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Sabine began to understand the incompatibility of discrete physics with relativity

Several years ago, the writers of The Big Bang Theory, the sitcom, may have been inspired by Sabine Hossenfelder when they ordered Leslie Winkle to present a ludicrous anti-string rant defending loop quantum gravity in front of Sheldon Cooper.

The episode S02E02 was actually the first one I ever watched because of that exchange.

Winkle would say the very same things as her real-world subpar physics counterparts. Things like "loop quantum gravity" are predictive and it's so great that they predict that the speed of light depends on the frequency due to the discrete character of the spacetime near the Planck scale.

In the last 10 years, you could have read dozens of blog posts on this weblog about the incompatibility of all theories of discrete physics with the established rules of the special relativity – laws first appreciated by Albert Einstein that have been increasingly demonstrated by the experiments (e.g. recently by the Fermi telescope) to be almost certainly exact.

The room for a fundamental violation of the Lorentz symmetry has been shrunk to such a small volume that all the fundamentally Lorentz-breaking theories that continue to be compatible with the experimental constraints are unavoidably unnatural, unmotivated, unjustifiable, and heavily fine-tuned cousins of a Lorentz-invariant theory. As long as a Lorentz-violating theory may be made compatible with the experiments at all, which is rare and usually requires lots of extra assumptions, you may always pick the Lorentz-invariant relative as a superior cousin and reduce discussions about the Lorentz-violating deformations to the status of completely uninteresting speculations about some numbers that seem to be strictly zero but with some tolerance, could be supertiny and nonzero, discussions that have nothing to do with the explanations of any phenomena in Nature.

If a theory fails to naturally explain the Lorentz symmetry, it is a huge problem for the theory – the theory becomes pretty much a falsified one – rather than something to boast about!

For example, you may look at this 2009 blog post about the misconceptions related to the "minimal length". When correctly interpreted, I was arguing, new physics phenomena that may be associated with the concept of the "minimal length" aren't incompatible and shouldn't be incompatible with the Lorentz invariance.

However, if you discretize the spacetime in a brutal way, e.g. if you attempt to embed a graph to the Minkowski space, you inevitably break the Lorentz invariance. I have shown this picture

and argued that these randomly sprinkled points are Lorentz-invariant when it comes to their distribution. When you boost a picture like that, it looks the same. On the other hand, if you boost a picture of the Minkowski spaces that has some edges or faces such as the U.S. map,

you get a very different picture that betrays that you have boosted a nicer picture to a less natural inertial frame:

You see that the areas are stretched in one of the diagonal directions but shrunk in the complementary direction. This is unavoidable for a simple reason I have explained as well. The edges have directions and in the Minkowski space, the space of possible directions is the hyperboloid composed of (e.g. timelike) vectors obeying\[

v^\mu v_\mu = +1.

\] But this hyperboloid, the coset \(SO(3,1)/SO(3)\), is non-compact. Lorentz invariance requires all elements of this hyperboloid to be equally frequently represented as "edges of your spin network". But no probability distribution may exist on the hyperboloid because it's noncompact – its volume is infinite – which means that a uniform distribution would inevitably have an infinite total probability. It couldn't be normalized.

The very same argument eliminates not just timelike edges of a graph but also spacelike edges of a graph – the space of their direction is also a non-compact hyperboloid. You may also eliminate null directions because their proper length is zero and all lengths must be equally allowed – but again, the space of possible lengths is non-compact. For the same reason, faces (or higher-dimensional shapes but not the maximally dimensional shapes) embedded into a Minkowski space inevitably obey a statistical distribution that violates the Lorentz symmetry, too.

I don't want to describe all these trivial things again because I have done so many times but it's nice to see that now, in 2014, Sabine Hossenfelder wrote her blog post

Evolving dimensions, now vanishing
that not only contains the same arguments why the discrete structures embedded into the Minkowski space inevitably pick a preferred reference frame and therefore violate relativity. She has included pretty much the same pictures, too! That's great because my previous blog posts about the very same argument were partially inspired by her misunderstanding in the past.

So 5 years were enough to forget that I should be credited with these arguments which made it possible and politically correct to mention them! ;-)

More generally, she talks about a paper by Dejan Stojkovič about some scale-dependent spacetime dimension. I am not 100% sure that Dejan fails to understand the simple argument above but I would bet that unlike Sabine, he still does misunderstand those things. The idea of a scale-dependent spacetime dimension is credible and intriguing and in some sense, it is correct. For example, at distances shorter than the Kaluza-Klein compactification scale, the spacetime dimensionality includes the small dimensions while the effective dimensionality drops if you go to longer distances.

This "loss of dimensions" associated with the flow towards longer distances is generalized in perturbative string theory by the Zamolodčikov \(c\)-theorem which proves that \(c\), effectively the number of dimensions, is decreasing. There are other ways to argue that the spacetime dimensionality is effectively depending on the scale. In holography, the spacetime dimension is pretty much lowered if the physics is well described by the physics of the event horizon (or the boundary of the AdS space). In perturbative string theory, the fundamental, generating "spacetime" dimension is really \(1+1\) because the spacetime is created out of the world sheet. And there exist more sophisticated approaches to see that the claim that "the effective spacetime dimension is scale-dependent" is at least morally correct.

According to a/the Lorentz-invariant measure, hyperboloids are non-compact and have infinite volumes which prohibits a Lorentz-invariant probability distribution on them. If something describing the structure of the vacuum takes values in a hyperboloid and the number of these patterns per unit spacetime hypervolume is finite, the violation of the Lorentz symmetry is inevitable.

What is completely wrong, however, is the idea that the spacetime of a relativistic theory may be described by any particular discrete structure connecting "points" or other localized objects – at any scale. Whenever you draw something like that into your spacetime, you are inevitably breaking the Lorentz symmetry because the probabilistic distribution for the directions of these edges or other discrete shapes must be "centered" around a direction that defines the preferred reference frame. It must be concentrated because a non-concentrated, uniform probability distribution wouldn't be a normalizable one.

After all, if a lattice-like structure were embedded in the vacuum, the configuration of the lattice/vacuum wouldn't be unique and would carry a huge entropy (like a liquid). This leads to many problems unrelated to relativity but it also violates relativity because the entropy density is the temporal component of a 4-vector, and if this 4-vector is nonzero, it picks a preferred frame, too. Too bad. The vacuum of a relativistic theory must have a vanishing entropy density! So every attempt to imagine that much like a crystal, the vacuum is made out of some connected visualizable pieces, is wrong.

This argument is extremely simple and obvious and eliminates pretty much all proposed "discrete theories" of physics that have ever been promoted by the naive people. What is amazing from a sociological viewpoint is how many people have invested their would-be creative thinking into this manifestly incorrect research program. It's not clear whether these people could have contributed something to science if they hadn't been caught by this trap. But even if the answer is "No", it's always good to see someone who finally sees the light, and Sabine apparently does.

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reader Giotis said...

BTW and somewhat related, have you seen this?

reader Giulio said...

Oh, I see, this shape writing is like the swype, that was bundled with my samsung android phone: very nice feature :-)

reader Josef said...

I am happy with my new iPhone, have even managed to text a first message,
which is not easy with my large fingers, ha ha. Will get a small pencil with
rubber eraser at the end, that might do the trick, rather then have surgically
narrowed my finger tips.. Well, all one would really need is to adjust one
finger, in my case I would use the middle one ;-)

reader Gene Day said...

Am I wrong in thinking that the uncertainty principle also rules out discrete physics? It has always seemed to me that quantum mechanics is incompatible with anything that is truly discrete.

reader Giulio said...

In the classical CFT, the trace of the stress tensor vanishes, but in the quantum theory it will no longer be true on a curved background. I think this is one of the main points discussed here, if I understood correctly...

reader kristan said...

hi lubos,


one minor nitpick/question: what precisely did you have in mind about the c-theorem and perturbative strings? the worldsheet theory is a CFT after all, rather than an RG flow.


reader Giotis said...

I think what Lubos is saying is that c in Zamolodčikov c-theorem represents basically the number of degrees of freedom

reader anony said...

Here's a thought of the day. Since observable events are a transfer of energy, we could just visualize each event as a mini explosion, just a little tiny super nova confined to some region of spacetime. Points in spacetime not via a real string but through metadata, eg through the quantum correlations that are only identifiable in the Hilbert space representation of events. Since the discreteness of Hilbert space is not in question, one can imagine that above a threshold, an observer of a high energy event will only be able to make statements about the dimensionality of the event in something less than 3+1 dimensions. It seems this isn't unreasonable since boosted objects are naturally compressed. It isn't that we need a full catalogue of space, but a full catalogue of events. However, the connection is that certain points do get preference if they are tied to am event. Each point in space, while gaining no preference as an observer, does get some preference in the underlying catalogue of events. It is simply more likely that certain histories, or future, are preferred given a set of initial conditions were the basic laws of physics are followed. If we see that our ordinary spacetime is a projection of the underlying Hilbert space then a lot of this confusion could be avoided. There simply isn't a spacetime there, just a projection of internal observers that are incapable of holding all the information. So a whole lot of tiny super novas that don't really exist

reader Uncle Al said...

"The room for a fundamental violation of the Lorentz symmetry has been shrunk to such a small volume" Anything postulate-allowed cannot find postulate violation. Falsifying experiments must offend physics. There are no triangles whose interior angles sum to fewer or more than 180 degrees within Euclid. Try a globe of the Earth. Here's your (admittedly small but measurable) Lorentz violation, Luboš
Two geometric Eötvös experiments. 0.113 nm^3 volume/alpha-quartz unit cell. 40 grams net as 8 single crystal test masses compare 6.68×10^22 pairs of opposite shoes (pairs of 9-atom enantiomorphic unit cells, the test mass array cube's opposite vertical sides).

NOBODY has tested vacuum isotropy with otherwise identical, opposite geometric chirality atomic mass distributions. Noether's theorems do not include absolutely discontinuous symmetry geometric chirality (parity). It is an allowed loophole bearing on the whole of physics - gravitation and particle. Look.

reader Eelco Hoogendoorn said...

Far from it. Infact, it is quantum mechanics that enables continuous motion over a discrete space. Think Feynman checkerboard. Without quantum mechanics, discrete physics would be truly hard to conceive; but I don't think relativity is such a problem.

I agree with lubbos that any discrete theory (that I can conceive of) will have a slightly preferred reference frame, and consequently slightly altered behavior at extreme energies. Indeed there is no experimental evidence of this, although it may emerge one day at scales which are as of yet unprobed. Kindof like super symmetry... But unlike our host, I wouldn't bet on either theory either way. Absence of evidence is not evidence of absence, and this calls for openmindedness rather than the Spanish inquisition.

reader kristan said...

hi giotis,

perhaps. let's see what he says. at the moment, I don't see how the c-theorem tells us anything about the effective number of X^{\mu}'s which may be excited below some energy scale, for the reason that worldsheet string theory is an exact CFT. but perhaps I am being insufficiently creative. ;)


reader Gordon said...

Interesting, but I don't think she has abandoned discrete spacetimes. On physicsstackexchange about 2 years ago, Carlo Rovelli was arguing that his version of LQG was not Lorentz Invariant...a version of the New Yorker "then a miracle occurs" cartoon...
When I suggested it was, he used the squid defense---disappeared in a cloud of ink.

reader lukelea said...

Somehow I missed this post originally but stumbled upon these lectures by chance. I thought they were great. Maybe you should post another link. I also watched another earlier series of his on topics in string theory. The last lecture had some highly speculative ideas in it, using the principle of insufficient reason among other things. I wondered what you thought of it?

reader QsaTheory said...

Lubos, anybody

I posted this question on Woit's blog(about string 2014) and it got deleted.

Cumrun Vafa says

"In fact, most of the mathematics needed for string theory is not even yet developed. String theorists thus have the exciting task of building new mathematics as tools to explore new laws of physics."

What do you think he means by that?

reader Gene Day said...

If you assume that space itself has no substructure, which it does not, I think I am right. Anything discreet implies discontinuities and these are strictly prohibited in quantum mechanics.

reader Luboš Motl said...

Cumrun is surely at least morally right. I am not certain whether he is literally right because I don't know how much maths is missing and how to measure whether it's "most", anyway. ;-) But lots remains to be built, I am sure by seeing e.g. his new papers. bringing new stuff so often, most recently about E+E=H combination of different strings. But even 10,000 papers like that could fail to be "most" of the stringy mathematics because the already existing one could be more fundamental or primary. It depends how one counts it.

reader Leo Vuyk said...

However Lorentz symmetry or not, there is still room to do experiments on the lightspeed variability.see;

Experiments to determine the mass related Lightspeed extinction volume around the Earth and around spinning objects in the Lab.

reader Leo Vuyk said...

the slightly preferred reference frame is also recognized by Simon Shnoll:

reader Dilaton said...

Triying to post on that troll site, even clicking it, is just a waste of time and energy ... ;-)

reader Dilaton said...

But scale dependence of stuff is described by the RG flow ...

reader Dilaton said...

Somewhat related: our first submission :-)

reader Colin M said...

So does this mean that Nokia Maps can now be downloaded direct to the SD card in Windows 8.1? If so this would be a great feature to have as their maps can sometimes take up many gigs.

reader Luboš Motl said...

I hope that the Nokia maps are no exception.

In Windows Phone (!!) 8.1, you may choose separately whether new downloads, new pictures, new videos, and new apps are stored on the phone or the external SD card.