Time discreteness and lorentz invariance

[unusualname]

Even at a naive level it is not hard to see how continuous symmetries may be emergent from discrete QM superpositions, eg 't Hooft as an argument for retrieving translational and rotational symmetry here:

Emergent Quantum Mechanics and Emergent Symmetries

and a simplistic argument in parts 2 and 3 of this presentation

 

[Careful]

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]

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.

 

[Careful]

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]

yes, amazingly we agree :-)

 

[Careful]

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]

OK, again we seem to agree :-)

 

[atyy]

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]

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]

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]

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.

 

[marcus]

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.

 

[Careful]

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 truly 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