Rosé and A-M: Geometrization of Quantum Mechanics

[torsten]

Well, in my opinion the H-constraint problem is unsolvable; something which is well known to be true in the old geometrodynamics approach and I fail to see how introducing new variables can lead to any substantial progress (since the problem is really of a geometrical nature).
Let's assume for sake of the argument that a change of DS gives new physics (I am not convinced yet). I repeat that I fail to see how differential equations; which live on one differentiable structure can give rise to a *change* of the latter.
Cheers,
Careful

I have a question about the Hamilton constraint:
In a paper of Kodama, he showed that the exponential of the Chern-Simons action solves the Hamilton constraint and he had a problem with the momentum constraint (or diffeomorphism constraint). Why does this apporach fails? Chern-Simons theory has a lot to do with knot theory or spin networks.

 

[Careful]

I have a question about the Hamilton constraint:
In a paper of Kodama, he showed that the exponential of the Chern-Simons action solves the Hamilton constraint and he had a problem with the momentum constraint (or diffeomorphism constraint). Why does this apporach fails? Chern-Simons theory has a lot to do with knot theory or spin networks.

I thought that the Kodama state is not normalizable with respect the physical inner product. I am not going to go into detail to this subject here, neither am I interested in topological field theory approaches to quantum gravity for obvious reasons. As I said, the game on this tread is over; there is nothing physical about changing differentiable structures. If you want to discuss the Hamiltonian constraint, you are free to open another thread.

 

[selfAdjoint]

Careful, I have been reviewing this thread, and I have a question about what you posted in #89: here it is (with tex tags around your codes):

I understand what is written above (actually this theorem of Harvey and Lawson is pretty easy to see) but let me treat some stuff in detail. In the paper dg-ga/9702017, E and F are (let's restrict to real) vector bundles of the same rank over one differentiable manifold X, \alpha being a bundle map. It is assumed that the bundle map is singular upon a submanifold \Sigma and that there is a Riemannian metric on each bundle which allows for the definition of the conjugate \alpha^*. I shall first comment upon these issues and then apply it to your paper.

D is defined as D = \alpha D_E \beta + D_F (1 - \alpha \beta) where \beta = (\alpha^* \alpha)^{-1} \alpha^*. In case both Riemannian metrics are the same (in the obvious sense), \alpha \beta is the orthogonal projection operator on the image of \alpha and therefore does depend upon the choice of Riemannian metric (unless im(\alpha) = 0); actually the entire expression D does. So, it is not fair to say that this holds for fibre bundles E and F; since there is lot's of more structure involved

Why pull in the Riemannian metric here? \alpha is a bundle map; a product is defined on the fibers (which are vector spaces, thus converted into algebras) to enable the adjoint \alpha^* to be defined fiberwise, and therefore also the bundle map \beta. The projection you describe seems to me to be defined in the fibers. None of this requires recourse to any Riemanian metric frame, or indeed any particular basis in the fibers. Neither does the connection or curvature which can all be defined at the bundle level. That these bundle-geometric definitions project down onto something that is expressible in Riemannian geometry is of course trivially true, but that does not constrain the bundle maps, etc. It is "downstream" from them.

But I am probably misunderstanding your meaning, so could you enlighten me?

 

[Careful]

**Careful, I have been reviewing this thread, and I have a question about what you posted in #89: here it is (with tex tags around your codes):
Why pull in the Riemannian metric here? \alpha is a bundle map; a product is defined on the fibers (which are vector spaces, thus converted into algebras) to enable the adjoint alpha^* to be defined fiberwise, and therefore also the bundle map \beta. **

The vector spaces over x in E (call it V), and F (call it W) are different, \alpha_x : V -> W. So, how do you define the adjoint of a linear transformation ? You choose a basis in V, one in W, write out \alpha_x as a matrix and take the Hermitian conjugate of the associated matrix. Choosing the bases is equivalent (up to the respective unitary transformations) to introducing Riemannian metrics in V and W. Now, it seems logical to me that such choice of base is made smoothly, otherwise \beta would look even nastier on \Sigma (\beta is already not smooth there). Mathematically speaking, the Riemannian metric is a smooth section in the bundle of frames defined by the respective vector bundles (at least one which covers \Sigma which is all we need for our purposes).


**
The projection you describe seems to me to be defined in the fibers. **


sure


**None of this requires recourse to any Riemanian metric frame, or indeed any particular basis in the fibers. **


Sure it does ! There does not exist something like a ``canonical´´ projection operator of a space on a subspace.


**Neither does the connection or curvature which can all be defined at the bundle level. **


That is true, but that has nothing to do with projections...

 

[selfAdjoint]

So, how do you define the adjoint of a linear transformation ? You choose a basis in V, one in W, write out \alpha_x as a matrix and take the Hermitian conjugate of the associated matrix. Choosing the bases is equivalent (up to the respective unitary transformations) to introducing Riemannian metrics in V and W.

Once you have the inner product on V, say <,> you know there is a unique map \alpha^* satisfying \alpha^*(v) = \langle v,\alpha(v) \rangle. This is independent of any basis. Of course you can exhibit it in any basis but that is not part of its definition. It is as smooth as \alpha but clearly no smoother; if \alpha is not injective at some point \sigma \in V then \langle \sigma,\alpha(\sigma)\rangle is clearly undefined. Smoothness of \beta is obtained by the \chi approximation.

 

[Careful]

Once you have the inner product on V, say <,> you know there is a unique map \alpha^* satisfying \alpha^*(v) = \langle v,\alpha(v) \rangle. This is independent of any basis. Of course you can exhibit it in any basis but that is not part of its definition. It is as smooth as \alpha but clearly no smoother; if \alpha is not injective at some point \sigma \in V then \langle \sigma,\alpha(\sigma)\rangle is clearly undefined. Smoothness of \beta is obtained by the \chi approximation.

Are you pretending now that you never heard of a correspondence between a certain class of frames and a Riemannian metric ?? BTW selfadjoint, the map \beta is NOT smooth at all - if you would care to look at it you will see that there is an infinity replaced by a zero.

**if \alpha is not injective at some point \sigma \in V then \langle \sigma,\alpha(\sigma)\rangle is clearly undefined. **

Utterly false: if we assume V=W then this expression is perfectly well defined for all \alpha.

I promise you that when I make a stupid mistake I shall notify you so that you may scold upon me. But please, stop this ridiculous game about first grade algebra.


Cheers,

Careful

 

[Mike2]

To wrap up this discussion, I shall give a very simple physical reason why the claim that a change of differentiable structure introduces matter is false. The obvious reason it that the Torsten-Helge ``construction´´ does not produce tidal effects outside the ``material body´´ (in either: no gravitational waves) . Consequently there is no gravitational force (and even no volume effect due to Ricci curvature - since no conical singularity is produced). I was hoping that I did not have to state it that explicitly, but this ``correction of technicalities´´ game has cost enough time.
Cheers,
Careful

Force is a change of energy/mass/(differential structure?) with distance, and he has not come up with how things are changing with time (as well as with distance, I suppose). So I don't see how you can conclued no gravitational waves, etc, when the dynamics has not been derived yet. Have you jumped ahead of Torsten to prove that no dynamics is possible in this programme?

I read half of Frankel, The Geometry of Pysics, and half of Nakahara, Geometry, Topology, and Physics. But what area of study would you say would be a prerequisite for this stuff of Torsten's. And what books would you recommend. Thanks.

 

[Careful]

**Force is a change of energy/mass/(differential structure?) with distance, and he has not come up with how things are changing with time (as well as with distance, I suppose). So I don't see how you can conclued no gravitational waves, etc, when the dynamics has not been derived yet. Have you jumped ahead of Torsten to prove that no dynamics is possible in this programme? **

Torsten claims that a change of differentiable structure introduces a source term in the Einstein equations. This, he does by adding a singular part to an originaly smooth connection through a change of differentiable structure which *lives* on a three dimensional manifold (that is what he claims - it is not true of course). Now, start out with the flat connection and perform a coordinate transformation which is singular on \Sigma, then Torsten claims that the resulting connection (combined with a suitable transformation of the metric) is a solution to the Einstein field eqn's with a distributional source of matter. But this means, that his matter does not produce tidal effects (no gravitational waves/force), moreover he does not obtain any ricci curvature effect (volume contration). Hence, his ``matter´´ does not influence the gravitational field and vice versa; contrary to Einsteins theory (in 2+1 dimensions you can do that for non spinning particles, but ok, there we have no gravitational force anyway).

I shall present the argument differently : how do we know matter is present ? (a) We observe tidal effects (on light) (b) we observe that matter induces focal points (volume effect). Neither of these things are present (since the flat solution remains unatered outside the object), therefore we observe nothing. One simply seems to have forgotten that adding matter induces a GLOBAL operation on the physical quantities of interest.


**I read half of Frankel, The Geometry of Pysics, and half of Nakahara, Geometry, Topology, and Physics. But what area of study would you say would be a prerequisite for this stuff of Torsten's. And what books would you recommend. **

As I said, there is no Torsten stuff; at least not in the field of QG.

Cheers,

Careful

 

[Mike2]

I shall present the argument differently : how do we know matter is present ? (a) We observe tidal effects (on light) (b) we observe that matter induces focal points (volume effect). Neither of these things are present (since the flat solution remains unatered outside the object), therefore we observe nothing. One simply seems to have forgotten that adding matter induces a GLOBAL operation on the physical quantities of interest.

What is "outside the object" when matter, according to Torsten, are singularity points, or Delta functions, whose support is 3D, IIRC? I think all that needs to be shown is that the curvature is greater when particles are closer together? Then this would be the "force" you are referring to, right?

 

[Careful]

What is "outside the object" when matter, according to Torsten, are singularity points, or Delta functions, whose support is 3D, IIRC? I think all that needs to be shown is that the curvature is greater when particles are closer together? Then this would be the "force" you are referring to, right?

But his 3-D delta functions are supposed to represent matter ! The whole point he wants to make is that a change of diff structure induces singularities (on a 3-D support) in a (flat) background metric (or connection). But outside this 3-D support nothing happens with the metric (connection), so there is no propagation, no gravitational waves!
Moreover, I have PROVEN that nothing happens (even on this 3-D surface!) and have shown where his examples are flawed. I do not know what I can do more.

 

[marcus]

...
Moreover, I have PROVEN that nothing happens (even on this 3-D surface!) and have shown where his examples are flawed. I do not know what I can do more.

Careful, I'd really value some of your critical comments about the Charles Wang and Johan Noldus papers mentioned in this "List" thread
https://www.physicsforums.com/showthread.php?t=102147

especially the Noldus, which is new for me (I just noticed it)
I give a link in the last post on that thread
https://www.physicsforums.com/showthread.php?p=852574#post852574

 

[Mike2]

But his 3-D delta functions are supposed to represent matter ! The whole point he wants to make is that a change of diff structure induces singularities (on a 3-D support) in a (flat) background metric (or connection). But outside this 3-D support nothing happens with the metric (connection), so there is no propagation, no gravitational waves!
Moreover, I have PROVEN that nothing happens (even on this 3-D surface!) and have shown where his examples are flawed. I do not know what I can do more.

I thought the whole point with the delta function is that it integrates to the same value no matter what the size of the volume (support?) you integrate over. So if his conclusion is valid for one size support, then it is valid for all sizes of support so that in effect it is valid for all space, since any point in space might just as easily be used in some support. But now I wonder what the delta function is a function of. What physical field is the integrand inside the support integral?

 

[Careful]

**I thought the whole point with the delta function is that it integrates to the same value no matter what the size of the volume (support?) you integrate over. So if his conclusion is valid for one size support, then it is valid for all sizes of support**

True (I never claimed otherwise), but the support needs to be compact (moreover, I wonder what the volume measure is he wants to use since the metric would change). But I repeat, this is all NOT happening. I do not think this conversation is useful given the fact that there is really no such effect.

** so that in effect it is valid for all space, since any point in space might just as easily be used in some support. But now I wonder what the delta function is a function of. **

You cannot reasonably claim that the universe should consist of one particle.

 

[Mike2]

**I thought the whole point with the delta function is that it integrates to the same value no matter what the size of the volume (support?) you integrate over. So if his conclusion is valid for one size support, then it is valid for all sizes of support**

*True (I never claimed otherwise), but the support needs to be compact (moreover, I wonder what the volume measure is he wants to use since the metric would change). But I repeat, this is all NOT happening. I do not think this conversation is useful given the fact that there is really no such effect. *

"no such effect" as what? You've kind of lost me here.

If the delta integrates to the same value no matter what "size" the support is, then isn't this just another way of saying that this value is invariant wrt the metric, at least as far as scale is concerned, right?

** so that in effect it is valid for all space, since any point in space might just as easily be used in some support. But now I wonder what the delta function is a function of. **

*You cannot reasonably claim that the universe should consist of one particle.*

How is this a "claim" that the universe consists of one particle? If you've already covered this area, please indicate what post number it was covered in, and maybe I can read it anew in this light. Thanks.

 

[Careful]

**"no such effect" as what? You've kind of lost me here.**

There is NOTHING happening at a change of differentiable structure, please read my posts which adress this at a technical level.

**If the delta integrates to the same value no matter what "size" the support is, then isn't this just another way of saying that this value is invariant wrt the metric, at least as far as scale is concerned, right? **

You have missed my remark about WHICH VOLUME MEASURE to use (the metric would also transform on \Sigma).

Please, I have discussed these matters in details already (read it up)
Again, if you drop a compact form of matter in spacetime according to the authors prescription, the gravitational field outside this region is NOT changing in the procedure. I do not understand what your problem is ??

 

[Mike2]

***"no such effect" as what? You've kind of lost me here.***

**There is NOTHING happening at a change of differentiable structure, please read my posts which adress this at a technical level. **

Please don't take my comments as a challenge. I'm sure you are a better mathematician than I am. And I am asking for help. I only have a brief acquaintence with the math concepts that you discuss. And I'd like to know more. But I'm not sure what areas of math to study. Perhaps you could help me with that.

If you could be just a little more generous, I might actually begin to understand your objections. I'm not sure which of your previous posts would address the issues I raise. If you could at least give the post number (located in the upper right of each post), I might gain insight to your objections. Otherwise, I be lost in a sea of symbols as to which post you are referring to.

But as to your remarks above. I don't know what you mean by "NOTHING happening". I suppose the only issue at hand is whether the change in "differential structure" produces the change in curvature and is equivalent to adding another singularity as claimed.

***If the delta integrates to the same value no matter what "size" the support is, then isn't this just another way of saying that this value is invariant wrt the metric, at least as far as scale is concerned, right? ***

**You have missed my remark about WHICH VOLUME MEASURE to use (the metric would also transform on \Sigma).
Please, I have discussed these matters in details already (read it up)
Again, if you drop a compact form of matter in spacetime according to the authors prescription, the gravitational field outside this region is NOT changing in the procedure. I do not understand what your problem is ??**

It should be an easy matter to scroll through the post and refer to the post that best describes your argument. Otherwise, I'm afraid I'd be lost.

But it would sound as if you are saying that the form on the 3D support responsible for the curvature in the 4D manifold does not map points outside the 3D support. But isn't it true that the support is of arbitrary size? So if the form is defined only inside the support, but the support can be of any size, then doesn't this mean it is applicable for all space that could just as easily be included in some support or another.

 

[Careful]

Hi Mike,

To understand what I say, you just need to have some good grasp on the basis of differential geometry (and GR). I don't know if you are a math or physics oriented person, but a good physics book is the one by Misner, Thorne and Wheeler; a bit more mathematical Nakahara (but I guess you were doing that) Wald or Hawking and Ellis (my preferred one).

I think posts 99 and 100 are good starters. The claim at your third paragraph is correct with negative answers to all.

I briefly mention the volume measure in post 113 (the ``problem´´ is that in the spirit of Torsten and Helge this measure should also transform on \Sigma which -again- it does not.).

I hope this clarifies things

 

[selfAdjoint]

I read half of Frankel, The Geometry of Pysics, and half of Nakahara, Geometry, Topology, and Physics. But what area of study would you say would be a prerequisite for this stuff of Torsten's. And what books would you recommend. Thanks.

Mike you can find a lot of background material on Careful's objections in Nakahara's Chapter 7, Riemannian Geometry, and Chapters 9 and 10, Fibre Bundles and Connections on Fibre Bundles. In Chapter 7 he introduces you to the Levi-Civita Connection and its use in defining curvature, which is at the root of the disagreement between what we might calll the Brans-Nair-Asselmeyer program and Careful. In Chapter 9 Nakahara defines principle bundles and their features and in Chapter 10 he shows how to define connection and curvature through strictly bundle operations in principle bundles. He then goes on to show that in the case that the principle bundle is actually the tangent bundle of a Riemannian manifold, the bundle connection coincides with the Levi-Civita connection, and you recover all the Riemannian results, including even the Bianchi Identities. So there is no conflict between bundle construction and traditional Riemannian methods.

If I may suggest, I believe Careful's criticisms rest on two major assertions. First that the mapping between bundles that is key to the Asselmeyer 1996 paper is not well defined, and then that the connections have to be supported at the manifold level by smooth (C^{\infty} ) diffeomorphisms, which would destroy the paper's claim that there is actually a failure of diffeomorphism, a lack of injectivity, on a "set of measure zero", which can however be approximated to any degree by smooth constructions and therefore meaningfully integrated into Einstein's field equations. I am still researching these issues for my own peace of mind, in the absence of any response from Torsten or Helge on these points.

 

[Careful]

If I may suggest, I believe Careful's criticisms rest on two major assertions. First that the mapping between bundles that is key to the Asselmeyer 1996 paper is not well defined, and then that the connections have to be supported at the manifold level by smooth (C^{\infty} ) diffeomorphisms, which would destroy the paper's claim that there is actually a failure of diffeomorphism, a lack of injectivity, on a "set of measure zero", which can however be approximated to any degree by smooth constructions and therefore meaningfully integrated into Einstein's field equations. I am still researching these issues for my own peace of mind, in the absence of any response from Torsten or Helge on these points.

To name a few mathematical criticisms:
(a) the bundle in Torsten's example is *not* the tangent bundle to the four manifold (so the connections are not spacetime connections)
(b) If you want to define the singular connection rigourously, the construction becomes trivial and there is no curvature effect at all (as I computed explicitely)
(c) There is nothing happening to the geometry outside \Sigma, so an observer outside the ``singular region´´ will not detect anything at all.

But, it is even much easier to see that nothing happens if you think about how to complete M - \Sigma.

 

[marcus]

... the Charles Wang and Johan Noldus papers mentioned in this "List" thread
https://www.physicsforums.com/showthread.php?t=102147

especially the Noldus, which is new for me (I just noticed it)
I give a link in the last post on that thread
https://www.physicsforums.com/showthread.php?p=852574#post852574

Careful has already taken the trouble to comment on the Noldus paper, which I mentioned in the "List" thread. Thanks Careful. It was just a brief comment, and may not be his last word on it.

I am thinking that even though we always need to be focused, and this thread has been very intensively focused on the Torsten Helge idea, perhaps we should start a thread about another approach----not to spend ALL the critical talent on one thread.

Maybe i will start a thread on this Noldus idea, and see if there is any comment. It might be a good idea, and also might possibly be no good at all---I certainly cannot tell at first sight, without some help from others here.

One thing I can say is that Noldus is willing to take risk. he does not merely play it safe. i think this at least is a clear plus. he says that he will try to rebuild QM, put it on a new footing, to make it compatible with GR. the newly founded QM is supposed to APPROXIMATE the old one, but new experiments could distinguish and falsify it, if it is wrong.

So as not to conflict with the topic of THIS thread, namely the Torsten Helge paper, I will make a Noldus thread (not a list of rebel QG approaches, but a thread focused just on the Noldus paper)

Here it is:
https://www.physicsforums.com/showthread.php?t=103750

 

[torsten]

To name a few mathematical criticisms:
(a) the bundle in Torsten's example is *not* the tangent bundle to the four manifold (so the connections are not spacetime connections)
(b) If you want to define the singular connection rigourously, the construction becomes trivial and there is no curvature effect at all (as I computed explicitely)
(c) There is nothing happening to the geometry outside \Sigma, so an observer outside the ``singular region´´ will not detect anything at all.
But, it is even much easier to see that nothing happens if you think about how to complete M - \Sigma.

I completely disagree with all these points but unfortunately I had an influenca and my head ache forbids me to answer...
But a first comment about the physical effect of different differential strucrures can be given: Based on the work of Taylor, Sladkowski shows that the exotic R^4 admits non-trivial solutions of Einsteins equation. Thus there must be an effect.
Furthermore as LeBrun proved, exotic 4-manifolds admits no metric of strictly positive Ricci curvature.

We need more time to react on all the points above. We don't think that careful has proven the converse.
Especially the example of an elliptic fibration mentioned in the paper is correct as I discussed with Terry Lawson long time ago.
But more later...

 

[Careful]

Sorry to hear that you were sick. Concerning your LeBrun remark, this result does not imply that something ``has to happen´´, it simply indicates that the family of *smooth, regular* Riemannian metrics has different qualitative features which sounds reasonable since the smoothness and regularity demand are very severe (as you can immediatly learn from the Schwarzschild example). The example of an elliptic fibration is correct within the framework of bundle connections. Unfortunately, this has nothing to do with spacetime connections which obey different transformation laws.

 

[selfAdjoint]

unfortunately I had an influenca

Golly, I hope it wasn't serious Torsten! Do you have public flu shots there? I used not to believe in them, but they have kept me influenza-free for several years now (as I have come to trust).

I do look forward to your reply to Careful, but it can all wait till you have recovered!

 

[marcus]

Golly, I hope it wasn't serious Torsten! Do you have public flu shots there? I used not to believe in them, but they have kept me influenza-free for several years now (as I have come to trust).
I do look forward to your reply to Careful, but it can all wait till you have recovered!

Me too, I used to avoid medicines whenever possible, but now I think always always always get your flu shot. Each fall, they have a different kind each year.

You will have a lot of work to catch up at Fraunhofer Institute, i think. We will keep your chair empty here at PF for whenever you can get back. Get well soon!

 

[torsten]

Sorry to hear that you were sick. Concerning your LeBrun remark, this result does not imply that something ``has to happen´´, it simply indicates that the family of *smooth, regular* Riemannian metrics has different qualitative features which sounds reasonable since the smoothness and regularity demand are very severe (as you can immediatly learn from the Schwarzschild example). The example of an elliptic fibration is correct within the framework of bundle connections. Unfortunately, this has nothing to do with spacetime connections which obey different transformation laws.

Dear Careful,

now after nearly two weeks in bed, I(=Torsten) feel much better and I
will try to react on your helpful remarks.
The following quote bothered us for a long time, because it implies
the vanishing of the additional curvature after the change of the
differential structure.

D is defined as D = \alpha D_E \beta + D_F (1 - \alpha \beta) where
\beta = (\alpha^* \alpha)^(-1) \alpha^*. In case both Riemannian
metrics are the same (in the obvious sense), \alpha \beta is the
orthogonal projection operator on the image of \alpha and therefore
does depend upon the choice of Riemannian metric (unless im(\alpha) =
0); actually the entire expression D does. So, it is not fair to say
that this holds for fibre bundles E and F; since there is lot's of
more structure involved.
Now, suppose for the moment that we can apply this in your case; on
\Sigma, your \alpha = 0 as is the \beta involved, therefore D = D_F .
On X - \Sigma, D= \alpha D_E \alpha^{-1}. As said, this is the
*only* exceptional case where D does not depend upon the metrics
involved. Now, a bundle covariant derivative depends upon three
indices, a covariant one depending upon the base manifold X and a
covariant and contravariant one depending upon the fiber; in
mathematical terminology, a bundle connection is a special map of the
smooth smooth sections from X to E, to the smooth sections of X to E
\tensor T*X. \alpha can only transform the E part and leaves the T*X
part invariant.

But we found the error: Our knowledge of singularity is limited.
We use a wrong definition (which I found in a book of Brcker/Jnich
about catastroph theory). So you were right in your first quote:

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.

Thus our singular set \Sigma is defined by
\Sigma=\{x\in M|rank(df_x)=3\}. Outside of the singular set
the rank of the map f is 4. According to the work of Stingley (under
the supervisor Lawson), generic (stable) maps f:M->N between
homeomorphic 4-manifolds have only rank 3 singularities.

Now, in your case you have TM and TN which are
different fibre bundles over *different* base differentiable manifolds
(M is not equal to N!). So, you still have to pull back differential
forms from N to M using f (which is trivial and this is where a
difference with the above construction is made !). Hence, your D
should be an object which maps smooth sections of f*(TN) over M to
smooth sections of M to f*(TN) \tensor T*M: (i) (D(V))(W)(x) = (df
(D_{M}(V(x))) (df)^{-1}) W(f(x)) on M - \Sigma (ii) (D(V))(W)(x) =
(D_{N} (df(V)(x)) ) ( W(f(x)) ) = 0 (!) on \Sigma for V \in TM and W
a section of M to f*(TN). Now if D_{M} is a connection of zero
curvature, then (ii) implies that D is also ! The calculation with
V,W \in TM and Z in f*(TN): (i) R(V,W)Z(x) = 0 for x \in M - \Sigma
(obviously) (ii) R(V,W)Z(x) = 0 since D_{N}(df(V)(x)) = D_{N}(0) is a
null transformation (which is generically not the case in the paper
dg-ga/9702017). It is true that (D(W))(Z) is not a smooth section of
M to f*(TN) but that does *not* affect the conclusion, the
corresponding operation (however you wish to define it) should still
be linear in df(V)(x) and hence zero. So, it seems to me that I have
proven that there is no curvature added. However, there is a further
point which I tried to make in the beginning: how are you going to
compare two connections on different manifolds (non equivalent
differentiable structures) in a way which is invariant with respect to
diffeo's on *both* manifolds as well as invariant wrt to any other
kind of structure involved? A similar question is: how to compare two
metrics on different manifolds?? A simple question which is
notoriously difficult to answer *without introducing a background
frame* and therefore by no means as simple as you suggest (books by
great people such as Gromov have been written about this) !

But now we are in the standard situation: \alpha is not
invertible and we using the conjugate \alpha^* to define a
formal inverse via \beta_s. We don't see any problems in
this definition. But we know that this approach bothers you. Thus we
propose to use another approach via the h-cobordism.

A smooth map f:M->N as described above can be described by a cobordism which is a h-cobordism in that particular case. Sorry but I have to tell you some technical details.
This h-cobordism is non-trivial and can be described by a theorem of Freedman et.al.: Let W be the h-cobordism between M and N. Then there is a subcobordism V between A\subset M and B\subset N which is nontrivial. But the cobordism between M-A and N-B is trivial, i.e. diffeomorphic to the product (M-A)\times [0,1]. Thus to understand the difference between the two differential structures is located into the subcobordism V. The 4-dimensional submanifolds A and B are contractable and having the boundary of a homology 3-sphere. The interior of the subcobordism V contains one or more 2-/3-handle pairs. It is possible to cancel these pairs by using a Casson handle. There are many possible description of such a handle and we choose the simplest one: a Casson handle is given by S^2\times S^2-(D^2\times D^2). Now we move the Casson handle into (w.l.o.g.) A. Obviously the Casson handle does not change the topology of A but by choosing the standard metric on the two punctured S^2 factors we get the obvious curvature change R_B=R_A+R_{S^2\times S^2}.
Thus we are able to define the change of a differential structure. But more is known: The boundaries of A and B are homeomorphic homology 3-spheres. Both homology 3-spheres have infinite fundamental groups and by Thurstons hyperbolization theorem, both homology 3-spheres carry a hyperbolic structure. The theorem of Freedman et.al. clarifies also the relation between the boundary of A and B. Both boundaries are related via an involution, i.e. a map t:M->M with t\circ t=id_M,t\not=\pm id_M. By a theorem of Thurston, an involution changes the hyperbolic structure and by Mostows theorem the curvature changes too.
We think that answers your question completely.

With the best wishes for the new year
Helge and Torsten

 

[selfAdjoint]

Torsten, how good to hear you are out of bed again! That must have been a terrible siege of flu! You have my deepest sympathy and great hopes for the New Year.

It was also fine to see you are now founding your proof upon the differential topology of three manifolds. I have been hoping for a long time to see this rich source of structure exploited in physics. Best of luck with the new approach!

 

[Careful]

** So you were right in your first quote:
Thus our singular set \Sigma is defined by
\Sigma=\{x\in M|rank(df_x)=3\}. Outside of the singular set
the rank of the map f is 4. According to the work of Stingley (under
the supervisor Lawson), generic (stable) maps f:M->N between
homeomorphic 4-manifolds have only rank 3 singularities. **

That seems already much more logical to me.

**
But now we are in the standard situation: \alpha is not
invertible and we using the conjugate \alpha^* to define a
formal inverse via \beta_s. We don't see any problems in
this definition. But we know that this approach bothers you. **

No, you are still not in the standard situation : f still operates between two different base manifolds while \alpha does no such thing (so you are still working in the pull back bundle I described before). Now, consider two Riemannian metrics g and h on M and N respectively and assume that the connections you consider are metric compatible (wrt to g and h). Then, you have a canonically defined conjugate and there is no problem as far as I am concerned (I also stressed this point before, but you kept on insisting working at the level of connections and independently of any metric):
h_{f(x)}(df_x(v_x) , w_f(x) ) = g_x (v_x , (df_x)*(w_f(x)))

So you should add this subtlety too.

Ok, so now provide us with a real example - I agree now that it is well defined - which I did not see yet. Note that such procedure can never produce COMPLEX curvatures as you once claimed.

However, my physical objections remain:

(a) \Sigma is a three dimensional surface, so where is the worldtube ?
(b) in the Lorentzian case, it seems to me that you have to put in by hand that \Sigma is spacelike wrt g
(c) your matter is a sitting duck, nothing changes to the curvature outside \Sigma : in particular there is no Weyl tensor in a Minkowski background. You might in the best case generate a ricci volume effect in ``space´´ but you still have to tell us what the *physical* space is (see (e))
(d) What is the dynamics of your function f - this is related in some way to (a).
(e) What is the relational context between different chumps of matter (in either different f's) ?

Anyway, that is enough for now.

Cheers,

Careful

 

[Helge Rosé]

Dear Careful,

h_{f(x)}(df_x(v_x) , w_f(x) ) = g_x (v_x , (df_x)*(w_f(x)))
So you should add this subtlety too.

we appreciate your comment. We are also happy to fix with your help this bad error in the definition of \Sigma.
Thats why we will not give a quick answer about your question related to **COMPLEX curvatures** now, give us some time to carefully check this.

Ok, so now provide us with a real example

Question: What would you accept as a REAL example? I think you agree that at this state of the theory we can not deduce real physical situations ("Given the electron the DS has the form ..."). In the paper we provide a concrete realization of a DS change by the example of the logarithmic transform. We think that is a generic case. But sure it is not a physical example - we missing the interpretation. I think to give an interpreation we need the field eq. of the DS - we try our best but it is not ready now.
Connected with this missing of the field eq. are your questions:

(a) \Sigma is a three dimensional surface, so where is the worldtube ?
(d) What is the dynamics of your function f - this is related in some way to (a).
Careful

At this stage of the theory there is no worldtube. The notion worldtube requested a spliting of the 4MF. We have a natural spliting in the theory: \Sigma \times R (local). But to determine which \Sigma is realized we have to determine the DS and for this we need a field eq.
We don't want to introduce a ad hoc spliting like in other approches, because such a split is meaningless. So we have to wait what the field eq. will say to answer your question.

(b) in the Lorentzian case, it seems to me that you have to put in by hand that \Sigma is spacelike wrt g
Careful

We have not say anything about \Sigma is spacelike. For such statement you need the metric as you say, but the DS don't determine the metric. For this we need need the two field eq. and have to solve them. This is related to the next point:

(c) your matter is a sitting duck, nothing changes to the curvature outside \Sigma : in particular there is no Weyl tensor in a Minkowski background. You might in the best case generate a ricci volume effect in ``space´´ but you still have to tell us what the *physical* space is (see (e))
Careful

At this stage we have only one field eq.: Einsteins eq. The DS is fixed (we have no field eq., we set the DS ad hoc). This fixed ad hoc DS leads to a source term T(g,\Phi) in the Einstein eq: E(g) = T(g,\Phi) with the Einstein-tensor E.

The \Phi is a field (not defined yet!) which represents the DS change (or the change given by f: M \to N). We think there is a field eq. for \Phi: D(g, \Phi) = 0 were g is the metric and D is some (not known yet) Differential operator expression.

Both eq. are coupled, you have to sovle them to determine g and \Phi. The fixed ad hoc \Phi has a \Sigma with curvature zero outside - but puting this source term in the E. eq. modifies g and by this the curvature outside \Sigma. On the other hand D(g, \Phi) = 0 depends on g and that's why you have to solve both eq. if you want to make statements like "the curvature outside Sigma is ...".

If you think that such a coupled system never can produce a curvature outside Sigma, please give me a little more detail explanation of your arguments.
Best regards
helge

 

[Careful]

**
Question: What would you accept as a REAL example? I think you agree that at this state of the theory we can not deduce real physical situations **

A detailed example which shows me how matter is generated and how gravitational waves are produced. As you know, I do not consider your example which produces complex curvature as correct : (a) it requires breaking down the diffeomorphism group by picking out a very special coordinate system associated to a fibration (b) any procedure involving real numbers cannot give you a complex result without violating coordinate invariance.

Concerning your final comments: take your singular covariant derivative

D_E = df D_M \beta + D_N (1 - df \beta)

in the pull back bundle. On \Sigma the stalks of this bundle become one dimensional, outside \Sigma , they are four dimensional, f is a diffeomorphism and D_E = df D_M (df)^{-1} and therefore has the same curvature as D_M. Hence, one does not obtain propagation of waves, the reason being the very definition of the singular connection itself (or singular metric if you want). As I said, you might get a volume effect of some (distributional) kind.

Cheers,

Careful

 

[Mike2]

**
Question: What would you accept as a REAL example? I think you agree that at this state of the theory we can not deduce real physical situations **
A detailed example which shows me how matter is generated and how gravitational waves are produced. As you know, I do not consider your example which produces complex curvature as correct : (a) it requires breaking down the diffeomorphism group by picking out a very special coordinate system associated to a fibration (b) any procedure involving real numbers cannot give you a complex result without violating coordinate invariance.

"waves"?... You are asking for dynamics, and they've admitted that they haven't got that far.

However, if singularity particles give additional curved spacetime, then what could a wave of curved spacetime passing by be except particles coming into and out of existence. In other words, there would have to be a quantum foam possible before waves could propagate. Would this effort prove the existence of the quantum foam of virtual particles?

One question I have is whether the Differential Structure is a mathematically necessary one if one has both particles and curved spacetime? Is this adding structure, or is this recognizing that this structure was there all along? Thanks.

 

[Helge Rosé]

One question I have is whether the Differential Structure is a mathematically necessary one if one has both particles and curved spacetime? Is this adding structure, or is this recognizing that this structure was there all along? Thanks.

Every diff. manifold has a DS - it is simply the equivalence class of its allowed atlases. So it is not a additional structure but in many cases you don't have to care about the DS. For instance in 3dim you have only one DS, i.e. there is no structure which we could use for physics. Only in 4dim there are infinite DS and this is mathematically necessary. From a physical fact like "particles and curved spacetime" you can not deduce a mathematical fact - but if you have a rich mathematical structure (like DS) you can use it to make a hypothesis explaining the physical fact.

 

[Careful]

**"waves"?... You are asking for dynamics, and they've admitted that they haven't got that far. **

It is not that straightforward in this case... They DO have a classical dynamics which is the Einstein equations (you can imagine the virtual particles to be real and not care about the (stochastic) mechanism which generates them). It seems to me that they better provide a clear picture at that level before they start thinking about a quantum dynamics.

 

[selfAdjoint]

It seems to me that they better provide a clear picture at that level before they start thinking about a quantum dynamics.

Careful, I hope you won't take this in the wrong spirit, but it seems to me that not only here but generally your criticisms come in two flavors. First is careful (yes!) critiquing of the mathematics, which in this case have had the enormously productive result that Helge and Torstein recast their proof on what seems to be a much more solid basis.

The other kind are more physicalist , and seem to me to be counsels of perfection, which would lead to rejection of almost any partial result because it did not do the complete job of connecting the mathematical model to the "clicks in the detector" as you expressed it on another thread. I don't say these concerns of yours are wrong - how could I? - but they do not seem to be completely productive. They might have a chilling effect on perfectly valuable research or distract researchers into trying to fulfill something that cannot at a given point in history be fulfilled.

I think the enterprise to connect "matter" to change of differential structure is potentially valuable in itself and worth your concern and Helge's and Torstein's effort to get it right. The "rest of physics" can wait till that first step is completed.

 

[Careful]

**The other kind are more physicalist , and seem to me to be counsels of perfection, which would lead to rejection of almost any partial result because it did not do the complete job of connecting the mathematical model to the "clicks in the detector" as you expressed it on another thread. **

Haha Well, this detector issue has been subject of quite some debate and many physicsts erroneously believe that calculating the two point functions or performing the coordinate transform are sufficient (actually I know of many misinterpretations of the effect, I was guilty myself once ). I just said it is not known yet, moreover it is a neglegible effect (as I once calculated in another thread).

** I don't say these concerns of yours are wrong - how could I? - but they do not seem to be completely productive. **

Well, not entirely because they gave me the opportunity to give some more detailed comments regarding QFT and GR. Moreover, I felt is important to understand the effect well (as you noticed there was some unclarity about this).

**I think the enterprise to connect "matter" to change of differential structure is potentially valuable in itself and worth your concern and Helge's and Torstein's effort to get it right.**

As I mentioned before, their main motivation rests upon the modification of the Einstein equations : so they have to get that right at least.

** The "rest of physics" can wait till that first step is completed **

I agree upon this, I am well aware of the enormous restrictions set upon ``deviant´´ papers and some referees really ask you to come up with a closed theory before you can publish anything at all. But at least they ought to clarify the mechanism ``classically´´. I mean, in every attempt to QG there has to be at least a clear connection with either GR or QM.

Cheers,

Careful

 

[selfAdjoint]

I am well aware of the enormous restrictions set upon ``deviant´´ papers and some referees really ask you to come up with a closed theory before you can publish anything at all. But at least they ought to clarify the mechanism ``classically´´. I mean, in every attempt to QG there has to be at least a clear connection with either GR or QM.

I fully agree with this (surprise!). But given the difficulty in doing anything much with the full Einstein equations without truncating something or linearizing something, I personally would cut them some slack on the details!

 

[Mike2]

**"waves"?... You are asking for dynamics, and they've admitted that they haven't got that far. **
It is not that straightforward in this case... They DO have a classical dynamics which is the Einstein equations (you can imagine the virtual particles to be real and not care about the (stochastic) mechanism which generates them). It seems to me that they better provide a clear picture at that level before they start thinking about a quantum dynamics.

But it seems to me that they would eventually have to transfer the dynamics of waves in Einstien's equation to dynamics of paritcle creation and annihilation through the intermediacy of Differential Structures. Like you say, we already have Einstien's equations. What is new is the how the DS's give us particles. So the classical waves of spacetime curvature will translate into particle creation as the wave crests and particle annihilation after the wave has passed by. So in any event we ARE talking about particle creation and annihilation, just as in QFT. The question is whether we need an underlying zero point foam whose average changes as the gravity wave passes by, or are we talking about particles whose existence is longer lived as longer waves pass by. So I wonder if there is a "bandwidth-pulsewidth product" to Einstein equation that might serve as an uncertainty relation to tell us how long particles exist as waves pass by. What is the shortest wave possible?

 

[Careful]

** But it seems to me that they would eventually have to transfer the dynamics of waves in Einstien's equation to dynamics of paritcle creation and annihilation through the intermediacy of Differential Structures. **

All they propose so far is a way to kinematically understand matter in terms of change of DS (well matter in the sense of a particle which instantaneously gets born and dies). This is all ``classical´´ as far as I see it, so it is legitimate to ask for a classical mechanism for a change of DS which couples to the Einstein field equations and produces GR waves. I do not see however, how the latter can be achieved in the context of the construction of the ``singular metric´´ proposed in their paper.

I am not asking for anything more...

Cheers,

Careful

 

[Mike2]

** But it seems to me that they would eventually have to transfer the dynamics of waves in Einstien's equation to dynamics of paritcle creation and annihilation through the intermediacy of Differential Structures. **
All they propose so far is a way to kinematically understand matter in terms of change of DS (well matter in the sense of a particle which instantaneously gets born and dies). This is all ``classical´´ as far as I see it, so it is legitimate to ask for a classical mechanism for a change of DS which couples to the Einstein field equations and produces GR waves. I do not see however, how the latter can be achieved in the context of the construction of the ``singular metric´´ proposed in their paper.
I am not asking for anything more...
Cheers,
Careful

A change in the gravitational field of Einstein's field equations is continuous, but the change in mass that causes the field is discrete if it complies with the algebra in their scheme. So you're asking how something discrete can be equated to something continuous, right? I suspect the only way to do this is to rely on the average of some discrete process for the continuously changing gravitational field of a wave. In other words, a quantum backgound foam.

 

[Careful]

A change in the gravitational field of Einstein's field equations is continuous, but the change in mass that causes the field is discrete if it complies with the algebra in their scheme. So you're asking how something discrete can be equated to something continuous, right? I suspect the only way to do this is to rely on the average of some discrete process for the continuously changing gravitational field of a wave. In other words, a quantum backgound foam.

As far as I see, there is nothing distinctively ``discrete´´ about their program. But let us first figure out my primary concerns (I still have pleanty of remarks about the algebra itself).

 

[Mike2]

As far as I see, there is nothing distinctively ``discrete´´ about their program. But let us first figure out my primary concerns (I still have pleanty of remarks about the algebra itself).

Excuse me? Wasn't their whole point to connect general relativity (which is conintuous) with quantum mechanics (which is discrete)? And the ONLY way to do this is to invoke an average of the discrete variable to come up with the continuous variables, right?

As I understand it, a single DS is associated with single curvature. Yet, GR allows the curvature to change continously in waves. But QM only allows discrete changes of matter. This much they already have, and this much doesn't involve dynamics. Instead, you want to see them come up with some mechanism for a change in the DS which is made discrete by its association with a particles Hilbert space. And you seem to want them to somehow make this a continuous change so that it can be consistent with continuous waves. But if you don't allow them to invoke an averaging of the DS's, then there is no way they can explain how gravitational waves continuously move by. The mechanism that you are looking for with a change in the DS is the existence of a quantum background foam whose average gives an equivalent curvature as waves pass by. If the equivalence of curvature and particles can be made through the DS, then the fact that gravitational waves are continuous would only prove the existence of the quantum foam.

 

[Careful]

** Excuse me? Wasn't their whole point to connect general relativity (which is conintuous) with quantum mechanics (which is discrete)? **

Quantum mechanics is not discrete a priori by any means. It are spectra for some physical observables in bound states which are discrete however (the rest is as continuous as it can be).

**
And the ONLY way to do this is to invoke an average of the discrete variable to come up with the continuous variables, right? **

No, usually people consider sending \hbar to zero as doing that, but averaging might also do a good job, yes. But I shall ask you a question for a change : why do you think quantum mechanics predicts discrete energies (even for bound states) ?


As a summary to the rest: I repeat, there is nothing discrete in their formalism so far (I am just reading the paper, not their thoughts). There is no \hbar involved even, and I disagree with the way they claim to get out complex numbers.

Cheers,

Careful

 

[Mike2]

** Excuse me? Wasn't their whole point to connect general relativity (which is conintuous) with quantum mechanics (which is discrete)? **
Quantum mechanics is not discrete a priori by any means. It are spectra for some physical observables in bound states which are discrete however (the rest is as continuous as it can be).

I may have misspoke. I meant to say that particles come in quantized values so that the DS's come in quantized value. Is this what their theory says?

A better question might be if a differential structure is the equivalent class of atlases, then do these equivalent classes come in discrete form? Or can you move continuously from one equivalence class to another?

**
And the ONLY way to do this is to invoke an average of the discrete variable to come up with the continuous variables, right? **
No, usually people consider sending \hbar to zero as doing that, but averaging might also do a good job, yes.

Actually, if you bring h-bar to zero then you totally eliminate the discrete value so that you are not using it to obtain a continuous average value from discrete values.

 

[Chronos]

What happens to the model if it 'bounces' at the Planck scale? I'm thinking along the lines of an equation of state.

 

[Careful]

** I may have misspoke. I meant to say that particles come in quantized values so that the DS's come in quantized value. Is this what their theory says? **

No, it does not, the mass formula they get depends upon the details of f (the volume/area ratio).

**
Actually, if you bring h-bar to zero then you totally eliminate the discrete value so that you are not using it to obtain a continuous average value from discrete values. **

Sure, I proposed both as *different* mechanisms to get classical behaviour out.

Cheers,

Careful

 

[Mike2]

I'm still curious: if a differential structure is the equivalent class of atlases, then do these equivalent classes come in discrete form? Or can you move continuously from one equivalence class to another? Are there any other examples of moving continuously from one equivalent class to another? Thanks.

 

[Careful]

I'm still curious: if a differential structure is the equivalent class of atlases, then do these equivalent classes come in discrete form? Or can you move continuously from one equivalence class to another? Are there any other examples of moving continuously from one equivalent class to another? Thanks.

Well I thought there was an uncountable number of different differentiable structures on 4-d manifolds which nevertheless come in discrete form - at least what concerns any notion of closeness based upon differentiable mappings (notions of closeness based upon continuous mappings cannot distinguish).

*Are there any other examples of moving continuously from one equivalent class to another?*

This is a problem I mentioned in the very beginning...

Cheers,

Careful

 

[selfAdjoint]

Careful, this thread has strayed a bit. Let me ask where we are between you and Helge & Torsten now. Has their recasting of their proof satisfied your objections on that score (leaving aside your questions about gravity waves, the algebra of quantization, etc.)? In other words can we move on to other features of their latest paper?

 

[Careful]

Careful, this thread has strayed a bit. Let me ask where we are between you and Helge & Torsten now. Has their recasting of their proof satisfied your objections on that score (leaving aside your questions about gravity waves, the algebra of quantization, etc.)? In other words can we move on to other features of their latest paper?

SelfAdjoint, I am simply waiting for Torsten to reply

 

[selfAdjoint]

in post #127 you stated fiveiticisms:

(a) \Sigma is a three dimensional surface, so where is the worldtube ?
(b) in the Lorentzian case, it seems to me that you have to put in by hand that \Sigma is spacelike wrt g
(c) your matter is a sitting duck, nothing changes to the curvature outside \Sigma : in particular there is no Weyl tensor in a Minkowski background. You might in the best case generate a ricci volume effect in ``space´´ but you still have to tell us what the *physical* space is (see (e))
(d) What is the dynamics of your function f - this is related in some way to (a).
(e) What is the relational context between different chumps of matter (in either different f's) ?

It seems to me that (a) and (b) are real criticisms, i.e assertions that the research is wrong, while (c), (d), and (e) are like suggestions for future directions. Is this what you are waiting for a response to?

 

[Mike2]

in post #127 you stated fiveiticisms:

(a) \Sigma is a three dimensional surface, so where is the worldtube ?
(b) in the Lorentzian case, it seems to me that you have to put in by hand that \Sigma is spacelike wrt g
(c) your matter is a sitting duck, nothing changes to the curvature outside \Sigma : in particular there is no Weyl tensor in a Minkowski background. You might in the best case generate a ricci volume effect in ``space´´ but you still have to tell us what the *physical* space is (see (e))
(d) What is the dynamics of your function f - this is related in some way to (a).
(e) What is the relational context between different chumps of matter (in either different f's) ?

It seems to me that (a) and (b) are real criticisms, i.e assertions that the research is wrong, while (c), (d), and (e) are like suggestions for future directions. Is this what you are waiting for a response to?

It seems to me that (a) and (b) are still asking for a time evolution that has admittedly not been developed yet. If you have any critizism of the static relation between differential structure and an instantaneous curvature, I would think it better to get that out of the way before addresssing any time evolutional process.

Toward dynamics I asked:
Quote:

Originally Posted by Mike2
I'm still curious: if a differential structure is the equivalent class of atlases, then do these equivalent classes come in discrete form? Or can you move continuously from one equivalence class to another? Are there any other examples of moving continuously from one equivalent class to another? Thanks.

to which you responded:

Well I thought there was an uncountable number of different differentiable structures on 4-d manifolds which nevertheless come in discrete form - at least what concerns any notion of closeness based upon differentiable mappings (notions of closeness based upon continuous mappings cannot distinguish).

*Are there any other examples of moving continuously from one equivalent class to another?*

This is a problem I mentioned in the very beginning...

I'd be very curious to learn more. It seems that discrete vs. infinitesimal changes in equivalence classes of atlases is at the root to the dynamical situation. If you could shed more light on the situation I'd be greateful. Can you point to the post number that address this? I know I'm asking you to repeat yourself. But now I'm beginning to see the relevance, which is the motivation I need to study it. Thanks.