I Superdeterminism and the hidden variable

[Sunil]

The two assumption lead to inequalities violated by QM formalism - which one is wrong?
1) There exists joint probability distribution Pr(ABC) ,
2) Kolmogorov axioms,
No, probability distribution does not have to predefine results, like for Pr(head)=Pr(tail)=1/2 coin toss.

The assumption of a joint probability distribution Pr(ABC) presupposes a lot. First, without any presuppositions, you would have to distinguish at least six things, $(A_A, A_B, B_A, B_B, C_A, C_B)$. And even this is far from sufficient, you also have to distinguish the experiments done, and their order. So,
$(A_A\to A_B, A_B \to A_A, A_A\to B_B, B_B \to A_A, \ldots, C_B \to C_A)$.

And even this is not all. If we know that $A_A\to B_B$, Alice measures A and after this Bob measures B, what are the values of the two instances of C? They have not been measured. If they would have been measured, that would have been a completely different experiment.

 

[Jarek 31]

Probability distribution just gives statistics e.g. of measurement results, for example if tossing 3 distinguishable coins, probability of getting (A,B,C) result is Pr(ABC), they have to sum to 1 ... it seems really tough to escape existence of such joint probability distribution.
And we are talking about just "tossing 3 coins, at least 2 give the same" - absolutely obvious example of Dirichlet principle.

To violate it, first of all it is crucial that we measure only 2 out of 3 variables - measuring all 3, there is joint probability distribution, this inequality has to be satisfied.
So the third variable is not only unknown, but unmeasured - for the Born rule: to add amplitudes, then square to get probability distribution.

So what is the difference between unknown and unmeasured variable?
Where does Born rule come from - required to violate such inequalities?

 

[PeterDonis]

We are here in the interpretation subforum.

The quantum mechanics intepretation subforum. Not the relativity interpretation subforum. As you yourself recognize.

all the realist interpretations of quantum theory require a preferred frame (I exclude here MWI and similar things sometimes claimed to be realist too), thus, the preferred frame hypothesis can be considered part of them, thus, can be discussed too.

If you provide a specific reference to a QM interpretation that requires a preferred frame and claims to be incompatible with relativity, we could potentially discuss that. Whether that discussion would be on topic for this thread is a separate question.

I disagree about the identification of the Lorentz ether with the preferred frame hypothesis

I didn't introduce the term "Lorentz ether" or equate it with "preferred frame". You did. If it's the wrong term for what you were actually talking about (which it is--see below), you shouldn't have used it.

If GR and SM are only effective field theories, then there will be in principle such a way, namely go down to the critical distance of that theory. One can, of course, hope that the underlying high energy theory will have the same relativistic symmetry, but there is nothing which suggests this. At least those straightforward regularizations which can pretend to look like more fundamental theories, like lattices, violate relativistic symmetry.

None of these suggestions involve "Lorentz ether", so you should not be using that term to refer to them.

 

[PeterDonis]

The possibility of violations of the Bell inequalities for spacelike separated events is, therefore, a prediction of the Lorentz ether which differs from the corresponding prediction of the spacetime interpretation.

Please provide a reference for this claim.

 

[Sunil]

If you provide a specific reference to a QM interpretation that requires a preferred frame and claims to be incompatible with relativity, we could potentially discuss that. Whether that discussion would be on topic for this thread is a separate question.

I have mentioned several of them, dBB, Nelsonian stochastics, Caticha's entropic dynamics, with links. They require a preferred frame if considered in a relativistic context. This is obvious because they all use the Bohmian velocity (even if, except dBB itself, only as an average velocity).

I didn't introduce the term "Lorentz ether" or equate it with "preferred frame". You did. If it's the wrong term for what you were actually talking about (which it is--see below), you shouldn't have used it.

Then I have somehow misinterpreted the following

Please note, first, that Lorentz Ether Theory is generally not discussed here at PF, since that term as it is usually understood refers, not to any theory in which Lorentz invariance is "only emergent", but to an interpretation of Special Relativity in which there is a preferred frame, but it is unobservable and does not affect the results of any experiments.

as equation the Lorentz ether with SR with a preferred frame.

I distinguish them, as explained, and both are necessary: The preferred frame is an actual property of the mentioned interpretations, the Lorentz ether in its modern interpretation is an effective field theory on a background with a preferred frame which allows to meet arguments against such interpretations of the following type:

... or claiming that Lorentz invariance is not fundamental but only emergent and so should be expected to be violated under appropriate conditions. The latter position, while it is logically consistent, has a heavy burden of proof since we have no experimental evidence whatever of Lorentz invariance being violated.

So, yes, I think Lorentz invariance is only emergent. And once objections against this are made, I have to refer to the Lorentz ether.
I have answered an argument made here, I have given a reference to a particular variant of the Lorentz ether compatible with GR/SM where Lorentz invariance is only emergent, and I have given more general (and therefore not mentioning the Lorentz ether) considerations which suggest that Lorentz symmetry is only emergent, which is necessary given that the number of supporters of the Lorentz ether is minimal. So I don't see the point of your objections.
There was also another claim in #21 which had to be answered by an explicit reference to the Lorentz ether:

The point is, we have one and the same principle explanation for the mysteries of SR and QM while both are still without consensus constructive counterparts after 115 and 85 years, respectively.

The possibility of violations of the Bell inequalities for spacelike separated events is, therefore, a prediction of the Lorentz ether which differs from the corresponding prediction of the spacetime interpretation.

Please provide a reference for this claim.

It is unclear to me what you think requires support with a reference. That classical causality, which is the causality which is relevant for the Lorentz ether, is not sufficient to prove the Bell inequalities for space-like separated events? The possibility of violations of the Bell inequalities for the Lorentz ether simply means that there is no proof. I cannot prove non-existence of proofs, the burden of proof is on the side of those who claim the existence of proofs. So, it is you (given that you claimed equivalence) to give a proof of Bell's theorem which works for the Lorentz ether (that means with classical causality in the preferred frame) with all other assumptions unchanged.

And we are talking about just "tossing 3 coins, at least 2 give the same" - absolutely obvious example of Dirichlet principle.

This is something you can trivially prove, no problem. That's not about violations of Bell inequalities.

To violate it, first of all it is crucial that we measure only 2 out of 3 variables - measuring all 3, there is joint probability distribution, this inequality has to be satisfied.

So the third variable is not only unknown, but unmeasured - for the Born rule: to add amplitudes, then square to get probability distribution.

Once your two measurements violate the Bell inequalities, you can be certain that the theory that these are three objectively existing coins is wrong.

So what is the difference between unknown and unmeasured variable?
Where does Born rule come from - required to violate such inequalities?

The Born rule does not violate any inequalities. You seem to think that fairy tale that quantum theory is incompatible with classical probability theory. It is not, the realist interpretations prove this.

 

[PeterDonis]

Then I have somehow misinterpreted the following

That was in response to the post of yours in which you said:

Which makes them incompatible with the spacetime interpretation of relativity, but not with the Lorentz ether interpretation which assumes a preferred frame.

 

[PeterDonis]

It is unclear to me what you think requires support with a reference.

The specific claims of yours that I quoted when I asked for one. So far, all you have done regarding references is throw out this laundry list of papers, with no explanation of what you think they say that supports your claims:

Ref:
Weinberg, S. (1997). What is Quantum Field Theory, and What Did We Think It Is. arxiv:hep-th/9702027
Donoghue, J.F. (1994). General relativity as an effective field theory: The leading quantum corrections. Phys Rev D 50(6), 3874-3888
Donoghue, J.F. (1996). The Quantum Theory of General Relativity at Low Energies, Helv.Phys.Acta 69, 269-275, arXiv:gr-qc/9607039
Schmelzer, I. (2012). A generalization of the Lorentz ether to gravity with general-relativistic limit, Advances in Applied Clifford Algebras 22(1) 203-242, arXiv:gr-gc/0205035
Schmelzer, I. (2009). A Condensed Matter Interpretation of SM Fermions and Gauge Fields, Found. Phys. 39(1), 73-107, arXiv:0908.0591

You did mention Schmelzer's theory more specifically; however, that theory has already been discussed extensively in prior PF threads (none very recently since there has been no recent work of his to discuss), and those prior threads were one of the things that contributed to the current PF policy of not allowing discussion of Lorentz ether theory.

 

[PeterDonis]

it is you (given that you claimed equivalence) to give a proof of Bell's theorem which works for the Lorentz ether

I have claimed no such equivalence and there is no burden on me to provide any such proof.

 

[PeterDonis]

Thread closed for moderation, to review whether (a) some posts are in violation of the PF policy on discussion of Lorentz ether theory, and (b) whether the thread is viable without those posts.

 

[PeterDonis]

After review, the thread has been reopened.

 

[Sunil]

The specific claims of yours that I quoted when I asked for one. So far, all you have done regarding references is throw out this laundry list of papers, with no explanation of what you think they say that supports your claims:

Look in the text abpve the references where the authors Weinberg, Donoghue and Schmelzer have been mentioned.

You did mention Schmelzer's theory more specifically; however, that theory has already been discussed extensively in prior PF threads (none very recently since there has been no recent work of his to discuss), and those prior threads were one of the things that contributed to the current PF policy of not allowing discussion of Lorentz ether theory.

Those threads would be interesting for me. But my search for "Schmelzer" has given me essentially only one thread where his theories have been explicitly discussed, namely "Generalization of Lorentz ether". An impressive number of 50 posts, and only roughly 50% of them about the theories itself. Everything else seem to be single posts in threads about something different and a welcome thread. Once this does not look like an "extensive discussion" in "threads" (plural), I would guess that I have simply not found those threads where the theories were extensively discussed. Can you give me the links to those threads?

Oh, I see that the author "Ilja" is striken through, so, AFAIU, he was (for whatever reasons) banned. I hope it is not some personal conflict which is the reason why discussing the Lorentz ether is forbidden here.

 

[PeterDonis]

Look in the text above the references where the authors Weinberg, Donoghue and Schmelzer have been mentioned.

I did. It didn't help me much. It might be more helpful if I read all the papers you referenced, but that's quite a bit of reading and I don't expect to be able to do so any time soon. The one paper on the list that I already have read is the first one, by Weinberg, and I don't recall anything in it that relates to any of the specific claims you've made, so that doesn't help me much either.

 

[RUTA]

Back to the OP, I just read and would recommend this paper: Bell’s Theorem, Quantum Probabilities, and Superdeterminism.

 

[Sunil]

The one paper on the list that I already have read is the first one, by Weinberg, and I don't recall anything in it that relates to any of the specific claims you've made, so that doesn't help me much either.

Strange. I have quoted Weinberg, and the quote is in the paper I have given. Similar with the other references, they contain the evidence for what I have claimed in the text which mentions them.

Back to the OP, I just read and would recommend this paper: Bell’s Theorem, Quantum Probabilities, and Superdeterminism.

Indeed, a good paper. Some interesting new arguments against alternatives which are, in my personal opinion, classified as "insane", like superdeterminism or a replacement of classical logic with "quantum logic".

But I'm afraid such arguments will not help much: I suspect that those who embrace such "insane" alternatives cannot be reached with rational arguments. Once somebody accepts giving up such fundamental concepts like realism and causality, and is ready to embrace even superdeterminism, despite the existence of simple common sense compatible alternatives, I cannot but suspect that rational argumentation will fail to reach them.

 

[PeterDonis]

I have quoted Weinberg, and the quote is in the paper I have given.

Your quote from Weinberg was this:

Weinberg claims that "The present educated view of the standard model, and of general relativity, is again that these are the leading terms in effective field theories"

That is not "evidence" for what you have claimed. You are not just claiming that the SM and GR are leading terms in effective theories, which is a common claim that many other physicists besides Weinberg have also made. You are claiming that they are leading terms in your particular preferred effective theory. Weinberg makes no such claim. The fact that the general claim Weinberg makes is mainstream does not imply that your far more particular claim is mainstream. Nor does it imply that Weinberg, or anyone else, had your particular preferred effective theory in mind when he made his general claim. Go look and see how many papers Weinberg has published on Lorentz ether theory.

 

[RUTA]

Indeed, a good paper. Some interesting new arguments against alternatives which are, in my personal opinion, classified as "insane", like superdeterminism or a replacement of classical logic with "quantum logic".

But I'm afraid such arguments will not help much: I suspect that those who embrace such "insane" alternatives cannot be reached with rational arguments. Once somebody accepts giving up such fundamental concepts like realism and causality, and is ready to embrace even superdeterminism, despite the existence of simple common sense compatible alternatives, I cannot but suspect that rational argumentation will fail to reach them.

It's clear something we cherish has to go (free will or causality or locality or realism or objectivity, etc.), since QM works. So, each interpretation is based on a property of Nature the author can live without :-)

 

[Sunil]

That is not "evidence" for what you have claimed. You are not just claiming that the SM and GR are leading terms in effective theories, which is a common claim that many other physicists besides Weinberg have also made. You are claiming that they are leading terms in your particular preferred effective theory. Weinberg makes no such claim. The fact that the general claim Weinberg makes is mainstream does not imply that your far more particular claim is mainstream. Nor does it imply that Weinberg, or anyone else, had your particular preferred effective theory in mind when he made his general claim.

There was no intention to claim that Schmelzer's theory is mainstream. There is a necessity to meet arguments of type "there is no viable version of the Lorentz ether", this is already all I need from Schmelzer's theory: It has a preferred frame but is nonetheless compatible with GR and SM, published, and I'm not aware of any decisive argument against it. The aim of the quote from Weinberg was to counter the argument "the preferred frame is unobservable in principle and therefore bad". If we introduce a preferred frame into an effective field theory, there is no base to assume that the result is more than an effective field theory. Thus, there is no base to assume that relativistic symmetry is fundamental and holds also below the critical system. The straightforward regularization of a field theory with preferred frame would be a spatial lattice regularization and it would not have relativistic symmetry below the critical distance. This is a general counterargument, completely independent of the particular example of such a theory given by Schmelzer. Schmelzer's theory is just an example resp. existence proof. And Donoghue shows that the field theory version of GR can be quantized as an effective field theory.

 

[PeterDonis]

If we introduce a preferred frame into an effective field theory, there is no base to assume that the result is more than an effective field theory.

If introducing a preferred frame into an effective field theory makes no difference, why would anyone bother doing it?

there is no base to assume that relativistic symmetry is fundamental and holds also below the critical system

I would phrase this a bit differently. Since we cannot currently experimentally test for Lorentz invariance below a finite length and time scale (about 1/10 the size of an atomic nucleus, or the corresponding light travel time, IIRC), we cannot say that we have experimentally confirmed that Lorentz invariance holds below that scale (or, if you prefer, above the corresponding energy and momentum scale). So if some theory claims that Lorentz invariance is not fundamental, but is an approximate symmetry that is broken above some energy and momentum scale beyond what we can currently test, that theory is consistent with currently known experimental data.

 

[PeterDonis]

The aim of the quote from Weinberg was to counter the argument "the preferred frame is unobservable in principle and therefore bad".

I fail to see how it does so, since Weinberg does not even mention any such argument. He simply says that he considers the SM and GR to be effective field theories. He says nothing whatever about preferred frames. Neither SM nor GR are preferred frame theories as they currently stand, so there is not even any reasonable inference that could be made that Weinberg was trying to say something about preferred frame theories. The fact that you propose some effective field theory that also happens to be a preferred frame theory is irrelevant to what Weinberg was or was not saying.

 

[PeterDonis]

There is a necessity to meet arguments of type "there is no viable version of the Lorentz ether", this is already all I need from Schmelzer's theory

Need for what? If your only point is to claim that we cannot rule out preferred frame theories based on our current experimental data, see my post #48.

 

[Sunil]

I fail to see how it does so, since Weinberg does not even mention any such argument.

Weinberg was quoted to show that SM and GR considered as effective field theories is mainstream. That's all. Once you have conceded this, there is no further need to consider Weinberg.

If introducing a preferred frame into an effective field theory makes no difference, why would anyone bother doing it?

Need for what?

To show the viability of all the realistic interpretations of quantum theory, given that they require a preferred frame in the relativistic context. Correspondingly, to show the viability of EPR realism (given that together with Einstein causality one could prove the Bell inequalities, while with the Lorentz ether one cannot prove it), the viability of Reichenbach's common cause principle (same reason).

Moreover, to argue that even thinking about superdeterminism, quantum logic and other such "theories" is unreasonable, given that much simpler realistic causal alternatives exist.

This is one point I have liked in that paper recommended by RUTA: The point against superdeterminism that it has to be a very complicate theory. But to reject superdeterminism because it is too complex means one needs simpler alternatives. They exist, in form of the realistic interpretations, but require a preferred frame. Thus, to present a viable alternative we need a viable theory with preferred frame able to replace GR + SM.

I would phrase this a bit differently. Since we cannot currently experimentally test for Lorentz invariance below a finite length and time scale (about 1/10 the size of an atomic nucleus, or the corresponding light travel time, IIRC), we cannot say that we have experimentally confirmed that Lorentz invariance holds below that scale (or, if you prefer, above the corresponding energy and momentum scale). So if some theory claims that Lorentz invariance is not fundamental, but is an approximate symmetry that is broken above some energy and momentum scale beyond what we can currently test, that theory is consistent with currently known experimental data.

That's correct but too weak. It would be valid also if there would be no GR and the QFT would be described by some AQFT. But the situation looks much worse for relativistic symmetry. There is not even a single viable candidate for an AQFT, and given the situation with gravity AFAIU we have even sufficient evidence that below Planck length existing field theory (GR + SM) fails. There exist simple, straightforward replacements for more fundamental theories - lattice regularizations. But they have no relativistic symmetry. So, there exist sufficiently simple viable theories with only effective relativistic symmetry but none with fundamental relativistic symmetry.

 

[PeterDonis]

It would be valid also if there would be no GR

I'm not sure what you mean by "if there would be no GR". What symmetries do and do not exist physically does not depend on what theories humans do or do not discover.

given the situation with gravity AFAIU we have even sufficient evidence that below Planck length existing field theory (GR + SM) fails

I don't know what evidence you are talking about.

there exist sufficiently simple viable theories with only effective relativistic symmetry but none with fundamental relativistic symmetry.

Your definition of "viable" is rather idiosyncratic.

 

[Sunil]

I'm not sure what you mean by "if there would be no GR". What symmetries do and do not exist physically does not depend on what theories humans do or do not discover.

I mean a hypothetical world, where a viable AQFT would exist, not the real world.

I don't know what evidence you are talking about.

In this case the theoretical one. The quantization of GR as an effective field theory (see Donoghue) does not seem to suggests that it works below Planck length. That's AFAIU quite typical for non-renormalizable theories, they work fine for some time until the counterterms one has to add become large.

Your definition of "viable" is rather idiosyncratic.

You think so? Any evidence for this?

 

[PeterDonis]

You think so? Any evidence for this?

The theories you are calling "viable" are not supported by any experimental evidence at all. They simply agree with your own theoretical preferences. You even admit that you are talking about hypothetical worlds instead of the real world.

 

[Sunil]

The theories you are calling "viable" are not supported by any experimental evidence at all. They simply agree with your own theoretical preferences. You even admit that you are talking about hypothetical worlds instead of the real world.

The reference to the hypothetical world was part of a side remark - some fantasy world were the hypothesis of fundamental Lorentz covariance would have been reasonable, to illustrate how this differs from the actual world where the hypothesis of fundamental Lorentz covariance is simply unreasonable. Given that your remark suggests something completely different, it is [self-censored].

Then, "viable" means viable, that means, no falsifying empirical evidence is known. "Empirical evidence" also means simply empirical evidence, independent of the time it was made (prediction or postdiction). So, all the empirical evidence supporting GR also supports Schmelzer's Lorentz ether, and all the empirical evidence supporting the SM supports also the part obtained from Schmelzer's model (fermion and gauge content of the SM).

The claim "not supported by any experimental evidence at all" is therefore plainly wrong. You can argue that postdictions are less valuable than predictions, but they are, nonetheless, some experimental evidence.

And even if you compare the empirical content, which is something much more subtle than simply support by some experimental evidence, this looks not that bad. The SM is essentially purely phenomenological, it predicted essentially nothing beyond what was observed. Ok, one can disagree here, given that quarks themselves have not been observed, but are theoretical objects. Whatever, they were a guess based on the experimental data. The properties of the quarks, their number, the number of colors and so on were essentially taken from observation. In Schmelzer's model they follow from the model. This model predicts the fermion and gauge content of the SM. Competitors like supersymmetry have predicted additional such particles at LHC and failed, other competitors (like strings) predict nothing. For gravity, Schmelzer introduces additional terms (- in empirical content) but restricts the topology to $\mathbb{R}^4$ and requires the existence of a global harmonic time coordinate (+ in empirical content), thus, no clear advantage.

 

[PeterDonis]

The reference to the hypothetical world was part of a side remark - some fantasy world were the hypothesis of fundamental Lorentz covariance would have been reasonable, to illustrate how this differs from the actual world where the hypothesis of fundamental Lorentz covariance is simply unreasonable.

Your definition of "unreasonable" is your personal opinion. Evidently my personal opinion differs from yours. But PF discussions should not be based on personal opinions. Discussions of what predictions various models make are fine. Claims that one model is "unreasonable" are not.

"viable" means viable, that means, no falsifying empirical evidence is known

By this criterion, lots of nonsense theories are "viable", such as Russell's teapot.

If different theoretical models make exactly the same predictions, so that there is no way to distinguish them experimentally, then the usual scientific criterion that is applied is Occam's razor.

If different theoretical models make different predictions, but the difference is not testable with our current technology and powers of observation, then we are in much the same position as above (i.e., most scientists will use Occam's razor) until our technology or our powers of observation improve to the point where we can test the different predictions.

 

[Sunil]

Your definition of "unreasonable" is your personal opinion. Evidently my personal opinion differs from yours. But PF discussions should not be based on personal opinions. Discussions of what predictions various models make are fine. Claims that one model is "unreasonable" are not.
...
By this criterion, lots of nonsense theories are "viable", such as Russell's teapot.

That empirical viability includes a lot of nonsense theories is simply fact. You cannot get rid of it (except with Orwellian renaming). Logical consistency allows even much more nonsense, nonetheless it remains a useful criterion (say, against MWI and similar nonsense). And that's why you have to allow the use of other criteria to get rid of the nonsense.

So what do you propose? Either you restrict yourself to empirical predictions and their agreement with observation, and reject arguments going beyond this as "personal opinions" with no place in science, then Russel's teapot is fine.

I allow them, and accept the usefulness of criteria like simplicity, predictive power, compatibility with fundamental principles like causality and realism, which go, indeed, beyond viability.

If different theoretical models make exactly the same predictions, so that there is no way to distinguish them experimentally, then the usual scientific criterion that is applied is Occam's razor.

The problem with Occam's razor is that it is not that well-defined as it should be. Say, the spacetime interpretation adds a whole dimension to what actually exists. But somehow the theory which uses only a three-dimensional space, where the past no longer exists and the future not yet exists, is rejected using Occam's razor. IMHO that's complete nonsense, but that's the problem with Occam's razor that one can reinterpret it in such a way that it somehow sounds acceptable.

Then, don't forget that "exactly the same predictions" is a more an urban myth than reality. Say:
EPR realism + SR spacetime interpretation ===> Bell's inequality for spacetime-separated events.
EPR realism + SR pref. frame interpretation =/=> Bell's inequality for spacetime-separated events.
There are other examples. So, there is the Wallstrom objection against some realistic interpretations of QM, which is that the realistic "interpretations" are, in fact, different theories which make different predictions (namely they exclude some QM solutions).

If different theoretical models make different predictions, but the difference is not testable with our current technology and powers of observation, then we are in much the same position as above (i.e., most scientists will use Occam's razor) until our technology or our powers of observation improve to the point where we can test the different predictions.

That's the problem of modern physics. The new devices which have given the SM have reached their limit, and all those speculations beyond the SM are evaluated in a completely unprofessional way. Because the evaluation with the standard methods, namely experiments, gives nothing, nada, nitschewo, nix. And the professional discussion of other ways to evaluate theories has been essentially suppressed as evil "metaphysics" or "philosophy". As a consequence, what is used is bad philosophy. Unprofessional because professional philosophy is already much better than the philosophy used by physicists.

And if we would follow your philosophy, there would be no discussion about philosophy in the future too, and as a consequence, I guess, the philosophy used to evaluate theories beyond the SM would remain as bad and unprofessional as it is today.

 

[PeterDonis]

And that's why you have to allow the use of other criteria to get rid of the nonsense.

I gave another criterion: Occam's razor. That's generally what scientists use.

That's the problem of modern physics.

It's not a problem, it's a fact of life: we always have more to learn. We might have different personal opinions about which direction we should go next to try to learn more, but, as I've already said, PF is not about personal opinions.

As a consequence, what is used is bad philosophy.

Philosophy is out of scope for this forum. Here we discuss physics.

 

[PeterDonis]

The OP question has been sufficiently discussed, and personal opinions are off topic. Thread closed.