Foundations of a theory of quantum gravity - Johan Noldus' book.

[MTd2]

Well, given that you put unfair words in my hands, I will forward you Noldus' answer:

"The problems which plague gauge theory are adressed from several novel perspectives:
(a) Haag's theorem, which obstructs a nonperturabive formulation of QFT, is circumvented by means of generalized Fock bundles and giving up causality on spacetime.
(b) The quantum gauge theory he constructs is not the ordinary quantization of classical gauge theories.
(c) The way the renormalization problem is attacked is by abandonning causality and cluster decomposition, there are no apriori classical lagrangians and coupling constants only arise when solving the equations of motion; not by merely putting them in by hand.

Defining an effective S matrix and therefore calculating energy shifts and cross sections depends upon how you coarse grain the local particle notions. This is a very difficult task which is not even accomplished in a satisfactory way in classical relativity.

Concerning the 100 parameters, some of your points needs to be addressed: (a) they do not need to be small, there exist large scales in cosmology where GR also needs modification (b) 100 free parameters are not much compared to the extra degrees of freedom introduced in the landscape in string theory or LQG.

What spaces are you talking about ? The necessary functional analysis is developped in section seven partially. The sections are not unstructured and perhaps you should give it a chance. All sections stick closely together and build upon each other sequentially.

Why is this so ? There have been plenty of good QG physicists writing about the philosophical problems: these include Chris Isham and Karel Kuchar.

The first six chapters are crucial and by no means side issues. They provide the justification and motivation for the work to come. Again, please, give it a chance :).

Concerning the axioms, it was explained you in private several nice reasons why he chose not to write them in a full mathematical form:
(a) because physical axioms are much more general and powerful than mathematical ones.
(b) Because I did not want to commit the same error Von-Neumann made by unnecessarily overspecifying the mathematical context.

Actually, this does not make the work any less precise because it is clearly states in section seven that hard calculations will have to show which limitations can be imposed and which are unwarranted. This book is not finished and it not claimed to be. This the reason why the title is ''Foundations of a theory of quantum gravity''.

And yes, the theory depends upon the choices made, some of them which have no experimental impact. For example, the Unruh effect holds or not depends upon local versus quasi local particle notions, the latter which would require higher bundles. Now, the unruh effect is not going to be tested in the next 50 years probably. Nevertheless, the theoretical implications are huge. So this is one example why it was not not specified the full mathematical context, because there would not be a good reason to do so.

Concerning axiom 0, it must be understood that these concepts cannot be explained in one simple axiom or not even in section 8, because even this generalized definition of a Nevanlinna space takes 10 pages to explain, which happens in section 7. Again, be a nice guy with the paper! :)

Sure, TM is usually regarded as a natural fiber bundle whose manifold stucture is not canonically given. Actually there exist several different constructions : one could passively lift coordinate charts from M to TM as is usually done, or one could use a dynamical object such as the vierbein to define a dynamical atlas, which is what is done. Both differentiable structures give rise to different non-linear connections, so this is not canonical at all.

By coincidence or not, the generalized Fock bundles was invented again... It could be worse not having heard about them but not knowing how to use!

There is no inconsistency whatsoever concerning Axiom 3 because the generators of the local Poincare groups have to be split into two parts, one which is generated from Noether's theorem and contains all the matter degrees of freedom and the second part which gives the graviton sector. This literally mandatory due to the Weinberg-Witten theorem, which is a concept that is not trivial.

Regarding the axioms of consciousness, it is hard to tell them right away so easily.

I think it must be clear that you seem to be personally angry towards the author and perhaps missed some fundamental points, given the fact that of the carefully explained issues discussed in private communication. In case this line of argumentation is not forfeit, the author sees no reason to keep a discretionary attitude."

 

[A. Neumaier]

Well, given that you put unfair words in my hands,

I had only forgotten a pair of quotation tags; this is now corrected.

I will forward you Noldus' answer

I'd have been more interested in _your_ response to the paper in the light of my remarks.
Anyway, here are some comments on Noldus' reply:

(c) The way the renormalization problem is attacked is by abandonning causality and cluster decomposition, there are no apriori classical lagrangians and coupling constants only arise when solving the equations of motion; not by merely putting them in by hand.

Writing equations of motions are a way to propose a theory, but as long as one hasn't begun to solve it, one doesn't know that renormalization problems are absent. For example, the equations of motions written down for QED in the early days of quantum mechanics appeared quite innocent but turned out to lead to divergences in second order perturbation theory, indicating the need for renormalization.

Noldus hasn't presented any evidence that his equations of motions can be solved without recourse to renormalization. But he states on p.95: ''the theory constructed here
is extremely ambitious, it does not only want to solve technical ”details” such as renormalizability but it also claims to address long standing conceptual issues
in quantum mechanics.'' -- Maybe he wants, but he doesn't do it. His paper is full of ideas and suggestions but far from having demonstrated the ''logically consistent and precise theory of quantum gravity'' the abstract promises.

Concerning the axioms, it was explained you in private several nice reasons why he chose not to write them in a full mathematical form:

The private exchange was so little convincing that I copied many of my email remarks on the prior 90 page preprint that we had discussed into the preceding post - none of my criticism had any influence on the current form of the manuscript.

This book is not finished and it not claimed to be.

I measured the paper on the preprint server by its abstract. If the current version does not satisfy what is promised in the abstract, the abstract is misleading.

I think it must be clear that you seem to be personally angry towards the author

I deliberately waited with my post assessing the whole paper till the emotionally heated part of the discussion was over.

 

[MTd2]

Alright, so this is an answer he sent me. I do not have an opinion on his work yet.

"You were not sent a 90 page preprint, bu one about 130 pages thick.

Noldus : But, on the other hand, as I said, I do not have a standard
dynamical system. I don't have a (global) Hamiltonian, my formulation is
much more like the standard 4-covariant formulation of general relativity.

Neumaier : Of course, one cannot have a global Hamiltonian in a diffeomorphism
invariant theory. Nevertheless, one still has a symmetric hyperbolic
structure, and this is what allows one to formulate an initial-value
problem in local covariant coordinates. Solving that will most likely
bring up renormalization issues.


Noldus : No, there are several ways of understanding this; let me give you two reasons why covariant formulations are better suited:
(a) in GR if you work in a physical gauge such as the Gaussian gauge, then usually you run into inconsistencies because of the physics of GR (focal points); any physical gauge in GR suffers from similar problems. Of course, if you take suitable coordinate systems, nothing happens. The best example is that of the original Schwartzschild solution which blows up at the event horizon, but the physics doesn't and indeed going to Eddington-Finkelstein coordinates gives you a very different picture. So, the idea here is that renormalization problems in QFT show up because of a bad ''physical gauge choice'' (and I will explain in a minute why my theory proves this to be the case) ; the Heisenberg equations of GR are generally covariant, but the Heisenberg commutation relations are not. They are attached to a physical ''gauge'', that is the hypersurfaces of equal ''time'' should be spacelike. Now, you may think of causality (the commutation relations) as kind of an ''initial value formulation'' of a fully relativistic dynamics. But my theory proves that is wrong. There is not so much freedom to pick initial data on an initial hypersurface, the only freedom is given by a data on a holographic two-sphere at infinity. So, my suspicion is that the ordinary formulation of interacting QFT is **overconstrained** and that's why you run into troubles, you ''gauge fix'' more than there are ''gauge'' degrees of freedom.
(b) Another way is a more technical one, and I have explained it already to you (but it boils down to the same thing). If you would fix causality **a priori** (the commutation relations), then you must go over to field theory (see Weinberg). In the interaction picture, this leaves you with a number of free coupling constants and you assume them to be small, so you expand order by order in the coupling constants and treat each order separately. There is *no* freedom here, the physics is fixed and must come out *finite*, but it doesn't. This leads you to two problems: the renormalization of the separate terms and the non-analyticity of the series expansion.
Now, it is long time well known that there exists ways to do the summation differently and mix different orders of the coupling constant (infinities cancel out this way much better, actually, this is the rationale behind the asymptotic freedom program of Saueressig, Benedetti and co). However, within standard field theory, there is no good motivation for this and you don't really know what you are doing physically. Now, in my theory, there are no coupling constants, but each term in front of a monomial of normal ordered creation and annihilation operators comes with free functions in terms of the Lorentz covariant four momenta, physical polarization vectors and spacetime coordinates. So, when calculating the unitary potential, the appropriate expansion is in monomials of the creation and annihilation operators (and all those terms are of the right type since the number of integrals equals the number of operators minus one - hence formally satisfying the cluster decomposition principle). So, if we ignore the clifford terms you have the same number of free functions as you have terms to make finite: indeed, delta functions do show up since the product of two terms of n -1 + m - 1 = n + m - 2 so, that is precisely one integration short (these terms give singular operators) - if you, by normal ordering replace a creation and annihilation operator by one delta function, you again have n + m - 3 integrals for n + m - 2 operators which is fine, n + m - 4 integrals for n + m - 4 operators is still well defined but more contractions give again singular operators. So, the point is that higher order coefficient functions in the Hamiltonian do get involved in the lower order coefficients of the unitary expansion while those terms are usually thought of as being higher order in the coupling constant. So, it is a very different scheme, with plenty of more freedom and no a priori constraints due to causality.

That is all I have to say."

 

[A. Neumaier]

You were not sent a 90 page preprint, bu one about 130 pages thick.

I was only discussing the 90 page version he sent me first (on December 12). I didn't read the later versions he sent me since it was clear that it was still work in progress and much more was to come. Since I didn't want to spend so much time on a half-baked manuscript, I preferred to wait for the public version now on the arXiv. Unfortunately, this online version is still only half-baked.

I grant that the paper contains lots of potentially interesting ideas. But it sells the plan of a house for the actual building. To advertise beliefs, hopes, or expectations as achievements is very poor scientific practice. The proper thing to do would be to modify the abstract, reducing the claims to what is actually proved, and to moderate in the bulk of the work the tone of superiority about unproved items.

I am looking forward to seeing on the arXiv a revised version v3 that is much improved in this respect...

Noldus : But, on the other hand, as I said, I do not have a standard
dynamical system. I don't have a (global) Hamiltonian, my formulation is
much more like the standard 4-covariant formulation of general relativity.

Neumaier : Of course, one cannot have a global Hamiltonian in a diffeomorphism
invariant theory. Nevertheless, one still has a symmetric hyperbolic
structure, and this is what allows one to formulate an initial-value
problem in local covariant coordinates. Solving that will most likely
bring up renormalization issues.

Noldus : No, there are several ways of understanding this; let me give you two reasons why covariant formulations are better suited:
(a) in GR if you work in a physical gauge such as the Gaussian gauge, then usually you run into inconsistencies because of the physics of GR (focal points); any physical gauge in GR suffers from similar problems. Of course, if you take suitable coordinate systems, nothing happens. The best example is that of the original Schwartzschild solution which blows up at the event horizon, but the physics doesn't and indeed going to Eddington-Finkelstein coordinates gives you a very different picture. So, the idea here is that renormalization problems in QFT show up because of a bad ''physical gauge choice'' (and I will explain in a minute why my theory proves this to be the case) ; the Heisenberg equations of GR are generally covariant, but the Heisenberg commutation relations are not. They are attached to a physical ''gauge'', that is the hypersurfaces of equal ''time'' should be spacelike. Now, you may think of causality (the commutation relations) as kind of an ''initial value formulation'' of a fully relativistic dynamics. But my theory proves that is wrong. There is not so much freedom to pick initial data on an initial hypersurface, the only freedom is given by a data on a holographic two-sphere at infinity. So, my suspicion is that the ordinary formulation of interacting QFT is **overconstrained** and that's why you run into troubles, you ''gauge fix'' more than there are ''gauge'' degrees of freedom.
(b) [...] Now, in my theory, there are no coupling constants, but each term in front of a monomial of normal ordered creation and annihilation operators comes with free functions in terms of the Lorentz covariant four momenta, physical polarization vectors and spacetime coordinates. [...] So, it is a very different scheme, with plenty of more freedom and no a priori constraints due to causality.

I don't think these comments improve the quality of the overall argument. My response (originally happening on December 15) was (spelling amended):

... to point (a) of this dialogue:
''I don't find QED overconstrained. If it were, it could make no predictions since it would produce immediate contradictions. This is the case for the old QED of the 1930s but not for the successful (renormalized) QED of today.
And renormalization has nothing to do with gauge fixing. You also have it in Phi^4 theory where there is no gauge group. And indeed, you have it in much simpler systems, e.g., in the system consisting of a single particle in an external delta function potential.''

... and to point (b):
''But how do you achieve any predictivity with so many degrees of freedom. QED is very predictive just because it is so restrictive: one one knows the electron mass m and charge e, you can predict everything of interest.
In your scheme, you apparently need to know many more degrees of freedom before you can predict anything specific (such as the Lamb shift).''

 

[MTd2]

Noldus sent me this email to show that he sent the 134 page version and not the 90page one:

"

Hi,

My way of doing nonperturbative QED would be based upon what you can find in sections 8,9,10 of the paper below. There is no renormalization at all, and nothing is based upon action principles. Note, this is a draft version of my work and nothing may be used or transmitted to third parties. If you are interested in having some more explanations/discussion about it, I would be happy to provide you with. This work has already been discussed partially with Rafael Sorkin and Arkadiusz Jadczyk.

Especially in section 10, you can understand how such computations would need to be done.
Could you please send me back a confirmation of receipt ?

All the best,

Johan Noldus

approach quantum gravity.pdf
707K Visualizar Baixar"


I opened the file and it is dated 12th december, 2010. It has 134 pages.

He later sent me this message:

"Concerning my promises; at least I give plenty of plausibility arguments. Rovelli for example writes a book ''Quantum Gravity'' and there is not even a theory inside it (no proposal for a dynamics even), Smolin writes ''three roads to quantum gravity'' and he does not even outline a single road in any detail. In version 3, there will be no word changed, on the contrary, more evidence will be provided that these claims are correct. Bold conjectures are OK as far as they are reasonable. Mathematics is full of them, see the Poncare or Fermat conjecture or the Riemann hypothesis... they are the driving force of the field."

 

[Fra]

Concerning my promises; at least I give plenty of plausibility arguments. Rovelli for example writes a book ''Quantum Gravity'' and there is not even a theory inside it (no proposal for a dynamics even), Smolin writes ''three roads to quantum gravity'' and he does not even outline a single road in any detail. In version 3, there will be no word changed, on the contrary, more evidence will be provided that these claims are correct. Bold conjectures are OK as far as they are reasonable. Mathematics is full of them, see the Poncare or Fermat conjecture or the Riemann hypothesis... they are the driving force of the field."

Too bad Careful doesn't post more in this thread.

If anyone spots the paper where the motivation for the continuum structures that is his startingpoitn please notify. I think it was suppose to be in some fqxi contest.

I intended to keep skimming his ideas, but to make it in the order I prefer to see the motivation for the starting points before it makes sense to study the constructions based on it.

/Fredrik

 

[Fra]

Smolin writes ''three roads to quantum gravity'' and he does not even outline a single road in any detail

I was some year ago HOPING that the book that Smoling and R. Unger was supposed to release, might outline a little more in detail the "philosophy" that Smolin has advocated in several talks and papers.

But for some reason...the book still not out... I'm starting to think that this book won't come, and if it comes, it will be a more another popular style book rather than reconstruction of new formalism. I fear that that in order to see such a book, I someone else just have to write it ,worse case I'll have to try write it myself.

I suppose I'm curious to find out if Carefuls book will make some contribution here. But more important than anything to start with IMHO is to try to understand the reasoning. This is why I see the next step as the motivation behind his starting point.

/Fredrik

 

[A. Neumaier]

Noldus sent me this email to show that he sent the 134 page version and not the 90page one:
[...] I opened the file and it is dated 12th december, 2010. It has 134 pages.

In this case, I read the 134 pages; I can't check it anymore since I delete attachments from my mailbox. In any case, this doesn't affect at all the content of my comments.

"Concerning my promises; at least I give plenty of plausibility arguments.

But they are misleadingly announced in the abstract as being ''logically consistent''.

Rovelli for example writes a book ''Quantum Gravity'' and there is not even a theory inside it (no proposal for a dynamics even), Smolin writes ''three roads to quantum gravity'' and he does not even outline a single road in any detail.

Did they claim to do that? If not, they are faithful to their promises.

 

[A. Neumaier]

Too bad Careful doesn't post more in this thread.

As you can see, he still does post, through the address of MTd2. But it is borderline to violating the rule of PF that states:

''(iii) Only one person per account/username--accounts are not to be shared.''

 

[MTd2]

This is why I see the next step as the motivation behind his starting point.

/Fredrik

Well, he posted an entry to this year's contest on FQXI:

http://www.fqxi.org/community/forum/topic/905

On that entry, you can post questions to the author, like in a forum's thread or blog post.

 

[MTd2]

As you can see, he still does post, through the address of MTd2.'

He doesn't use my account. Any admin can check that all IP acesses of my account comes just from my country, Brazil. I just forward part of what he answers, which I heavily edit, that is, I remove all strong words and observations that he likes to use here.

 

[Fra]

Well, he posted an entry to this year's contest on FQXI:

http://www.fqxi.org/community/forum/topic/905

On that entry, you can post questions to the author, like in a forum's thread or blog post.

Thanks! Will check it out.

I really suck at keeping track of news. That's in fact one of the things I like with PF. Alot of people, you included + Marcus and lots of other do a nice job at spotting new papers and bringing them up for discussion.


/Fredrik

 

[A. Neumaier]

He doesn't use my account. Any admin can check that all IP acesses of my account comes just from my country, Brazil. I just forward part of what he answers, which I heavily edit, that is, I remove all strong words and observations that he likes to use here.

Yes, through your mediation he comes across much more civilized. Thanks for filtering that out!

 

[MTd2]

Alright, I more or less understand what Noldus wants to say. I chatted daily with him for the last few weeks and I formed a personal view or opinion about what he is dealing with.

What he wants to do can be summarized by Feynman's view on quantum mechanics, but applied to gravity.

"Thirty-one years ago [1949], Dick Feynman told me about his "sum over histories" version of quantum mechanics. "The electron does anything it likes," he said. "It just goes in any direction at any speed, forward or backward in time, however it likes, and then you add up the amplitudes and it gives you the wave-function." I said to him, "You're crazy." But he wasn't.
Freeman J. Dyson, in a statement of 1980, as quoted in Quantum Reality : Beyond the New Physics (1987) by Nick Herbert"

So, we have to apply the most general structures possible within what is reasonable in gravity.

The tangent space must not only be just Lorentz invariant, but Poincaré invariant.
The geometry must not only be Riemann, but Finsler.
The quantum vector space must not only be Hilbert (definite norm), but Nevanlinna (indefinite norm).
The wave function must not only be complex, but consist of Clifford numbers.
There must be covariance, but just not only have curvature on space-time, but have torsion on the tangent space.
The statistics must be of the most general type, since it is not constrained anymore by coleman-mandula, because there is no restriction for causality when something is not observed.

 

[A. Neumaier]

Alright, I more or less understand what Noldus wants to say.
[...]
So, we have to apply the most general structures possible within what is reasonable in gravity.

The tangent space must not only be just Lorentz invariant, but Poincaré invariant.
The geometry must not only be Riemann, but Finsler.
The quantum vector space must not only be Hilbert (definite norm), but Nevanlinna (indefinite norm).
The wave function must not only be complex, but consist of Clifford numbers.
There must be covariance, but just not only have curvature on space-time, but have torsion on the tangent space.
The statistics must be of the most general type, since it is not constrained anymore by coleman-mandula, because there is no restriction for causality when something is not observed.

OK, this is a proposal of where to start. But the question is - if one does all this, does he end up with a theory that has a well-defined dynamics so that computations do not result in divergences?

Proposing a new dynamics is easy. But its possible virtues can be seen only after one has tried to solve it to some nontrivial approximation. Noldus hasn't done this - so his claims to success are not better founded than QED was in 1929.

Note that a proposed dynamics for QED was written down in 1929, but it turned out to be not solvable in perturbation theory because of divergences at second order. It took almost 20 more years to find out how to modify the dynamics through renormalization.

For gravity, we know already a number of schemes that lead to divergences,

We know how to renormalize canonical gravity - though with an infinite number of parameters - most of which are however suppressed by high powers of the Planck mass.

We know nothing at all about Noldus gravity - not even whether computations can be done at all. To claim success, any alternative to canonical gravity must be better than the latter. In particular, to show Noldus' method promising the least that must be shown is that some version of perturbation theory is finite without introducing infinitely many renormalization constants.

Since this hasn't been shown, Noldus' work is currently not more that a proposal of a new direction where one could hope to be successful one day.