Bounds On Discreteness In Quantum Gravity From Lorentz Invariance

Sabine Hossenfelder has an excellent little blog post discussing the limitations that the empirical reality of Lorentz Invariance to an extreme degree of precision places upon quantum gravity theories with a minimum unit of length.  Go read it.

Key takeaway points:

1.  The post defines some closely related bits of terminology related to this discussion (emphasis added):

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.

2.  A minimum unit of length is attractive because it tames infinities in quantum field theories associated with extremely high momentum.  These infinities in the quantum field theory are particularly hard to manage mathematically if you naively try to create a quantum field theory that mimics Einstein's equations of general relativity.

In general, neither the lattice methods used to describe low energy QCD systems (which are used because the usual approach based upon "renormalization" requires too many calculations to get an accurate result to be practical to use to make low energy calculations), nor the "renormalization" technique that is used to make calculations for all three of the Standard Model forces (although usually only in high energy systems in QCD), work for quantum gravity.

A naive lattice method applied to quantum gravity is attractive because it makes it possible to do calculations without having to rely on the "renormalized" infinite series approximation used for QED, the weak nuclear force and high energy QCD calculations which are too hard for scientists to calculate in the case of low energy QCD. But, in the case of quantum gravity, unlike the Standard Model QFTs, using a naive lattice method doesn't work because it gives rise to significant Lorentz invariance violations that are not observed in nature.

Renormaliation also doesn't work for quantum gravity either for reasons discussed below in the footnote.

3.  There are a couple of loopholes to the general rule that a minimum length scale implies a Lorentz invariance violation. One forms a basis for string theory which is derivative of some of the special features of the Planck length.  The other of involves "Causal Sets" which is one of the subfields of "Loop Quantum Gravity" approaches to quantum gravity.

Guess where the action is in quantum gravity research.

4.  A couple of experimental data points that strongly constrain any possible Lorentz invariance violations in the real world are provided:

A good example is vacuum Cherenkov radiation, that is the spontaneous emission of a photon by an electron. This effect is normally – i.e. 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.

Footnote on Renormalization In Quantum Gravity

Gravity is also not renormalizable, for reasons that are a bit more arcane.  To understand this, you first have to understand why quantum field theories use renormalization in the first place.

Quantum Field Theory Calculations

Pretty much all calculations in quantum mechanics involve adding up an infinite series of path integrals (which sums up the values of a function related to the probability of a particle taking a particular path from point A to point B) which represent all possible paths from point A to point B with the simpler paths (with fewer "loops") generally making a larger contribution to the total than the more complicated paths (with more "loops").  

In practice, you calculate the probability you're interested in using as many terms of the infinite series as you can reasonably manage to calculate and then make an estimate of the uncertainty in the final result that you calculate as a result of leaving out all of the rest of the terms in the infinite series.

It turns out that when you do these calculations for the electromagnetic force (quantum electrodynamics) and the weak nuclear force, that calculating a pretty modest number of terms provides an extremely accurate answer because the terms in the series quickly get smaller.  As a result these calculations can usually be done to arbitrary accuracy up to or beyond the current limits of experimental accuracy (beyond which we do not have accurate measurements of the relevant fundamental constants of the Standard Model making additional precision in the calculations spurious).

But, when you do comparable calculations involve the quarks and gluons of QCD, calculating a very large number of terms still leaves you with a wildly inaccurate calculation because the terms get smaller much more slowly.  There are a variety of reasons for this, but one of the main ones is that gluon-gluon interactions that don't exist in the analogous QED equations make the number of possible paths that contribute materially to the ultimate outcome much greater.

(Of course, another possibility is that QCD calculations are hard not because they are inherntly more difficult, but instead mostly because lots of terms in the infinite series cancel out in a manner that is not obvious given the way that we usually arrange the terms in the infinite series we are using to calculate our observables.  If that is true, then if we cleverly arranged those terms in some other order, we might discern a way to cancel many more of them out against each other subject to only very weak constraints. This is the basic approach behind the amplituhedron concept.)

While this isn't the reason that quantum gravity isn't renormalizable, even if it was renormalizable, renormalization still wouldn't be a practical way to do quantum gravity calculations in all likelihood because the fact that gravitons in quantum gravity theories can interact with other gravitons, just as gluons can interact with other gluons in QCD, means that the number of terms in the infinite series that must be included to get an accurate result is very, very large.

Renormalization

The trouble is that almost every single one of the path integrals you have to calculate produces an answer that is non-sense unless you employ a little mathematical trickery that works for Standard Model physics calculations even though it isn't entirely clear under what circumstances this trick works in general, because it hasn't been proven in a mathematically rigorous way to work for all conceivable quantum field theories.

Naively, even the simplest quantum field theories (QFT) are useless because the answer to almost any calculation is infinite. . . .  The reason we got this meaningless answer was the insistence on integrating all the way to infinity in momentum space. This does not make sense physically because our quantum field theory description is bound to break at some point, if not sooner then definitely at the Planck scale (p ∼ M Planck). One way to make sense of the theory is to introduce a high energy cut off scale Λ where we think the theory stops being valid and allow only for momenta smaller than the cutoff to run in the loops. But having done that, we run into trouble with quantum mechanics, because usually, the regularized theory is no longer unitary (since we arbitrarily removed part of the phase space to which there was associated a non-zero amplitude.) We therefore want to imagine a process of removing the cutoff but leaving behind “something that makes sense.” A more formal way of describing the goal is the following. We are probing the system at some energy scale ΛR (namely, incoming momenta in Feynman graphs obey p ≤ ΛR) while keeping in our calculations a UV cutoff Λ (Λ ≫ ΛR because at the end of the day we want to send Λ → ∞.) If we can make all physical observables at ΛR independent of Λ then we can safely take Λ → ∞.

It turns out that there is a way to make all of the physical observables in the quantum field theories of the Standard Model (and a considerably broader generalization of possible theories that are similar in form to the QFTs of the Standard Model).

A side effect of renormalization which gives us confidence that this is a valid way to do QFT calculations is that when you renormalize, key physical constants used in your calculations take on different values depending upon the momentum scales of the interacting particles. These mathematical tools are not energy scale invariant.

Amazingly, this unexpected quirk arising from the mathematical shortcut that we invented strictly for the purpose of making an otherwise intractable calculation, we aren't even really sure is mathematically proper if you are being rigorous to use in making these calculations, and which we initially assumed was a bug, turns out to be a feature. Because, in real life, we actually do observe the basic physical constants of the laws of nature change based upon the energy scale of the particles involved in an experiment.

Why Quantum Gravity Isn't Renormalizable

But, back to the task at hand: the unavailability of renormalization as a tool in quantum gravity theories.

Basically, the problem is that in the case of QED, the weak force and QCD, there is nothing inherently unique about an extremely high energy particle, so long as it doesn't have infinite energy. It isn't different in kind from a similar high energy particle with just a little bit less energy. The impact of the extremely high energy states that are ignored in these calculations on the overall result of the calculation is tiny and has a magnitude that is easily determined to set a margin of error on the calculation, because these extremely high energy states simply correspond to possible paths of the particles whose behavior you are calculating that are so extremely unlikely to occur during the lifetime of the universe so far, that it is basically artificial to even worry about these possibilities because they are so ephemeral and basically never happen anyway, if you set the energy cutoff to a high enough arbitrary number.

In contrast, with a naive set of quantum gravity formulas based directly upon general relativity, the concentration of extremely high energies into very small spaces have a real physical effect which is very well understood in classical general relativity. 

If you pack too much energy into too small a space in a quantum gravity equation derived naively from the equations of general relativity, you get a black hole, which is a different in kind physical state that generates infinities even in the non-quantum version of the theory and corresponds to a physical reality that is a different physical state from which no light can escape (apart from "Hawking radiation").

For reasons related to the way that the well known formula for the entropy of a black hole affects the mathematical properties of terms in the calculations that are performed that correspond to very high energy states in quantum gravity, it turns out that you can't simply ignore the contribution of high energy black hole states in quantum gravity calculations in the way that you can in ordinary Standard Model quantum field theories and similar theories with the same mathematical properties (mostly various "Yang-Mills theories" in addition to the Yang-Mills theories found in the Standard Model).

If you could ignore these high energy states, you could then estimate the loss of precision price that you pay for ignoring them based upon the energy scale cutoff you use to renormalize the way that you can for calculations of the other three Standard Model forces.  

But, it turns out that black hole states in quantum gravity are infinite in a physically real way that can't be ignored, rather than in a mathematically artificial way that doesn't correspond to the physical reality as is the case with the other three Standard Model forces. And, of course, black holes don't actually pop into existence each of the myriad times that two particles interact with each other via gravity.  

So, something is broken, and that something looks like it needs to use either the string theory loophole related to the Planck length, or the Causal Sets loophole, to solve the problem with a naive quantum gravity formulation based on Einstein's equations.  The former is the only known way to get a minimum length cutoff, and the latter is another way to use discrete methods.

One More Potential Quantum Gravity Loophole

Of course, if Einstein's equations are wrong in some respect, it is entirely possible that renormalizing the naive quantum gravity generalization of the correct modification of Einstein's equations is renormalizable and that the fact that the math doesn't work with Einstein's equations is a big hint from nature that we have written down the equations for gravity in a way that is not quite right even though it is very close to the mark.

The trick to this loophole, of course, is to figure out how to modify Einstein's equations in a way that doesn't impair the accuracy of general relativity in the circumstances where it has been demonstrated to work experimentally, while tweaking the results in some situation where there are not tight experimental constraints, ideally in a respect that would explain dark matter, dark energy, inflation, or some other unexplained problems in physics as well.

My own personal hunch is that Einstein's equations are probably inaccurate as applied to high mass systems that lack spherically symmetry because the way that these classical equations implicitly model graviton-graviton interactions and localized gravitational energy associated with gravitons is at odds with how nature really works.

It makes sense that Einstein's equations, which have as a fundamental feature the fact that the energy of the gravitational field can't be localized, can't be formulated in a mathematically rigorous way as a quantum field theory in which gravity arises from the exchange of gravitons, which are the very embodiment of localized gravitational energy.  And, it similarly makes sense that in a modification of Einstein's equations in which the energy of the gravitational field could be localized, that some of these problems might be resolved.

Furthermore, I suspect that this issue is (1) producing virtually all of the phenomena that have been attributed to dark matter, (2) much (if not all) of the phenomena attributed to dark energy, (3) makes the quantum gravity equations possible to calculate with, (4) might add insights into cosmology and inflation, and (5) when implemented in a quantum gravity context may even tweak the running of the three Standard Model gauge coupling constants with energy scale in a manner that leads to gauge unification somewhere around the SUSY GUT scale.  (I don't think that this is likely to provide much insight into the baryon asymmetry in the universe, baryogenesis or leptogenesis.)

But, I acknowledge that I don't have the mathematical capacity to demonstrate that this is the case or to evaluate papers that have suggested this possibility with the rigor of a professional.