Von Neumann QM Rules Equivalent to Bohm?

[Ilja]

Now I don't understand what you are talking about. The renormalized n-point functions and thus the S-matrix elements, cross sections, etc. calculated from them are manifestly covariant.

The point is that, if we take Haag's theorem seriously, the only really consistent theories are similar to lattice theories (at least in their main property, being not relativistically covariant). If we have a lattice theory as a fundamental theory, the large distance limit is a continuous theory, which can have relativistic covariance. In a simple case, the lattice defined by atoms of a crystal gives a wave equation with constant speed of sound, and the symmetry of this simplest wave equation is, of course, the Poincare group. But this can be, by construction (once obtained from a lattice theory) only an approximate symmetry, the whole continuous theory is only an approximation.

 

[TrickyDicky]

The point is that, if we take Haag's theorem seriously,

We should. For some reason people often dismiss it using as argument the accuracy of the results obtained in QED when all the theorem says is those results are actually obtained in a way that is not exactly the way the idealized theory says. It should be looked as a real good hint instead of trying to ignore it.

the only really consistent theories are similar to lattice theories (at least in their main property, being not relativistically covariant).

This I disagree that is the only really consistent modification. It is too strong an assumption as departure point.

 

[TrickyDicky]

Nevertheless the S-matrix elements as calculated in renormalized perturbation theory and which are compared to observations are Lorentz invariant (and gauge invariant).

Say you take a regularization method like DR, you can say that since it's physical limit is d→4, it is in that sense trivially Lorentz invariant as the limit is 4-spacetime. But then you still have that as ε→0 the chiral(axial) and conformal symmetries are broken in general.

 

[stevendaryl]

I'm not sure what you think the problem is. Suppose I took a pair of socks, one red the other blue. Put one in a box, and send it to Alpha Centauri. If I waited some time and opened the box, I would instantly know the state of the other pair. This is completely classical, and shows that there is nothing wrong with nonlocal correlations, indeed almost every correlation is nonlocal.

But classically, a nonlocal correlation can always be explained in terms of local correlations plus ignorance of the physical state. So in this case, the two possible physical states for the pair socks are:

1. A = red sock to Alpha Centauri, blue sock to Earth
2. B = blue sock to Alpha Centauri, red sock to Earth

The "mixed state" of state A with probability 1/2, state B with probability 1/2 reflects, not anything nonlocal about the world, but merely reflects human ignorance.

Classically--and I should say, quantum-mechanically as well, in any situation not involving entanglement--a nonlocal correlation is always evidence of local correlations involving as yet unknown parameters.

 

[Haelfix]

I completely agree, again the only difference with quantum mechanics is that in so far as the analogy goes (say in the Bell thought experiment experiment), the classical mixed state is promoted to a pure entangled state. You still require preparation of the initial state within a local setting.

So what we have is a fundamentally new object (an entangled state) where we can have maximal knowledge about both of the subsystems by themselves but know nothing about the composite system, or viceversa. That's the difference with classical physics!

All that Bell says is that you can either make a big issue about locality, or you can accept the very pedestrian notion that there is nothing wrong with locality but rather that quantum mechanics requires a new object that is not describable in terms of classical physics.. I think that's all the minimal intepretation actually says, and there is really nothing wrong or mysterious with that point of view.

 

[stevendaryl]

I completely agree, again the only difference with quantum mechanics is that in so far as the analogy goes (say in the Bell thought experiment experiment), the classical mixed state is promoted to a pure entangled state. You still require preparation of the initial state within a local setting.

So what we have is a fundamentally new object (an entangled state) where we can have maximal knowledge about both of the subsystems by themselves but know nothing about the composite system, or viceversa. That's the difference with classical physics!

Yes, but when people argue about whether quantum mechanics is local or not, people are using slightly different definitions of "local". Here's a definition: A theory is local if every maximally informative correlation is local. By that definition, quantum mechanics is nonlocal.

But your way of putting it is an interesting one. In classical probability, complete knowledge of a composite system implies complete knowledge of each component. It doesn't in quantum probability. A quantum measurement of a component of a composite system amounts to creating new information about the universe, not simply revealing hidden information.

 

[atyy]

I'm not sure what you think the problem is. Suppose I took a pair of socks, one red the other blue. Put one in a box, and send it to Alpha Centauri. If I waited some time and opened the box, I would instantly know the state of the other pair. This is completely classical, and shows that there is nothing wrong with nonlocal correlations, indeed almost every correlation is nonlocal. There would only be a problem if I could wiggle a sock on Alpha Centauri, such that it would instantenously cause a wiggle on the earth. Then I would have acausal observable physics.

The difference between quantum mechanics and this classical example lies encoded in the fact that when we do GHZ or EPR experiments, we can infer that the original state was NOT either red or blue, but rather something new eg a state that is in a weird mixture, like half red and half blue. That further the dynamics must allow interference of states and the noncommutativity of operators. However all these things stay manifestly local, in the sense that the lagrangian stays written in a certain form and that operators at spacelike separation don't commute.

The sense of "local" you are talking about is "no faster than light communication of classical information". This sense of locality is preserved by quantum mechanics.

vanhees71 claims that collapse cannot be physical because that would violate Einstein causality, according to the argument of EPR. Einstein causality, defined in this way, is gone in quantum mechanics. It is either empty if the wave function is not real, or violated if the wave function is real. So Einstein causality is no argument against a physical collapse.

 

[Ilja]

This I disagree that is the only really consistent modification. It is too strong an assumption as departure point.

I have written only "similar too", and further reduces this to the main property of not being relativistically covariant.

Anyway yet too strong? I think not - a consistent regularization, which is relativistically covariant, would be, in its essence, a counterexample to Haag's theorem, not?

 

[atyy]

I have written only "similar too", and further reduces this to the main property of not being relativistically covariant.

Anyway yet too strong? I think not - a consistent regularization, which is relativistically covariant, would be, in its essence, a counterexample to Haag's theorem, not?

No, because there are rigourous relativistic quantum field theories constructed in 2 and 3 spacetime dimensions. Haag's theorem applies, and it just means that the usual derivation of the formulas is wrong. However, the theories are able to construct a Hilbert space etc and the usual formulas by other methods.

There is a discussion of this issue on p15 of http://rivasseau.com/resources/book.pdf.

 

[TrickyDicky]

No, because there are rigourous relativistic quantum field theories constructed in 2 and 3 spacetime dimensions.

If you take a look at the original 1955 paper by Haag, in the section about general assumptions (page 6) it is quite obvious he is referring to 4-spacetime and its ten dimensional group of symmetry. So I don't understand your insisting use of 2 and 3 dimensional constructions as counterexamples.

 

[TrickyDicky]

I have written only "similar too", and further reduces this to the main property of not being relativistically covariant.

Anyway yet too strong? I think not - a consistent regularization, which is relativistically covariant, would be, in its essence, a counterexample to Haag's theorem, not?

For what's physically relevant which is what concerns us here yes.
I only disagreed with the lattice assumption. I agree with the qualification you wrote between parenthesis, which is basically my point.

 

[vanhees71]

Say you take a regularization method like DR, you can say that since it's physical limit is d→4, it is in that sense trivially Lorentz invariant as the limit is 4-spacetime. But then you still have that as ε→0 the chiral(axial) and conformal symmetries are broken in general.

Sure, but that's not a bug but a feature since these symmetries are violated in nature. You can show that they do not survive quantization of the classical theory, no matter how you regularize the theory. BTW that it's the $U_A(1)$ and not the $U(1)$ that is violated is dictated by gauge invariance. Thus it's unique which of the U(1) symmetries has to be violated, leading to the special rule for the $\gamma^5=\mathrm{i} \gamma^0 \gamma^1 \gamma^2 \gamma^3$ matrix, which is assumed to be anticommuting with $\gamma^{\mu}$ for $\mu \in \{0,1,2,3 \}$ and commuting for $\mu \geq 4$ ('t Hooft-Veltman convention). In any case, you have to choose which linear combination of the currents must be non-conserving, and gauge symmetry uniquely tells you that it must be the axial current, and that's good, because otherwise the pion decay to photons would come out wrong by several orders of magnitude.

 

[vanhees71]

I'm not sure this is right, but let me see if this argument can persuade you that what you say is at odds with the Bell theorem.

The locality of the interactions in the Lagrangian ultimately becomes the locality in the Hamiltonian. The Hamiltonian in the Heisenberg picture evolves according to classical local equations of motion and deterministically. The Bell theorem forbids local deterministic explanations of the nonlocal correlations. So the local deterministic Hamiltonian cannot explain the nonlocal quantum correlations - it needs other things like the quantum state, the nonlocal observables, and the Born rule - but by the time we calculate that, it is hardly clear that the calculation is local.

But that in fact IS what's meant by saying the interaction is local. That there are non-local correlations in relativistic QFT is clear. We are discussing an example right now!

 

[vanhees71]

The point is that, if we take Haag's theorem seriously, the only really consistent theories are similar to lattice theories (at least in their main property, being not relativistically covariant). If we have a lattice theory as a fundamental theory, the large distance limit is a continuous theory, which can have relativistic covariance. In a simple case, the lattice defined by atoms of a crystal gives a wave equation with constant speed of sound, and the symmetry of this simplest wave equation is, of course, the Poincare group. But this can be, by construction (once obtained from a lattice theory) only an approximate symmetry, the whole continuous theory is only an approximation.

Yes that may be, but so far it's not proven that Poincare invariance is broken in this way. So if this is so, then the scales are much smaller than resolved by our experiments yet. Of course, the relativistic QFTs in the continuum limit are effective low-energy theories of whatever more comprehensive theory we don't know yet (we don't even know whether such a thing exists at all).

 

[martinbn]

OT: Sorry for the side question, but what exactly is Einstein causality? It seems, from the posts above, that it is something different than the usual relativity theory causality that comes from no faster than light signals.

 

[TrickyDicky]

Sure, but that's not a bug but a feature since these symmetries are violated in nature. You can show that they do not survive quantization of the classical theory, no matter how you regularize the theory. BTW that it's the $U_A(1)$ and not the $U(1)$ that is violated is dictated by gauge invariance. Thus it's unique which of the U(1) symmetries has to be violated, leading to the special rule for the $\gamma^5=\mathrm{i} \gamma^0 \gamma^1 \gamma^2 \gamma^3$ matrix, which is assumed to be anticommuting with $\gamma^{\mu}$ for $\mu \in \{0,1,2,3 \}$ and commuting for $\mu \geq 4$ ('t Hooft-Veltman convention). In any case, you have to choose which linear combination of the currents must be non-conserving, and gauge symmetry uniquely tells you that it must be the axial current, and that's good, because otherwise the pion decay to photons would come out wrong by several orders of magnitude.

I know those are not symmetries of nature, that is my argument, that we should follow nature, rather than artificial symmetries. But we were talking about QED and its Poincare invariance and chiral symmetry is part of that symmetry, the fact that it is broken in nature as shown by experiments of parity violation just restates my point about the extraordinary results of perturbative QED(wich doesn't exist mathematicaly being rigorous about the symmetry axioms demanded to the free field theory, i.e. what Haag's theorem proves) taken as a nice hint to look for a theory in which Poincare invariance is not a fundamental symmetry, just an approximate one.

Haag's theorem is nicely stated and proven besides the original paper by Haag in 1955 in the book by Streater and Wightman "PCT, spin and statistics, and all that".

 

[TrickyDicky]

OT: Sorry for the side question, but exactly is Einstein causality? It seems, from the posts above, that it is something different than the usual relativity theory causality that comes from no faster than light signals.

It is. Einstein causality is what is violated by Bell type quantum exeriments, it is also called local causality and any quantum theory(so that includes QFT) breaks it. However the concept of no possible FTL signals is strictly respected by construction of the QFT theories, by something called microcausality(fields (anti)commuting at spacelike separation) and that survives violations of Bell inequalities.

 

[martinbn]

It is. Einstein causality is what is violated by Bell type quantum exeriments, it is also called local causality and any quantum theory(so that includes QFT) breaks it. However the concept of no possible FTL signals is strictly respected by construction of the QFT theories, by something called microcausality(fields (anti)commuting at spacelike separation) and that survives violations of Bell inequalities.

My question is what exactly is Einstein causality since it is something else than faster than light signals?

 

[TrickyDicky]

My question is what exactly is Einstein causality since it is something else than faster than light signals?

Well, it seems it means different things to different people, for instance in this thread vanhees71 identified Einstein causality with microcausality and thus the disagreemnet with atyy. For others it means the causal structure of Minkowski space. I was identifying it with local causality so I refer you to https://en.wikipedia.org/wiki/User:Tnorsen/Sandbox/Bell's_concept_of_local_causality

 

[atyy]

But that in fact IS what's meant by saying the interaction is local. That there are non-local correlations in relativistic QFT is clear. We are discussing an example right now!

That's fine as a definition. The problem is that you say that physical collapse is not allowed because it would violate Einstein causality, by the EPR argument. The local interactions of QFT do not save Einstein causality.

 

[atyy]

My question is what exactly is Einstein causality since it is something else than faster than light signals?

Einstein causality is defined in post #102, and it is basically the causality of classical spacetime. One way to see that quantum mechanics violates it is that if you treat the wave function as real, the collapse of the wave function is clearly not Einstein causal, yet there is no faster than light communication of classical information.

 

[vanhees71]

That's fine as a definition. The problem is that you say that physical collapse is not allowed because it would violate Einstein causality, by the EPR argument. The local interactions of QFT do not save Einstein causality.

We are discussing in circles. Clearly the local interactions are sufficient for the linked-cluster theorem to hold, and this precisely preserves Einstein causality.

 

[vanhees71]

I know those are not symmetries of nature, that is my argument, that we should follow nature, rather than artificial symmetries. But we were talking about QED and its Poincare invariance and chiral symmetry is part of that symmetry, the fact that it is broken in nature as shown by experiments of parity violation just restates my point about the extraordinary results of perturbative QED(wich doesn't exist mathematicaly being rigorous about the symmetry axioms demanded to the free field theory, i.e. what Haag's theorem proves) taken as a nice hint to look for a theory in which Poincare invariance is not a fundamental symmetry, just an approximate one.

Haag's theorem is nicely stated and proven besides the original paper by Haag in 1955 in the book by Streater and Wightman "PCT, spin and statistics, and all that".

Poincare symmetry is defined as symmetry under the proper orthochronous Lorentz group, and that's precisely what's preserved up to the energies available to us today.

 

[atyy]

We are discussing in circles. Clearly the local interactions are sufficient for the linked-cluster theorem to hold, and this precisely preserves Einstein causality.

It does not, because you still have the collapse of the wave function which violates Einstein causality.

Either that, or you are using the statement of the theorem in Weinberg's book which is wrong. See the discussion regarding the sloppy statement in Weinberg's book https://www.physicsforums.com/threads/cluster-decomposition-and-epr-correlations.409861/.

 

[TrickyDicky]

Poincare symmetry is defined as symmetry under the proper orthochronous Lorentz group, and that's precisely what's preserved up to the energies available to us today.

Physically it may be what's preserved by available energies but I've never seen the Poincare group defined as the identity component of its fixed origin subgroup(Lorentz), I'm pretty sure that is not what is meant when referring to the Poincare symmetry say in Weinberg's TQF vol. 1. Otherwise I don't see why one should additionally insist on the commutation condition and the cluster decomposition, since the proper orthocronous Lorentz group already preserves causality(preserves orientation both spatially and in time, unlike the Lorentz group).

 

[Haelfix]

Well, it seems it means different things to different people, for instance in this thread vanhees71 identified Einstein causality with microcausality and thus the disagreemnet with atyy. For others it means the causal structure of Minkowski space. I was identifying it with local causality so I refer you to https://en.wikipedia.org/wiki/User:Tnorsen/Sandbox/Bell's_concept_of_local_causality

See this is why I dislike discussions about the foundations of QM, b/c it invariably goes in a circle and we end up redefining words. Post 102's definition of Einstein Causality, or Bell's local causality criteria given in the wiki article is a bit weasly. The problem, as correctly pointed out in 102, is that it presupposes notions of beables and classical probability theory. So I would call it Einstein Causality + classical concepts about what a state really *is* and how it is allowed to combine. Aspects experiment shows that this is wrong, but that just means that you have to give up one, but not necessarily both concepts.

 

[TrickyDicky]

See this is why I dislike discussions about the foundations of QM, b/c it invariably goes in a circle and we end up redefining words. Post 102's definition of Einstein Causality, or Bell's local causality criteria given in the wiki article is a bit weasly. The problem, as correctly pointed out in 102, is that it presupposes notions of beables and classical probability theory. So I would call it Einstein Causality + classical concepts about what a state really *is* and how it is allowed to combine. Aspects experiment shows that this is wrong, but that just means that you have to give up one, but not necessarily both concepts.

Sorry but I fail to see what's weaselly about the definitions, and I don't find anything about beables in #102. I found #102 quite clear and right and know close to nothing about beables. It is a provable fact that "Einstein causality" is a concept often confusingly used but tend to think that it is the kind of thing that can be corrected by openly discussing what one means by it.

 

[Haelfix]

The problem for instance in the wiki definition is that it starts writing down statements about classical probability. Statements where the logic goes like (A or B) = 1 or (A and B) = 0, allowing for operations where truth values commute around the place etc.

Thats very dangerous in quantum mechanics, and indeed is what historically tripped up John Bell when he was writing down his theorem. I mean even Bells theorem itself is absolutely trivial, and essentially a tautological statement within classical probability theory. But it shouldn't come as too much of a surprise to realize that quantum mechanics violates this, quite independently of the interpretations.

So you see, definitions where multiple different concepts are conflated at the same time are not necessarily very useful, and part of the reason why there might be confusion. So perhaps can we agree to just refer to that statement as "Einstein Causality + classical probability theory"?

 

[TrickyDicky]

The problem for instance in the wiki definition is that it starts writing down statements about classical probability. Statements where the logic goes like (A or B) = 1 or (A and B) = 0, allowing for operations where truth values commute around the place etc.

Thats very dangerous in quantum mechanics, and indeed is what historically tripped up John Bell when he was writing down his theorem. I mean even Bells theorem itself is absolutely trivial, and essentially a tautological statement within classical probability theory. But it shouldn't come as too much of a surprise to realize that quantum mechanics violates this, quite independently of the interpretations. So you see, definitions where multiple different concepts are conflated at the same time are not necessarily very useful, and part of the reason why there might be confusion.

Yes, this is an interesting point and I actually wrote a similar warning in some old post.

So perhaps can we agree to just refer to that statement as "Einstein Causality + classical probability theory"?

It actually includes what in terms of the 1964 Bell theorem was called local hidden variables, i.e. Locality+determinism.

 

[TrickyDicky]

I refer in any case martinbn to #102 which he must have missed.

 

[stevendaryl]

Yes, this is an interesting point and I actually wrote a similar warning in some old post.
It actually includes what in terms of the 1964 Bell theorem was called local hidden variables, i.e. Locality+determinism.

As Bell explained, determinism is not an assumption behind his analysis. It's a conclusion.

Alice and Bob decide ahead of time to measure the spins of twin particles along the same axis. Suppose that Alice measures her particle before Bob measures his. Then at the moment that Alice gets her result, she knows, with 100% probability, what Bob's result will be. So from that point on, Bob's result is deterministic.

 

[TrickyDicky]

As Bell explained, determinism is not an assumption behind his analysis. It's a conclusion.

Alice and Bob decide ahead of time to measure the spins of twin particles along the same axis. Suppose that Alice measures her particle before Bob measures his. Then at the moment that Alice gets her result, she knows, with 100% probability, what Bob's result will be. So from that point on, Bob's result is deterministic.

Determinism in its more generic conception is not an assumption of Bell's analysis. It actually does assume something more restricted, a linear form of determinism.

 

[atyy]

See this is why I dislike discussions about the foundations of QM, b/c it invariably goes in a circle and we end up redefining words. Post 102's definition of Einstein Causality, or Bell's local causality criteria given in the wiki article is a bit weasly. The problem, as correctly pointed out in 102, is that it presupposes notions of beables and classical probability theory. So I would call it Einstein Causality + classical concepts about what a state really *is* and how it is allowed to combine. Aspects experiment shows that this is wrong, but that just means that you have to give up one, but not necessarily both concepts.

The problem for instance in the wiki definition is that it starts writing down statements about classical probability. Statements where the logic goes like (A or B) = 1 or (A and B) = 0, allowing for operations where truth values commute around the place etc.

Thats very dangerous in quantum mechanics, and indeed is what historically tripped up John Bell when he was writing down his theorem. I mean even Bells theorem itself is absolutely trivial, and essentially a tautological statement within classical probability theory. But it shouldn't come as too much of a surprise to realize that quantum mechanics violates this, quite independently of the interpretations.

So you see, definitions where multiple different concepts are conflated at the same time are not necessarily very useful, and part of the reason why there might be confusion. So perhaps can we agree to just refer to that statement as "Einstein Causality + classical probability theory"?

I agree that Einstein causality, as defined for the Bell inequalities, does include classical probability.

There is also a second sense of locality that quantum mechanics does fulfill, which is "no faster than light transmission of classical information".

So there are two widely agreed upon definitions of locality, one of which is empty or violated by quantum mechanics, the other which is fulfilled by quantum mechanics.

Is there a third definition of locality that corresponds to "Einstein causality without classical probability?" Maybe, but then why call it Einstein causality, which is the causality of classical relativity? Some candidates are information causality http://arxiv.org/abs/1112.1142 and macroscopic locality http://arxiv.org/abs/1011.0246.

 

[atyy]

See this is why I dislike discussions about the foundations of QM, b/c it invariably goes in a circle and we end up redefining words. Post 102's definition of Einstein Causality, or Bell's local causality criteria given in the wiki article is a bit weasly. The problem, as correctly pointed out in 102, is that it presupposes notions of beables and classical probability theory. So I would call it Einstein Causality + classical concepts about what a state really *is* and how it is allowed to combine. Aspects experiment shows that this is wrong, but that just means that you have to give up one, but not necessarily both concepts.

The problem for instance in the wiki definition is that it starts writing down statements about classical probability. Statements where the logic goes like (A or B) = 1 or (A and B) = 0, allowing for operations where truth values commute around the place etc.

Thats very dangerous in quantum mechanics, and indeed is what historically tripped up John Bell when he was writing down his theorem. I mean even Bells theorem itself is absolutely trivial, and essentially a tautological statement within classical probability theory. But it shouldn't come as too much of a surprise to realize that quantum mechanics violates this, quite independently of the interpretations.

So you see, definitions where multiple different concepts are conflated at the same time are not necessarily very useful, and part of the reason why there might be confusion. So perhaps can we agree to just refer to that statement as "Einstein Causality + classical probability theory"?

I commented on this in post #208. But here is another reason why I don't think your suggestion of "Einstein causality without classical probability" saves vanhees71's argument. Let us define "Einstein causality without classical probability" as quantum mechanics. Now, how do the predictions of quantum mechanics differ depending on whether collapse is physical or not?

 

[vanhees71]

Physically it may be what's preserved by available energies but I've never seen the Poincare group defined as the identity component of its fixed origin subgroup(Lorentz), I'm pretty sure that is not what is meant when referring to the Poincare symmetry say in Weinberg's TQF vol. 1. Otherwise I don't see why one should additionally insist on the commutation condition and the cluster decomposition, since the proper orthocronous Lorentz group already preserves causality(preserves orientation both spatially and in time, unlike the Lorentz group).

Of course, that's precisely what's meant in Weinbergs book. In nature the discrete symmetries P, T, PT, and CP are all independently verified to be violated by the weak interaction. Due to the CPT theorem, valid for local microcausal QFTs, then also C must be violated. So what's the space-time symmetry of nature is the proper orthochronous Poinare group and not larger subgroups of O(1,3).