How useful are indefinite state spaces?

[A. Neumaier]

Didn't I just explicitely explain the meaning of dead before?

If you want to be understood, use the terms with the meaning everyone uses them.

 

[Careful]

Now that you revoke your superiority, we are again on equal footing.

I didn't Where did I write that again ? I asked you for a quotation where I said this, nothing more. I have nothing to hide and nothing to boast about, I only make my point as I know it to be likely. If you have this subjective impression of my book, please address specific points as I do in your approach (and which you constantly ignore). Your Christian moral severly clashes here with my Nietzschian philosophy.

Furhermore, I do not think that your judgement is equal to mine. You are a mathematician who has until so far displayed little or no knowledge of relativity, quantum gravity, modern approaches to the foundations of quantum mechanics and axiomatic approaches of QFT in curved spacetime. I am not going to tell either that my judgement about Riemannian geometry is as good as that of Misha Gromov because I have studied Petersen, Gromov's own book and Alexandrov on metric geometry, am I ? That would not be very humble of me.

You throw all the stuff I value in quantum physics into your thrashbin, and I reciprocate by throwing your book into mine. Nothing is proved by each other's thrashbin, and it is far too early to find out whose view will be more justified by history.

It is indeed to early to tell wether I will succeed (however likely I think it is). It is however not to early to tell that your chances are extremely slim.

 

[A. Neumaier]

I do not think that your judgement is equal to mine. You are a mathematician who has until so far displayed little or no knowledge of relativity, quantum gravity, modern approaches to the foundations of quantum mechanics and axiomatic approaches of QFT in curved spacetime. I am not going to tell either that my judgement about Riemannian geometry is as good as that of Misha Gromov because I have studied Petersen, Gromov's own book and Alexandrov on metric geometry, am I ? That would not be very humble of me.

But comparing yourself with Gromov is humble!?? Where is your array of prizes??


Anyway, back to the topic of the thread:

Can you explain to us - without requiring us to read your 160 page book - how your indefinite spaces lead to positive probabilities in simple situations?
And how to interpret the negative probabilities if they arise?

 

[Careful]

But comparing yourself with Gromov is humble!?? Where is your array of prizes??

It is not the array of prizes that counts, actually I am pretty sure that would not even come to Gromov's mind; it clearly shows you do not understand how people who have done something of value think. They don't care about prizes, they care about brains. But now I am confused Gromov doesn't write anything down according to your standards of rigor and leaves quite some gaps to the reader; are you saying now that while he does not satisfy your criteria of a proper mathematician, he still is in your high regard because other people have given him ''prizes''?

Anyway, back to the topic of the thread:

Can you explain to us - without requiring us to read your 160 page book - how your indefinite spaces lead to positive probabilities in simple situations?
And how to interpret the negative probabilities if they arise?

As I said, these schemes are being worked out fully at this moment; there are actually inequivalent approaches one can take, one more robust than the other. And no, I cannot explain these things even on one full A4 page since it requires many subtle considerations and additional concepts. But even then you miss the point, it often happened that physicists have developped a mathematical tool they deemed necessary for nature without fully grasping at first the interpretation. The very rule you are questioning now is a prime example of this ! There has been a long debate initially about several competing interpretations going from matter waves to probabilities. But here, even this is not a problem, there however are some real results which need to be developped first before I know which interpretation is preferred.

 

[A. Neumaier]

It is not the array of prizes that counts, actually I am pretty sure that would not even come to Gromov's mind; it clearly shows you do not understand how people who have done something of value think. They don't care about prizes, they care about brains. But now I am confused Gromov doesn't write anything down according to your standards of rigor and leaves quite some gaps to the reader

The work for which Gromov received the prizes is commonly agreed to be rigorous according to the standards of today. (I don't care about prizes, but they reflect the recognition of the community, and hence are useful for comparison. Those who select the prize winners don't care about prizes either, they care about brains.) Gaps do not matter as long as the community agrees that they can be filled. Hardly any mathematician aspires to rigor in the sense of the logicians, where even the smallest gap is filled.

As I said, these schemes are being worked out fully at this moment;

So not even on that you have a definite account, not even at the level of rigor of theoretical physics.

Let me try again:

In which sense is the conventional quantum mechanics in Hilbert spaces an approximation or limiting case of your indefinite theory?

 

[Careful]

The work for which Gromov received the prizes is commonly agreed to be rigorous according to the standards of today. (I don't care about prizes, but they reflect the recognition of the community, and hence are useful for comparison. Those who select the prize winners don't care about prizes either, they care about brains.) Gaps do not matter as long as the community agrees that they can be filled. Hardly any mathematician aspires to rigor in the sense of the logicians, where even the smallest gap is filled.

So perhaps then, it is just a matter of people actually having studied his work and thought about it in great detail for a long while? Note that Gromov often doesn't even care about giving proper definitions. God praise those people who have actually taken the effort for that and not just thrown away the manuscript because not everything was written down in what they conceive as a very precise way. So, what was our discussion again about? Nevanlinna spaces which are not rigorous, unbounded operators which are troublesome or eh new equations whose integrability has not been formally shown yet ? Let me tell you that even in Clifford analysis, similar, but much easier, types of equations than the ones I construct have been investigated and plenty of solutions have been found. But alas, no rigorous integrabilty theorem has been constructed yet even in those simple cases as far as I know.

So not even on that you have a definite account, not even at the level of rigor of theoretical physics.

If you continue to make such idiotic and manifestly false comments, our discussion is over. There is still discussion in our days about the probability interpretation of QM and the equations have been constructed 90 years ago. You constantly show that you do not understand how theoretical physics works; perhaps you should remain with those things you are educated in: mathematics. For example, none of the other quantum gravity programs suggested so far has even attempted to construct a coherent interpretation. So would you also piss on the capacities of say string theorists in that way? You would not even dare so, because you may hide in some dusty corner of your office when Edward Witten comes to you to complain. Moreover, I do have working interpretations, some of which are similar to those proposed in the literature, but you are not aware of those either as far as I understand.

Let me try again:

In which sense is the conventional quantum mechanics in Hilbert spaces an approximation or limiting case of your indefinite theory?

That is a different question which I do have a definite answer for (and which is also provided in the book); it is not the same one you asked before. The negative norm originates from the indefinite character of the Clifford numbers which are only turned on when there is a nonzero gravitational field. So, in case gravity vanishes, the dynamical sector leaves a Hilbert space invariant and you have the standard interpretation.

 

[A. Neumaier]

new equations whose integrability has not been formally shown yet ?

Has it been shown at least perturbatively to second order in the gravitational constant?
If not, the status of your theory is like that of QED in 1930, maybe with all the problems ahead that QED had to face. Or like that of string theory around 1980, where the problems still aren't overcome.

There is still discussion in our days about the probability interpretation of QM and the equations have been constructed 90 years ago.

But this discussion is irrelevant for the practice of quantum mechanics. The probabilistic aspect of QM is very well understood; only its origin is somewhat in the clouds.

The negative norm originates from the indefinite character of the Clifford numbers which are only turned on when there is a nonzero gravitational field. So, in case gravity vanishes, the dynamical sector leaves a Hilbert space invariant and you have the standard interpretation.

What happens when the gravitational field is small, bulk matter is far away, and energies are far below the Planck scale? Do you get a Hilbert space and an external gravitational field?

 

[Careful]

Has it been shown at least perturbatively to second order in the gravitational constant?
If not, the status of your theory is like that of QED in 1930, maybe with all the problems ahead that QED had to face. Or like that of string theory around 1980, where the problems still aren't overcome.

My computations (which are not in the draft yet) so far reveal no problems on the perturbative level, the hard question is whether it closes nonpertubatively. Moreover, again, you are extremely unreasonable. Your contributions to this discussion are nihil because you put me in the situation where I would have to present a fully closed theory as a single person in a single effort. Either you comment upon the proposal as it stands now (and which is much more precisely formulated than other approaches going on for many years), or you shut up. What you do not want to see is that my comments regarding your work are embedded in a historical series of failed attempts while my approach is fully fresh and has no evidence against it at this moment in time. If you refuse to understand this dynamics and important difference in your next post, I will report it as obstructive. My patience with your nonsensical comments is over.

But this discussion is irrelevant for the practice of quantum mechanics. The probabilistic aspect of QM is very well understood; only its origin is somewhat in the clouds.

Again, you do not understand the point here. It is alas like that with professors who think that God likes them in all aspects. In a theory of quantum gravity, the probability aspect becomes different because you do not dispose anymore of the classical observer. Hence, what you calculate is an ''absolute'' probability which has nothing to do with observation which requires relative probabilities. This aspect is just an example of the very many things which are alive still. You may also wish to study the work of Sorkin who tries to construct dynamical probability interpretations starting from the quantum measure. Here you actually have to prove some decoherence for ''macroscopic'' bodies in order for the Born rule to emerge.

What happens when the gravitational field is small, bulk matter is far away, and energies are far below the Planck scale? Do you get a Hilbert space and an external gravitational field?

Probably not, the dynamics will go slightly outside any fixed sub-Hilbert space, but that is not a problem is it ? Point is, that the pure Hilbert space picture does not exist because G is nonzero.

 

[Careful]

To summarize, you are only pleased with Noldus when he performs the work of Heisenberg, Von Neumann, Wigner (regarding a fully complete and rigorous foundation of a new quantum theory) and Feynman, 't Hooft (showing that the theory makes sense - although there is no renormalization issue in my approach) in one book. And then, I did not even speak about the gravitational theory. Otherwise, you deem it unworthy of theoretical physics. Please, go back to your mathematical desk and fill in epsilon's and delta's in proofs of people who have real ideas and don't bother people who do work hard to get a genuine new insight.

You constantly refuse to see or understand or acknowledge the point I am making to you. The level of irrationality (and this is a very mild word) which you display here (in many ways) is beyond comprehension for someone who has an actual responsability towards the intellectual capital he teaches to every year.

Careful

 

[A. Neumaier]

you put me in the situation where I would have to present a fully closed theory as a single person in a single effort.

Great and radical claims are measured by much higher standards than minor main stream contributions.

And asking for a finite second order perturbation theory is far from asking for a closed theory.

If [...] I will report it as obstructive. My patience with your nonsensical comments is over.

Your derisive comments would be worthy of reporting, too. I'd never challenge a humble mind like I challenge you!

It is alas like that with professors who think that God likes them in all aspects.

Well, why shouldn't he like his children? He gives them freedom of thought and power to think independently.

In a theory of quantum gravity, the probability aspect becomes different because you do not dispose anymore of the classical observer.

Even in flat space quantum field theory, one doesn't have (or need) a classical observer.
As is well-known, all known observers are quantum objects, though macroscopic ones.

Hence, what you calculate is an ''absolute'' probability which has nothing to do with observation which requires relative probabilities.

What should an absolute probability mean, if it cannot be observed and hence tested?

Probably not, the dynamics will go slightly outside any fixed sub-Hilbert space, but that is not a problem is it ? Point is, that the pure Hilbert space picture does not exist because G is nonzero.

One needs to be able to recover to first order in G the standard Hilbert space quantum mechanics in an external gravitational field, under the assumption that an observer inside the system observes another subsystem - since this is what we observe as real observers.

I hope you are at least able to do first order perturbation theory...

I am looking for _some_ things that you can explain without a 160 page overhead...

 

[A. Neumaier]

The level of irrationality (and this is a very mild word) which you display here (in many ways) is beyond comprehension for someone who has an actual responsibility towards the intellectual capital he teaches to every year.

Well, it seems that you are the only one who perceives me as being irrational.
Others here appreciate a lot my way of imparting understanding:

BTW it'd be really nice to have a teacher like you : )

^ That was a surprisingly good (and concise) explanation.

 

[Hurkyl]

If you think a post deserves to be reported, you should report it, rather than responding to it...

 

[Careful]

Great and radical claims are measured by much higher standards than minor main stream contributions.

Yes, it requires 5 nobels in one book

Your derisive comments would be worthy of reporting, too. I'd never challenge a humble mind like I challenge you!

There are two points:
A. I am actually more humble than you are it seems to me.
B. You did not challenge me at any instant: all you did was mentioning projects which I did not finish yet for very understandable reasons. This is fun in the beginning, but gets very annoying in the end.

Even in flat space quantum field theory, one doesn't have (or need) a classical observer.
As is well-known, all known observers are quantum objects, though macroscopic ones.

Nonsense, in that case you need a super observer and the state of the system is not directly discribing the probabilities of measurements you make. Actually, this superobserver would first have to measure you and then the system under study and apply the Bayesian rule.

One needs to be able to recover to first order in G the standard Hilbert space quantum mechanics in an external gravitational field, under the assumption that an observer inside the system observes another subsystem - since this is what we observe as real observers.

Sure, this is the standard analytical perturbative argument no? If you shut off G, you get standard QFT with all it's limitations, if G is turned on, then first order corrections arise.

I am looking for _some_ things that you can explain without a 160 page overhead...

It is difficult, of similar order than explaining general relativity to Newtonian physicists. You must accept that some ideas are not trivial and that 160 pages is sometimes not too much. For example, I once send you a summary and you found it full of buzzwords, while for QG physicists it was very clear what I wrote. You think you can decouple gravity from QM and what I have learned is that both need each other for a consistent formulation.

 

[Careful]

Well, it seems that you are the only one who perceives me as being irrational.
Others here appreciate a lot my way of imparting understanding:

Well, I thought the same when I got to know you. And indeed, you explain standard stuff impartial and in a good way - we had never a dispute about that (we are both intelligent enough for this). However, you get irrational when some bold, new ideas are launched; you understand very well the process of repetition, but alas not of innovation.

Let us quit here with our little fight. I guess we both understand that battles can be fought without any need for casualties ... an art which is not too well understood.

 

[A. Neumaier]

You did not challenge me at any instant: all you did was mentioning projects which I did not finish yet for very understandable reasons. This is fun in the beginning, but gets very annoying in the end.

I challenged your patience to the point where you threatened to report me.

Nonsense, in that case you need a super observer and the state of the system is not directly describing the probabilities of measurements you make. Actually, this superobserver would first have to measure you and then the system under study and apply the Bayesian rule.

Nonsense. There are no such superobservers.

But we routinely observe as quantum systems other quantum systems at energies where quantum corrections to gravity are completely negligible.

Sure, this is the standard analytical perturbative argument no? If you shut off G, you get standard QFT with all it's limitations, if G is turned on, then first order corrections arise.

Of course. But since you start with indefinite space, the question is whether or not you end up in a Hilbert space - as it must be in this case, since it is very well known how to describe this situation.[/QUOTE]

I once send you a summary and you found it full of buzzwords, while for QG physicists it was very clear what I wrote.

It is very easy to claim clarity. It is much more difficult to make it believable.

Please have some QG physicist comment here in PF on your book, to confirm your claim that for QG physicists it was very clear what you wrote.

 

[Careful]

I challenged your patience to the point where you threatened to report me.

Yes that is true, you managed to raise my bloodpressure !

Nonsense. There are no such superobservers.

Sure there are, if you have only quantum systems and the observers are quantum themselves, you need something which breaks the superpostion of this observer. You may call this a second classical observer, but it will only lead to the conclusion of a superobserver.

But we routinely observe as quantum systems other quantum systems at energies where quantum corrections to gravity are completely negligible.
Of course. But since you start with indefinite space, the question is whether or not you end up in a Hilbert space - as it must be in this case, since it is very well known how to describe this situation.

Didn't we have this whole discussion that you should not end up in Hilbert spaces in the first place because gauge theories in 4-D strongly appear to resist such formulation ? You are right in the sense that the dynamics should approximately leave some Hilbert space invariant, but not exactly.

It is very easy to claim clarity. It is much more difficult to make it believable.

Please have some QG physicist comment here in PF on your book, to confirm your claim that for QG physicists it was very clear what I wrote.

I know this due to personal communication. Why would I say this if this were not the case ? Have I ever tried to conceal a fact from you? I think all my responses were fair... what content is concerned. You may actually check the forum and you will see that tom.stoer found an almost complete copy of this document helpful and actually requested for me to put it on te web (but this is not the communication I was talking about).

 

[A. Neumaier]

Sure there are, if you have only quantum systems and the observers are quantum themselves, you need something which breaks the superposition of this observer.

Only if one subscribes to von Neumann's very idealized description of the measurement process. I don't, because it is very inadequate to account for the practice of measurement. It applies only to certain idealized model measurements capable of treatment in the 1930's.

Didn't we have this whole discussion that you should not end up in Hilbert spaces in the first place because gauge theories in 4-D strongly appear to resist such formulation ?

This doesn't relieve you from the need to recover the traditional framework (which adequately expresses almost everything we can measure at ordinary distances and energies) in some very good approximation. If you can't, your predictions will be inconsistent with experiment.

 

[Careful]

Only if one subscribes to von Neumann's very idealized description of the measurement process. I don't, because it is very inadequate to account for the practice of measurement. It applies only to certain idealized model measurements capable of treatment in the 1930's.

Von Neumann's is the only sensible one. Moreover, there exists no interpretation which can do what you want it to do. If you think there is, please provide all details; I will be happy to shoot them down.

 

[A. Neumaier]

Von Neumann's is the only sensible one.

Von Neumann's cannot be sensible since for consistency it requires a hierarchy of bigger and bigger superobservers, which is nonsense.

Moreover, there exists no interpretation which can do what you want it to do. If you think there is, please provide all details; I will be happy to shoot them down.

See Sections 7.3 - 7.5 of http://lanl.arxiv.org/abs/0810.1019

 

[Maui]

Von Neumann's cannot be sensible since for consistency it requires a hierarchy of bigger and bigger superobservers, which is nonsense.

You have a viable second option that doesn't resort to miracles?

 

[A. Neumaier]

You have a viable second option that doesn't resort to miracles?

Yes. See the reference given in #49.

 

[Maui]

Yes. See the reference given in #49.

You have a no go theorem that explicitly restricts deterministic models. My layman opinion says that if you are proposing another non-local HV theory, that'd be just another case of magic.

 

[Careful]

See Sections 7.3 - 7.5 of http://lanl.arxiv.org/abs/0810.1019

There is no way in conventional QM do to what you claim, there are actually theorems about this. If you think you have something, please summarize in a few lines. I am sure it will be great fun.

 

[A. Neumaier]

There is no way in conventional QM do to what you claim, there are actually theorems about this. If you think you have something, please summarize in a few lines. I am sure it will be great fun.

Oh, I thought you'd understand the need for more than a few lines to do something beyond what conventional QM can do:

no, I cannot explain these things even on one full A4 page since it requires many subtle considerations and additional concepts.

 

[A. Neumaier]

You have a no go theorem that explicitly restricts deterministic models. My layman opinion says that if you are proposing another non-local HV theory, that'd be just another case of magic.

It requires no magic to avoid a no-go theorem by not satisfying its assumptions or conclusions.

 

[Maui]

It requires no magic to avoid a no-go theorem by not satisfying its assumptions or conclusions.

The locality assumption or the realism(determinsim) assumption? Or do you propose some caveat? You are either too good or too naive(we most are anyway).

 

[A. Neumaier]

The locality assumption or the realism(determinsim) assumption? Or do you propose some caveat? You are either too good or too naive(we most are anyway).

Let me propose that you first look at the reference and try to understand it. Unlike Careful's treatise, it is not technical.

 

[Hurkyl]

Von Neumann's cannot be sensible since for consistency it requires a hierarchy of bigger and bigger superobservers, which is nonsense.

Accepting for the sake of argument it requires a hierarchy of superobservers, I really don't see why such a thing should be nonsensical.

I could easily imagine an argument that in reality that such a hierarchy would contain much more information than is accessible to us, but that's a rather normal state of affairs for physical theories, rather than being nonsense.

 

[A. Neumaier]

Accepting for the sake of argument it requires a hierarchy of superobservers, I really don't see why such a thing should be nonsensical.

I could easily imagine an argument that in reality that such a hierarchy would contain much more information than is accessible to us, but that's a rather normal state of affairs for physical theories, rather than being nonsense.

Let the first order observer observe a single particle, and for k>1, let the k-th order observer observe the (k-1)-st order observer.

We first fix the situation von Neumann is talking about: ideal measurements that put the system instantaneously into a well-defined eigenstate, hence measure a complete set of observables of the system.

An observer (together with the observing equipment) must therefore be much larger than the object it observes, since it must have essentially classical pointer readings for each particular observable in the complete set. If the observed system consists of n quantum particles, it has 3n continuous quantum degrees of freedom. hence the observer must have 3n classical pointers, each comprising at least N atoms, where for a reasonable accuracy, N>>100. Therefore the observer consists of a system consisting of n particles consists of at least 300n particles. But this is an extremely conservative estimate, since we ignore all the logistics that is necessary to make such complete measurements with some reliability.

The k-th order observer of a single particle (n=1) therefore contains at least 300^k particles. Assuming the number of particles in the universe to be U, we see that the existence of the k-th observer implies
$$k\le log U/log 300 < 33$$
when U=10^81. Even allowing lots of unknown dark matter will not change the resulting upper bound by much.

The only escape to this argument is to assume that the number of particles is infinite.
But even then there appear to be unsurmountable problems with the k-th observer measuring all details approximately instantaneously, given that the single particle observed by the lowest order observer is on the Earth and we know quite well the distribution of particles close enough to the Earth to be able to neglect relativistic delays.

 

[Careful]

Dear Arnold,

Your scheme needs a superobserver as well, you cannot escape that. Let me hint you were the devil is: you uncritically assume that a ''macroscopic'' ''apparatus'' behaves more or less classically. This goes straight against Schrödinger's cat argument of course; whatever way you chose to escape, I guarantuee you upfront that you won't escape Von Neumann's conclusion. It appears to me that you did not follow the logical implications of your own writings far enough.

Moreover, your reasoning about the tower of observations is ''ganz falsch''.

Johan