Rosé and A-M: Geometrization of Quantum Mechanics

[marcus]

mmmm hmmm
agreed

 

[Careful]

I am still waiting ... but as a response to the above. No, I don't see why the solved Poincare conjecture (by Freedman in 1982) is important for physics. This conjecture basically says that any compact 4 manifold homotopic to the 4 sphere is homeomorphic to it which limits therefore the number of exotic differential (as well as topological) structures. Can someone explain me WHY one should be interested in compact 4 - D manifolds homotopic to the 4 -sphere anyway ?

 

[Kea]

Quoting from the conclusions of the Brans paper (reference 3):

"The example [of the Schwarzschild singularity and Kruskal coordinates] helps to illustrate that in General Relativity our understanding of the physical significance of a particular metric often undergoes an evolution as various coordinate representations are chosen. In this process, the topology and differentiable structure of the underlying manifold may well change. In other words, as a practical matter, the study of the completion of a locally given metric often involves the construction of the global manifold structure in the process."

Interestingly, Penrose had a great intuition for the importance of these modern methods before they were developed. See for example the book Techniques of Differential Topology in Relativity (1972) Soc. Indust. Appl. Math.

 

[Mike2]

The particles generate source-terms (engery-momentum-tensor) in the grav. field eq. and by this changing the curvature of space-time.

So the addition of each new particle changes the DS and adds a new source-term to the GR equations and increases the curvature. But a given number of particles has a determined curvature of a given DS which may have many different metrics. So what this says is that the curvature is an intrincis property independent of metric? But I thought curvature was determined by the metric. What am I missing?

And the algebra among the many different DS's is a Hilbert space. So if a Hilbert Space exists, then there must simultaneously exist all these DS's and with it the various curvatures. So it would seem that if the zero point energy exists, then there is a Hilbert Space, and so there must be a superposition of DS's and with each a superposition of curvatures. So does the Hilbert space algebra of the DS's translate into a Hilbert Space algebra for the curvature/metrics? If so, it would seem that we now have a quantum theory of geometry/gravity, right?

PS. Are there both positive and negative curvature in this programme?

 

[Helge Rosé]

So the addition of each new particle changes the DS and adds a new source-term to the GR equations and increases the curvature. But a given number of particles has a determined curvature of a given DS which may have many different metrics. So what this says is that the curvature is an intrincis property independent of metric? But I thought curvature was determined by the metric. What am I missing?

If we only consider the 4-MF with its DS (mathematically) - then the DS is compatible with different metrics. But if you also demand the Einstein eq. and putting the DS as a source term in it then you get one metric as solution of E. eq. You may change that metric by diffeomorphisms but this does not change the physics: Einstein eq and DS are invariant wrt diffeomorphisms.

And the algebra among the many different DS's is a Hilbert space. So if a Hilbert Space exists, then there must simultaneously exist all these DS's and with it the various curvatures.

At the end of section 3:

The completion of the algebra $$(\mathcal{T},Tr)$$ is a complex Hilbert space. By fixing the parameter $$\tau$$ of the algebra $$\mathcal{T}$$ to be $$\tau=1/2$$, the completion of $$\mathcal{T}$$ corresponds to the Fock space of fermions in quantum field theory (see [PlyRob:94], chapter 2). That is a remarkable result: The self-adjoint projectors $$e_{k}$$ generate the creation and annihilation operators of the fermions. That means, for $$\tau=1/2$$ the algebra $$\mathcal{T}$$ is the standard Clifford algebra of anti-commutative operators. For the case $$\tau\not=1/2$$, $$\mathcal{T}$$ extends the standard quantum field algebra to a Temperley Lieb algebra.So you have superpositions, yes. (see eq. (19,20))

So it would seem that if the zero point energy exists, then there is a Hilbert Space, and so there must be a superposition of DS's and with each a superposition of curvatures. So does the Hilbert space algebra of the DS's translate into a Hilbert Space algebra for the curvature/metrics? If so, it would seem that we now have a quantum theory of geometry/gravity, right?
PS. Are there both positive and negative curvature in this programme?

Im not sure. The change of DS is expressed by change of connection - not curvature or metric.

The particles are the transition (the difference) between two DS. It is like in "Einsteins lift": the force I feel is because the accelerated frame or grav. field - accelerted frames and grav. field are equivalent.

In our case: curvature is because of matter or transition of DS - matter and DS-transition are equivalent.

So, a DS-Transition - or call it particle creator - modifies the curvature but as source term via the Einstein eq. At the moment, I don't know if there is a direct mapping between DS and curvature, so that the algebraic structure of DS maps into a algebra of curvatures.

Maybe some other can explain this better? (Torsten?)

 

[Careful]

Quoting from the conclusions of the Brans paper (reference 3):
"The example [of the Schwarzschild singularity and Kruskal coordinates] helps to illustrate that in General Relativity our understanding of the physical significance of a particular metric often undergoes an evolution as various coordinate representations are chosen. In this process, the topology and differentiable structure of the underlying manifold may well change. In other words, as a practical matter, the study of the completion of a locally given metric often involves the construction of the global manifold structure in the process."
Interestingly, Penrose had a great intuition for the importance of these modern methods before they were developed. See for example the book Techniques of Differential Topology in Relativity (1972) Soc. Indust. Appl. Math.

A change of differentiable structure when going from Schwarzschild to Kruskal coordinates, mmmm ?? That is not how you should see it: the manifold for the Schwartzschild differentiable structure does not contain the event horizon while the manifold for the Kruskal coordinates does (on the overlap, both differentiable structures are perfectly compatible). However, nothing physical is involved here (and I guess nothing physical happens in Helge's paper either) ! The physical interpretation on the black hole horizon can be equally made using the Schwarzschild coordinates by taking suitable limits of the metric invariants towards the cut-out horizon. Anyway, I pointed out that albeit the pull back of the connection on N is a singular connection on M (which is invariant under coordinate transformations on N); the splitting they make in a ``regular´´ and ``singular´´ part is not intrinsic (with respect to N) at all !

 

[torsten]

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

Hallo, I'm the second author Torsten and try to answer your question.
At first one has to divide the manifolds into 2 classes: compact and non-compact. The table in our paper is only true for compact n-manifolds. In the non-compact case the number of structures may differ. For example, all trivial R^n have only one differential structure for $$n \neq 4$$. The case n<4 is more or less trivial. The higher-dimensional case n>4 was covererd by Stallings 1962 by using the non-compact version of the h-cobordism theorem, the so-called engulfing theorem. By using the h-cobordism theorem, Kervaire and Milnor are able (around 1963) to classify the exotic spheres in dimension n>4. In a serie of papers, Kirby and Siebenman (in the 1970's) extend the work of Lashof, Mazur, Hirsch etc. to show that the number of differential structures of a compact manifold is the same as for the corresponding sphere. Thus, our table is true for all compact manifolds. The case n=4 is open for some "trivial" compact manifolds like $$S^4,S^2\times S^2,{\mathbb C}P^2,...$$. It is verified for more complex manifolds like the K3 surface.
I hope that will resolve your confusion.

 

[torsten]

I am still waiting ... but as a response to the above. No, I don't see why the solved Poincare conjecture (by Freedman in 1982) is important for physics. This conjecture basically says that any compact 4 manifold homotopic to the 4 sphere is homeomorphic to it which limits therefore the number of exotic differential (as well as topological) structures. Can someone explain me WHY one should be interested in compact 4 - D manifolds homotopic to the 4 -sphere anyway ?

Hi, before I react on your other remarks, I will say some words about the Poincare conjecture in dimesnion 4:
Freedman proved in 1982 that any manifold which is homotopic to the 4-sphere then this manifold is homeomorphic to the 4-sphere. As Donaldson showed by a counterexample, the smooth variant of this theorem breaks in dimension 4. Thus, there is the possibility that the 4-sphere has an infinite number of differential structures. By the so-called Gluck construction such possible candidates were constructed but a suitable invariant is missed to distinguish them.

 

[Careful]

Hi, before I react on your other remarks, I will say some words about the Poincare conjecture in dimesnion 4:
Freedman proved in 1982 that any manifold which is homotopic to the 4-sphere then this manifold is homeomorphic to the 4-sphere. As Donaldson showed by a counterexample, the smooth variant of this theorem breaks in dimension 4. Thus, there is the possibility that the 4-sphere has an infinite number of differential structures. By the so-called Gluck construction such possible candidates were constructed but a suitable invariant is missed to distinguish them.

I never claimed otherwise! I simply said that this theorem implies that the number of differentiable structures on any topological compact four manifold homotopic to the 4 sphere is the same as that for the topological four sphere itself (meaning that homotopy is not going to add any other forms of exotism).

 

[torsten]

Hi, I got to page five and have already loads of technical questions/remarks. The authors start by noticing that a differentiable structure carries lot's of topological information and provides as well the necessary mathematical setting to write out the Einstein Field equations. That is certainly correct, ONE differentiable structure actually determines all Betti numbers (by studying critical points of vectorfields). However, the authors are not pleased with the knowledge of the number of multidimensional handles and want to include exotic differentiable structures associated to a topological manifold. Any good motivation for this is lacking; string theorists would actually jump out of the roof since in ten dimensions, only six inequivalent differential structures exist. It would be instructive to UNDERSTAND why in dim 2 and 3 (one is easy to proof) only one differentiable structure exists and what makes four so special, but no such insight is provided. For example: one should know if an explicit algorithm exists for creating such inequivalent types. The authors do suggest in that respect the use of surjective, smooth (between two inequivalent differentiable structures) but not injective mappings, but this is by far not sufficient.

That is difficult to answer and hopefully the following is not to technical. In dimension 2 and 3 the uniqueness of the differential structure can be shown where the problem is attributed to the 1-dimensional case. In 1982 Freedman classifies all topological, simply-connected manifolds to show that this classification mimics the higher-dimensional case. Thus, it is better to look at the higher-dimensional classification of differential structures by using the h-cobordism theorem. The failure of the smooth h-cobordism (Donaldson, 1987) opens the way to show that there are more than one possible differential structure on simply-connected 4-manifolds. For sufficient complicated 4-manifolds there is an explicite construction by Fintushel and Stern using knots and links (see the pages 9 and 10 of our paper for a description). Now, why is dimension 4 so special? The interior of a h-cobordism between two topologically equivalent 4-manifolds M,N consists of 2-/3-handle pairs. All other handles can be killed by using Morse theory (see Milnor, Lectrures on the h-cobordism theorem). These 2-/3-handle pairs can be killed if and only if there is a special embedded disk (the Whitney Disk). But if the disk has self-intersection then this disk ist not embedded. But that happens in dimension 4 by dimensional reasons. In higher dimensions there is no self-intersections and thus such a Whitney disk always exists. In our paper we use these results to consider a smooth map f:M->N between M,N with different differential structures. The singular set is by definition the set where the map f is not invertable or where df=0. So, we start with a mathematically given situation: two topological equivalent 4-manifolds with different differential structures. In the paper we are not dealing with the question to decide wether two 4-manifolds are diffeomorphic or not. That question has to be addressed later.

Some technical comments regarding section II: the definition of a singular set is very strange, one would expect df_x to have rank < 4 and not df_x=0.
Section III deals with pulling back the tangent structures from a differentiable structure N to a differentiable structure M by a mapping f. The authors define the singular CONNECTION one form G associated to f There is not given any rigorous definition of G = f_{*}^{-1} d f_{*} since this expression is meaningless where df_x has rank < 4 (since
f_{*}^{-1} does not exist there), so at least one should do this in the distributional sense wrt to a volume form determined by an atlas in the differentiable structure.

That is correct. The paper is written for physicists and we are not dealing with the theory of currents which is necessary to understand such singular objects. I know that problem. The whole idea based on the work of Harvey and Lawson about singular bundle maps. The map df:TM->TN is such a singular bundle map and Harvey and Lawson define a current Div(df) which they call a divisor. It is true that the expression in the given form as G = f_{*}^{-1} d f_{*} is mathematically not rigorous but it can be defined by the theory of Harvey and Lawson (see HL, A theory of characteristic currents associated with a singular connection, Asterisque 213, 1993). In the following we consider the cohomology class associated to the current (in the sense of Federer, Geometric measure theory). Thus, a diffeomorphism of M AND/OR N does not change this class. Secondly, by a result of Freedman, two homotopy-equivalent 4-manifolds are homeomorphic. Thus, the cohomology classes are connected to the differential strcuture. That agrees also with the results of Seiberg-Witten theory where special cohomology classes (called basic classes) determine the differential structure. What I want to say is that one can give the experssion G a sense by using the theory of Harvey and Lawson. I see that we forgot to write it.

A second comment is that G is not anything intrinsic - it is just a (distributional) gauge term and NOT a one form. Therefore, it is an uninteresting object related to a specific mapping f and to a choice of coordinate systems on M AND N (and especially this last property is very bad) - admittedly, it depends slightly upon the change of differentiable structure (through f) and does give rise to a distributional source in the energy momentum tensor. Nevertheless, the authors want to do something with it and give two inequivalent definitions for G; one based on nontrivial connections and one on the flat connection.

Yes it is right that the G in that form depends on the differential structure. That is the reason why we take the trace of the connection or curvature to exclude the dependence of the diffeomorpism. We always use in the paper the fact that a cohomology class can be associated to a current and vice versa.

The definition of the support is fine (since one wants to single out the singular part). With the definition of the product, something strange happens: the authors seem to consider G as a ONE FORM (which it isn't) and POSTULATE that the singular support of G is a three manifold and want to associate a specific generator of the first fundamental group to it. Poincare duality as far as I know is a duality between cell complexes of dimension k and n-k or homology classes of dimensions k and n-k, and this is clearly not the case. What the authors seem to allude to is the duality between the first homotopy class and the first homology class, which is the de Rahm duality and this could be only appropriate in case the singular support of G is a three manifold but still there is NO CANONICAL ONE FORM given, which is the other essential part of de Rahm theory. The same comment applies to the use Seifert theory; this COULD be only meaningful when the singular support of G is a three manifold, which is NOT necessarily the case (for a generic surjective, non injective, smooth f, the singular support could not even be a manifold) - the authors should provide a theorem that this is so. The latter is necessary since the theory of knots makes only sense in three dimensions (and M is a four dimensional manifold).
I think these issues need clarification otherwhise it seems to go wrong from the beginning...

It is not necessary to consider G as a one form. You can also consider G as a current with support $$\Sigma$$. Let f:M->N be a singular map. Now I will say some words about the structure of $$\Sigma$$. For that purpose, we have to say some words about the theory of singular maps. In topology, we are only interested in such topological characteristics like intersection points which are coupled to the question when two sub-manifolds intersect transversal. In most cases that happens and
we are done, but according to Sard's theorem a smooth mapping $$f:X\to Y$$ between smooth manifolds $$X,Y$$ has a set of critical
values of f of measure zero. That means, there are some (countable
many) cases where we don't get a transversal intersection between
sub-manifolds represented by some map $$f:X\to Y$$. The question is
now: which deformation of the smooth map f to $$\tilde{f}$$ given by
a deformation of the smooth manifolds $$X,Y$$ eliminates the critical
(or singular) values of f. Such a procedure is called unfolding of f and Hironaka proves the general theorem that for every singular map f there is a
sequence of operations which unfolds f. These operations are
usually called blow-up and blow-down. In our case $$f:M\to M'$$ a
blow-up leads to a map $$\tilde{f}:M\#{\mathbb C}P^2\to M'$$ and a
blow-down to $$\tilde{f}:M\#\overline{{\mathbb C}P}^2\to M'$$.
Hironakas theorem means that the unfolding of f leads to a
diffeomorphism
$$\tilde{f}:M\underbrace{\#{\mathbb C}P^2\#\cdots\#{\mathbb C}P^2}_n\# \underbrace{\overline{{\mathbb C}P}^2\cdots\#\overline{{\mathbb C}P}^2}_m \to M'$$
and by using the diffeomorphism (see Kirby, Topology of 4-manifolds)
$$(S^2\times S^2)\#{\mathbb C}P^2={\mathbb C}P^2\#\overline{{\mathbb C}P}^2\#{\mathbb C}P^2$$
we obtain a diffeomorphism
$$\tilde{f}:M\underbrace{\#S^2\times S^2\cdots\#S^2\times S^2}_m\# \underbrace{\#{\mathbb C}P^2\#\cdots\#{\mathbb C}P^2}_{n-m}\to M'$$
where we assume w.l.o.g. $$m<n$$. But that is nothing than a weaker
version of the famous theorem of Wall about diffeomorphisms between
4-manifolds (see Kirby). A very important concept is the
stable mapping. Let $$f\in C^\infty(X,Y)$$ be a smooth mapping $$f:X\to Y$$. Then f is stable if there is a neighborhood $$W_f$$ of f in
$$C^\infty(X,Y)$$ (we use the compact-open topology for that space)
such that each $$f'$$ in $$W_f$$ is equivalent to f. According to
Mather stable smooth mappings between 4-manifolds are
dense in the set of smooth mappings. Thus according to Stingley
(see the phd thesis under supervisition of Lawson) one has to focus on that particular subset to study
maps between homeomorphic but non-diffeomorphic 4-manifolds.
Locally such maps are given by stable maps between $${\mathbb R}^4\to{\mathbb R}^4$$, where there is two types: 2 maps (rank 2
singularities) with a 2-dimensional singular subset and 5 maps
(Morin singularities or rank 3 singularities) with a 3-dimensional
singular subset. Stingley extends this result to
smooth 4-manifolds and shows that the rank 2 singularities can be
killed by an isotopy for maps $$f:M\to M'$$ between two homeomorphic
but non-diffeomorphic 4-manifolds. Thus we are left with the rank 3 singularities. Furthermore the corresponding manifold is closed. That supports the use of Seifert theory.
That agrees with a result of Freedman, Hsiang and Stong. They analyse the failure of the smooth h-cobordism and prove a structure theorem. Then the h-cobordism can be divided into two parts: a trivial h-cobordism inducing the homeomorphism between the two manifolds and a subcobordism between two contractable submanifolds A1, A2 of M and N, respectively. The boundary of this submanifolds A1,A2 are homology 3-spheres (see Freedman, 1982). By the usual association between critical points of Morse functions and cobordism, it was shown (I forgot the reference, maybe Milnor) that a singular map and the cobordism are associated to each other.

Some words about the Poincare duality. Yes you are right. I use a combination of the Poincare duality to relate the k form to an n-k cycle. Then I use the duality of an k cycle and an n-k cycle for a compact manifold. The element of the fundamental group is related to homology class by using the Hurewicz isomorphism, i.e. I can only relate the elements of the fundamental group which are not belong to the commutator subgroup.
Hopefully you are satisfied with that explanation. Otherwise please write.

 

[Careful]

** In our paper we use these results to consider a smooth map f:M->N between M,N with different differential structures. The singular set is by definition the set where the map f is not invertable or where df=0. **

?? Not invertible means rank df_x < 4 and not necessarily df_x = 0.

**I know that problem. The whole idea based on the work of Harvey and Lawson about singular bundle maps. The map df:TM->TN is such a singular bundle map and Harvey and Lawson define a current Div(df) which they call a divisor. **

And you can define this current without introducing a background metric on M (I do not believe that) ?? Please give this definition (I do not have easy acces to the book of Federer).

** It is true that the expression in the given form as G = f_{*}^{-1} d f_{*} is mathematically not rigorous but it can be defined by the theory of Harvey and Lawson (see HL, A theory of characteristic currents associated with a singular connection, Asterisque 213, 1993). **

Why even bother defining G since it depends upon a choice of coordinates on N anyway ? Shouldn't one concentrate on the pull back of the covariant derivative ?

** In the following we consider the cohomology class associated to the current (in the sense of Federer, Geometric measure theory). Thus, a diffeomorphism of M AND/OR N does not change this class. **

Sure but you still need to tell me how to define the current.

**What I want to say is that one can give the experssion G a sense by using the theory of Harvey and Lawson. I see that we forgot to write it. **

But still G depends on the particular coordinate chart in N, even if I trace it in M (that is actually easily seen on the regular part of G- I do not even need to bother about the singular part). The rest of the message sounds acceptable (though I did not know many of these details).

Cheers,

Careful

 

[torsten]

People may be interested in an old thread on the spin foam connection
http://www.lns.cornell.edu/spr/2003-10/msg0055272.html

Hi, I'm the second author of the paper. Yes I find that work interesting. The spin foam approach is not so far away from our approach. For instance, Rovelli and Pietri showed by using Recoupling theory that the scalar product of the Loop quantum gravity is the trace of the Temperley-Lieb algebra.
Furthermore, one can remark that a PL structure in 4 dimensions is equivalent to a DIFF structure. That means that two non-diffeomorphic, but homeomorphic 4-manifolds differ also by the combinatorical structure. Thus, the spin-foam model of a 4-manifold describes the differential structure. But I think I don't tell anything new. (see Pfeiffers paper)

 

[Kea]

The spin foam approach is not so far away from our approach.

Hi Torsten

I am very pleased to meet you. I am one of the authors of
http://www.arxiv.org/abs/gr-qc/0306079

 

[torsten]

** In our paper we use these results to consider a smooth map f:M->N between M,N with different differential structures. The singular set is by definition the set where the map f is not invertable or where df=0. **
?? Not invertible means rank df_x < 4 and not necessarily df_x = 0.
**I know that problem. The whole idea based on the work of Harvey and Lawson about singular bundle maps. The map df:TM->TN is such a singular bundle map and Harvey and Lawson define a current Div(df) which they call a divisor. **
And you can define this current without introducing a background metric on M (I do not believe that) ?? Please give this definition (I do not have easy acces to the book of Federer).
** It is true that the expression in the given form as G = f_{*}^{-1} d f_{*} is mathematically not rigorous but it can be defined by the theory of Harvey and Lawson (see HL, A theory of characteristic currents associated with a singular connection, Asterisque 213, 1993). **
Why even bother defining G since it depends upon a choice of coordinates on N anyway ? Shouldn't one concentrate on the pull back of the covariant derivative ?
** In the following we consider the cohomology class associated to the current (in the sense of Federer, Geometric measure theory). Thus, a diffeomorphism of M AND/OR N does not change this class. **
Sure but you still need to tell me how to define the current.
**What I want to say is that one can give the experssion G a sense by using the theory of Harvey and Lawson. I see that we forgot to write it. **
But still G depends on the particular coordinate chart in N, even if I trace it in M (that is actually easily seen on the regular part of G- I do not even need to bother about the singular part). The rest of the message sounds acceptable (though I did not know many of these details).
Cheers,
Careful

OK I see the point. You are right. The pure definition of the current needs a metric for M and N but the definition of the sum and product don't depend on the particular metric. The intersection between sets and the linking of the curves don't depend on the metric.

But I have also a question: Why do you think that the differential structure on a 4-manifold has nothing to do with physics?
I think it is interesting for you that we are able to derive the Temperley-Lieb algebra by using the h-cobordism of 4-manifolds and the theory of Casson handles. The connection approach is not the only way to quantum mechanics.

 

[torsten]

Hi Torsten
I am very pleased to meet you. I am one of the authors of
http://www.arxiv.org/abs/gr-qc/0306079

Hi Kea,
I am also very pleased to meet you. I think I know your work and find it very interesting. It remembers me on a construction in singularity, called the cone of a singularity. In that construction, the singularities of a function look like a cone. Thus, the change of a 3-manifold (visualized as a 4-dimensional cobordism) cane be visualized as a conical singularity of some function (related to the Morse function of the cobordism). The resolution of the singularity should end with a smooth 4-manifold but with non-trivial topology.
Unfortunately, I can't fill in all details.

But maybe more later

Torsten

 

[Careful]

OK I see the point. You are right. The pure definition of the current needs a metric for M and N but the definition of the sum and product don't depend on the particular metric. The intersection between sets and the linking of the curves don't depend on the metric.
But I have also a question: Why do you think that the differential structure on a 4-manifold has nothing to do with physics?
I think it is interesting for you that we are able to derive the Temperley-Lieb algebra by using the h-cobordism of 4-manifolds and the theory of Casson handles. The connection approach is not the only way to quantum mechanics.

As a general comment, I really think you should consider rewriting the paper: you are trying to convey many ideas and the reader has virtually no chance at all to judge *fairly* wether they make sense or not (unless he/she reads a bunch of technical papers). A few important concepts should be made clear (some of which you explained already): (a) why considering only maps f which a singular support which is a closed three manifold (and why this particular notion of singular) (b) why do you insist upon G while it is only the pull back of the entire *covariant* derivative which makes sense as a distributional covariant derivative on M ? (c) What is the precise definition of a singular connection on M? (d) give a simple detailed example which makes this all clear ! (e) How do these one forms show up which you attach to f (I guess you could take the trace of the pull back of the covariant derivative - but this is a singular object again on the entire 3 - manifold, how does Poincare duality apply for this ?)? These are to my feeling things which need to be made more precise. I believe the rest follows then more naturally, but these things form the crux of your approach and they should be clear (and I would like to see points b,c,d and e answered one day). Another remark is: you have distributional connections; but how does this translate in the energy momentum tensor? Is there really a physical part added to the Einstein equations (see my remark in a previous post)? It is possible to have a bad choice of coordinates for the connection, but still have perfectly well defined (smooth) curvature invariants in the same coordinate system (Friedmann versus Kruskal). Are you really adding a PHYSICAL singularity here in the background differentiable structure? (M)

Why should a change of differentiable structure have something to do with physics ?? A bunch of remarks:
(a) you can obtain your singularities without considering changes of diff structure (moreover, your singularities have a volume - in contrast to the familiar black hole singularities)
(b) where, in your formalism do you obtain that the singular 3 manifolds are SPACELIKE (an essential ingredient in LQG?)
(c) assuming that you can solve (b) and that you have singular spacelike three manifolds; but how does this fit the picture that matter cuts out a four dimensional singular worldTUBE in your framework? (even classically)
(d) It seems to me that even classically you will need to have equations which allow for a change of differentiable structure (for example two blobs of matter clutting together); how is this possible within the framework of differential equations which live on ONE differentiable structure?
(e) Let me note that in LQG : (i) the Hamiltonian constraint is still an unsolved (unsolvable) problem (ii) therefore it is not known at all whether area, volume and length operators have a discrete spectrum on the PHYSICAL Hilbert space (iii) it is not known in my knowledge how to get (spatial) curvature out on spin networks
(f) still lots of comments, will come back later

 

[torsten]

As a general comment, I really think you should consider rewriting the paper: you are trying to convey many ideas and the reader has virtually no chance at all to judge *fairly* wether they make sense or not (unless he/she reads a bunch of technical papers). A few important concepts should be made clear (some of which you explained already): (a) why considering only maps f which a singular support which is a closed three manifold (and why this particular notion of singular) (b) why do you insist upon G while it is only the pull back of the entire *covariant* derivative which makes sense as a distributional covariant derivative on M ? (c) What is the precise definition of a singular connection on M? (d) give a simple detailed example which makes this all clear ! (e) How do these one forms show up which you attach to f (I guess you could take the trace of the pull back of the covariant derivative - but this is a singular object again on the entire 3 - manifold, how does Poincare duality apply for this ?)? These are to my feeling things which need to be made more precise. I believe the rest follows then more naturally, but these things form the crux of your approach and they should be clear (and I would like to see points b,c,d and e answered one day). Another remark is: you have distributional connections; but how does this translate in the energy momentum tensor? Is there really a physical part added to the Einstein equations (see my remark in a previous post)? It is possible to have a bad choice of coordinates for the connection, but still have perfectly well defined (smooth) curvature invariants in the same coordinate system (Friedmann versus Kruskal). Are you really adding a PHYSICAL singularity here in the background differentiable structure? (M)
Why should a change of differentiable structure have something to do with physics ?? A bunch of remarks:
(a) you can obtain your singularities without considering changes of diff structure (moreover, your singularities have a volume - in contrast to the familiar black hole singularities)
(b) where, in your formalism do you obtain that the singular 3 manifolds are SPACELIKE (an essential ingredient in LQG?)
(c) assuming that you can solve (b) and that you have singular spacelike three manifolds; but how does this fit the picture that matter cuts out a four dimensional singular worldTUBE in your framework? (even classically)
(d) It seems to me that even classically you will need to have equations which allow for a change of differentiable structure (for example two blobs of matter clutting together); how is this possible within the framework of differential equations which live on ONE differentiable structure?
(e) Let me note that in LQG : (i) the Hamiltonian constraint is still an unsolved (unsolvable) problem (ii) therefore it is not known at all whether area, volume and length operators have a discrete spectrum on the PHYSICAL Hilbert space (iii) it is not known in my knowledge how to get (spatial) curvature out on spin networks
(f) still lots of comments, will come back later

Time flies and thus I don't had the time to answer carefully but at first some comments:
(a) The support is a 3-manifold as shown in the first reply by using singularity theory.
(b) Yes, you are right that's the idea behind the singular connection: see it as pullback connection.
(c) see (b)
(d) the example will be given later
(e) accroding to Harvey and Lawson the form is L^1_loc integrable

Some words about the motivation: We were looking for a principle as an extension of the general relativity principle which can explain the appearance of matter too. All the other proposals have to introduce something like the fibration of space-time and connectiosn on them or strings etc. We need only one principle and obtain the 3+1 splitting, the field operator algebra etc.

More later Torsten

 

[Mike2]

Time flies and thus I don't had the time to answer carefully but at first some comments:
(a) The support is a 3-manifold as shown in the first reply by using singularity theory.
(b) Yes, you are right that's the idea behind the singular connection: see it as pullback connection.
(c) see (b)
(d) the example will be given later
(e) accroding to Harvey and Lawson the form is L^1_loc integrable
Some words about the motivation: We were looking for a principle as an extension of the general relativity principle which can explain the appearance of matter too. All the other proposals have to introduce something like the fibration of space-time and connectiosn on them or strings etc. We need only one principle and obtain the 3+1 splitting, the field operator algebra etc.
More later Torsten

Perhaps these issues are covered in your book, I hope? What would be the prerequistes for your book anyway? Thanks.

 

[Careful]

Perhaps these issues are covered in your book, I hope? What would be the prerequistes for your book anyway? Thanks.

I am sceptic; I zapped through the publications of Brans since 1992 on the Arxiv and guess what?? In 13 years of speculation about the possible relevance of exotic differentiable structures in relativity, NOBODY even managed to produce a SINGLE example which produces a PHYSICAL source term in the Einstein equations (this was my objection in my last two posts). Sorry that I say this, but GOOD math and phys ALWAYS start with a solid example; Thorston is throwing mathematical concepts around our ears in a paper which has a sloppy style, does not produce one single example, does not even *define* the main concepts and doesn't provide examples to illustrate these as well (which do exist I presume in the literature)... How are we supposed to make sense of this ?

 

[selfAdjoint]

I am sceptic; I zapped through the publications of Brans since 1992 on the Arxiv and guess what?? In 13 years of speculation about the possible relevance of exotic differentiable structures in relativity, NOBODY even managed to produce a SINGLE example which produces a PHYSICAL source term in the Einstein equations (this was my objection in my last two posts). Sorry that I say this, but GOOD math and phys ALWAYS start with a solid example; Thorston is throwing mathematical concepts around our ears in a paper which has a sloppy style, does not produce one single example, does not even *define* the main concepts and doesn't provide examples to illustrate these as well (which do exist I presume in the literature)... How are we supposed to make sense of this ?

Bottom line of this rant: no examples. The fact that they show the changes of differential structure form an algebra, and they can quantize this with the GNS procedure, and even show fermion behavior, counts as nothing for you because they don't give an example. Bah!

 

[Careful]

Bottom line of this rant: no examples. The fact that they show the changes of differential structure form an algebra, and they can quantize this with the GNS procedure, and even show fermion behavior, counts as nothing for you because they don't give an example. Bah!

Sorry, but this is not serious anymore ! In my humble opinion, the theory is void in the sense that changing differentiable structure does not give rise to any physical effect. Moreover, there is a serious gap in the presentation which I outlined already. The claimed results of the paper are actually NOT surprising if I were to start from ``singular´´ connections associated to particular knots in closed three spaces since these existed already (by the way, notice that in LQG these results are obtained WITHOUT the introduction of matter). What standard of science is one proclaiming when it is even too much to ask from an author that (a) s/he presents the material in a sufficiently self contained way (b) s/he can show that her/his theory is nontrivial by presenting an example after 13 YEARS of speculation ?
If I am wrong then I hope that the authors take my suggestions seriously which is IMO also for the benifit of promotion of their work.
Bah !

 

[torsten]

I am sceptic; I zapped through the publications of Brans since 1992 on the Arxiv and guess what?? In 13 years of speculation about the possible relevance of exotic differentiable structures in relativity, NOBODY even managed to produce a SINGLE example which produces a PHYSICAL source term in the Einstein equations (this was my objection in my last two posts). Sorry that I say this, but GOOD math and phys ALWAYS start with a solid example; Thorston is throwing mathematical concepts around our ears in a paper which has a sloppy style, does not produce one single example, does not even *define* the main concepts and doesn't provide examples to illustrate these as well (which do exist I presume in the literature)... How are we supposed to make sense of this ?

Dear careful,
I understand your comments and agree with you in some points but it is not fair if you mix stylistic and content related arguments. We appreciate the constructive part of your critique.
The examples can be found in section 4 of our paper (pages 9-10). The calculation of explicite expressions for the source term can only be done by using more sophisticated methods. To get a view at these terms look at my paper gr-qc/9610009.
Now to your reproval that there is no physics in the paper. The misunderstanding between us is maybe rooted in your remark that we have a gauge term. If that were true then you are right and we don't describe nothing. Our paper is *sloppy* because we don't make clear enough the difference between general relativity and gauge theory (what you have in your mind).
In a principle bundle P over M with structure group G, the connection is globally defined by the splitting between the vertical and horizontal subspace of the tangent bundle TP. By using a section you can pullback that connection to the manifold and one obtains the gauge potential A(x). A gauge transformation g is a map M->G which transforms A by $$g^{-1} A g + g^{-1}dg$$. This map g has nothing to do with the underlying manifold. Especially the curvature (or field strength) $$F=dA+A\wedge A$$ changes by $$g^{-1}Fg$$. Ok fine. In contrast in general relativity we have to consider gauge transformations which are diffeomorphisms of the underlying manifold. Let $$G$$ be a diffeomorphism of the manifold then the Levi Civita connection $$\Gamma$$ transforms as $$G^{-1}\Gamma G+G^{-1}dG$$. By the relation $$d(G^{-1}dG)+G^{-1}dG\wedge G^{-1}dG=0$$, the diffeomorphism don't contribute to the curvature which corresponds to the connection. It is possible to prove that all changes of the form $$G^{-1}dG$$ which don't produce an additional curvature are induced by a diffeomorphism of the underlying manifold. Thus the diffeomorphism can be seen as something like a gauge transformation but it isn't. In an ordinary gauge transformation we change the coordinate description of the bundle but don't change the structure of the bundle itself. Here we change the coordinates of the manifold which changes the description of the tangent bundle.
Now we take two inequivalent bundles A,B which are tangent bundles of two 4-manifolds, A=TM,B=TN. By definition the two manifolds are not equal. Now if we assume that M and N are homeomorphic then the two bundles correspond to two different differential strcutures. Then the bundle map a:A->B is connected to the smooth map f:M->N. Now we can use the map a to pullback the connection on N to a connection on M. A simple calculation was done in the paper to construct an expression which looks like $$f^{-1}df$$. BUT that expression is not a gaue transformation because f is not a diffeomorphism and it cannot be made to a diffeomorphism. All the critical points of f cannot be removed by a diffeomorphism of M and/or N. But then we don't have the relation $$d(f^{-1}df)+f^{-1}df\wedge f^{-1}df\not=0$$ instead we produce an additional curvature whioch cannot be *gauged* away by a diffeomorphism. So, what we obtain is a physical, measurable effect and that effect has to be discussed.

 

[Careful]

Now to your reproval that there is no physics in the paper. The misunderstanding between us is maybe rooted in your remark that we have a gauge term. If that were true then you are right and we don't describe nothing. Our paper is *sloppy* because we don't make clear enough the difference between general relativity and gauge theory (what you have in your mind).
In a principle bundle P over M with structure group G, the connection is globally defined by the splitting between the vertical and horizontal subspace of the tangent bundle TP. By using a section you can pullback that connection to the manifold and one obtains the gauge potential A(x). A gauge transformation g is a map M->G which transforms A by $$g^{-1} A g + g^{-1}dg$$. This map g has nothing to do with the underlying manifold. Especially the curvature (or field strength) $$F=dA+A\wedge A$$ changes by $$g^{-1}Fg$$. Ok fine. In contrast in general relativity we have to consider gauge transformations which are diffeomorphisms of the underlying manifold. Let $$G$$ be a diffeomorphism of the manifold then the Levi Civita connection $$\Gamma$$ transforms as $$G^{-1}\Gamma G+G^{-1}dG$$. By the relation $$d(G^{-1}dG)+G^{-1}dG\wedge G^{-1}dG=0$$, the diffeomorphism don't contribute to the curvature which corresponds to the connection. It is possible to prove that all changes of the form $$G^{-1}dG$$ which don't produce an additional curvature are induced by a diffeomorphism of the underlying manifold. Thus the diffeomorphism can be seen as something like a gauge transformation but it isn't. In an ordinary gauge transformation we change the coordinate description of the bundle but don't change the structure of the bundle itself. Here we change the coordinates of the manifold which changes the description of the tangent bundle.
Now we take two inequivalent bundles A,B which are tangent bundles of two 4-manifolds, A=TM,B=TN. By definition the two manifolds are not equal. Now if we assume that M and N are homeomorphic then the two bundles correspond to two different differential strcutures. Then the bundle map a:A->B is connected to the smooth map f:M->N. Now we can use the map a to pullback the connection on N to a connection on M. A simple calculation was done in the paper to construct an expression which looks like $$f^{-1}df$$. BUT that expression is not a gaue transformation because f is not a diffeomorphism and it cannot be made to a diffeomorphism. All the critical points of f cannot be removed by a diffeomorphism of M and/or N. But then we don't have the relation $$d(f^{-1}df)+f^{-1}df\wedge f^{-1}df\not=0$$ instead we produce an additional curvature whioch cannot be *gauged* away by a diffeomorphism. So, what we obtain is a physical, measurable effect and that effect has to be discussed.

Sigh, I obviously understood all that from the beginning. What I was asking you all along was to DEFINE this singular connection and curvature. If you would be so kind?

 

[Helge Rosé]

Sorry, but this is not serious anymore ! In my humble opinion, the theory is void in the sense that changing differentiable structure does not give rise to any physical effect.
Moreover, there is a serious gap in the presentation which I outlined already. The claimed results of the paper are actually NOT surprising if I were to start from ``singular´´ connections associated to particular knots in closed three spaces since these existed already (by the way, notice that in LQG these results are obtained WITHOUT the introduction of matter). What standard of science is one proclaiming when it is even too much to ask from an author that (a) s/he presents the material in a sufficiently self contained way (b) s/he can show that her/his theory is nontrivial by presenting an example after 13 YEARS of speculation ?
If I am wrong then I hope that the authors take my suggestions seriously which is IMO also for the benifit of promotion of their work.

Dear careful,

I hope Torstens reply could help to resolve some misunderstandings. I want to add some thoughts.

I think the main reason for the confusion is the mix-up of two concepts: gauge-invariance and covariance.

In all gauge-theories we find non gauge-invariant objects which are nevertheless very fundamental for this theory - the connection A represents interaction and is not gauge-invariant. Would you say also in this case the theory is unphysically and blame the QFT?

I think it is very difficult to decide what is physical and what not - or would you say the gauge-principle is very intuitive and has a direct physical meaning? I think it has lead to a successful theory and that's the reason why you and other people are not in doubt about it - but this is not a physical reason.

As Torsten has written our paper deals with 4-manifold and its tangential bundles and not with gaugeing and fiber bundels. In space-time we require covariance of the equations and not gauge-invariance. Unlike gauge-invariance, covariance is better understandable from a physical point of view: It represents the independency from the reference frames and reference frames are changed by diffeomorphisms.

All solution of Einsteins are invariant wrt diffeomorphisms, i.e. the physically solutions are the equivalence classes wrt diffeomorphisms of space-time. You can make a transition from one class of solutions to another by putting a source term in this equation - i.e. the source terms - matter - causes transitions between the classes of solutions.

Now, what's about the reference frames? You can also build equivalence classes of reference frames - the differential structures of space-time. As torsten has shown in his first paper a transition between DS produces an additional connection and this a source term in Einsteins eq.

So we have the following situation:

_______________ Solutions of E-eq._______________Diff. structures of space-time
invariance: ______classes of physically solutions ____classes of diffm. reference frames
transtion:________source term in E-eq.____________addition connection (= source term)
physical meaning:_matter_______________________ ?

We think that the physical meaning of the transitions of DS is the same as the transition between classes of solutions of E-eq.: Matter
This is no presumtion of the paper this is a consequence.

We describe this additional connection by a form and call it singular. Maybe you are missleaded by this term. It is not a singularity like in QFT - which you totally accept, I guess, although it causes very unphysically conclusions. The form has a support consitsts of the critical points (also called singular) of the map f - we could this form also call critical form - if you prefer.

The point is, this form $$\phi$$ is only non-vanishing on the critical set and represents by this the difference of DS between N, M. If N, M are diffeomorph there is no critical set of map f and $$\phi$$ vanishing erverywhere. But if N, M have different DS then the critical set of f cannot be empty and $$\phi$$ has a non-empty support. You can change $$\phi$$ by a coordinate transformation but you can not make the support empty - in no coordinate system.

This fact is expressed by the curvature of $$\phi$$: if $$\phi$$ is generated by a non-diffeomorphism f between N,M then the curvature is not zero and you can not make it zero by no coordinate transformation. I.e. $$\phi$$ expressed the a difference in DS of N, M - independent from the choice of coordinate systems (i.e. if you don't leave the respective equivalence class - DS).

 

[selfAdjoint]

Thank you for this clear explanation, Helge. I don't know what's got careful's underwear in a tangle; he seems to have more animus to the very idea of diffeomeorphism change -> curvature -> matter than just hostility to your paper. His refusal to address your fine achievements and insistence on criticising from the meta level (no examples..) suggests that.

I am sure you and Torsten don't object to having your presentation in the paper critiqued. That's what a referee would do after all, and it's generally considered a positive if painful aspect of publishing. But this almost foaming at the mouth reaction is over the top.

 

[marcus]

Hi Helge, your and torsten posts are helpful. I like the words "additional connection" and also that this extra connection is a source term---like the matter in classical Einst. eqn.----and in my own head I am calling it the "extra connection" or the "critical connection" because it lives on the critical set. I still don't adequately understand but that is all right, these things take time. Nevertheless the ideas are new and exciting. I will try to improve your table using the simple "CODE" symbol we have at this forum, where you put [kode] and [/kode] around what you want to be in the table----but spell it code.
Helge you say

The form has a support which consists of the critical points of the map f - we could call this form also critical form - if you prefer.

I like this way to call it, and I wish to try it out, to hear how it sounds----so I will tentatively EDIT your post, as an experiment in rewording.

... You can make a transition from one class of solutions to another by putting a source term in this equation - i.e. the source term - matter - causes transitions between the classes of solutions.
Now, what about the reference frames? You can also build equivalence classes of reference frames - the differential structures of space-time. As torsten has shown in his first paper, a transition between DS produces an additional connection and this [is analogous to?] a source term in Einstein's eq.
So we have the following situation:

basic objects:      solutions of E-eq.          diff. structures of space-time
invariance:       classes of physical solutions     classes of  DS
transition:         source term in E-eq.        critical connection 
physical meaning:   matter                      ?

We think that the physical meaning of the transitions of DS is the same as the transition between classes of solutions of E-eq.: Matter
This is no presumption of the paper this is a consequence.
We describe this additional connection by a form and call it critical. The form has a support which consists of the critical points of the map f.
The point is, this form $$\phi$$ is only non-vanishing on the critical set and represents by this the difference of DS between N, M. If N, M are diffeomorphic, there is no critical set of map f and $$\phi$$ vanishes everywhere. But if N, M have different DS then the critical set of f cannot be empty and $$\phi$$ has a non-empty support. You can change $$\phi$$ by a coordinate transformation but in no coordinate system can you make the support empty.
This fact is expressed by the curvature of $$\phi$$: if $$\phi$$ is generated by a non-diffeomorphism f between N,M then the curvature is not zero and you can not make it zero by any coordinate transformation. I.e. $$\phi$$ expresses the difference in DS of N, M - independent from the choice of coordinate systems.

 

[marcus]

Thank you for this clear explanation, Helge. I don't know what's got careful's underwear in a tangle; he seems to have more animus to the very idea of diffeomeorphism change -> curvature -> matter than just hostility to your paper. His refusal to address your fine achievements and insistence on criticising from the meta level (no examples..) suggests that.
I am sure you and Torsten don't object to having your presentation in the paper critiqued. That's what a referee would do after all, and it's generally considered a positive if painful aspect of publishing. But this almost foaming at the mouth reaction is over the top.

this is one sign of an interesting fresh idea
that it shocks some people and excites frantic resistance
this already makes this thread worthwhile---and I'm really glad that H. and T. showed up to explain their paper!

BTW selfAdjoint, do you think the alternative wording "critical connection" sounds better or is more transparent than "singular connection"? Helge at some point was using this term "critical" as an alternative and it seemed to me less confusing---but I would like to know your impression.

 

[Helge Rosé]

Thank you for this clear explanation, Helge. I don't know what's got careful's underwear in a tangle; he seems to have more animus to the very idea of diffeomeorphism change -> curvature -> matter than just hostility to your paper. His refusal to address your fine achievements and insistence on criticising from the meta level (no examples..) suggests that.
I am sure you and Torsten don't object to having your presentation in the paper critiqued. That's what a referee would do after all, and it's generally considered a positive if painful aspect of publishing. But this almost foaming at the mouth reaction is over the top.

Dear selfAdjoint,

thanks for your encouragement. Indeed, we are appreciate the critique - this damned topic is complicated enough and so every comment is welcome. Careful ask why it needs 15 year - I can only say, it needs time to understand this difficult math-stuff and extract the things which could be meaningful for physics.

Our credo is simple: Try to understand nature without introduction of the "Äther" - and Äther is: 4 space-time + 6 Äther dimensions, Higgs-fields - you can't measure but it is everywhere, background metrics ... Ok careful thinks the critical form is Äther - but this is not true.

If the critical form describes a transition of DS and transition of DS is a source term in Einsteins eq. then it is physics like GRT. And more important, it is physics without additional entities - only a 4-manifold is needed. But I feel you know this and there is no need to argue.

I like to say that the feeling and discussion here is very fruitful and a great deal of inspiration- and I like to thank very much all folks here for that.

 

[Helge Rosé]

Hi Helge, your and torsten posts are helpful. I like the words "additional connection" and also that this extra connection is a source term---like the matter in classical Einst. eqn.----and in my own head I am calling it the "extra connection" or the "critical connection" because it lives on the critical set. I still don't adequately understand but that is all right, these things take time. Nevertheless the ideas are new and exciting. I will try to improve your table using the simple "CODE" symbol we have at this forum, where you put [kode] and [/kode] around what you want to be in the table----but spell it code.
Helge you say

I like this way to call it, and I wish to try it out, to hear how it sounds----so I will tentatively EDIT your post, as an experiment in rewording.

basic objects:      solutions of E-eq.          diff. structures of space-time
invariance:       classes of physical solutions     classes of  DS
transition:         source term in E-eq.        critical connection 
physical meaning:   matter                      ?

Dear marcus,

thanks for your tentative improvements and inspiring activity. I am a follower of Popper and that's why words mean nothing for me. But I appreciate it very much if you introduce terms which leads to a less level of confusion. Maybe the "singular" attribute was a mistake - generates more trouble then information. I remember a sentence of Einstein where he notes that the "singularities of the field are the particles" and so we like this term - but in this meaning it describes matter and not unphysical blow ups. I think your recommendation of "critical connection" is very good.

I think the important point is that it introduced a new kind of equivalence principle:
One one hand you may think there is only one DS and all transitions between the physical classes of solutions (of E.eq.) are caused by matter - that's the way of Einstein. On the other hand you can also think there is no matter and the transition between the diffenerent physical solutions of E.eq are caused by changing the DS of space-time. The equivalence principle is: change of DS = matter.

But indepentent of this principle (maybe it is wrong, like Careful thinks), the changes of DS are building an algebra - a Temperley-Lieb algebra - and for a special case (tau=1/2) this is the Clifford algebra of fermions. The first reason to write the paper was because this amazing fact and not because the association between transitions of DS and matter suggested by Einsteins eq.

 

[Helge Rosé]

this is one sign of an interesting fresh idea
that it shocks some people and excites frantic resistance
this already makes this thread worthwhile---and I'm really glad that H. and T. showed up to explain their paper!

BTW selfAdjoint, do you think the alternative wording "critical connection" sounds better or is more transparent than "singular connection"? Helge at some point was using this term "critical" as an alternative and it seemed to me less confusing---but I would like to know your impression.

Thanks!
-------------------

 

[selfAdjoint]

BTW selfAdjoint, do you think the alternative wording "critical connection" sounds better or is more transparent than "singular connection"? Helge at some point was using this term "critical" as an alternative and it seemed to me less confusing---but I would like to know your impression.

There does seem to be a legitimate problem between their use of singular to mean lacking an inverse and the sense of singlular as in singularity ("going to infinity") which is common in physics. Critical is perhaps a better word. The important thing is tho get to a language that clearly expresses this new insight in physics research.

 

[selfAdjoint]

I think the important point is that it introduced a new kind of equivalence principle:
One one hand you may think there is only one DS and all transitions between the physical classes of solutions (of E.eq.) are caused by matter - that's the way of Einstein. On the other hand you can also think there is no matter and the transition between the diffenerent physical solutions of E.eq are caused by changing the DS of space-time. The equivalence principle is: change of DS = matter.

But indepentent of this principle (maybe it is wrong, like Careful thinks), the changes of DS are building an algebra - a Temperley-Lieb algebra - and for a special case (tau=1/2) this is the Clifford algebra of fermions. The first reason to write the paper was because this amazing fact and not because the association between transitions of DS and matter suggested by Einsteins eq.

I think this aspect is why I myself respond to positively to your research: The ability to derive an "Einsteininan" account of quantized matter, using his materials plus just the new mathematics of exotic differential structures that was unavailable to him. Beyond that, a road to quantization that comes from the "analytical" tradition, as a complement to the one from the lattice/triangulation tradition is welcome.

A question (perhaps stupid). As it actually worked out, Einstein's equations concerned not "matter" but the momentum energy tensor. Does your extra curvature term caused by the critical shift lend itself to this represenation? Or has there been any work in this direction that you know of?

 

[Mike2]

I think the important point is that it introduced a new kind of equivalence principle: One one hand you may think there is only one DS and all transitions between the physical classes of solutions (of E.eq.) are caused by matter - that's the way of Einstein. On the other hand you can also think there is no matter and the transition between the diffenerent physical solutions of E.eq are caused by changing the DS of space-time. The equivalence principle is: change of DS = matter.

But indepentent of this principle (maybe it is wrong, like Careful thinks), the changes of DS are building an algebra - a Temperley-Lieb algebra - and for a special case (tau=1/2) this is the Clifford algebra of fermions. The first reason to write the paper was because this amazing fact and not because the association between transitions of DS and matter suggested by Einsteins eq.

Perhaps Careful's objection is not that a change in DS is accomplished with additional matter terms in Einstein's eq. But his objection may be that there does not seem to be a reasonable mechanism explained in the paper for the change in the DS in the first place to give rise to matter - just that IF there were changes in the DS's, then it has a Hilber space algebra, etc. What then is the mechanism for these changes in the DS to begin with?

I don't understand it all yet, but perhaps there is a mechnism that does give rise to matter as singularities such that then this business with DS give the right algebra and matter terms. I consider that there may be an overriding entropy prinicple involved. Then by arXiv:math.DS/0505019 v1 2 May 2005, singularities appear in an expanding universe. Obviously, the universe must expand if it has a beginning. So it is accompanied by singularities/matter, which must have the right algebra and curvatures.

 

[selfAdjoint]

don't understand it all yet, but perhaps there is a mechnism that does give rise to matter as singularities such that then this business with DS give the right algebra and matter terms. I consider that there may be an overriding entropy prinicple involved. Then by arXiv:math.DS/0505019 v1 2 May 2005, singularities appear in an expanding universe. Obviously, the universe must expand if it has a beginning. So it is accompanied by singularities/matter, which must have the right algebra and curvatures.

As far as I could see, Mike, that paper concerns piecewise affine maps, not differential structures. Is there something I am missing?

 

[Helge Rosé]

A question (perhaps stupid). As it actually worked out, Einstein's equations concerned not "matter" but the momentum energy tensor. Does your extra curvature term caused by the critical shift lend itself to this represenation? Or has there been any work in this direction that you know of?

Yes, in E.eq matter is represented by the momentum energy tensor. (If I speak about matter in this context I mean the M-E-Tensor)
The additional term in the E.eq. is obtaint as follows (see torsten, gr-qc/9610009)
The change of DS is expressed by the change of connection $$\phi$$
$$\omega^\prime = \omega + \phi$$
This connection change causes a change in the Ricci-Tensor $$\Delta RIC$$ and scalar curvature $$\Delta R$$ (see gr-qc/9610009, eq. (22), (23))
i.e. you get a additional term in E.eq $$\Delta RIC + \Delta R$$.
This term represents the change of geometrie (curvature) caused by the DS change. You may bring this term on the right hand side of the vacuum E.eq and identify it as source term of the E.eq. , i.e as a expression for the M-E-tensor.
This identification is a hypothesis: "the M-E-Tensor is explainable by a geometry change caused by a DS transition" which has the consequence that you don't need second theory (like Einstein, e.g. Electrodynamics) to explain the origin of M-E-Tensor - the origin of M-E-Tensor is the geometry of space-time itself.