Time discreteness and lorentz invariance

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atyy

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I think in the case of spin networks, it is only spatial diffeomorphisms that are involved.

Part of the semiclassical limit of a Lorentzian spinfoam model has been calculated in http://arxiv.org/abs/0907.2440
 
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Three. If there are only finitely many vertices, how do you expect the Hilbert space to furnish a representation of an infinite dimensional noncompact group?
I don't know if this is a problem; the original nonseparable Hilbert space actually had the right dimension since the diffeomorphism group has cardinality \aleph_1^{\alpha_1} and therefore a separable Hilbert space would be way too small. Actually, one can easily see that the finite number of vertices is sufficient to encode all analytic diffeomorphisms (I think this is actually proven somewhere); the more generic class of C^{infinity} diffeomorphisms might not be captured by these structures.

Four. I was also under the impression that most Spinfoam work was done in Euclidean signature, and thus the analytic continuation was always a persistent nag. Here the problem is particularly virulent, b/c if you boost the system, you run into contradictions with the finiteness of the volume measure.
True, the same thing holds by the way for the area spectrum and so on.

Careful
 

marcus

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Two: I think this non-separable Hilbert space was an intermediate step only and one was able to resolve this issue?
...
Yes, in earlier years there was a non-separable Hilbert space that occurred at one point, and that was resolved by factoring it down in various fashions. But I have not seen that non-sep space in recent presentations of the theory.

For example this year there have been 3 review papers that describe and present the current formulation of of LQG. As i recall none of them had a non-sep Hilbert space. It is just not an issue any more.

The best way to check would be to look at the concise presentation of LQG in the most recent (December) review paper: http://arxiv.org/abs/1012.4707

Does anyone need us to go through that and explain why no non-sep Hilbert arises there. The definition of the theory is short and clear, so it should not be too difficult to explain.
 
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Does anyone need us to go through that and explain why no non-sep Hilbert arises there. The definition of the theory is short and clear, so it should not be too difficult to explain.
Well, you are not adressing the subtlety I posed, I don't know what Healfix had in mind. These constructions all work for analytical diffeomorphisms, I remember that John Baez tried to generalize it to C^{infinity} ones but I am not sure whether he obtained a conclusive answer. As a LQG specialist, you must surely know this ''detail''. I think this paper http://arxiv.org/PS_cache/q-alg/pdf/9708/9708005v2.pdf may be helpful for you.

Careful
 
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marcus

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...These constructions all work for analytical diffeomorphisms,..
Careful, I was talking to Tom Stoer. Neither you nor Haelfix seemed to me to be talking about LQG as it is defined today and I don't wish to argue with either of you. Criticisms of a 2004 or 1997 version just do not seem relevant.

I recall that analytic diffeomorphisms played a vital role in the version of LQG presented in 2004 by Ashtekar and Lewandowski.
http://arxiv.org/abs/gr-qc/0404018
A hefty review article, 125 pages :biggrin:

Rovelli's December 2010 gives a nice exposition of the theory, three equations and around 3 pages---no "analytical diffeomorphisms" or "non-separable" or anything remotely like that.

Each graph Hilbert is clearly separable, and the overall H is the projective limit.

Of course the graphs are finite but each individual graph Hilbert is countably infinite dimensional.

The construction can be made explicitly and obviously Lorentz invariant.

I really do not want to address preconceptions people have about some vintage 2004 version they read about or heard about. That is their business. I suppose anyone who is genuinely interested in this approach to QG will have read http://arxiv.org/abs/1012.4707 by now, and read it carefully, if they are "careful". :biggrin:
 
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Careful, I was talking to Tom Stoer. Neither you nor Haelfix seemed to me to be talking about LQG as it is defined today and I don't wish to argue with either of you. Criticisms of a 2004 or 1997 version just do not seem relevant.
They are, if they have not been adressed. :biggrin: You don't solve problems by stopping to talk about them.

Rovelli's December 2010 gives a nice exposition of the theory, three equations and around 3 pages---no "analytical diffeomorphisms" or "non-separable" or anything remotely like that.

Each graph Hilbert is clearly separable, and the overall H is the projective limit.

Of course the graphs are finite but each individual graph Hilbert is countably infinite dimensional.

The construction can be made explicitly and obviously Lorentz invariant.
Sure, but is it general enough to recuperate the larger group of C^{infinity} diffeomorphisms ? I don't care what people say and how many review papers they write, I care about what they do. So, does Rovelli adress this issue or not ?

BTW: why do you have the preconception that nonseparable Hilbert spaces are a bad thing?
 

marcus

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I'm sorry. Your questions do not make sense. It is off-topic to ask about a non-sep H because one does not arise. It is irrelevant to talk about "addressing problems" which do not occur.
There are no diffeomorphisms because there is no manifold involved in formulating the theory.

Your questions are apparently based on an out-of-date notion. You may not realize it but you are harrassing me with questions which are meaningless in the present LQG context.

All I can say is read http://arxiv.org/abs/1012.4707 and make up some new questions that really connect with LQG. It should be easy for you to baffle me---there are some real, hard, and still-unanswered questions. The theory is unfinished. If you want to point that out, it would be easy for you to make meaningful points, that actually connect with current work by the Lqg community.
 
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There are no diffeomorphisms because there is no manifold involved in formulating the theory.
Didn't you say some 3 weeks ago that LQG needs the manifold while I was pointing out to you it doesn't (in the modern version) ! :wink:

Your questions are apparently based on an out-of-date notion. You may not realize it but you are harrassing me with questions which are meaningless in the present LQG context.
Well, let me not ask you a question then and simply respectfully say that you are wrong. All present LQG does is just take the point of view they derived for analytical diffeomorphisms as a new starting point and say: hey this is our theory. It is of course extremely easy to do that and one may pretend in this way that the issue goes away but sadly it never does. Let me give you a logical argument why this may pose a problem: if the manifold does not matter at all for the formulation of the theory (and it shouldn't), then the kind of ''labelling'' invariance you should have must be extremely general. Having only invariance with respect to the analytical diffeomorphisms is not good because these are nonlocal (in the sense that they are determined everywhere if you know them between two patches). The ''meaning'' of the equivalence principle is in the non-analytical ones, they allow you to locally change the equivalence class without fixing it everywhere else. So, yes, they are damn important I would say and the point I wanted to make (and I think also Healfix was suggesting this) is that if you try to include them, the mathematics suddenly gets a lot more complicated. This I believe, is what the Baez paper was pointing out.

All I can say is read http://arxiv.org/abs/1012.4707 and make up some new questions that really connect with LQG.
Are you suggesting that I do not know or understand what the real problems are ? Likewise, you might assume that I do know what is written in that paper.

It should be easy for you to baffle me---there are some real, hard, and still-unanswered questions. The theory is unfinished. If you want to point that out, it would be easy for you to make meaningful points, that actually connect with current work by the Lqg community.
Like I said, I am not interested in where the latest fad is leading the troop. My concern lies with the foundations of the theory and all my posts concerning LQG try to adress these issues as I feel they are central.

Moreover, I am directly adressing Healfix concerns so I don't think this is off topic at all!

Careful
 
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I was thinking about what it would mean for space to be continuous, but time to be discrete, and wondering if such ideas have been explored, or even how the math on such a vector space would work. In my mind, this would make the physics rotationally invariant still, since time as a single dimension can only undergo one rotation that will always map onto a previous time on an evenly spaced metric. Or is it that this discontinuity in the translational symmetry for time would violate the invariance too?
I seem to remember that 't Hooft has written a paper about 2+1 quantum gravity where time turns out to be discrete and space is continuous; wait a moment, http://www.phys.uu.nl/~thooft/gthpub/canonical_quant_point_part_2plus1_dim.pdf
here it is. The mathematical reason is that the Hamiltonian is only well defined upon a multiple of two pi; this implies that time must be discrete while space isn't.

Cheers,

Careful
 

tom.stoer

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I think this discussion between marcus and Careful is somewhere related to the Nicolai-Thiemann debate some years ago. I agree that not all issues have been resolved since.
- it is not clear how the correct Hamiltonian including regularization looks like
- it is not clear how to prove that the constraint algebra is anomaly-free
- deriving a separable Hilbert space via the spatial diffeomorphisms seems to be problematic

What Careful and marcus are discussing is that Rovelli's step from a bottom-up "quantization" approach (manifolds - loops - constraints - Hilbert space - Hamiltonian - ...) to a top-down "axiomatic" approach ("this IS the quantum theory BY DEFINITION, no let's see what we can calculate") is still not fully justified.

From the bottom-up quantization perspective it seems that this program has not been completed. From the top-down perspective it is clear that a full derivation of the correct classical theory (which is the starting point of the bottom-up approach) is still missing. So I agree that there is still a gap.

My arguments regarding discrete structures were not about this gap, but about the fact that discrete structures need not automatically mean that certain continuous symmetries are violated. So YES, there may be a violation of Lorentz invariance due to some anomaly, dynamical mechanism, etc., but NO, this is not simply due to the fact that one uses spin networks at the kinematical level. This conclusion would be too hasty.
 
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There are too very short arguments against "discretized spacetime violates Lorentz-invariance":
- the operator algebra respects the Lorentz-algebra
- discrete eigenvalues do not signal any such violation

Compare the situation with standard rotational symmetry: the angular momentum has a discrete spectrum and leads to a discrete basis of eigenvectors; nevertheless the rotational symmetry is not violated.
But the argument does hold for the Poincare algebra (unless you average over a special ensemble of discrete possibilities such as happens in causal set theory, but then an individual causet has no meaning). It is long known that discrete space time is not in conflict with Lorentz invariance, Hartland Snyder has written about that in the 1940 ties as far as I remember.
 

tom.stoer

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It is long known that discrete space time is not in conflict with Lorentz invariance ...
But it's a common misconception arising in many discussions. The problem is that violation of Lorentz invariance in theories with discrete XYZ is (by many people) always related to "the trivial fact that one discretizing ABC to XYZ violates Lorentz invariance". This is simply wrong. A related misconception is "the fact that gauge fixing breaks gauge invariance - b...sh..". Gauge fixing reduced the (unphysical) gauge symm. to the identity within the physical sector (using the c.o.m. frame in a two-particle system and restricting to the c.o.m. momentum P~0 sector does not beak translational incariance).

Unfortunately people stop thinking about violaton of Lorentz invariance at all once they have discovered these "obvious facts".

Afaik it is still unclear whether and how (both local and global) Poincare invariance and diffeomorphism invariance can be checked within the LQG framework b/c the Hamiltonian H and the question of anomalies cannot be addressed sufficiently. w/o knowing H these questions cannot be answered. (even worse in the new setup one concludes that these questions need not be answered - which is in my opinion wrong as well; simply b/c you are no longer able to ask these questions does not mean that answers have been found).

My conclusion is that one should think about "gauge anomalies" in the LQG framework (even if they do NOT follow from the simple fact of discretization). Nicolai raised these questions some years ago and I still do not understand Thiemann's reply.
 
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Afaik it is still unclear whether and how (both local and global) Poincare invariance and diffeomorphism invariance can be checked within the LQG framework b/c the Hamiltonian H and the question of anomalies cannot be addressed sufficiently. w/o knowing H these questions cannot be answered. (even worse in the new setup one concludes that these questions need not be answered - which is in my opinion wrong as well; simply b/c you are no longer able to ask these questions does not mean that answers have been found).

My conclusion is that one should think about "gauge anomalies" in the LQG framework (even if they do NOT follow from the simple fact of discretization). Nicolai raised these questions some years ago and I still do not understand Thiemann's reply.
So, we agree.

Careful
 

tom.stoer

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yes, amazingly we agree :-)
 
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yes, amazingly we agree :-)
It is not amazing: as far as I know, we never had a disagreement about the current status of LQG. It is hard to quarrel about facts if both parties are sufficiently educated in (the technicalities of) this business. We have very different ''ideas'' however what conclusions to draw from this and how to move on, but that is fine.

Careful
 

tom.stoer

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OK, again we seem to agree :-)
 

atyy

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Afaik it is still unclear whether and how (both local and global) Poincare invariance and diffeomorphism invariance can be checked within the LQG framework b/c the Hamiltonian H and the question of anomalies cannot be addressed sufficiently. w/o knowing H these questions cannot be answered. (even worse in the new setup one concludes that these questions need not be answered - which is in my opinion wrong as well; simply b/c you are no longer able to ask these questions does not mean that answers have been found).
No, in fact these are being investigated by groups associated with Kaminski (directly) and with Dittrich (indirectly). Kaminski is asking what is the relationship of the new models to the canonical formulation, while Dittrich is looking not specifcally at LQG but at how asymptotic safety would imply the Hamiltonian constraint.

While I prefer an emergent gravity (if the spinfoam formalism means anything, its relation to classical spacetime cannot be so naive) - I believe Kaminski and Dittrich are pursuing logically open questions.
 

tom.stoer

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OK; so what are the operators H, P, L and K (generators of the Poincare group) in this setup? How is the regularization defined?
 

atyy

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Oh, I don't know. I'm just saying that not everyone has concluded that in the new setup the questions no longer need to be answered.
 

marcus

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Oh, I don't know. I'm just saying that not everyone has concluded that in the new setup the questions no longer need to be answered.
That's true and before dismissing what they have to say one should at least try to understand the answers which the researchers have given.

I recall Renate Loll addressed the issue of spatial diffeo invariance in CDT in one of their review papers. If anyone is interested, I could try to find the reference. Or perhaps someone already knows the argument she gave.

Rovelli addresses both the issues of diffeo invariance and Lorentz invariance on page 8 of the April review paper http://arxiv.org/abs/1004.1780

And of course there was last month's paper "Lorentz Covariance of LQG"
http://arxiv.org/abs/1012.1739
One cannot truthfully say that these questions are dismissed by saying they "no longer need to be answered".

And if an argument is presented as to why they no longer need to be answered in the same way then it's incumbent on us to show we understand the argument.
 
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marcus

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This seems like a good statement. It sheds some light on the discussion. You posted after 11 PM in the evening, pacific time, and I did not see it until just now.
I think this discussion between marcus and Careful is somewhere related to the Nicolai-Thiemann debate some years ago. I agree that not all issues have been resolved since.
- it is not clear how the correct Hamiltonian including regularization looks like
- it is not clear how to prove that the constraint algebra is anomaly-free
- deriving a separable Hilbert space via the spatial diffeomorphisms seems to be problematic

What Careful and marcus are discussing is that Rovelli's step from a bottom-up "quantization" approach (manifolds - loops - constraints - Hilbert space - Hamiltonian - ...) to a top-down "axiomatic" approach ("this IS the quantum theory BY DEFINITION, so let's see what we can calculate") is still not fully justified.

From the bottom-up quantization perspective it seems that this program has not been completed. From the top-down perspective it is clear that a full derivation of the correct classical theory (which is the starting point of the bottom-up approach) is still missing. So I agree that there is still a gap.

My arguments regarding discrete structures were not about this gap, but about the fact that discrete structures need not automatically mean that certain continuous symmetries are violated. So YES, there may be a violation of Lorentz invariance due to some anomaly, dynamical mechanism, etc., but NO, this is not simply due to the fact that one uses spin networks at the kinematical level. This conclusion would be too hasty.
That is a nice characterization of the "top-down axiomatic" approach. And truly it is still not fully justified! Since one can only justify such an approach to the extent that one DOES the calculations. Only a first order "2-point" function for the graviton has been calculated. A propagator of one graviton from here to there. They will have to calculate n-point functions.
However Battisti et al have a cosmology result using this result which may be falsifiable. Admittedly slender, but progress nonetheless.

No substitute for actually reading the 2010 review papers, but i will risk giving my own interpretation. Hopefully I will not misrepresent: The fundamental premise of LQG is that there is nothing between adjacent events. "If you take away the gravitational field you do not have empty space left—you have nothing." Therefore every criterion, every logical test, comes down to scrutinizing the vertex amplitude.

There is nothing in between network nodes for a diffeo to stir and push and play around with. So an evolution process is fully characterized by what it does to nodes---that is, by what happens at vertices.

Diffeo invariance—general covariance—is respected precisely by adopting this viewpoint. That points of spacetime not defined by events are non-existent and should not be represented mathematically since they can play no physical role.
 
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And if an argument is presented as to why they no longer need to be answered in the same way then it's incumbent on us to show we understand the argument.
That is playing with words and one could easily turn this argument around. The status of spatial diffeomorphism invariance in CDT, for example, is admittedly not a simple one and I doubt whether there has been said something truely deep about it. Concerning the timelike diffeomorphisms however, it is no coincidence that research in this field has made contact with Horava gravity. In LQG, the situation is simple and I believe the argument which I have given is an essentially correct one; at least Baez used to think likewise.

Sorry to say, but Rovelli does not treat the issues of diffeo and Lorentz invariance at all; he just mentions them in a casual way.

Careful
 

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