How useful are indefinite state spaces?

  • #1
A. Neumaier
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Main Question or Discussion Point

I am continuing here the discussion of side issues from another thread, quoting a number of Careful's promotional posts for indefinite spaces, negative probabilities, and other unconventional ideas for a generalized quantum mechanics. (A Nevanlinna space is a vector space equipped with an indefinite sesquilinear form, and perhaps further properties which Careful can surely point out if they are relevant for the discussion. A Krein space
is a vector space equipped with an indefinite sesquilinear form and extra topological structure described, e.g., in http://en.wikipedia.org/wiki/Krein_space)

if you generalize away to Nevanlinna space, unitary operators also become unbounded (on some subspace of zero norm states) so here, the picture of bounded operators completely evaporates. [...] So, I think C^{*} algebra's (as well as Von Neumann algebra's) are (a) not natural and (b) too limited.
So, in my opinion, bounded operators are dead.
I am afraid you confuse negative energy with negative probability. They are different things. It is very easy to define a Lorentz covariant positive energy Hamiltonian with positive and negative norm particles. The interpretation of course happens on a sub-hilbertspace but (a) this one is dynamical and by no means invariant under the Hamiltonian and (b) observer dependent. [...]

Right, there is no substance behind bounded operators. The best proof is that we never use them.
For example, hermitian operators can have a complex spectrum (on the ''ghost'' states) which is totally unbounded. [...]
What would be interesting from the point of view of ''C* algebra's'' is that you try to extend the GNS construction to non-positive states, so that you will get Nevanlinna space representations. This requires of course a change in the C* norm identities in the first place, but it might be good to define such generalized algebra's.
Hilbert space is not only unsuitable because it has only positive norm
Actually on Nevanlinna space, there is no natural algebraic criterion which gives only operators with a real spectrum.

It is possible of course to define bounded operators on Krein space, but it is not the natural class of operators (since their very definition requires a Hilbert space construction!)
All I am pointing out is that from where I stand and how I know quantum gravity to work out, bounded operators have evaporated.
But what I want to do is pull this discussion away from some silly textbook prejudices people have to situations where it really matters. For example to QFT or quantum gravity: that is where these issues really show their teeth, not in standard QM.
No speculation, operators in Krein space have been rigorously studied as well as spectral decompositions and so on. It is just much less known obviously.
And here a few comments:

(i) All of quantum information theory happens in finite-dimensional Hilbert spaces, in which all observables are bounded.
(ii) Most successes of quanrtum mechanics (with thousands of applications) are on the level of nonrelativistic quantum mechanics.
(iii) All free quantum field theories (in any dimension) and all quantum field theories which are known to exist rigorously (in d<4) - in that the solvability of the dynamics can be rigorously shown, were constructed in the Hilbert space framework.
(iv) Nobody knows about the rigorous status of interacting field theories in 4 or higher dimensions, no matter which techniques are employed.

Indefinite state spaces haven't made the slightest impact in cases (i)-(iii). They have had important heuristic successes in case of (iv), where ghost states are an important feature of gauge theories. But even there, the physics happens in the centralizer of the BRST charge, which is a standard Hilbert space.

Thus your language seems to be far too strong, and your assertions far too speculative.
 

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  • #2
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(i) All of quantum information theory happens in finite-dimensional Hilbert spaces, in which all observables are bounded.
(ii) Most successes of quanrtum mechanics (with thousands of applications) are on the level of nonrelativistic quantum mechanics.
(iii) All free quantum field theories (in any dimension) and all quantum field theories which are known to exist rigorously (in d<4) - in that the solvability of the dynamics can be rigorously shown, were constructed in the Hilbert space framework.
(iv) Nobody knows about the rigorous status of interacting field theories in 4 or higher dimensions, no matter which techniques are employed.

Indefinite state spaces haven't made the slightest impact in cases (i)-(iii).
Sure not because all these things you mention do not belong to proper Lorentz covariant physics. So your comment is entirely irrelevant here.

They have had important heuristic successes in case of (iv), where ghost states are an important feature of gauge theories. But even there, the physics happens in the centralizer of the BRST charge, which is a standard Hilbert space.
Sure, because of gauge invariance. But err, we haven't found a rigorous quantization procedure of gauge theories, have we ? What I suggest is to abandon field theory and therefore also gauge invariance, then Nevanlinna spaces will show their true power. Actually, if you want to understand why even complex Nevanlinna spaces are not sufficient, read my book: sections 6-8.

Thus your language seems to be far too strong, and your assertions far too speculative.
They are not, since I actually managed to construct a nonperturbatively defined quantum gravity theory within the language of Clifford-Nevanlinna modules (and constructed some easy solutions). And this matematical language is a minimum minimorum: of course you may utter that I still have to perform computations for the interacting theories. If you can construct a QG theory without it, you have a right to speak. But even the QG aspect is irrelevant: I actually manage to surpass Haag's theorem and Coleman-Mandula in this way which is very relevant for GUT's.
 
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  • #3
A. Neumaier
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Sure not because all these things you mention do not belong to proper Lorentz covariant physics. So your comment is entirely irrelevant here.
Most uses of QM are not concerned with Lorentz covariant physics.

And _all_ Lorentz covariant QFTs that are known to exist were constructed via Hilbert space methods. Mathematically, the others only have the status of conjectures. The impact of indefinite methods on turning conjectures into theorems has so fare been nil.


Sure, because of gauge invariance. But err, we haven't found a rigorous quantization procedure of gauge theories, have we ?
No, but this is the goal of mathematical (as opposed to heuristic) techniques.
It is worth a million of dollars.

What I suggest is to abandon field theory and therefore also gauge invariance, then Nevanlinna spaces will show their true power. Actually, if you want to understand why even complex Nevanlinna spaces are not sufficient, read my book: sections 6-8.
This will make the rigorous perspective even worse. I very much doubt that indefinite methods (which have much poorer mathematical structure) will be more powerful than Hilbert space methods for proving the existence of gauge theories in 4D.

Show the true power for rigorous results not by easily made promises for the future but by successes in the past!

I actually managed to construct a nonperturbatively defined quantum gravity theory within the language of Clifford-Nevanlinna modules. And this mathematical language is a minimum minimorum. If you can construct a QG theory without it, you have a right to speak. But even the QG aspect is irrelevant: I actually manage to surpass Haag's theorem and Coleman-Mandula in this way which is even relevant for GUT's.
This is not a construction in the mathematical sense, only a plausibility consideration.
A construction would show in particular the existence of a well-defined dynamics -
not only equations of motion, but a proof that these can be satisfied!

Equations of motion for QED were known many years before people found out how to make sense of them perturbatively. The latter was worth a Nobel prize, while writing down the equation of motions was not considered a big achievement.

And your approach lacks the re-derivation of the standard theories in appropriate limits, which is a necessity if you want to claim to do better than tradition.

At least you need to reproduce some of their most conspicuous results, such as a good nonrelativistic limit where the Hilbert space formulation of the standard atomic Hamiltonian approach (on which all of quantum chemistry is based) must reappear, and some relativistic gems such as the color of gold or the numerical value of the Lamb shift.

Before you can do that, your theory is a mere possibility....
 
  • #4
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Most uses of QM are not concerned with Lorentz covariant physics.

And _all_ Lorentz covariant QFTs that are known to exist were constructed via Hilbert space methods. Mathematically, the others only have the status of conjectures. The impact of indefinite methods on turning conjectures into theorems has so fare been nil.
Again, all unphysical ones :biggrin: It is getting quite boring to be honest, you don't care about nature, I do.

No, but this is the goal of mathematical (as opposed to heuristic) techniques.
It is worth a million of dollars.
No, this is not the goal of mathematics. Why don't you try to prove that such construction does not exist? That would be equally mathematical ! The goal of mathematics is to construct natural languages and theories, and gauge theories do not belong to this class since the standard construction is actually the most natural one and it doesn't work ! What you are trying to do is bad mathematics (and physics), you insist upon fitting something where it most likely does not exist. And people give a million dollars for all kinds of silly things, so I am surprised that you are blinded by the silly prize awarded by the Clay institute.

This will make the rigorous perspective even worse. I very much doubt that indefinite methods (which have much poorer mathematical structure) will be more powerful than Hilbert space methods for proving the existence of gauge theories in 4D.

Show the true power for rigorous results not by easily made promises for the future but by successes in the past!
I hope you can read; I said that it is not my aim to recover fied theory on Hilbert space because that does not exist. What I am doing is constructing a theory which behaves like QED to the extend that gravitation is weak and it is not a field theory since causality is violated weakly as is the cluster decompostion principle. So gauge theories in 4 D do NOT exist, period ! The million dollar prize you are running after is a hoax. As concerning the weight of my words: well rationally there is a lot more to say for my position than for yours, at least I do not have 50 years of failure behind me :biggrin:


This is not a construction in the mathematical sense, only a plausibility consideration.
A construction would show in particular the existence of a well-defined dynamics -
not only equations of motion, but a proof that these can be satisfied!

Equations of motion for QED were known many years before people found out how to make sense of them perturbatively. The latter was worth a Nobel prize, while writing down the equation of motions was not considered a big achievement.
Sure not, because the hard work was done before: quantum mechanical equations had to be constructed first and that was worth a few nobels ! Now you tell me that someone who invents a new quantum theory surpassing all difficulties of traditional QFT in one stroke is a ''nobody'' ?? Because he does not only have to do the work of one generation, but of two generations in one book ??? :surprised

And your approach lacks the re-derivation of the standard theories in appropriate limits, which is a necessity if you want to claim to do better than tradition.
Well at least all free theories are rederived until now... but sure you are right, there is a lot of work to do! Why don't you pull up your sleeves instead of making silly comments like this?
 
  • #5
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So gauge theories in 4 D do NOT exist, period !
Your proof for this statement shows that what you consider to be rigor is very different from the customary view of rigor.

Proving the nonexistence of gauge theories in 4D would also earn you the Clay prize!

Why don't you pull up your sleeves instead of making silly comments like this?
Because I work on what _I_ find promising, and my standards are very different from yours.
 
  • #6
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Your proof for this statement shows that what you consider to be rigor is very different from the customary view of rigor.

Proving the nonexistence of gauge theories in 4D would also earn you the Clay prize!
Really, did I ever say I had a proof of this ??? Please quote me :grumpy: If the Clay institute also rewards money for that, I advise you to change tactics. What you have tremendous problems with are deep plausibility arguments and physical reasoning. You think you have done something worthwile when you can express things Bourbaki style. Well even most mathematicians are not that limited in their understanding; ever heard of the Russian school ? Ever cared to read work of Gromov, Perelman and giants like that ? I guess you would spit on them too !


Because I work on what _I_ find promising, and my standards are very different from yours.
Sure, they are obviously much lower. I actually develop new mathematics because all evidence I know of is pointing into the direction that we need this while you keep on being stuck with tools developed at the beginning of the 20'th century which until now, have proven to be insufficient.
 
  • #7
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Really, did I ever say I had a proof of this ???
Why do you claim it if you have no proof for it?
 
  • #8
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Why do you claim it if you have no proof for it?
Simple, because you do not understand the dynamics of physics. Your attitude is similar to those professors looking for ad-hc corrections to Newtonian gravity before and after Einstein came along with general relativity. They also needed a proof;well as far as I know, there doesn't even exist something which comes close to that standard. Nevertheless, Newtonian gravity is safely relegated to the trashbin and everybody agrees upon the fact that it is dead. Likewise, it will be so with your quest; you will be frustrated because nobody will prove you wrong but everybody will agree that something else is right. I am sure you would have spit on Einstein too because integrability of his equations was only formally proven 40 years later. It just shows that your attitude is wrong; intelligence consists of more than playing flawlessly logic games, you know. It also consists in knowing which games to play.
 
  • #9
dextercioby
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Nevertheless, Newtonian gravity is safely relegated to the trashbin and everybody agrees upon the fact that it is dead. Likewise, it will be so with your quest; you will be frustrated because nobody will prove you wrong but everybody will agree that something else is right. I am sure you would have spit on Einstein too because integrability of his equations was only formally proven 40 years later. It just shows that your attitude is wrong
You haven't justified that gauge theory is dead. You have no mathematical proof for it, your claim is based on the fact that the theory of newtonian gravity was superseded by the general theory of relativity. But this actually happens (and should normally happen) to any physical theory. Physicsts will have always have to make a living in proving that someone else before them was wrong and didn't see the picture they were seeing.

With this in mind, almost everyone jumped to rearch generalizations for the standard model at the beginning of the 1970', even when this was not only 100% experimentally confirmed (and even today it isn't), nor 100% mathematically accurate (even today it isn't).

You have chosen to work along with the very many. That's ok for you. Others are still interested in laying rigorous foundations for the things we already know.

I think both parties deserve the same credit for their work and should not conflict with one another.
 
  • #10
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You haven't justified that gauge theory is dead.
Oh but I have to Arnold on many occasions. The way Arnold tries to cure gauge theory is actually very mainstream amongst mathematical physicists and can be found in textbooks of Streater and co. Those people lie the blame with Fock space, but Fock space is natural and almost canonical for everyone who understands the issue of Poincare invariance and statistics. Therefore, the perturbative quantization is the natural physical quantization and it fails. The logical conclusion is that something else fails than Fock space but Arnold is not willing to even consider this.

You have no mathematical proof for it, your claim is based on the fact that the theory of newtonian gravity was superseded by the general theory of relativity. But this actually happens (and should normally happen) to any physical theory. Physicsts will have always have to make a living in proving that someone else before them was wrong and didn't see the picture they were seeing.
Also problems with reading skills I presume ? I said that in phyiscs nothing is proved wrong according to the standards of classical logic employed in mathematics. Physicists actually are far more intelligent and understand the subtle art of accepting better and more plausible theories.

With this in mind, almost everyone jumped to rearch generalizations for the standard model at the beginning of the 1970', even when this was not only 100% experimentally confirmed (and even today it isn't), nor 100% mathematically accurate (even today it isn't).
So what ? It was the natural thing to do and even that fails. It is by no means an argument for your case.

You have chosen to work along with the very many. That's ok for you. Others are still interested in laying rigorous foundations for the things we already know.
This is too silly to even react upon. My ideas are so far those of a singleton; 99,9% doesn't even dare to touch quantum physics.

I think both parties deserve the same credit for their work and should not conflict with one another.
No, they don't because Arnold's approach is very old and the axioms he refers to (the Wightman axioms) also have been relegated to the trashbin already.

Careful
 
  • #11
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Newtonian gravity is safely relegated to the trashbin and everybody agrees upon the fact that it is dead.
Strange - your community of people whom you call ''everybody'' must be marginally small.

I find it discussed in many physics book.

Also it is still used every day everywhere in the engineering world (and they know best what reality is!) - whether it is kitchen scales, simulations of car behavior on slippery roads, calculating the stability of buildings, whether forecast, computing space craft missions, ....

The only major application area where it is a little too inaccurate is the GPS.
 
  • #12
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The way Arnold tries to cure gauge theory is actually very mainstream amongst mathematical physicists and can be found in textbooks
Indeed, my purpose here on PF is to explain the mainstream views in intelligible terms, with slight corrections in interpretational matters but not in the substance. I wait with explaining more radical attempts to cure the known defects of the main tradition (most of which fail though they are announced with trumpets) until they have become mainstream.

Physicists actually are far more intelligent.
Do you have sociological data to support this, or is it only as well proved as the announced ''fact'' that 4D gauge theories don't exist?

Arnold's approach is very old and the axioms he refers to (the Wightman axioms) also have been relegated to the trashbin already.
Your trashbin seems to contain everything of value except for the speculative theory of Johan Noldus, which is supposed to supersede everything - although no hard conclusion (that could be compared with experiment) has been derived from it so far.
 
  • #13
dextercioby
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Your trashbin seems to contain everything of value except for the speculative theory of Johan Noldus, which is supposed to supersede everything - although no hard conclusion (that could be compared with experiment) has been derived from it so far.
My point exactly. The <trashbin> metaphor is what's really bothering in his statements. If the physics community was sharing his drastic view, there would be nothing taught to students in colleges around the world, because everything would be wrong, or inaccurate. We wouldn't be teaching students classical electrodynamics, because this is superseded by the quantum one, which is not even 100% mathematically rigorous, so we wouldn't be teaching that as well. And the examples can continue. If I was mean, I would say to teach only his theory. But the question would be then: Professor, which are the prerequisites for the quantum gravity course you're teaching ? :wink:
 
  • #14
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Strange - your community of people whom you call ''everybody'' must be marginally small.

I find it discussed in many physics book.

Also it is still used every day everywhere in the engineering world (and they know best what reality is!) - whether it is kitchen scales, simulations of car behavior on slippery roads, calculating the stability of buildings, whether forecast, computing space craft missions, ....

The only major application area where it is a little too inaccurate is the GPS.
There is another major difference between both of us which is that I do not use such cheap diversion tactics. :grumpy: If you want to continue like this, you know, why don't you study the theory of the old Greeks that the universe is inside a platonic solid. You may actually find a lot of friends in the spin foam or LQG community by doing this. :biggrin:
 
  • #15
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There is another major difference between both of us which is that I do not use such cheap diversion tactics. :grumpy:
I know. Your diversion tactic uses instead a very expensive period:
So gauge theories in 4 D do NOT exist, period !
 
  • #16
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Indeed, my purpose here on PF is to explain the mainstream views in intelligible terms, with slight corrections in interpretational matters but not in the substance. I wait with explaining more radical attempts to cure the known defects of the main tradition (most of which fail though they are announced with trumpets) until they have become mainstream.
A. Your view is not mainstream, it has been largely given up by the theoretical physics community.
B. Your attitude shows you cannot separate something which has potential from something that has not. Conservatism is good as far as it remains plausible. If it is not, then you need to dare something.
C. Nothing is announced here with trumpets, I am merely explaining the virtues of my approach as it has been developed so far. Overadvertising consists in saying things which you did not proof yet. I have never done such thing. So, your psychological problems are elsewhere, perhaps in your own creativity ?

Do you have sociological data to support this, or is it only as well proved as the announced ''fact'' that 4D gauge theories don't exist?
Sure there exists such data, for example the conversation between me and you is exemplary to prove my assertion.

Your trashbin seems to contain everything of value except for the speculative theory of Johan Noldus, which is supposed to supersede everything - although no hard conclusion (that could be compared with experiment) has been derived from it so far.
First of all, it is not a speculative theory. It appears to be logically consistent framework which has a good gravity limit, gives free QFT and surpasses all traditional no go theorems. Moreover, there are good indications renormalization issues are solved. You say it is speculative because you have never taken the effort to understand it. So, it matches experiment already to a considerable degree. Your golden box however, appears to contain everything theoretical physicists know to be inadequate since 60 years. It appears logical to believe in something which has potential instead of defending those things which have limited application.
 
  • #17
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I know. Your diversion tactic uses instead a very expensive period:
This is not a diversion tactic : it is most likely true; but yes, people who only understand yes or no have trouble with understanding matters of probability.
 
  • #18
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My point exactly. The <trashbin> metaphor is what's really bothering in his statements. If the physics community was sharing his drastic view, there would be nothing taught to students in colleges around the world, because everything would be wrong, or inaccurate. We wouldn't be teaching students classical electrodynamics, because this is superseded by the quantum one, which is not even 100% mathematically rigorous, so we wouldn't be teaching that as well. And the examples can continue. If I was mean, I would say to teach only his theory. But the question would be then: Professor, which are the prerequisites for the quantum gravity course you're teaching ? :wink:
You confuse a few important issues here:
A. I see the roll of PF as dual, where people like Arnold can come and explain old undergraduate level stuff. I, on the other hand, am interested in modern physics where the battles are fierce and the arguments heated. This can also be treated here.
B. I have never said students shouldn't learn classical physics or old quantum physics although I suspect that this won't be the case anymore in 100 years. However, once you have gone through the boring obligate stuff you have to do to earn a degree, you have to go to a higher level and see the gaps in what you learned.
C. Considering those points, I think a discussion about C* algebra's as if it were relevant for QM is a bad one. It neither serves the educational purpose, nor is it relevant at the cutting edge of physics. It is only a subject which some dull mathematical physicists talk about because they simply hate infinity (and the mathematical physicists community is very small indeed) and Lorentz covariance. Indeed, those people usually don't even know relativity and Arnold is a clear cut example of that culture (which started 90 years ago, but then for different reasons).

So, you should not try to be unfair to me, while the logical inconsistency is in your line of argumentation. Not mine.
 
  • #19
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I see the roll of PF as dual, where people like Arnold can come and explain old undergraduate level stuff. I, on the other hand, am interested in modern physics where the battles are fierce and the arguments heated. This can also be treated here.
For this reason, PF has several physics forums.

In particular, the forum ''Quantum Physics'' is for discussing the established QM stuff, while the forum ''Beyond the Standard Model'' is for discussing what goes beyond it.

Your superior intelligence should have no difficulties finding out where your interest should place your postings.
 
  • #20
dextercioby
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A. I see the roll of PF as dual, where people like Arnold can come and explain old undergraduate level stuff. I, on the other hand, am interested in modern physics where the battles are fierce and the arguments heated. This can also be treated here.
I see you point. However, in the quantum physics subforum, people discuss issue pertaining to quantum mechanics and QFT of the standard model, even though, according to you, they both are to be <relegated to the trashbin>. For modern theories we've got a subforum which I'm sure you're very familiar with.

I see the purpose of this particular subforum more as a pedagogical one, rather than one focused on research. Most of the discussions here appear because people have problems understanding things you probably consider obsolete.

That being said, I just opened a thread on axioms of quantum mechanics. You are welcome to join it, even though you might think that the quantum mechanics founded 85 years ago is a dead theory, thus it shouldn't have any open interesting points (I think it has).

careful said:
B. I have never said students shouldn't learn classical physics or old quantum physics although I suspect that this won't be the case anymore in 100 years. However, once you have gone through the boring obligate stuff you have to do to earn a degree, you have to go to a higher level and see the gaps in what you learned.
I agree. That's why I spoke above of two sides: one thinking that what they learn in college must be superseded by a new theory and go searching for it, while other would work only to set what they learn in school on rigorous mathematical foundations. You're part of the first category, Arnord apparently supports the second.
 
  • #21
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Your trashbin seems to contain everything of value except for the speculative theory of Johan Noldus, which is supposed to supersede everything - although no hard conclusion (that could be compared with experiment) has been derived from it so far.
it is not a speculative theory. It appears to be logically consistent framework
Careful alias Johan Noldus seems to be the only one who rates Noldus' work so highly.
We ordinary mortals apparently don't have enough intelligence to recognize the quality and rigor of his work.
 
  • #22
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In particular, the forum ''Quantum Physics'' is for discussing the established QM stuff, while the forum ''Beyond the Standard Model'' is for discussing what goes beyond it.
Could you actually refer me to this rule? Or is this your personal interpretation ?
 
  • #23
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Careful alias Johan Noldus seems to be the only one who rates Noldus' work so highly.
We ordinary mortals apparently don't have enough intelligence to recognize the quality and rigor of his work.
If you have any specific comment, you might want to discuss it in the appropriate place, I would be happy to do so. All I see now is a pathetic attempt to degrade something you have shown plenty of misunderstandings about. Btw, you repeatedly acknowledged not to have sufficient understanding of, neither to care about quantum gravity. As far as I see, you do not even care about special relativity, so I guess your set of mere mortals at least contains one element :biggrin:
 
  • #24
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Could you actually refer me to this rule? Or is this your personal interpretation ?
Is your intelligence too advanced to interpret the titles of the forums in the usual, conservative way?

You say that Quantum Mechanics is dead and must be generalized, so discussion based upon this insight is beyond the ''Quantum Mechanics'' forum, and that gauge theories in 4 D do NOT exist, period !, so your insights, comments, and serious work are ''Beyond the Standard Model''.
 
  • #25
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All I see now is a pathetic attempt to degrade something you have shown plenty of misunderstandings about.
I only try to deflate the conceit of someone who boasts shamelessly about his own superior intelligence that throws everything valuable into the trashbin - mathematical physics, Hilbert space, positive probabilities, quantum mechanics, causality, gauge theories - and expects to be taken seriously with his speculative replacements.
 
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