantic nature.

Thanks
Bill

 

[bhobba]

I like it so far...(I'm halfway through)

I have read it before - it just reinforces my current view - a lot of it is semantics. But it is good to know how it is viewed using one lot of semantics, and contrasting it to another.

Thanks
Bill

 

[Tendex]

I'm not sure if the previous discussion was under the assumption that QFT was a deterministic(hidden variables) theory or not. If the premise is that it's not deterministic then Bell's quantum non-locality doesn't follow necessarily since it includes determinism as premise to conclude "EPR locality" is impossible.

With respect to the semantics of "local", I think that the sense in which QFT is usually said to be local is part of the "EPR locality" in Bell's theorem. So I guess the above commenters don't consider QFT as a deterministic theory.

 

[vanhees71]

Standard local relativistic QFT is a QT and not a deterministic HV theory. Locality in relativistic QFT means that the Hamilton density is built by field operators and their derivatives at one spacetime point and that all local observables commute with it a space-like distances of the arguments (microcausality condition).

 

[Tendex]

Standard local relativistic QFT is a QT and not a deterministic HV theory.

Fine, I guessed right in your case then.

Of course, justifying that QFT is not deterministic as a mathematical theory is a tall order(doing it on the grounds that it includes probabilities as some people do doesn't seem right). But maybe for another thread.

Locality in relativistic QFT means that the Hamilton density is built by field operators and their derivatives at one spacetime point and that all local observables commute with it a space-like distances of the arguments (microcausality condition).

Yes, this is consistent with EPR locality's assumption of separability.

 

[vanhees71]

I don't understand the 1st part. As any QT also local relativistic QFT is not deterministic, because the state provides probabilities for the outcome of measurements not determined values of all observables of the quantum system. Why do you think that the probabilistic interpretation of the state is not right? There's not the slightest evidence for such an idea to hold true.

 

[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.