Jfy4 comment sparked by Torsten and Helge's new paper

[marcus]

A new paper by Helge Rose and Torsten A-M was recently added to the bibliography and sparked comment. This thread is in case there is any further related comment or discussion.

Torsten and Helge have discussed their idea some with us here at PF Beyond. It is a radical and high-risk idea.
http://arxiv.org/abs/1006.2230
On the geometrization of matter by exotic smoothness
Torsten Asselmeyer-Maluga, Helge Rose
17 pages
(Submitted on 11 Jun 2010)
"Clifford's hypothesis is investigated: A particle is made up of nothing but a distinct type of a space manifold, differing from the surrounding manifold of empty space. It is shown that this distinct space manifold representing matter differs from the surrounding vacuum by the exotic smoothness of its spacetime. The smoothness structure of spacetime can be described by a tree-like subset -- the Casson handle -- consisting of immersed discs and connecting tubes between them. The Weierstrass representation shows that the immersed discs are represented by spinors fulfilling the Dirac equation and leading to a mass-less Dirac term in the Einstein-Hilbert action. The connecting tubes between the discs realize an action term of a gauge field. Both terms are purly geometrical and characterized by the mean curvature of the components of the Casson handle. This gives a good support to Clifford's conjecture that matter is nothing more but an exotic kind of space."

The authors had an interesting conjecture, right at the very end of the paper, on page 15:
==sample excerpt==
"At the end we want to give another interpretation of the Casson handle. Connes [37] showed that by means of the non-commutative geometry the action of the standard model can be reproduced. His model is based on the space M×F where the additional space F is ad hoc and has no relation to the spacetime M.
In our model the space F could be interpreted as an expression of the Casson handle and so of the smoothness of spacetime establishing a deep relation between quantum matter and space."
==endquote==

One of us, Jfy4, commented as follows.

Ahh! recently I have been considering to myself whether it would be possible to construct matter from space using a transformation of sorts. in fact it was 2 days ago that this idea hit me, and then yesterday I come to PF only to find a Paper on that very idea, and a long history of interest in the possibility forming matter out of geometry. My intuition told me that using a discrete set of space-time quanta, one could assemble a wave in much the same way a discrete transform takes quanta inputs and out-puts a continuous wave. A very alluring idea.

This is actually a very broad topic. I guess nearly everybody is aware of the work of Sundance Bilson-Thompson and 4 or 5 others along entirely different lines----still it's getting matter out of spatial geometry/topology features, in that case it was braids and knots in networks.
For instance, several papers by Yidun Wan:
http://arxiv.org/find/hep-th/1/au:+Wan_Y/0/1/0/all/0/1

 

[marcus]

Here's a 2009 paper by Sundance and others:

http://arxiv.org/abs/0903.1376
Particle Topology, Braids, and Braided Belts
Sundance Bilson-Thompson, Jonathan Hackett, Louis H. Kauffman
21 pages, 16 figures
(Submitted on 7 Mar 2009)
"Recent work suggests that topological features of certain quantum gravity theories can be interpreted as particles, matching the known fermions and bosons of the first generation in the Standard Model. This is achieved by identifying topological structures with elements of the framed Artin braid group on three strands, and demonstrating a correspondence between the invariants used to characterise these braids (a braid is a set of non-intersecting curves, that connect one set of N points with another set of N points), and quantities like electric charge, colour charge, and so on. In this paper we show how to manipulate a modified form of framed braids to yield an invariant standard form for sets of isomorphic braids, characterised by a vector of real numbers. This will serve as a basis for more complete discussions of quantum numbers in future work."

Several people here follow related work by other authors, such as X-G Wen at MIT. If someone made a serious effort they could probably come up with several more names of researchers who have investigated the possibility of matter arising as "kinks" in the geometry of space, or as some other feature of geometry or topology.

Anybody should feel free to comment, suggest other names, and give links to other papers along these lines.

 

[qsa]

this thread discuss similar ideas

https://www.physicsforums.com/showthread.php?t=410789

 

[tom.stoer]

I think some comments are in order.

The original idea of braiding was inspired but not based on some known spacetime structure. Only recent papers now try to harmonize braiding with "deformed spin networks", but I still would not agree that braiding can be derived from spin networks.

w/o having studied the Torsten and Helge paper in detail, the approach seems to be totally different; they are using the idea of exotic smoothness. The difference is exactly that braiding uses non-smooth structures.

Regarding NCG I have the impression (= I do not understand enough) that it is based on a somehow effective action approach, saying that spacetime geometry and particle interactions are described by an effective action leaving aside the question what the fundanmental degrees of freedom really are. Perhaps this is not even necessary; perhaps the effective action approach works for arbitrary small but still finite "distances".

Last but not least there are attempts to introduce gravity w/o referring to spacetime (emergent spacetime) whereas here we are discussing exactly the opposite idea namely emerging particles and (gauge) interactions from geometry.

There is now a web of interesting ideas which seem to be somehow related. Currently the combination of these approaches seems to be too complicated. Every single approach is not rich enough to explain all structures whereas the combination of different approaches is not simple enough as there is always some overlapping input: both LQG and NGC have an own picture of spacetime; LQG allows to put matter on top whereas braiding says that matter emerges from some algebraic structures; exotic smoothness still relies on smoothness whereas in LQG a smooth manifold emerges from an algebraic structure in a low-energy limit; ...

A new idea would be to try to understand if exotic smooth structures can emerge from spin networks or spin foams.

 

[Berlin]

I think it is related to Christoph Schiller much criticised work on the "strand model". with reidemeister moved and all. I like his intuitive approach because it can be linked to Verlindes idea on entropic gravity. Not yet thourough physics yet, i know.

berlin

 

[MTd2]

Exotic smoothness just exists in a non trivial way in 4 dimensions and it is in that dimension that happens special things ( http://en.wikipedia.org/wiki/4-manifold ), and which makes it the most complicated to describe among all dimensions.

There are "3 levels" of complications (not exclusive to 4 manifolds, but infinitely more complicated):

-Easy:

2 manifolds can be (topology) homeomorphic and diffeomorphic.

-Medium:

A manifold can be (topology) homeomorphic and but not diffeomorphic to any smooth structure, because one cannot triangulate that manifold.

-Hard:

2 manifolds are homeomorphic but not diffeomorphic, but both of them have differential smooth.

Torsten Asselmeyer-Maluga considers the hard case for a very good reason. General Relativity is a theory about diffeomorphic manifolds. So the Hard case, as the Easy case, are about diffeomorphic manifolds, but unlike this, it can have an infinite variety of differential structures for each topological structure.

The key words important for this thread are "infinite" and "triangulable". If that is triangulable, you can define a spin foam, and infinite, it means that one can use an infinite number of different triangulations for every topology (take as an example the infinite types of topology of a scattering amplitude). So, for example, if you are calculating the transition amplitude of a spin foam, or any other theory that aims to describe quantum gravity, if you ignore exotic smoothness, you are doing it wrong (and people are not even aware of that!).

Now, comparing exotic smoothness with braidings is not so straightforward. There are 2 cases, and just one of them is directly linked to a kind of braidings, which is the one that describe the exotic smoothness as appearing from the failure of h-corbodism (http://en.wikipedia.org/wiki/H-cobordism) You can build that one out of a casson handle. See the description:

http://en.wikipedia.org/wiki/Casson_handle

Braidings involve making tricks with 1d strings with algebra. A casson handle involve trying to do smoothout a 4 manifolds using 2d-strings. So, it is like doing computation with topological string (theory).

There are several more points to this, but it boils down to the sufficient and necessary need of using simultaneously spin foams, LQG, string theory and general relativity at some point and at various levels in order to describe nature.

 

[tom.stoer]

Exotic smoothness just exists in 4 dimensions and it is in that dimension that happens a special things ( http://en.wikipedia.org/wiki/4-manifold ), and which makes it the most complicated to describe among all dimensions.

There exist homeomorphic but not diffeomorphic manifolds in D>4 as well, a famous example are the differential structures of the 7-sphere.

 

[jfy4]

I guess one of my first uncertainties has to do with the existence of matter. Can matter be considered separate from space-time?

Consider the existence of a particle. That particle has a given wave-length/period. Both of these properties are defined using the gravitational field (length,time). And, the observables of this particle (length,period), at least prescribing to quantum space-time, must be composed of discrete quanta of space-time. Therefore, it makes me ask, are there any restrictions on the "shape" of space-time from the continuity of a given wave-form? or, are there restrictions of the wave-forms of matter from the spin-network solutions which may only allow wave-forms of various forms to become manifest?

 

[MTd2]

There exist homeomorphic but not diffeomorphic manifolds in D>4 as well, a famous example are the differential structures of the 7-sphere.

They are not infinite though, and only restricted to spheres, as far as I know. Let me correct the text.




T

 

[tom.stoer]

They are not infinite though, and only restricted to spheres, as far as I know. Let me correct the text.

You are referring especially to exotic Rⁿ. This is possible only for n=4.

The question is if it is somehow possible to embed (compact) exotic manifolds into a non-compact, non-exotic manifold. Or if one would observe a standard smooth manifold locally where the exotic smoothness is "hidden beyond a horizon" or "localized in a certain sense".

 

[MTd2]

You are referring especially to exotic Rⁿ. This is possible only for n=4.

Not really. Casson handles, which are frequently mentioned in Torsten and Helge`s papaers, is exotic and homeomorphic to D2xR2.

http://en.wikipedia.org/wiki/Exotic_R4

 

[tom.stoer]

Not really. Casson handles, which are frequently mentioned in Torsten and Helge`s papaers, is exotic and homeomorphic to D2xR2.

http://en.wikipedia.org/wiki/Exotic_R4

Yes, but it's not Rⁿ. Afaik R⁴ is the only Rⁿ where more than one differential structure exists.

 

[MTd2]

It says in the paragraph that Casson handle are exotic. On this other article:

"Freedman's main theorem about Casson handles states that they are all homeomorphic to D2×R2; or in other words they are topological 2-handles. In general they are not diffeomorphic to D2×R2 as follows from Donaldson's theorem, and there are an uncountable infinite number of different diffeomorphism types of Casson handles"

http://en.wikipedia.org/wiki/Casson_handle

 

[tom.stoer]

Are we talking about the same manifold? R⁴ vs. R²*D²

 

[MTd2]

Not "vs." but one more example that exotic smoothness in 4-manifolds is not trivial.

 

[torsten]

Thanks for the interest on our new paper.

First some things about exotic smoothness:
In higher dimensions (>4) there is only a finite number of exotic smoothness
structures. for instance d=7 with 28 or d=9 with 8
In dimension 4 one has infinitely many: for compact 4-manifolds countable infinite and
for non-compact 4-manifolds uncountable infinite
For compact 4-manifolds one has a procedure to get exotic smoothness by using a knot or link.
In a recent paper I studied the effect of this procedure for quantum gravity
(one can show a quantization of the surface area without LQG)
see http://arxiv.org/abs/1003.5506" (accepted in Class. Quant. Grav.)

The non-compact case was settled by Jerzy Krol and me (see http://arxiv.org/abs/0904.1276" ).
We got a relation between codimension-1 foliations and exotic R^4's.

In dimension 4 there is a surprising result again: there is a one-to-one relation between
triangulations and smoothness structures. Thus exotic smoothness corresponds to some kind
of exotic triangulations. Or, non-equivalent triangulations are able to represent different
smoothness structures. Thus, exotic smoothness is note purely connected with the contiuum but it
can also appear in discrete structures.
Currently I try to understand this and relate them to spin foams etc.

Furthermore we will study the influence of our result for cosmology. There is one problem with geometrical
models of matter: the cosmological expansion has no influence on matter, or matter don't scale
(that's the reason why we notice the expansion). But if matter is a part of the space why it don't scale?
We got the answer by Mostow rigidity: our matter are knot complements (3-sphere minus knot) having the
structure of a hyperbolic 3-manifold. But the scale of a hyperbolic manifold don't change or the
volume of a hyperbolic 3-manifold is a topological invariant (volume quantization?)
Thus our matter model is consistent with this observation.
So, again: fermions are knot complements and bosons are the connecting tubes between the complements.
A knot determines a fermion. Secondly, there are three types of connecting tubes which have to corresponds
to the three gauge interactions.

Now some words about quantization:
Jerzy and me studied a model where we are able to show that exotic smoothness implied quantization.
see http://arxiv.org/abs/1001.0882"
So, our models are possibly quantized...


That's all for now

Torsten

 

[qsa]

So, again: fermions are knot complements and bosons are the connecting tubes between the complements.
A knot determines a fermion. Secondly, there are three types of connecting tubes which have to corresponds to the three gauge interactions.


Torsten

Have you concidered how these tubes interact with other tubes of other particles?

Also from your paper

That means one can give a physical meaning to the components of a fermionic spinor field: The spinor components are the coordinates of spacetime.

Does that mean the spacetime coordinates have no meaning without particles(wave components)?

What about gravity, could it be a tube or what?

 

[tom.stoer]

In dimension 4 there is a surprising result again: there is a one-to-one relation between triangulations and smoothness structures. Thus exotic smoothness corresponds to some kind of exotic triangulations. Or, non-equivalent triangulations are able to represent different smoothness structures. Thus, exotic smoothness is note purely connected with the contiuum but it can also appear in discrete structures. Currently I try to understand this and relate them to spin foams etc.

This is a fascinating idea. I always thought that the naive combination of a geometry-inspired model with a particle inspired model combines too many structure to be simple enough to be true; e.g. LQG plus matter on top is not as elegant as matter emerging from geometry alone.

As far as I understand you the idea is to study spin foams in order to understand exotic triangulations (which I never understood in detail). Does that mean that the naive picture where a triangulation (which is mapped to cells with faces which is then mapped to dual graphs with vertices and edges) is generalized in order to allow graphs which do not correspond to this naive picture = graphs that violate this duality?

 

[torsten]

Have you concidered how these tubes interact with other tubes of other particles?

Also from your paper

That means one can give a physical meaning to the components of a fermionic spinor field: The spinor components are the coordinates of spacetime.

Does that mean the spacetime coordinates have no meaning without particles(wave components)?

What about gravity, could it be a tube or what?

There are two kind of tubes:
the tubes between the fermions are torus bundles and one has three principal types. The isometry classes (i.e. the connecting components of the isometry group) are the bosons.
These tubes corespond to three Thurstons Geometrization classes (Eucledian, NIL, SOL).
The first two classes have each two isometry classes and one has the bosons: Photon, Z0 as well W+ and W-
The last class (SOL) has eight isometry classes corresponding to the 8 gluons. Unfortunately we don't find any further interaction (except gravity).
Gravity is exceptional and corresponds to sphere bundles explaining its universality (one can aways add such a bundle). We try to understand this now.

Does that mean the spacetime coordinates have no meaning without particles(wave components)?:
Honestly, I don't know. But your interpretation is very interesting. Matter is part of the space and thus no space, no matter and vice verse.

 

[torsten]

This is a fascinating idea. I always thought that the naive combination of a geometry-inspired model with a particle inspired model combines too many structure to be simple enough to be true; e.g. LQG plus matter on top is not as elegant as matter emerging from geometry alone.

As far as I understand you the idea is to study spin foams in order to understand exotic triangulations (which I never understood in detail). Does that mean that the naive picture where a triangulation (which is mapped to cells with faces which is then mapped to dual graphs with vertices and edges) is generalized in order to allow graphs which do not correspond to this naive picture = graphs that violate this duality?

Yes something like that.
It is true that not every graph induces a triangulation. What I miss in the spin foam approach is the global look. Usually one argues locally and changes some vertices or edges but the real effect is a more global one. Then one has two classes of changes:
1. the change of the triangulation but the corresponding manifold is topologically the same
2. the change of the topology
the first change transforms a given triangulation into an (what I call) exotic triangulation, i.e. a triangulation corresponding to an exotic smooth manifold. Currently I try to understand these transforms.

The aim must be to (naturally) derive a spin foam model from these data. I'm rather shure to get such a model.

 

[tom.stoer]

Afaik the (local) moves are still not yet fully defined. I remember a recent paper from Rovelli where he discusses a new regularization of the Hamiltonian in LQG which induces new moves (not just the moves Thiemann considered).

The next question is what your "vacuum state" looks like. You can construct spin networks by local moves just as you create the states in the harmonic oscillator. Questions:
- is it possible to "change" the topology induced by the SN?
- is it possible to create a SN that is not dual to a triangulation?
- is it possible to create (or define) exotic triangulations (exotic SNs)?
- are there "superselection sectors" regarding the "exotic-ness" of the trianglations or the topology

 

[jfy4]

Does that mean the spacetime coordinates have no meaning without particles(wave components)?:
Honestly, I don't know. But your interpretation is very interesting. Matter is part of the space and thus no space, no matter and vice verse.

This would be quite a new observation no? We have free fields, and if the gravitational field did not exist without matter, that would certainly not be a classical interpretation. the last statement doesn't seem to go both ways. the first way makes sense to me, if there is no space, there is no matter. but visa verse, if there is no matter, there is no space, at least classically with the free field solutions, does not seem to fit...

I realize you have not committed to anything yet of course, and you are still working on ideas, but would this last statement be a new sort of QG interpretation?

 

[torsten]

Afaik the (local) moves are still not yet fully defined. I remember a recent paper from Rovelli where he discusses a new regularization of the Hamiltonian in LQG which induces new moves (not just the moves Thiemann considered).

The next question is what your "vacuum state" looks like. You can construct spin networks by local moves just as you create the states in the harmonic oscillator. Questions:
- is it possible to "change" the topology induced by the SN?
- is it possible to create a SN that is not dual to a triangulation?
- is it possible to create (or define) exotic triangulations (exotic SNs)?
- are there "superselection sectors" regarding the "exotic-ness" of the trianglations or the topology

Yes to all question. I'm currently occupied by the details.
The first two questions (as far as I understand them) are trivial (SN is rather general graphs).
The third question is my current research.
The last question is by definition: two exotic triangulations are different and not connected by moves (called the Pachner moves)

 

[torsten]

This would be quite a new observation no? We have free fields, and if the gravitational field did not exist without matter, that would certainly not be a classical interpretation. the last statement doesn't seem to go both ways. the first way makes sense to me, if there is no space, there is no matter. but visa verse, if there is no matter, there is no space, at least classically with the free field solutions, does not seem to fit...

I realize you have not committed to anything yet of course, and you are still working on ideas, but would this last statement be a new sort of QG interpretation?

Yes it must be a new interpretation of QG. Exotic smoothness has many to do with QG. We think that it is the main contribution to QG.

 

[tom.stoer]

The last question is by definition: two exotic triangulations are different and not connected by moves (called the Pachner moves)

Thanks! I am not able to answer these questions - but it seems that I am able to ask the right questoions :-)

Regarding the superselection sectors: Question is if they stay disconnected due to some superselction rule, if there is something like "theta vacua" or whatever.

 

[tom.stoer]

Yes it must be a new interpretation of QG. Exotic smoothness has many to do with QG. We think that it is the main contribution to QG.

And it could guide us to answer the question "why four dimensions?"

 

[torsten]

And it could guide us to answer the question "why four dimensions?"

That was my starting motivation to study 4-manifolds and exotic smoothness.

 

[torsten]

Thanks! I am not able to answer these questions - but it seems that I am able to ask the right questoions :-)

Regarding the superselection sectors: Question is if they stay disconnected due to some superselction rule, if there is something like "theta vacua" or whatever.

The questions are very helpful, Thanks a lot!
Yes, there must be something like a theta vacua which is conntected with an invariant for exotic smoothness. I'm currently looking for one.
I have an idea to use a non-linear spinor equation (a la Heisenberg-Pauli). But nothing more is known.

 

[inflector]

I know this thread is six months old, but it is relevant to my research so I thought this would be the most appropriate place for this question:

exotic smoothness

I've googled a bit and read a few papers but I still can't get a good handle on what "exotic smoothness means." The papers seem to assume an understanding of topology and related mathematics that I don't yet have.

Does anyone have a relatively simple definition for "exotic smoothness" suitable for digestion by a layman?

 

[tom.stoer]

Not really.

Exotic smoothness means that you can create a set of diffeomorphic structures each consisting of a set of maps covering the whole manifold where
1) different sets of maps create homeomrphisms such that one can talk about ONE topological manifold but where
b) these different sets of maps create DIFFERENT differentiable manifolds such that one can no longer talk about one differentiable manifold.

You start with a manifold M.
You create different sets of maps M = {m1, m2, ...}; N = {n1, n2, ...}, ...
You construct homeomorphisms between m's n's etc., such that the manifold described by M is topologically identical with the manifold described by N etc.
But there are diffeomorphism within M, N, etc. such that M and N are no longer diffeomorphic.

But I think you already knew this ...