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-   -   Rosť and A-M: Geometrization of Quantum Mechanics (http://www.physicsforums.com/showthread.php?t=101540)

marcus Nov26-05 03:59 PM

Asselmeyer-Maluga and Rosť: Geometrization of Quantum Mechanics
 
this paper was mentioned by selfAdjoint in another thread.
people there seemed to think it should be studied/discussed
so maybe this paper should have its own thread, besides
being included in our list of new QG/matter ideas

http://arxiv.org/abs/gr-qc/0511089
Differential Structures - the Geometrization of Quantum Mechanics
Torsten Asselmeyer-Maluga, Helge Rosť
13 pages, 2 figures
"The usual quantization of a classical space-time field does not touch the non-geometrical character of quantum mechanics. We believe that the deep problems of unification of general relativity and quantum mechanics are rooted in this poor understanding of the geometrical character of quantum mechanics. In Einstein's theory gravitation is expressed by geometry of space-time, and the solutions of the field equation are invariant w.r.t. a certain equivalence class of reference frames. This class can be characterized by the differential structure of space-time. We will show that matter is the transition between reference frames that belong to different differential structures, that the set of transitions of the differential structure is given by a Temperley-Lieb algebra which is extensible to a C*-algebra comprising the field operator algebra of quantum mechanics and that the state space of quantum mechanics is the linear space of the differential structures. Furthermore we are able to explain the appearance of the complex numbers in quantum theory. The strong relation to Loop Quantum Gravity is discussed in conclusion."

my comment: this looks interesting. I would not have caught it. selfAdjoint flagged it.
http://www.physicsforums.com/showthr...906#post834906
in post #7 of the Garrett Lisi thread.

what is impressing me most is that right now seems to be a time of new ideas. a lot of new ideas are appearing that connect different mathematical pictures of spacetime, all having to do with Quantum Gravity

Kea Nov26-05 04:27 PM

initial reaction
 
Thanks selfAdjoint and Marcus

Well I guess you know what I'm going to say! All roads lead...

The references to Krol on page 2 are interesting. A recent and related paper by Krol is

Model Theory and the AdS/CFT correspondence
http://arxiv.org/PS_cache/hep-th/pdf/0506/0506003.pdf

Quote from the abstract: ...though explicit calculations refer to the would be noncompact smooth 4-invariants based on the intuitionistic logic.

I think it is just great that Rose' and A-M have spelled this out carefully.
:smile:

selfAdjoint Nov26-05 04:30 PM

Quote:

Quote by Marcus
what is impressing me most is that right now seems to be a time of new ideas. a lot of new ideas are appearing that connect different mathematical pictures of spacetime, all having to do with Quantum Gravity

Yes, I am excited about these new deep results. For example Asselmeyer and Rose' show that their approach restricts the underlying coefficient module to be the complex numbers, which would answer that puzzled about quantum physics. Schroedinger brough complex numbers in from nineteenth century theoretical optics, but they were never shown to be required before, AFAIK.

Kea Nov26-05 04:37 PM

Helge is here! Hi!

Kea Nov26-05 04:40 PM

Remember

Quantum general relativity and the classification of smooth manifolds
Hendryk Pfeiffer
http://arxiv.org/abs/gr-qc/0404088

Might be useful.

marcus Nov26-05 04:45 PM

these two people are at what I think is a semi-private contract Research and Development organization. the byline says Fraunhofer- Gesellschaft

http://www.fraunhofer.de/fhg/EN/company/index.jsp
http://fraunhofer-society.biography.ms/


the byline says FIRST FhG, Berlin.
FIRST must be an acronym for some department at Fraunhofer

Yes, FIRST means FRAUNHOFER INSTITUTE COMPTERARCHITECTURE SOFTWARE TECHNOLOGY

we would say FICST, but for them a computer is a "Rechner" (because it Reckons stuff) and so they say FIRST. I was wondering.
http://www.first.fraunhofer.de/

Maybe it is like being at an IBM Lab.

the FhG centers----there are many all over Europe----do CONTRACT research for both private companies and governments, they say they are the biggest organization for APPLIED research in Europe

Anyway these two young people Helge Rosť and Torsten A-M must likely be
BEGINNING researchers, because i dont find many previous papers by them,
only I think one by Torsten.
[EDIT: with Kea's help I found more papers by Torsten, he is more senior, has been working in this field 10 years, co-authored with Brans, is writing a book]

Well I didnt know about the Fraunhofer Institutes. You learn something new everyday.

I guess we all know about the famous Fraunhofer who was born in 1787 and invented spectroscopy---what much of atomic physics and astronomy is based on.
http://www.biography.ms/Joseph_von_Fraunhofer.html
It says he was orphaned at age 11, in 1798, so he had to go to work in a workshop, which however collapsed in 1801. Therefore as a young 14-year old Fraunhofer was buried in the remains of a badly constructed Munich lens-grinding factory. This however worked to his advantage, since he was rescued by the Prince of Bavaria who later became Maximilian Joseph the King of Bavaria. This prince was leading the crew digging people out, and he later helped Fraunhofer get time and books to study physics.

marcus Nov26-05 05:06 PM

Quote:

Quote by Kea
Helge is here! Hi!

:smile:

yeah, I saw Helge was online here at PF, so that was what prompted me to start this thread. but it was sA who twigged the paper

here are snapshots of the two guys who wrote the paper, Torsten and Helge
http://mmm.first.fraunhofer.de/de/team/

Kea Nov26-05 07:37 PM

Quote:

Quote by marcus
Anyway these two young people Helge Rosť and Torsten Asselmeyer-M ....

Actually, it seems that Torsten has only recently appended the second part of his surname. See:

http://www.arxiv.org/find/grp_physic...4d475087264912

:smile:

and also, one of the references in the paper

Generation of Source Terms in General Relativity by differential structures
T. Asselmeyer
14 pages
http://www.arxiv.org/abs/gr-qc/9610009

Mike2 Nov26-05 08:42 PM

Are these differential structure related to topologial quantum field theories? Quantum Mechanics is derived from Quantum field theory where we have creation and anhiliation operators for particles. The double slit experiment tell us that there must be something global that influences the path of particles, such that a single particle going through one of the slits seems to take into account whether the other slit is covered or not. But before you can have a particle trajectory you must have particles. Virtual particles seem to pop into and out of existence as part of the zero point energy. They can be made real particles if one of the pair is captured by a horizon. So it seems we are looking for some global mechanism for particle creation in the first place. And the same global topological concerns that give rise to virtual particles to begin with should incorporate some dynamics to account for trajectories of real particles to end with. So I consider what topological entities might give rise to particles. I think in terms of an index theorm or some AdS/CFT effect going on.

When I think of the first particles arising from the tiny, expanding universe, it seems that whatever the mechanism of virtual particle creation, it must some how proceed in a smooth way from a singularity. The first fluxuations would be in the size and location of the entire, tiny spacetime of the universe. There would not be room enough, yet for virtual particles, and the fluxuation would simply be in some degree of freedom in the boundary or overall size of the tiny universe. Perhaps this is the same mechanism that forces the universe to expand. Then as the universe becomes large enough, these fluxuations can include particles that pop in and out of existence as these overall topological entities change.

marcus Nov26-05 09:48 PM

Quote:

Quote by Kea
Actually, it seems that Torsten has only recently appended the second part of his surname.

Good. so he has some halfdozen papers on arxiv, and they go back to 1995.

Helge said that Torsten was writing a book.
Exotic Structures and Physics: Differential Topology and Spacetime Models
I can make better sense of that now that I know he has been working in this general field for 10 years.

Helge Rosť Nov27-05 02:09 AM

Thats right, we think about the topic nearly 15 years. Torstens DS idea is younger (10 years). Since 2 years he could show that the DS should build a Hilbert-space. The Temperley-Lieb-algebra structure of the changes is from this year.

marcus Nov27-05 02:22 AM

Quote:

Quote by Helge Rosť
Thats right, we think about the topic nearly 15 years. Torstens DS idea is younger (10 years). Since 2 years he could show that the DS should build a Hilbert-space. The Temperley-Lieb-algebra structure of the changes is from this year.

it is after midnight and i have to sleep
I hope you return here tomorrow. I will try to have some questions.
I don't understand how a 3D submanif. can represent a particle
but no use explaining now, I will give it another try tomorrow

Helge Rosť Nov27-05 02:34 AM

This (post #6) is correct. We are work at FIRST in Berlin. We have published papers about different topics (Evolutionary Algorithms, Quantum-Hall-effect, Quantum Computing, Computer-stuff like simulation of complex systems ...) e.g. http://www.first.fhg.de/helge.rose/publications
or my home-page http://www.first.fhg.de/helge.rose
We have not published our ideas to QG (except Torstens 1996 paper) until now because - well it is not an easy topic and very explosive. We would like to make the ideas save to form a whole picture. The last steps (Temperly-Lieb-algebra of DS transitions) are appeared this year. This paper is only a first step - so to say the "kinematik" of the theory. We have ideas to the next steps - the dynamics, i.e. the field equation. Only this will complete the picture and hopefully get a new usefull theory. But we think the results from the 0506067-paper are promissing. I think the interessting discussion here will be very fruitfull for this.

Helge Rosť Nov27-05 12:20 PM

Quote:

Quote by marcus
I don't understand how a 3D submanif. can represent a particle

this is non trivial. Could you explain a little more your question.

marcus Nov27-05 12:43 PM

Quote:

Quote by Helge Rosť
this is non trivial. Could you explain a little more your question.

it is too early to ask that question and I should back up a little and pick something earlier, like on page 4, beginning of section III.

"In the introduction we have shown that there is a close relation between the transition of the DS and a singular connection with 3D supports. Such connections are expressed by singular 1-forms with 3D supports."

I am struggling at the very beginning of understanding this. I am familiar with connection being expressed by a 1-form (with values in a somewhat arbitrarily chosen Lie algebra)----unless I am confusing something, this is very usual.

But there is a lot that is new here. I think of an earthquake and a "FAULT-LINE" which is actually a fault surface going deep into the earth, and I try to imagine a 4D analog.

So there is a 3D "FAULT" hypersurface. And somehow the change in DS is closely related to a connection (or a 1-form) defined on this 3D "fault". This 3D thing is the SUPPORT of the 1-form.

It is a set. And when you make the algebra, you are using what looks like it might be ordinary set operations, like UNION and INTERSECTION of these support sets.

It becomes very urgent for me to try to understand how the support set of the 1-form can, in some way, characterize the earthquake that happens when you go from one DeeEss to another DeeEss.

I am a slow learner, it may take days before my brain stops smoking and making sparks and begins to understand this idea of transition of DeeEss.

Helge Rosť Nov27-05 02:24 PM

Quote:

Quote by marcus
Do you have any suggestions of papers to read as preparation for your paper with Torsten?

So far I just see the citations to papers by Brans and by Sladowski (including one that Torsten co-authored with Brans)
are these the best to read or are there also others you might suggest?

The work is distibuted over the math. literature. Torsten and Carl Brans wrote the book to give a review about this field. I think torsten should collect some papers, I will ask him.

I will try to answer your post about 3d-support next morning (at night of your time).

garrett Nov27-05 02:40 PM

This discussion started on another thread, but I thought it best to bring it over here:
Quote:

Quote by Helge Rosť
Hi garrett, thanks for your interest in our paper.
In the physical point of view an atlas is set of reference frames which are needed to describe measurements at different space-time regions. With Einstein all reference frames are physically equal if the charts can be transformed by diffeomorphisms - the charts are compatible.
In 1,2,3 dimensions all charts (reference frames) are compatible. In 4 dimensions you can find one set S1 of charts which are compatible with all other charts in this set. But you can also find a further set S2, were all charts compatible in S2 but with no chart in S1. S1 and S2 are two different representants of two atlases. S1 and S2 belong to two different differential structures.
In mathematics the differential structures are called exotic smooth strctures. It can be shown that for a manifold (e.g. Dim=7 like Milnor) atlases exist which are not compatible (transfromable by diffeomorphisms). It is also known that for a compact 4-manifold the number of non-compatible altlases are countable infinite. But the structure of the set of differential structures was unknown. We have shown that the set of the changes of a differential structure is a Temperley-Lieb algebra and the set of differential structures is a Hilbert-space (Dim H = inf). This is a mathematical fact like: "the number of integers is countable infinite". Torsten is writing a book about Exotic Structures and Physics: Differential Topology and Spacetime Models
At the mean time you may looking for the mathematical papers about "Exotic Structures", but this is hard to cover.

Hi Helge, I want to get something straight that's confusing me. I'm still just learning this stuff. I have a question about what you say above, and from this quote from your paper:
Quote:

Quote by Helge Rosť
As an important fact we will note that there is only one differential structure of any manifold of dimension smaller than four. For all manifolds larger than four dimensions there is only a finite number of possible differential structures, Diff_{dim M}. The following table lists the numbers of differential structures up to dimension 11.
1 1 1 inf 1 1 28 2 8 6 992

Are you really saying these are the number of differential structures for ALL manifolds of these dimensions?
This does agree with the wikipedia entry:
http://en.wikipedia.org/wiki/Differential_structure
Is this what you're saying? Because I think it's either not true, there's some miscommunication, or I'm really messing up.
(And it was a friend who pointed out to me this was a potential problem with your paper, I was just bumbling around confused.)
The table you quote is only true for spheres. Except in n=4 the sphere is not known to have an infinite number of DS, though it might. As a counter example, I read in Brans latest paper that the number of DS is 1 for R^n when n>4.
Could you help clear this up? Or maybe you need to fix your paper? I do really like the main idea.

Kea Nov27-05 03:05 PM

Quote:

Quote by marcus
it may take days before my brain stops smoking and making sparks and begins to understand...

Only days? Marcus, you're a wizard! :wink:


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