Friday, April 24, 2015

The problem with Poincaré-invariant networks

Following up on the discussions on these two previous blogposts, I've put my argument why Poincaré-invariant networks in Minkowski space cannot exist into writing, or rather drawing. The notes are on the arxiv today.

The brief summary goes like this: We start in 1+1 dimensions. Suppose there is a Poincaré-invariant network in Minkowski space that is not locally infinitely dense, then its nodes must be locally finite and on the average be distributed in a Poincaré-invariant way. We take this distribution of points and divide it up into space-time tiles of equal volume. Due to homogeneity, each of these volumes must contain the same number of nodes on the average. Now we pick one tile (marked in grey).

In that tile we pick one node and ask where one of its neighbors is. For the network to be Poincaré-invariant, the distribution of the links to the neighbor must be Lorentz-invariant around the position of the node we have picked. Thus the probability for the distribution of the neighboring node must be uniform on each hyberbola at equal proper distance from the node. Since the hyperbolae are infinitely long, the neighbor is with probability one at infinite spatial distance and arbitrarily close to the lightcone. The same is the case for all neighbors.

Since the same happens for each other node, and there are infinitely many nodes in the lightlike past and future of the center tile, there are infinitely many links passing through the center tile, and due to homogeneity also through every other tile. Consequently the resulting network is a) highly awkward and b) locally infinitely dense.

This argument carries over to 3+1 dimensions, for details see paper. This implies there aren't any finite Poincaré-invariant triangulations, since their edges would have to form a network which we've just seen doesn't exist.

What does this mean? It means that whenever you are working with an approach to quantum gravity based on space-time networks or triangulations, then you have to explain how you want to recover local Lorentz-invariance. Just saying "random distribution" doesn't make the problem go away. The universe isn't Poincaré-invariant, so introducing a preferred frame is not in and by itself unreasonable or unproblematic. The problem is to get rid of it on short distances, and to make sure it doesn't conflict with existing constraints on Lorentz-invariance violations.

I want to thank all those who commented on my earlier blogposts which prompted me to write up my thoughts.


  1. I believe Cris Moore showed something similar in
    Physical Review Letters 60 (1988) 655.

    Here is a pdf:

  2. Wolfgang,

    Thanks! In fact it's pretty much the same argument. Except that Lorentz-invariance isn't enough, you also need homogeneity, otherwise it's easy to circumvent the conclusion. I will add this reference in an update. Best,


  3. In the paper you write, "Because all the links are arbitrarily close to the lightcones, we must count all the
    points in the volume-tiles that can send light-signals to or receive light signals from the volume we considered."

    Why do we need to count all the points that are lightlike separated? I understand that the neighbors are a subset of the lightlike separated points. But why do all the points on the light cone have to be connected by links to the node?

  4. Sudip:

    They're not necessarily connected with the center tile, I'm counting all links that 'pinch the surface of'. Best,


  5. "you have to explain how you want to recover local Lorentz-invariance" "The universe isn't Poincaré-invariant" "doesn't conflict with existing constraints on Lorentz-invariance violation" Lorentz-invariance may not be empirically exact. Derivation cannot detect incomplete postulates, Newton versus relativity and QM. Look.

    Quantum field theories (QFT) with Hermitian Hamiltonians are invariant under the identity component Poincaré group containing spatial reflections. This subgroup excludes parity and time reversal. Parity is a spatial reflection but not a QFT symmetry. All Hermitian Hamiltonians will contain a symmetry and an observable with the properties of parity, but the Hamiltonian will not be symmetric under spatial reflection. QFT with non-Hermitian Hamiltonians can have real and positive energy spectra with PT invariance, but do not contain parity invariance alone. Test spacetime geometry with geometry, physics and chemistry.

  6. This comment has been removed by the author.

  7. Space-time seems to do a wonderful job to conceal its quantum properties.I have always thought that to achieve such a feat a Fourier superposition was at work.

  8. Hi Sabine,

    In your paper you state "It has been known for some while however that there are Poincaré-invariant random distributions of points in Minkowski-space that are locally finite [3]. They are invariant in the stochastical sense, so that averaged over a large number of repetitions (or a large sample of volumes respectively) they are invariant, even though any single point distribution on its own is not."

    However from [3]: "Based on the theorem, we can assert the following. Not only is the Poisson process in Minkowski space Lorentz invariant, but the individual realizations of the process are also Lorentz invariant in a definite and physically important sense."

    This seems to me to be in contradiction with your assertion above (and with your theorem).

    Or am I missing something?


  9. Dear Bee,
    So it seems classically discretizing space-time - in terms of lines, points, surfaces, etc., is a somewhat quixotic venture.

    Is there some purely "quantum" way of discretizing space-time? One would avoid a analogous classical discretization by requiring that the limit of zero length scale and the classical limit don't commute.

  10. Michael,

    Sorry, I don't see what you think is in contradiction, I agree with the quote. I believe what they mean with 'physical' is what I referred to as averaged over large volumes, which makes sense. Best,


  11. Dear Arun,

    Well, I suppose one could say that this is exactly what LQG tries to do? Best,


  12. Dear Bee,

    But LQG results in a deformed Poincare invariance, no? If so, does that work?


  13. Arun: This has never been proved. These deformations are problematic for other reasons, but they don't suffer from the density problem that I alluded to here, if that is what you mean, yes. LQG itself isn't actually based on a space-time network so the argument doesn't apply to it either. The recovery of Lorentz-invariance in LQG though is to my knowledge an unsolved problem. Best,


  14. I believe the claim in [3] is stronger than that: in Minkowski space a realization of the Poisson process cannot even determine a preferred direction locally (no reference to averaging over volumes). As opposed to, say, the Euclidean analog, where a realization of the Poisson process can determine a direction at any point that breaks the rotational symmetry, while it still respects the Euclidean symmetries in an average/coarse-grained sense.

    Regarding the perceived contradiction, you're right, I misunderstood your use of the word "locally finite". I only knew it in the context of posets, where it just means finite interval cardinality (as opposed to finite degree in the graph context). That a LI graph associated with a sprinkling in Minkowski space cannot have finite degree is of course true and agrees with the claims of [3].

  15. Would this argument still apply in doubly-special relativity, if the nearest-neighbor links are of the order of the invariant plank scale?

  16. DSR doesn't work in position space to begin with, you can do it in momentum space, yes.



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