Rosé and A-M: Geometrization of Quantum Mechanics

[Helge Rosé]

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.

Dear Mike, I hope I understand the meaning of your question: What is the reason (the cause) for a DS change?

Physics can not explain the reason for that like it can not explain: why is matter or why is space-time. The point is:

GRT says: if there is matter then it causes a curvature of space-time, i.e. a grav. field.

We say: if there is DS change then it causes a curvature of space-time, i.e. a grav. field.

But if we agree in the principle:

the meaning of term "matter" is: a entity that causes a grav. field

then a DS change is such a grav. generating entity and we can say: a DS change is identical (not mere caused by) with matter. (the same situation as in: energy = mass, grav field = accelerated reference frame)

 

[Careful]

Dear Mike, I hope I understand the meaning of your question: What is the reason (the cause) for a DS change?

I think you understood me (and probably mike too) wrong. I asked about the possibility for a dynamics to be constructed which *generates* change of differentiable structure. This question was connected to your claim that you generate singular three surfaces which need the interpretation (by HAND) of a spacelike three surface associated to a matter distribution (say elementary particle). At that moment, I suggested : ``but what about the singular world 4 tube ? ´´, ``how to find a determinstic mechanism which allows for change of differentiable structure ?´´ since classically you would expect this already to be necessary (clutting of matter and so on). These objections have been left unanswered so far

Concerning the reference to the 96 torston paper (I note that I still did not receive a definition of a singular connection here), let me ask some silly questions. For example at page 3, a map h: M -> N is constructed where M is an exotic 7 sphere and N the ordinary S^7, which is singular at one point, say x_0. You endow M and N with smooth Riemannian metrics, choose smooth frames e (in M) and f (in N) and claim that you can select the Riemannian metrics in such a way that dh(e)(x) = a(x) .f(h(x)) where a(x) is a S0(7) transformation in the specific orthogonal bundle over N related to coordinates in M. This seems wrong to me since dh(e)(x_0) = 0 and hence a(x_0) = 0 (could you clarify this??) which would lead to a zero curvature contribution (if I were to believe formula 9). Perhaps, I missed something but anyway...

Concerning the ``complex´´ curvature the authors get on page 10 in their example. It seems to me that they forgot to take the complex conjugate expression (a tangent basis in D^2 consists of d/dz and d/dz* which leads to matrix ( pz^{p-1} 0 )
( 0 pz*^{p-1})

and a^{-1} da = ( (p-1)dz/z 0 )
( 0 (p-1)dz*/z*)
(notice that there is NO division through p - these factors cancel out.) Since the authors are only interested in the trace, this gives:

(p-1)dz/z + (p-1)dz*/z*

which (in polar coordinates) gives : 2(p-1) dr/r which gives rise to zero curvature (at least when I would naively take the line integral of this around a circle).

I hope I made it more clear now why I insist upon a rigorous definition and example of a distributional connection related to a change of differentiable structure! I think this hardly classifies as ``frantic resistance´´.

Cheers,

Careful

 

[selfAdjoint]

I think you understood me (and probably mike too) wrong. I asked about the possibility for a dynamics to be constructed which *generates* change of differentiable structure. This question was connected to your claim that you generate singular three surfaces which need the interpretation (by HAND) of a spacelike three surface associated to a matter distribution (say elementary particle). At that moment, I suggested : ``but what about the singular world 4 tube ? ´´, ``how to find a determinstic mechanism which allows for change of differentiable structure ?´´ since classically you would expect this already to be necessary (clutting of matter and so on). These objections have been left unanswered so far

There are obviously questions raised by this research, such as: according to theorems there are countably many differential structures, but we observe only a finite, and very specific, particle spectrum. What gives? This has been the bete noir of the string theorists. The mechanism for the origin of these curvatures can be pushed back to the big bang, or whatever, but we need a particle dynamics. All this will be a research program for the future.

Concerning the reference to the 96 torston paper (I note that I still did not receive a definition of a singular connection here), let me ask some silly questions. For example at page 3, a map h: M -> N is constructed where M is an exotic 7 sphere and N the ordinary S^7, which is singular at one point, say x_0. You endow M and N with smooth Riemannian metrics, choose smooth frames e (in M) and f (in N) and claim that you can select the Riemannian metrics in such a way that dh(e)(x) = a(x) .f(h(x)) where a(x) is a S0(7) transformation in the specific orthogonal bundle over N related to coordinates in M. This seems wrong to me since dh(e)(x_0) = 0 and hence a(x_0) = 0 (could you clarify this??) which would lead to a zero curvature contribution (if I were to believe formula 9). Perhaps, I missed something but anyway...
Concerning the ``complex´´ curvature the authors get on page 10 in their example. It seems to me that they forgot to take the complex conjugate expression (a tangent basis in D^2 consists of d/dz and d/dz* which leads to matrix ( pz^{p-1} 0 )
( 0 pz*^{p-1})
and a^{-1} da = ( (p-1)dz/z 0 )
( 0 (p-1)dz*/z*)
(notice that there is NO division through p - these factors cancel out.) Since the authors are only interested in the trace, this gives:
(p-1)dz/z + (p-1)dz*/z*
which (in polar coordinates) gives : 2(p-1) dr/r which gives rise to zero curvature (at least when I would naively take the line integral of this around a circle).

I am sure these questions can be addressed. As I said before, careful critique is not at all objectionable.

I hope I made it more clear now why I insist upon a rigorous definition and example of a distributional connection related to a change of differentiable structure! I think this hardly classifies as ``frantic resistance´´.

It wasn't the requests for clarification but the insults ("sloppily written") that grated. The paper is up to the usual standards of preprint writing. Yes it will benefit by reconstructing some parts based on your remarks, but some of your comments seemed to demand they conduct a decade-long research program before publishing. Some physicists may work that way (Veltzmann comes to mind), but it is certainly not the community norm.

 

[Helge Rosé]

I think you understood me (and probably mike too) wrong. I asked about the possibility for a dynamics to be constructed which *generates* change of differentiable structure. This question was connected to your claim that you generate singular three surfaces which need the interpretation (by HAND) of a spacelike three surface associated to a matter distribution (say elementary particle). At that moment, I suggested : ``but what about the singular world 4 tube ? ´´, ``how to find a determinstic mechanism which allows for change of differentiable structure ?´´ since classically you would expect this already to be necessary (clutting of matter and so on). These objections have been left unanswered so far

Dear Careful,

thanks, this was a very constructive comment and I understand your intension much better. You are absolutely right, a dynamics, i.e. a field eq. is needed - but the truth is: we don't have it, yet. As I mention before the paper only deals with the "kinematics".

We have ideas to the field eq. which will determine the dynamics of the DS but that's are only ideas, we have wait for a settle down. Sorry but you know, also in very good developed and intuitive LQG the solving of the Hamilton-constrain is the ultimative goal and hard work.

We have the feeling it is possible to derive a field eq. from a variation principle und get a quantum field eq. and also the Einstein eq. from one DS -action - but at the moment I have no words to explain it.

 

[Careful]

**There are obviously questions raised by this research, such as: according to theorems there are countably many differential structures, but we observe only a finite, and very specific, particle spectrum. **

Well, I wouldn't worry about that ! I expect this particle spectrum to grow and grow when we get to higher and higher energies and I certainly DO welcome any effort which looks for ONE mechanism behind all substance (I myself do entertain such thoughts also). However, the questions I adress now are of a TECHNICAL level and the authors should have a good idea how to give a plausible explanation to these.


** I am sure these questions can be addressed. As I said before, careful critique is not at all objectionable. **

I am not so sure about that, but I would welcome any good explanation which I did not get so far.

** It wasn't the requests for clarification but the insults ("sloppily written") that grated. The paper is up to the usual standards of preprint writing. Yes it will benefit by reconstructing some parts based on your remarks, but some of your comments seemed to demand they conduct a decade-long research program before publishing. Some physicists may work that way (Veltzmann comes to mind), but it is certainly not the community norm **


Sorry, but that is exactly the reason why so much crap is written and we should not encourage this at all. Moreover, torsten and Helge face the more difficult task that they have to introduce a new (for physicists) technique. For example: sometimes I have the feeling that they bring in too much technicalities without explaining (and this does not need to take more than a few lines) why such line of thought is preferred. In my view, they should really consider doing things more stepwise, rigorous and example oriented and even sometimes leave away some generalizations. This would at most double the size of the paper. A good example in this respect is the style taken by Milnor (extremely clear, concise and deep).

Moreover, there are still the difficulties I raised.

Cheers,

Careful

 

[Careful]

We have ideas to the field eq. which will determine the dynamics of the DS but that's are only ideas, we have wait for a settle down. Sorry but you know, also in very good developed and intuitive LQG the solving of the Hamilton-constrain is the ultimative goal and hard work.

We have the feeling it is possible to derive a field eq. from a variation principle und get a quantum field eq. and also the Einstein eq. from one DS -action - but at the moment I have no words to explain it.

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

 

[selfAdjoint]

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).

Have you noticed Thiemann's Master Constraint Program? If you have, and care to comment on it, perhaps we could start a new thread.

 

[Careful]

Have you noticed Thiemann's Master Constraint Program? If you have, and care to comment on it, perhaps we could start a new thread.

Yes, I have noticed; although I did not read up with it (was it not replacing H by ``M = integral over H^2/volume ´´ and noticing that M commutes with itself and the spatial diffeo constraints and that there is an a.e equivalence between the observables *on* shell ?). It seems a cute trick (apart from the sacrifice that the constraint is fourth order in the connection) to obtain an a.e. equivalent set of constraints which *classically* forms an algebra which closes of shell. Notice however that the latter achievement for the *diffeomorphism* constraint algebra was already obtained (albeit in a more complicated way) in the old paper of Isham and Kuchar (1984). But, I admit that at first sight, the problems associated with the Kuchar proposal as well as the traditional H - constraint proposal are circumvented (and I was only commenting upon these).
If someone could tell me in a new thread if this program made any substantial progress already on a technical level, then I would be happy to learn about this.

However, it remains to be seen wether spacetime covariance is recovered in the classical limit. Moreover, I have many troubles with the way observables are treated: it seems you will only be able to speak about global spacelike averaged quantities (such as the average spatial curvature, volume, black hole area) unless you invoque a (arbitrary ?) couloring of ``points´´ in different spaces (there exist no ``points´´ in LQG as well as in causal dynamical triangulations - and there averaged observables are all you have access to). Unnecessary to say that this is entirely *unrealistic* (you give up locality on *any* scale) and it seems an almost impossible task to rigorously define what you mean with a semiclassical state which has a suitable notion of locality in it *without* making use of some or another background (some of these are very old problems even in standard quantum theory).

 

[selfAdjoint]

This is OT; I'll get some updated info on the Master Constraint Program and start a new thread. But I have to say about "no points" that I regard that as a feature not a bug. Given the problems that microscopic locality has caused, and the strange postures theoretical physicsts have got themselves into to circumvent its consequences, I think that regarding locality, like covariance, as emergent is an advantage. After all, neither you nor any experiment has ever demonstrated a point!

 

[Helge Rosé]

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

As I said, we have no eq. at the moment - only conjectures - but I'm sure you would say to that: speculative, no examples, etc... So it is better we first complete the next paper which will deal with the dynamics of DS and then start the discussion. I hope we will manage your good advice and put more examples in it.

Some words to the term "dynamics":

The problem with a covariant 4d description, like Einsteins eq., is that there is no dymanics - i.e. no time-evolution of a state. Here we have only a field eq. - and a given matter-distribution gives a metric as solution. By this past and future are determined. Popper call this the Parmenidis block universe and in a discusion Einstein agreed - there is no time-evolution.

But quantum mechanics has a concept of time: the future is only determined up to probabilities, past is determined by the result of measurements, i.e. by certain values of observables.

I think it is important to involve this time-concept of QM without lost the covariance. If you ad hoc global split the 4-MF (like in LQG) you get a time but you forget the covariance.

What we need is a natural splitting and the DS seems to support it: the critical curvature (from the critical 1-from) is a covariant 4d entity and determines the DS. But you can also use the 3d-support and by this you have a natural 3+1 split - but not a global split, you have a local split given for every 3d-support along the time line. The dynamics is that evolution of the 3d-support in the "4d tube".

I believe there is also a covariant 4d field eq. for the DS-curvature (and the E.eq is special case) but this eq. determines (without a meaning of time) the 4d-global DS - it only determines the probability part of information of the whole future - like a wavefunction. But measurement happens local and in 3d and here we have the spliting, time and a dynamics of 3MF (the particles).

 

[Careful]

This is OT; I'll get some updated info on the Master Constraint Program and start a new thread. But I have to say about "no points" that I regard that as a feature not a bug. Given the problems that microscopic locality has caused, and the strange postures theoretical physicsts have got themselves into to circumvent its consequences, I think that regarding locality, like covariance, as emergent is an advantage. After all, neither you nor any experiment has ever dmonstrated a point!

That is not the issue: call it ``point´´, Planck scale ``volume´´ or whatever but we can make that discussion in the other thread.

Cheers,

Careful

 

[Careful]

** in Einstein equations, there is no dynamics, no evolution of state **

Classically, one is in realistic solutions able to find dynamically preferred time functions (of course I am aware that this is not the generic situation, but our universe is after all very special ! ) and associated slices of constant time. So, in those situations, there is a clear Hamiltonian evolution of the spatial metric (you choose a *physical* gauge and thereby fix the lapse and the shift vector).

There is no consensus what to do quantum mechanically. In causal dynamical triangulations for example, a kinematically generated candidate T exists (the rescaled counting time associated to the ``hypersurfaces´´) and the Hamiltonian is non vanishing (the wave function depends on T and so forth) which begs for the question wether covariance is recovered in ``the classical limit´´´. In LQG the H constraint is unsolvable and attention has now turned towards the M constraint in which a preferred notion of simultaneity is present (since the hypersurfaces are fixed).

**
I think it is important to involve this time-concept of QM without lost the covariance. If you ad hoc global split the 4-MF (like in LQG) you get a time but you forget the covariance.
**

?? Not at all: classically, covariance is in the Dirac algebra; quantum mechanically the H-constraint cannot be solved, but the M constraint is solvable. It remains to be seen whether covariance is recuperated in a suitable classical limit (since it is not eminently present in the M constraint quantized theory).

About the rest: I am still waiting for a definition of singular connection and an explicit comment on my questions/remarks concerning *your* examples. I do not see yet how additional curvature source terms are generated (and certainly *no* complex ones ) but that is probably my mistake I presume.

 

[Kea]

...by the way, notice that in LQG these results are obtained WITHOUT the introduction of matter...

Er...on a new thread, would you mind explaining how a theory without matter is more physical than the Torsten and Helge work? Your criticism of the Brans' papers seems pertinent. Have you looked at the papers of Krol?

Reiterating others' remarks: your constructive remarks are greatly appreciated.

 

[Careful]

Er...on a new thread, would you mind explaining how a theory without matter is more physical than the Torsten and Helge work? Your criticism of the Brans' papers seems pertinent. Have you looked at the papers of Krol?
Reiterating others' remarks: your constructive remarks are greatly appreciated.

Just gave it a quick look (especially for you ) and he refers to Torsten concering this issue (of showing that a change of diff structure generates curvature). So, we are turning in ``loops´´; but I am sure that Torsten is going to clarify my silly objections concerning his examples in the 1996 and 2005 paper.

 

[Mike2]

We say: if there is DS change then it causes a curvature of space-time, i.e. a grav. field.

If GR is derived from the changes in DS, then should there be a mechanism for negative curvature as well as positive? You've seem to have covered matter with a point particle, but what about photons?

I wonder... they might be invariant wrt to diffeomorpism of different frames of velocity, but what about acceleration? The Unruh effect predicts a temperature with acceleration (and therefore the particles that produce the temperature). So if changes in accelerated frames produce particles, then perhaps changes in accelerations can produce the changes in the DS's which cause matter. And since acceleration can be both positive and negative, this might produce both positive and negative curvature so that we can still consider all possible cosmologies. Then again, it might be possible that I don't know what I am talking about. Good luck.

 

[torsten]

Just gave it a quick look (especially for you ) and he refers to Torsten concering this issue (of showing that a change of diff structure generates curvature). So, we are turning in ``loops´´; but I am sure that Torsten is going to clarify my silly objections concerning his examples in the 1996 and 2005 paper.

Hallo Careful,
at first thanks for your time in waiting on the definition of a singular connection. I need the time to look into all the relevant papers. But here is the extract:
Let E,F be vector bundles of equal rank over the manifold M and a smooth bundle map $$\alpha:E\to F$$. Let $$D_E,D_F$$ be connections on E and F, respectively. Furthermore we introduce a metric on the fibers so that we are able to define the adjoint $$\alpha^*$$. The map $$\alpha$$ admits singularities along a subset $$\Sigma\subset M$$, i.e. the set where the map $$\alpha$$ is not injective. Outside of the singular set we can define the inverse map $$\alpha^{-1}$$ or a map $$\beta=(\alpha^* \alpha)^{-1}\alpha^*:F\to E$$ which serve as the inverse. Instead of finding the singular connection directly one considers an approximation. By an approximate one we mean a function $$\chi$$ which satisfies the following properties: $$\chi:[0,\infty]\to [0,1]$$ which is $$C^\infty$$ on $$[0,\infty]$$ and satisfies $$\chi(0)=0,\chi(\infty)=1$$ and $$\chi'\ge 0$$. Now we consider the parameter family $$\beta_s=(\alpha^*/s^2)\rho(\alpha\alpha^*/s^2)$$ where $$\rho(t)=\chi(t)/t$$. Now Harveay and Lawson proof that $$\beta_s$$ converges for $$s\to 0$$ to $$\beta$$ uniformly in $$M-\Sigma$$ and pointwise in $$\Sigma$$. Obviously every diffeomorphism changes the function $$\chi$$ but don't change the properties of that function. The singular connection is the connection $$D_0$$ coming from the limit of the connection $$D_s=\alpha D_E\beta_s+D_F(1-\alpha\beta_s)$$. A good reference is the paper by Nair in dg-ga/9702017.
That is enough for today. I will try to give an example later.
Torsten

 

[selfAdjoint]

From the Nair paper, "singular on $$\Sigma$$" means "many-to-one" (i.e., not injective) on $$\Sigma$$.

 

[torsten]

From the Nair paper, "singular on $$\Sigma$$" means "many-to-one" (i.e., not injective) on $$\Sigma$$.

Ok right, try the original paper of Harvey and Lawson math.DG/9407216.

 

[Careful]

The singular connection is the connection $$D_0$$ coming from the limit of the connection $$D_s=\alpha D_E\beta_s+D_F(1-\alpha\beta_s)$$. A good reference is the paper by Nair in dg-ga/9702017.
That is enough for today. I will try to give an example later.
Torsten

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.
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.

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) !

 

[selfAdjoint]

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.

? Surely the inner product in a Riemannian geometry is tensorial? So the definition of conjugacy is too? And therefore the statements remain true even though the elements in them change with diffeomorphism equivalent frames. And when you lift these covariant statements to the tangent spaces in the two frames you get cvorrsponding geometric statemnts: two vectors and a projection are correct in both cases though the vectors are different. Not so?

 

[Careful]

? Surely the inner product in a Riemannian geometry is tensorial? So the definition of conjugacy is too? And therefore the statements remain true even though the elements in them change with diffeomorphism equivalent frames. And when you lift these covariant statements to the tangent spaces in the two frames you get cvorrsponding geometric statemnts: two vectors and a projection are correct in both cases though the vectors are different. Not so?

HUH ? Did I somewhere claim that the definition of conjugacy is non tensorial ? What I said is that in the general case outlined in dg-ga/9702017, the definition of D depends upon the notion of conjugacy (although this does not apply to the special case Torsten uses) and not only upon D_E, D_F and the bundle map \alpha !

 

[Kea]

Careful

If you type [ itex ]\alpha[ /itex ] for $\alpha$ and [ tex ]\alpha[ /tex ] for $$\alpha$$ (but WITHOUT the spaces in the brackets) your posts would be much easier for people to read.

Cheers

 

[Careful]

Careful

If you type [ itex ]\alpha[ /itex ] for $\alpha$ and [ tex ]\alpha[ /tex ] for $$\alpha$$ (but WITHOUT the spaces in the brackets) your posts would be much easier for people to read.

Cheers

I am sure you can figure it out It took me already more than one hour to type that in.

 

[selfAdjoint]

HUH ? Did I somewhere claim that the definition of conjugacy is non tensorial ? What I said is that in the general case outlined in dg-ga/9702017, the definition of D depends upon the notion of conjugacy (although this does not apply to the special case Torsten uses) and not only upon D_E, D_F and the bundle map \alpha !

Well, I think this case in Nair is handled by well-known theorems about complex bundles over manifolds, say the holomorphic theorem. But I don't want to keep jumping around to different papers. Right now I am getting Harvey and Lawson's characteristic currents (math.DG/9407216) under my belt (this paper is a little gem!), and then I'll come forward to its consequent, the Nair paper.

I agree that Torsten's 1996 paper is expressed too tersely, perhaps because writing in English was then a labor for him. He might not have wanted to write more paragraphs for the same reason that you don't want to put [ tex ] [ /tex ] around your LaTeX constructions .

 

[Careful]

Well, I think this case in Nair is handled by well-known theorems about complex bundles over manifolds, say the holomorphic theorem. But I don't want to keep jumping around to different papers. Right now I am getting Harvey and Lawson's characteristic currents (math.DG/9407216) under my belt (this paper is a little gem!), and then I'll come forward to its consequent, the Nair paper.
I agree that Torsten's 1996 paper is expressed too tersely, perhaps because writing in English was then a labor for him. He might not have wanted to write more paragraphs for the same reason that you don't want to put [ tex ] [ /tex ] around your LaTeX constructions .

I never said that the Torsten paper was too condense, I merely stated that some claim he made is incorrect (notice that I never got any answer to this; neither did I get any answer to my complaints about ``the´´ example in his recent paper). I also do not dispute the content of the dg-ga/9702017 and math.DG/9407216 papers and you do *not* need to read these to understand what Torsten tries to say. These papers are actually dealing with a mathematically *different* situation from the one adressed by Torsten (as I explained). Perhaps you should better try to figure out what might be wrong in my *proof* that there is no curvature added in Torsten's case. It is rather obvious that in the generic situation of the above mentioned papers curvature is going to be added (I do not have to read the papers to know that) since the extra subtlety which kills it off in Torsten's case is not present there.

 

[Kea]

Right now I am getting Harvey and Lawson's characteristic currents (math.DG/9407216) under my belt...

Hey, guys, if you can easily follow Harvey and Lawson, it would be much appreciated if you could clarify some points (any of them!).

For instance, they mention a similarity with the Quillen formalism that lies behind the localisation theorems of Witten et al in TFTs. See page 7 where they mention several options for the approximation mode $\chi$, such as $\chi \equiv 1$ for $t$> 1 giving approximations supported near the singular set.

 

[selfAdjoint]

Hey, guys, if you can easily follow Harvey and Lawson, it would be much appreciated if you could clarify some points (any of them!).

For instance, they mention a similarity with the Quillen formalism that lies behind the localisation theorems of Witten et al in TFTs. See page 7 where they mention several options for the approximation mode $\chi$, such as $\chi \equiv 1$ for $t$> 1 giving approximations supported near the singular set.

Since I don't know anything about the Quillen formalism I don't think I can help you. The definitions of $\chi$ seem clear enough and I can just follow their transgression arguments. Is there anything there you need?

 

[torsten]

I never said that the Torsten paper was too condense, I merely stated that some claim he made is incorrect (notice that I never got any answer to this; neither did I get any answer to my complaints about ``the´´ example in his recent paper). I also do not dispute the content of the dg-ga/9702017 and math.DG/9407216 papers and you do *not* need to read these to understand what Torsten tries to say. These papers are actually dealing with a mathematically *different* situation from the one adressed by Torsten (as I explained). Perhaps you should better try to figure out what might be wrong in my *proof* that there is no curvature added in Torsten's case. It is rather obvious that in the generic situation of the above mentioned papers curvature is going to be added (I do not have to read the papers to know that) since the extra subtlety which kills it off in Torsten's case is not present there.

At first some reaction on your last proof:
I know the difficulties in comparing the metrics of non-diffeomorphic manifolds. But in dimension 4 the situation is more friendly then in other dimensions: It is known (mostly the work of Quinn) that two homeomorphic 4-manifolds are diffeomorphic apart from a (possible collection of ) contractable 4-dimensional submanifold having the boundary of a homology 3-sphere. That can be considered as a kind of localisation. Thus a map $$f:M\to N$$ is a diffeomorphism apart from that contractable piece. But that means we have to understand the special structure of the tangent bundle.
Now to your example: We consider a special class of 4-manifolds known as elliptic surfaces (=complex surfaces) as classified by Kodaira. Such 4-manifolds are fibrations over a Riemannian surface S where the fibers are tori except for a finite number of cases. All possibe cases for these exceptional fibers were classified by Kodaira. A logarithmic transformation is the local modification of $$N(T^2)=D^2\times T^2$$ in the 4-manifold M by using a cluing map $$\partial N(T^2)\to M-\partial N(T^2)$$ which is a pair of maps $$(\phi,T)$$ with $$\phi:S^1\to S^1$$ given by $$z\mapsto z^p$$. The tangent bundle over $$D^2$$ is a complex line bundle. Now we are in the sitation of Harvey and Lawson: the map $$\phi$$ induces a map between the complex line bundles which is singular in z=0. Then we obtain a singular connection associated to this map. That modifies the trivial fibration $$N(T^2)=D^2\times T^2$$ to a non-trivial one with an exceptional fiber in z=0. That creates a cohomology class which agrees with the class of the exceptional fiber. (see the work of Gompf on the nucleus of elliptic surfaces)
In principle one can take the two coordinates z,z^*. But you can also take the one form dz/z and construct the other components by using the complex structure, i.e. z=x+iy. What is wrong with that approach?
In my 1996 gr-qc paper I discuss the exotic S^7 case but in the Class. Quant. Grav. paper I ommit it because of the known problems.
Secondly I don't understand why in your proof the connection in point (ii) (i.e. on \Sigma) vanishes. Maybe I'm to stupid to see it. Can you illuminate me?
Some more general words about the additional curvature. The work of LeBrun (based on Seiberg-Witten theory) showed that on an exotic 4-manifold there is NO metric of strictly positive scalar curvature. Thus the exotic structure has to change something on the manifold which modifies the curvature. That was the original motivation for our work.

 

[Careful]

**At first some reaction on your last proof:
I know the difficulties in comparing the metrics of non-diffeomorphic manifolds. But in dimension 4 the situation is more friendly then in other dimensions: It is known (mostly the work of Quinn) that two homeomorphic 4-manifolds are diffeomorphic apart from a (possible collection of ) contractable 4-dimensional submanifold having the boundary of a homology 3-sphere. That can be considered as a kind of localisation. **

?? You merely outline that you can ``clump´´ the ``non diffeomorphic´´ properties of both *manifolds* in four dimensions. This is *not* the issue I was reffering to (for my part: just start with two metric tensors on one and the same manifold). As you should know, manifolds by themselves are entirely uninteresting for gravitational physics: the only thing which matters are the causality and curvature properties of the metric. It is a very old issue how to compare two *different* metrics.


**Thus a map $$f:M\to N$$ is a diffeomorphism apart from that contractable piece. But that means we have to understand the special structure of the tangent bundle. In principle one can take the two coordinates z,z^*. But you can also take the one form dz/z and construct the other components by using the complex structure, i.e. z=x+iy. What is wrong with that approach?**
**

Ah, but in this example you are assuming an identification between M and N has been made. Let me explain what the difference is between this example and what you said before. In your theory, you consider a map f between two different manifolds M and N and you try to define a difference D between the covariant derivative on N and that on M. This is not easy since M and N are two different manifolds, so on what bundle does D have to live? Now, you want to consider an expression of the form:

(df) D_{M} (df)^{-1} on M - \Sigma
D_{N} on \Sigma

Normally, (in the dg-ga/9702017) paper you would have a mixed expression (suppose df is not equal to zero on \Sigma) :

D = df D_{M} \beta + D_{N} (1 - df \beta) (++)

where \beta does now depend upon the chosen Riemannian structures. We want to have an expression of the form (D(V))(Z)(x) so it seems appropriate to me to put V in TM, Z in f*(TN) and define:

(D(V))(Z)(x) = df D_{M}(V(x)) \beta (Z(f(x))) + [D_{N}( df(V(x)) ) ] (1 -
df \beta ) Z(f(x))

This leads to the trouble I mentioned (now first let me answer your following question and then come back to the example).

** Secondly I don't understand why in your proof the connection in point (ii) (i.e. on \Sigma) vanishes. Maybe I'm to stupid to see it. Can you illuminate me? **

On \Sigma : (D(V))(Z)(x) = [D_{N} ( df(V)(x) )] (W(f(x)) where V is a (smooth) section of M in TM and W is a (smooth) section of M in f*(TN). The point is that df = 0 on \Sigma therefore df(V)(x) = 0 hence (D(V))(Z)(x) = 0. The difference with the references you quote is that there both bundles live on one manifold (and you can limit yourself to the bundle map). Here you cannot, and the only way to make sense of this is to pull back the cotangent bundle T*N to M. But the latter is a trivial operation on \Sigma (since you assume df to be null).


Now, let me go on with your example where you regard the four dimensional manifold as a fibre bundle (with 2-D fibers, the tori T^2) over a Riemann surface. The map $\phi$ leaves the fibres T^2 invariant and can be undone on D^2 so I do not see how it ``induces´´ a different differentiable structure (neither do I see how an active diffeomorphism $\phi$ on D^2 \subset Riemann surface can be reduced to a bundle map of a complex line bundle over the Riemann surface). But anyway, you seem to be saying that one should compare different BUNDLE connections over thîs particular Riemann surface. This means you collapse the four dimensional diffeomorphism group to the subgroup which leaves the particular fibration invariant (in either acts only on the base = Riemann surface). Moreover, a spacetime connection on the four manifold has *nothing* to do with a bundle connection over the Riemann surface, so this approach would be obviously flawed. It is easy to see that there are less degrees of freedom in the bundle connection and moreover both connections live on different structures and obey different transformation laws!

For your information, there is an an approach to 2+1 quantum gravity ('t Hooft, Deser, Jackiw et al) based upon classical solutions to the field equations with a distributional energy momentum tensor source (corresponding to spinning particles) which are everywhere locally Minkowski (except where the particle is - there you have a conical singularity). The singularity in the metric is *not* generated by applying a singular coordinate transformation,but is made visible by it (sorry for first mentioning otherwise). When you simply apply a singular coordinate transformation, you have nothing : no conical singularity and no tidal effects.

 

[Careful]

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

 

[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.