Wednesday, April 20, 2016

Dear Dr B: Why is Lorentz-invariance in conflict with discreteness?

Can we build up space-time from
discrete entities?
“Could you elaborate (even) more on […] the exact tension between Lorentz invariance and attempts for discretisation?



Dear Noa:

Discretization is a common procedure to deal with infinities. Since quantum mechanics relates large energies to short (wave) lengths, introducing a shortest possible distance corresponds to cutting off momentum integrals. This can remove infinites that come in at large momenta (or, as the physicists say “in the UV”).

Such hard cut-off procedures were quite common in the early days of quantum field theory. They have since been replaced with more sophisticated regulation procedures, but these don’t work for quantum gravity. Hence it lies at hand to use discretization to get rid of the infinities that plague quantum gravity.

Lorentz-invariance is the symmetry of Special Relativity; it tells us how observables transform from one reference frame to another. Certain types of observables, called “scalars,” don’t change at all. In general, observables do change, but they do so under a well-defined procedure that is by the application of Lorentz-transformations.We call these “covariant.” Or at least we should. Most often invariance is conflated with covariance in the literature.

(To be precise, Lorentz-covariance isn’t the full symmetry of Special Relativity because there are also translations in space and time that should maintain the laws of nature. If you add these, you get Poincaré-invariance. But the translations aren’t so relevant for our purposes.)

Lorentz-transformations acting on distances and times lead to the phenomena of Lorentz-contraction and time-dilatation. That means observers at relative velocities to each other measure different lengths and time-intervals. As long as there aren’t any interactions, this has no consequences. But once you have objects that can interact, relativistic contraction has measurable consequences.

Heavy ions for example, which are collided in facilities like RHIC or the LHC, are accelerated to almost the speed of light, which results in a significant length contraction in beam direction, and a corresponding increase in the density. This relativistic squeeze has to be taken into account to correctly compute observables. It isn’t merely an apparent distortion, it’s a real effect.

Now consider you have a regular cubic lattice which is at rest relative to you. Alice comes by in a space-ship at high velocity, what does she see? She doesn’t see a cubic lattice – she sees a lattice that is squeezed into one direction due to Lorentz-contraction. Who of you is right? You’re both right. It’s just that the lattice isn’t invariant under the Lorentz-transformation, and neither are any interactions with it.

The lattice can therefore be used to define a preferred frame, that is a particular reference frame which isn’t like any other frame, violating observer independence. The easiest way to do this would be to use the frame in which the spacing is regular, ie your restframe. If you compute any observables that take into account interactions with the lattice, the result will now explicitly depend on the motion relative to the lattice. Condensed matter systems are thus generally not Lorentz-invariant.

A Lorentz-contraction can convert any distance, no matter how large, into another distance, no matter how short. Similarly, it can blue-shift long wavelengths to short wavelengths, and hence can make small momenta arbitrarily large. This however runs into conflict with the idea of cutting off momentum integrals. For this reason approaches to quantum gravity that rely on discretization or analogies to condensed matter systems are difficult to reconcile with Lorentz-invariance.

So what, you may say, let’s just throw out Lorentz-invariance then. Let us just take a tiny lattice spacing so that we won’t see the effects. Unfortunately, it isn’t that easy. Violations of Lorentz-invariance, even if tiny, spill over into all kinds of observables even at low energies.

A good example is vacuum Cherenkov radiation, that is the spontaneous emission of a photon by an electron. This effect is normally – ie when Lorentz-invariance is respected – forbidden due to energy-momentum conservation. It can only take place in a medium which has components that can recoil. But Lorentz-invariance violation would allow electrons to radiate off photons even in empty space. No such effect has been seen, and this leads to very strong bounds on Lorentz-invariance violation.

And this isn’t the only bound. There are literally dozens of particle interactions that have been checked for Lorentz-invariance violating contributions with absolutely no evidence showing up. Hence, we know that Lorentz-invariance, if not exact, is respected by nature to extremely high precision. And this is very hard to achieve in a model that relies on a discretization.

Having said that, I must point out that not every quantity of dimension length actually transforms as a distance. Thus, the existence of a fundamental length scale is not a priori in conflict with Lorentz-invariance. The best example is maybe the Planck length itself. It has dimension length, but it’s defined from constants of nature that are themselves frame-independent. It has units of a length, but it doesn’t transform as a distance. For the same reason string theory is perfectly compatible with Lorentz-invariance even though it contains a fundamental length scale.

The tension between discreteness and Lorentz-invariance appears always if you have objects that transform like distances or like areas or like spatial volumes. The Causal Set approach therefore is an exception to the problems with discreteness (to my knowledge the only exception). The reason is that Causal Sets are a randomly distributed collection of (unconnected!) points with a four-density that is constant on the average. The random distribution prevents the problems with regular lattices. And since points and four-volumes are both Lorentz-invariant, no preferred frame is introduced.

It is remarkable just how difficult Lorentz-invariance makes it to reconcile general relativity with quantum field theory. The fact that no violations of Lorentz-invariance have been found and the insight that discreteness therefore seems an ill-fated approach has significantly contributed to the conviction of string theorists that they are working on the only right approach. Needless to say there are some people who would disagree, such as probably Carlo Rovelli and Garrett Lisi.

Either way, the absence of Lorentz-invariance violations is one of the prime examples that I draw upon to demonstrate that it is possible to constrain theory development in quantum gravity with existing data. Everyone who still works on discrete approaches must now make really sure to demonstrate there is no conflict with observation.

Thanks for an interesting question!


  1. Hi Bee:
    Nice summary of conflict of discreteness with Lorentz invariance!Then the question arises, why would anyone continue working on LQG, which I understand depends on discretization? I am assuming that you yourself are not working on LQG.

  2. kashyap,

    No, I'm not working on LQG. Yes, it relies on discretization. That doesn't necessarily mean it has a problem with Lorentz-invariance, but it certainly means that it's an issue that should be paid attention to. Rovelli argues it can be made Lorentz-covariant. My dim recollection of the paper is that it argues violations of Lorentz-invariance come from the embedding (of the network) and hence if one averages over (a suitable choice of) such embeddings, Lorentz-invariance can be recovered. I actually think that's both possible and reasonable. But I'd think in this case the discrete spectra should become continuous. Pullin and Gambini argue the issue is serious. Best,


  3. Can you describe a little about what exactly is meant by "discretizaton", I vaguely understand it's related to the idea of quantizing continuous variables. Am I heading in the right direction if I conceptualize discretization as a generic process creating theories of objects that can be placed in correspondence with the set of integers or some subset along with appropriate algebraic operations combining and describing suitable relationships between the objects, as opposed to theories based on sets of reals? Or are there examples of discretization that use reals?

  4. Simple cubic lattice sphere packing fills 52.36%. Face-centered cubic and hexagonal close-packed fill 74.05%. Diagonally shift alternate layers by a radius for sqrt(2) factor more density (hcp)[1]. Smaller spheres fill hcp tetrahedral holes as larger spheres fill space[2].

    Discrete lattice achiral symmetry (quantum spacetime) fails[3]. Periodic lattices plus quantum spacetime attack vacuum isotropy[4]. Lattice treatments are defective - or validated - on a desktop as geometric Eötvös experiments opposing visually and chemically identical, single crystal test masses in enantiomorphic space groups, e.g., P3(1)21 (#152)versus P3(2)21 (#154) alpha-quartz, chiral emergence scale 0.113 nm^3. Look, then theorize.

    [2] doi:10.1063/1.1698327
    [3] doi:10.1016/0550-3213(81)90361-8, doi:10.1016/0550-3213(82)90011-6
    [4] doi:10.1016/j.disc.2013.02.010, arXiv:1109.1963

  5. Always impressed by how subtle these issues get in expert hands. Thanks Dr. H!

    P.S. why does everyone address you as "Dr. B" when your last name begins with 'H'?

  6. Disclaimer: I am not a physicist.

    If my understanding is correct, you are talking about 3-dimensional lattices: sets of points in 3-dimensional space, that are rendered as sets of parallel word lines in 4-dimensional space-time. Indeed, such lattices are not Lorentz-invariant, you explained it clearly.

    But what if we consider lattices of *4-points* (events)? Quite surprising to me, it appears that such lattices can be Lorentz-invariant! This page (which I don't understand fully) shows a nice animated example for 2-dimensional space-time:

    Are there any physical theories that apply such kind of discretization?

  7. Alexander,

    It basically means that instead of a continuous interval you only use certain supporting points. This doesn't necessarily have to be an interval in space-time, you could also do this for some observable (like volumes or momenta).

  8. Dmitry,

    As I wrote in my post, you can indeed do it with random distributions of points as long as you don't connect the points.

  9. Matthew,

    I wrote this blog under the pseudonym "Bee" for many years - until Google forced Bloggers to join accounts with G+. Since them my posts appear under my real name. I still sign with B out of nostalgia. It's an ancient nickname which I keep using because pretty much everybody mispronounces my name. "Bee" seems as easy as it gets. Best, Bee ;)

  10. Thank you, a cute story!FWIW I have no problem pronouncing your last name!

  11. "It's an ancient nickname which I keep using because pretty much everybody mispronounces my name. "Bee" seems as easy as it gets."

    For non-German readers: Sabine is a common name in Germany, particularly for women of Bee's age and a bit older. It derives from the Sabines, who were a tribe in Italy. (It is not related to Sabrina, which is pronounced similarly (Sabrina probably comes from the Celtic word for "river", and is also the Latin name for the river Severn).) The German pronunciation is something like Zuhbeenuh, so the second syllable is "bee". The last two syllables, "bine", are pronounced exactly like "Biene", which is the German word for "bee". (In essentially all languages besides English, "i" is pronounced similarly to an English long "e", but you need to smile a bit more to get the pronunciation more correct.) If you pronounce it like "Sabrina" but without the "r" you're pretty close.

  12. Keep in mind that dynamical triangulation, Regge gravity etc. consider the sum over all possible lattices and this sum and the expectation values calculated with it can be Lorentz- and actually Diff. invariant even if the individual lattices are not.

    One example is dynamical triangulation in 2D which reproduces Polyakov gravity correctly, another would be Ponzano-Regge in 3D which gives an invariant result (at least under some conditions).
    The 4D case is of course open, but we know (as you mention) that causal nets can be well defined there.

  13. Hi kashyap,

    the issue of Lorentz invariance in LQG is more complicated than often put, and it is currently not understood whether it is a feature of LQG, or to which extend it is broken. To figure it out, one would essentially need to solve the theory and check how matter propagates. This is currently out of reach for technical reasons.

    There are many pitfalls when thinking about this issue which might lead to premature conclusions. In particular, it turns out that discrete eigenvalues of geometric operators are no obstacle to Lorentz invariance, see for example here.

    Also, the statement that LQG is based on discretisation is somewhat misleading in this context, as the Hilbert space is constructed by quantising continuum GR. However, it happens to have a basis where individual elements can be interpreted as discrete geometries, i.e. truncations of the theory on given lattices. This however doesn’t mean that there cannot be continuum states in this Hilbert space, which a priori can be infinite superpositions of such lattices, and as such very well be Lorentz invariant.

    A more in depth explanation of the current status can be found here.

  14. Wolfgang,

    My understanding was that in CDT the lattice spacing goes to zero? Best,


  15. Norbert,

    On that issue about the discrete spectrum being compatible with a Lorentz-invariant transformation of observables: has someone actually computed this, or is this just a guess based on the comparison to the angular moment? Best,


  16. Hello Sabine, thank you for this elaborate explanation.Could you clarify more on this : Lorentz contractions can convert smaller momenta into larger ones,, ok. And that gives a conflict with cutting off momentum integrals. Could you specify that conflict ?

    My thanks

    Best, Noa

  17. Noa,

    Suppose you cut off momenta at p=p_cut in one frame. Now you make a boost into another frame. Where is the cutoff? Best,


  18. Sabine, as usual nice post. OT question. I would be interested in your comments on Stanley Deser's latest paper arXiv:1604.04015 where he dissects the LIGO limit on graviton mass (since you are an expert on massive graviton models)
    and whether you agree with him (or not). Thanks

  19. >> in CDT the lattice spacing goes to zero

    In lattice field theory people usually think about it the other way around and look for a (2nd order) phase transition so that correlation lengths become large compared to the lattice spacing.
    The real question for DT, CDT and others is therefore if such a phase transition can be identified.

  20. As a layman, I still don't quite understand this :)
    You've explained why having some pre existing lattice background can't work.
    But I assumed that discreteness would be purely relational anyway: something like that the space time interval was always quantised. And the only observables arise from propagators over those intervals
    Maybe it is difficult to get the Lorentz boosted quantised value, to be quantised too? With the frame velocity also subject to the uncertainty principle, exactly enough etc
    Are there obvious observables this type of theory violates?

  21. James,

    Yes, look at the propagator. Can or can it not go up to arbitrarily large values of p_\mu (the components). This enters observables eg through cross-sections. Best,


  22. A,

    There is a distinction to be drawn here between preferred frames which come about by matter content (eg a condensed matter system with a certain restframe) and preferred frames which are fundamental (do not come about from matter content but from the structure of space-time itself). The former doesn't violate Lorentz-invariance, the latter does. I didn't say it "violates the principle of relativity". In fact you can introduce a preferred frame so that it's covariant. It isn't in conflict with GR. But if it couples to particles of the standard model (which it should generally do if it's a feature of space-time itself), it will lead to deviations from the standard model predictions. Best,


  23. A,

    Sorry, my bad. I should have written observer independence, I will fix that. A preferred frame can transform covariantly, no problem with this. I'm afraid I'm not using the phrase 'principle of relativity' consistently, apologies. But no, the lattice I was referring do is *not* matter content. It's supposed to come about by discretizing space-time. Best,


  24. Hi Sabine,

    there exists an explicit computation in a simplified toy model related to Euclidean 3d LQG. It is not full 4d LQG, but it proofs the point that Lorentz invariance is not in logical conflict with the discrete quantum geometry that one finds in LQG.

    To cite from the conclusion:

    “We found that the compatibility of quantum discreteness of geometric spectra and continuous Lorentz invariance is possible due to the unitary action of the Lorentz boost operators on quantum states and distance operators, and a non-commutativity of these and their boosted counterparts. This results in the fact that the state of a localized system for a given observer turns into a de-localized one for another observer boosted with respect to the first. Our result then confirms, in this simplified context, but in full detail, the argument for resolving the apparent contrast between discrete quantum geometry and Lorentz invariance presented in [11].”

    [11] is the paper cited above. There, it is shown at the classical level that an area at rest and its boosted version should not Poisson-commute, but no explicit realisation of this at the quantum level was given.

  25. Hi Sabine, as a ''causal set person'' (it is one of my faces), I do not think that Lorentz invariance or any continuous symmetry poses a problem in discrete ''translations''. All these matters depend upon how you discretize and take the limit (look for example at such discretization dependencies in the definition of the path integral). From what I know, these things usually turn out fine if you care enough about them. What is not so trivial however is the rationale to do discrete physics at all: the infinities in Quantum Field Theory do not constitute a good motivator in my opinion and one quickly finds oneself into the philosophical debates about irreducibility, atomisticity and (local) finiteness. Fact of the matter is that discrete spacetime sugggests a very different physics than the continuum does and it is here that some problems and opportunities may arise. It is is an interesting but largely unexplored world and it will take many more years before its mysteries start to be uncovered. I don't know if you are aware but you may use other statistical distributions than the Poisson distribution; the latter however does not induce correlations between points which others do.

  26. Are the infinities in high momentum components of quantum gravity integrals that can't be eliminated with renormalization related in some way to the fact that in GR the energy of the gravitational field is not localized?

  27. A,

    It is rather pointless to pick around on how words are being used, the important thing is that a fundamental preferred frame that couples to matter fields has observational consequences which haven't been seen. This goes in the literature as Lorentz-invariance-violation, whether or not that nomenclature makes much sense.

    I would say a condensed matter system (say, a solid or a fluid) defines a preferred frame and is hence not Lorentz-invariant. It doesn't violate the symmetry though because there's a material basis for the frame, hence, if you calculate any interaction processes between particles that involve the lattice, these will not be in conflict with the standard model.

    Yes, in this terminology, Lorentz-invariance is almost always used to mean what should actually be Lorentz-covariance, which is what I was trying to explain.



  28. Dear Bee, if it's not too late to join this item, might I ask you to elaborate further on how a preferred frame might couple to a matter field, and how this might have observable consequences. Dan


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